A Stroll through Quantum Fields (antecedent of Quantum Field Theory - From Basics to Modern Topics) 978-1108480901

Table of contents :
Special relativity......Page 7
Free scalar fields, Mode decomposition......Page 12
Interacting scalar fields......Page 17
LSZ reduction formulas......Page 20
From transition amplitudes to reaction rates......Page 23
Generating functional......Page 28
Perturbative expansion and Feynman rules......Page 33
Calculation of loop integrals......Page 39
Källen-Lehmann spectral representation......Page 42
Ultraviolet divergences and renormalization......Page 44
Spin 1/2 fields......Page 53
Spin 1 fields......Page 58
Abelian gauge invariance, QED......Page 63
Charge conservation, Ward-Takahashi identities......Page 66
Spontaneous symmetry breaking......Page 69
Perturbative unitarity......Page 77
Path integral in quantum mechanics......Page 91
More functional machinery......Page 95
Path integral in scalar field theory......Page 102
Functional determinants......Page 104
Quantum effective action......Page 107
Two-particle irreducible effective action......Page 113
Euclidean path integral and Statistical mechanics......Page 120
Grassmann variables......Page 125
Path integral for fermions......Page 131
Path integral for photons......Page 133
Schwinger-Dyson equations......Page 136
Quantum anomalies......Page 139
Non-Abelian gauge symmetry......Page 149
Non-abelian Lie groups and algebras......Page 150
Yang-Mills Lagrangian......Page 158
Non-Abelian gauge theories......Page 163
Spontaneous gauge symmetry breaking......Page 168
theta-term and strong-CP problem......Page 174
Non-local gauge invariant operators......Page 182
Introduction......Page 193
Gauge fixing......Page 195
Fadeev-Popov quantization and Ghost fields......Page 197
Feynman rules for non-abelian gauge theories......Page 199
On-shell non-Abelian Ward identities......Page 203
Ghosts and unitarity......Page 205
Ultraviolet power counting......Page 217
Symmetries of the quantum effective action......Page 218
Renormalizability......Page 224
Background field method......Page 229
Callan-Symanzik equations......Page 237
Correlators containing composite operators......Page 240
Operator product expansion......Page 243
Example: QCD corrections to weak decays......Page 247
Non-perturbative renormalization group......Page 254
Effective field theories......Page 265
General principles of effective theories......Page 266
Example: Fermi theory of weak decays......Page 270
Standard model as an effective field theory......Page 273
Effective theories in QCD......Page 280
EFT of spontaneous symmetry breaking......Page 290
Axial anomalies in a gauge background......Page 301
Generalizations......Page 313
Wess-Zumino consistency conditions......Page 320
't Hooft anomaly matching......Page 324
Scale anomalies......Page 326
Localized field configurations......Page 333
Domain walls......Page 334
Skyrmions......Page 337
Monopoles......Page 339
Instantons......Page 349
Shortcomings of the usual approach......Page 363
Colour ordering of gluonic amplitudes......Page 364
Spinor-helicity formalism......Page 370
Britto-Cachazo-Feng-Witten on-shell recursion......Page 380
Tree-level gravitational amplitudes......Page 391
Cachazo-Svrcek-Witten rules......Page 401
Worldline representation......Page 413
Quantum electrodynamics......Page 419
Schwinger mechanism......Page 423
Calculation of one-loop amplitudes......Page 426
Lattice field theory......Page 437
Discretization of bosonic actions......Page 438
Fermions......Page 443
Hadron mass determination on the lattice......Page 447
Wilson loops and confinement......Page 448
Gauge fixing on the lattice......Page 452
Lattice worldline formalism......Page 456
Canonical thermal ensemble......Page 463
Finite-T perturbation theory......Page 464
Large distance effective theories......Page 483
Out-of-equilibrium systems......Page 498
Introduction......Page 507
Expectation values in a coherent state......Page 509
Quantum field theory with external sources......Page 515
Observables at LO and NLO......Page 516
Green's formulas......Page 522
Mode functions......Page 533
Multi-point correlation functions at tree level......Page 537

Citation preview

A Stroll Through QUANTUM FIELDS FRANC¸ OIS GELIS I NSTITUT DE P HYSIQUE T H E´ ORIQUE CEA-S ACLAY

Contents 1 Basics of Quantum Field Theory 1.1 Special relativity . . . . . . . . . . . . . . . . 1.2 Free scalar fields, Mode decomposition . . . . 1.3 Interacting scalar fields . . . . . . . . . . . . . 1.4 LSZ reduction formulas . . . . . . . . . . . . . 1.5 From transition amplitudes to reaction rates . . 1.6 Generating functional . . . . . . . . . . . . . . 1.7 Perturbative expansion and Feynman rules . . . 1.8 Calculation of loop integrals . . . . . . . . . . 1.9 K¨allen-Lehmann spectral representation . . . . 1.10 Ultraviolet divergences and renormalization . . 1.11 Spin 1/2 fields . . . . . . . . . . . . . . . . . . 1.12 Spin 1 fields . . . . . . . . . . . . . . . . . . . 1.13 Abelian gauge invariance, QED . . . . . . . . 1.14 Charge conservation, Ward-Takahashi identities 1.15 Spontaneous symmetry breaking . . . . . . . . 1.16 Perturbative unitarity . . . . . . . . . . . . . .

1 1 6 11 14 17 22 27 33 36 38 47 52 57 60 63 71

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2 Functional quantization 2.1 Path integral in quantum mechanics . . . . . . . 2.2 Classical limit, Least action principle . . . . . . . 2.3 More functional machinery . . . . . . . . . . . . 2.4 Path integral in scalar field theory . . . . . . . . 2.5 Functional determinants . . . . . . . . . . . . . . 2.6 Quantum effective action . . . . . . . . . . . . . 2.7 Two-particle irreducible effective action . . . . . 2.8 Euclidean path integral and Statistical mechanics

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85 . 85 . 89 . 89 . 96 . 98 . 101 . 107 . 114

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

3 Path integrals for fermions and photons

119

3.1

Grassmann variables . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.2

Path integral for fermions . . . . . . . . . . . . . . . . . . . . . . . 125

3.3

Path integral for photons . . . . . . . . . . . . . . . . . . . . . . . 127

3.4

Schwinger-Dyson equations . . . . . . . . . . . . . . . . . . . . . 130

3.5

Quantum anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4 Non-Abelian gauge symmetry

143

4.1

Non-abelian Lie groups and algebras . . . . . . . . . . . . . . . . . 144

4.2

Yang-Mills Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 152

4.3

Non-Abelian gauge theories . . . . . . . . . . . . . . . . . . . . . 157

4.4

Spontaneous gauge symmetry breaking . . . . . . . . . . . . . . . 162

4.5

θ-term and strong-CP problem . . . . . . . . . . . . . . . . . . . . 168

4.6

Non-local gauge invariant operators . . . . . . . . . . . . . . . . . 176

5 Quantization of Yang-Mills theory

187

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.2

Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.3

Fadeev-Popov quantization and Ghost fields . . . . . . . . . . . . . 191

5.4

Feynman rules for non-abelian gauge theories . . . . . . . . . . . . 193

5.5

On-shell non-Abelian Ward identities . . . . . . . . . . . . . . . . 197

5.6

Ghosts and unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6 Renormalization of gauge theories

211

6.1

Ultraviolet power counting . . . . . . . . . . . . . . . . . . . . . . 211

6.2

Symmetries of the quantum effective action . . . . . . . . . . . . . 212

6.3

Renormalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.4

Background field method . . . . . . . . . . . . . . . . . . . . . . . 223

7 Renormalization group

231

7.1

Callan-Symanzik equations . . . . . . . . . . . . . . . . . . . . . . 231

7.2

Correlators containing composite operators . . . . . . . . . . . . . 234

7.3

Operator product expansion . . . . . . . . . . . . . . . . . . . . . . 237

7.4

Example: QCD corrections to weak decays . . . . . . . . . . . . . 241

7.5

Non-perturbative renormalization group . . . . . . . . . . . . . . . 248

CONTENTS 8 Effective field theories

iii 259

8.1

General principles of effective theories . . . . . . . . . . . . . . . . 260

8.2

Example: Fermi theory of weak decays . . . . . . . . . . . . . . . 264

8.3

Standard model as an effective field theory . . . . . . . . . . . . . . 267

8.4

Effective theories in QCD . . . . . . . . . . . . . . . . . . . . . . . 274

8.5

EFT of spontaneous symmetry breaking . . . . . . . . . . . . . . . 284

9 Quantum anomalies

295

9.1

Axial anomalies in a gauge background . . . . . . . . . . . . . . . 295

9.2

Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.3

Wess-Zumino consistency conditions . . . . . . . . . . . . . . . . . 314

9.4

’t Hooft anomaly matching . . . . . . . . . . . . . . . . . . . . . . 318

9.5

Scale anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

10 Localized field configurations

327

10.1 Domain walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 10.2 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 10.3 Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10.4 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 11 Modern tools for tree level amplitudes

357

11.1 Shortcomings of the usual approach . . . . . . . . . . . . . . . . . 357 11.2 Colour ordering of gluonic amplitudes . . . . . . . . . . . . . . . . 358 11.3 Spinor-helicity formalism . . . . . . . . . . . . . . . . . . . . . . . 364 11.4 Britto-Cachazo-Feng-Witten on-shell recursion . . . . . . . . . . . 374 11.5 Tree-level gravitational amplitudes . . . . . . . . . . . . . . . . . . 385 11.6 Cachazo-Svrcek-Witten rules . . . . . . . . . . . . . . . . . . . . . 395 12 Worldline formalism

407

12.1 Worldline representation . . . . . . . . . . . . . . . . . . . . . . . 407 12.2 Quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . 413 12.3 Schwinger mechanism . . . . . . . . . . . . . . . . . . . . . . . . 417 12.4 Calculation of one-loop amplitudes . . . . . . . . . . . . . . . . . . 420

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13 Lattice field theory

431

13.1 Discretization of bosonic actions . . . . . . . . . . . . . . . . . . . 432 13.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 13.3 Hadron mass determination on the lattice . . . . . . . . . . . . . . 441 13.4 Wilson loops and confinement . . . . . . . . . . . . . . . . . . . . 442 13.5 Gauge fixing on the lattice . . . . . . . . . . . . . . . . . . . . . . 446 13.6 Lattice worldline formalism . . . . . . . . . . . . . . . . . . . . . 450 14 Quantum field theory at finite temperature

457

14.1 Canonical thermal ensemble . . . . . . . . . . . . . . . . . . . . . 457 14.2 Finite-T perturbation theory . . . . . . . . . . . . . . . . . . . . . 458 14.3 Large distance effective theories . . . . . . . . . . . . . . . . . . . 477 14.4 Out-of-equilibrium systems . . . . . . . . . . . . . . . . . . . . . . 492 15 Strong fields and semi-classical methods

501

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 15.2 Expectation values in a coherent state . . . . . . . . . . . . . . . . 503 15.3 Quantum field theory with external sources . . . . . . . . . . . . . 509 15.4 Observables at LO and NLO . . . . . . . . . . . . . . . . . . . . . 510 15.5 Green’s formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 15.6 Mode functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 15.7 Multi-point correlation functions at tree level . . . . . . . . . . . . 531

Chapter 1

Basics of Quantum Field Theory 1.1 Special relativity 1.1.1

Lorentz transformations

Special relativity plays a crucial role in quantum field theories1 . Various observers in frames that are moving at a constant speed relative to each other should be able to describe physical phenomena using the same laws of Physics. This does not imply that the equations governing these phenomena are independent of the observer’s frame, but that these equations transform in a constrained fashion –depending on the nature of the objects they contain– under a change of reference frame. Let us consider two frames F and F ′ , in which the coordinates of a given event ′ are respectively xµ and x µ . A Lorentz transformation is a linear transformation such 2 that the interval ds ≡ dt2 − dx2 is the same in the two frames2 . If we denote the coordinate transformation by x′µ = Λµ ν xν ,

(1.1)

1 An exception to this assertion is for quantum field models applied to condensed matter physics, where the basic degrees of freedom are to a very good level of approximation described by Galilean kinematics. 2 The physical premises of special relativity require that the speed of light be the same in all inertial frames, which implies solely that ds2 = 0 be preserved in all inertial frames. The group of transformations that achieves this is called the conformal group. In four space-time dimensions, the conformal group is 15 dimensional, and in addition to the 6 orthochronous Lorentz transformations it contains dilatations as well as non-linear transformations called special conformal transformations.

1

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

the matrix Λ of the transformation must obey gµν Λµ ρ Λν σ = gρσ

(1.2)

where gµν is the Minkowski metric tensor

gµν

 +1  −1  ≡  −1



   . 

(1.3)

−1

Note that eq. (1.2) implies that
Λµ ν = Λ−1 ν µ .

(1.4)

If we consider an infinitesimal Lorentz transformation, Λµ ν = δµ ν + ωµ ν

(1.5)

(with all components of ω much smaller than unity), this implies that ωµν = −ωνµ

(1.6)

(with all indices down). Consequently, there are 6 independent Lorentz transformations, three of which are ordinary rotations and three are boosts. Note that the infinitesimal transformations (1.5) have a determinant3 equal to +1 (they are called proper transformations), and do not change the direction of the time axis since Λ0 0 = 1 ≥ 0 (they are called orthochronous). Any combination of such infinitesimal transformations shares the same properties, and their set forms a subgroup of the full group of transformations that preserve the Minkowski metric. c sileG siocnarF

1.1.2

Representations of the Lorentz group

More generally, a Lorentz transformation acts on a quantum system via a transformation U(Λ), that forms a representation of the Lorentz group, i.e. U(ΛΛ ′ ) = U(Λ)U(Λ ′ ) .

(1.7)

For an infinitesimal Lorentz transformation, we can write i U(1 + ω) = I + ωµν Mµν . 2 3 From

eq. (1.2), the determinant may be equal to ±1.

(1.8)

1. BASICS OF Q UANTUM F IELD T HEORY

3

(The prefactor i/2 in the second term of the right hand side is conventional.) Since the ωµν are antisymmetric, the generators Mµν can also be chosen antisymmetric. By using eq. (1.7) for the Lorentz transformation Λ−1 Λ ′ Λ, we arrive at U−1 (Λ)Mµν U(Λ) = Λµ ρ Λν σ Mρσ ,

(1.9)

indicating that Mµν transforms as a rank-2 tensor. When used with an infinitesimal transformation Λ = 1 + ω, this identity leads to the commutation relation that defines the Lie algebra of the Lorentz group  µν  M , Mρσ = i(gµρ Mνσ − gνρ Mµσ ) − i(gµσ Mνρ − gνσ Mµρ ) . (1.10)

When necessary, it is possible to divide the six generators Mµν into three generators Ji for ordinary spatial rotations, and three generators Ki for the Lorentz boosts along each of the spatial directions: Ji ≡ 21 ǫijk Mjk ,

Rotations :

Ki ≡ Mi0 .

Lorentz boosts :

(1.11)

In a fashion similar to eq. (1.9), we obtain the transformation of the 4-impulsion Pµ , U−1 (Λ)Pµ U(Λ) = Λµ ρ Pρ , which leads to the following commutation relation between Pµ and Mµν ,  µ  P , Mρσ = i(gµσ Pρ − gµρ Pσ ) ,  µ ν P ,P = 0 .

1.1.3

(1.12)

(1.13)

One-particle states

 Let us denote p, σ a one-particle state, where p is the 3-momentum of that particle, and σ denotes its other quantum numbers. Since this state contains a particle with a definite momentum, it is an eigenstate of the momentum operator Pµ , namely p   Pµ p, σ = pµ p, σ , with p0 ≡ p2 + m2 . (1.14)  Consider now the state U(Λ) p, σ . We have    Pµ U(Λ) p, σ = U(Λ) U−1 (Λ)Pµ U(Λ) p, σ = Λµ ν pν U(Λ) p, σ . (1.15) {z } | Λµ ν P ν

 Therefore, U(Λ) p, σ is an eigenstate of momentum with eigenvalue (Λp)µ , and we may write it as a linear combination of all the states with momentum Λp,  X  U(Λ) p, σ = Cσσ ′ (Λ; p) Λp, σ ′ . (1.16) σ′

4

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.1.4

Little group

Any positive energy on-shell momentum pµ can be obtained by applying an orthochronous Lorentz transformation to some reference momentum qµ that lives on the same mass-shell, pµ ≡ Lµ ν (p) qν .

(1.17)

The choice of the reference 4-vector is not important, but depends on whether the particle under consideration is massive or not. Convenient choices are the following: • m > 0 : qµ ≡ (m, 0, 0, 0), the 4-momentum of a massive particle at rest, • m = 0 : qµ ≡ (ω, 0, 0, ω), the 4-momentum of a massless particle moving in the third direction of space. Then, we may define a generic one-particle state from those corresponding to the reference momentum as follows   p, σ ≡ Np U(L(p)) q, σ , (1.18)

where Np is a numerical prefactor that may be necessary to properly normalize the states. This definition leads to 


(1.19) U Λ p, σ = Np U L(Λp) U L−1 (Λp)ΛL(p) q, σ . {z } | Σ

−1

Note that the Lorentz transformation Σ ≡ L (Λp)ΛL(p) maps qµ into itself, and therefore belongs to the subgroup of the Lorentz group that leaves qµ invariant, called the little group of qµ . Thus, when U(Σ) acts on the reference state, the momentum remains unchanged and only the other quantum numbers may vary  X  U(Σ) q, σ = (1.20) Cσσ ′ (Σ) q, σ ′ . σ′

Moreover, the coefficients Cσσ ′ (Σ) in the right hand side of this formula define a representation of the little group, X Cσσ ′ (Σ2 Σ1 ) = (1.21) Cσσ ′′ (Σ2 ) Cσ ′′ σ ′ (Σ1 ) . σ ′′

Massive particles : In the case of a massive particles, the little group is made of the Lorentz transformations that leave the vector qµ = (m, 0, 0, 0) invariant, which is the group of all rotations in 3-dimensional space. The additional quantum number σ is therefore a label that enumerates the possible states in a given representation of SO(3). These representations correspond to the angular momentum, but since we are in the rest frame of the particle, this is in fact the spin of the particle. For a spin s, the dimension of the representation is 2s + 1, and σ takes the values −s, 1 − s, · · · , +s.

5

1. BASICS OF Q UANTUM F IELD T HEORY

Massless particles : In the massless case, we look for Lorentz transformations Σµ ν that leave qν = (ω, 0, 0, ω) invariant. For an infinitesimal transformation, Σµ ν ≈ δµ ν + ωµ ν , this gives the following general form

ωµν



0 −α  1 = −α2 0

α1 0 θ −α1

α2 −θ 0 −α2

 0 α1    , α2  0

(1.22)

where α, β, θ are three real infinitesimal parameters. Thus, an infinitesimal transformation U(Σ) reads 20 12 10 − M23}) . U(Σ) ≈ 1 − iθ M + M31}) − iα2 (M |{z} −iα1 (M {z {z | | J3

K1 +J2 ≡B1

(1.23)

K2 −J1 ≡B2

Thus, the little group for massless particles is three dimensional, with generators J3 (the projection of the angular momentum in the direction of the momentum) and4 B1,2 . Using eq. (1.10), we have 

 J3 , B1 = i B2 ,



 J3 , B2 = −i B1 ,



 B 1 , B2 = 0 .

(1.24)

The last commutators implies that we may choose states that are simultaneous eigenstates of B1 and B2 . However, non-zero eigenvalues for B1,2 would lead to a continuum of states with the same momentum, that are not realized in Nature. The remaining transformation, generated by J3 , can be viewed as a rotation about the direction of the momentum, and the corresponding group is SO(2). Therefore, the only eigenvalue that labels the massless states is that of J3 ,   J3 q, σ = σ q, σ ,

 U(Σ) q, σ

=

α1,2 =0

 e−iσθ q, σ .

(1.25)

The number σ is called the helicity of the particle. After a rotation of angle θ = 2π, the state must return to itself (bosons) or its opposite (fermions), implying that the helicity must be a half integer: bosons : fermions :

σ = 0, ±1, ±2, · · · σ = ± 12 , ± 23 , · · ·

(1.26)

4 The generators B1,2 are the generators of Galilean boosts in the (x1 , x2 ) plane transverse to the particle momentum, i.e. the transformations that shift the transverse velocity, vj → vj + δvj . The physical reason of their appearance in the discussion of massless particles is time dilation: in the observer’s frame, the transverse dynamics of a particle moving at the speed of light is infinitely slowed down by time dilation, and is therefore non relativistic (this intuitive idea can be further substantiated by light-cone quantization).

6

1.1.5

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Scalar field

A scalar field φ(x) is a (number or operator valued) object that depends on a spacetime coordinate x and is invariant under a Lorentz transformation, except for the change of coordinate induced by the transformation: U−1 (Λ)φ(x)U(Λ) = φ(Λ−1 x) .

(1.27)

This formula just reflects the fact that the point x where the transformed field is evaluated was located at the point Λ−1 x before the transformation. The first derivative ∂µ φ of the field transforms as a 4-vector, U−1 (Λ)∂µ φ(x)U(Λ) = Λµ ν ∂ν φ(Λ−1 x) ,

(1.28)

where the bar in ∂ν indicates that we are differentiating with respect to the whole argument of φ, i.e. Λ−1 x. Likewise, the second derivative ∂µ ∂ν φ transforms like a rank-2 tensor, but the d’Alembertian φ transforms as a scalar. c sileG siocnarF

1.2 Free scalar fields, Mode decomposition 1.2.1

Quantum harmonic oscillators

Let us consider a continuous collection of quantum harmonic oscillators, each of them corresponding to particles with a given momentum p. These harmonic oscillators can be defined by a pair of creation and annihilation operators a†p , ap , where p is a 3-momentum that labels the corresponding mode. Note that the energy of the particles is fixed from their 3-momentum by the relativistic dispersion relation, p p0 = Ep ≡ p2 + m2 . (1.29)

The operators creating or destroying particles with a given momentum p obey usual commutation relations,       ap , ap = a†p , a†p = 0 , ap , a†p ∼ 1 . (1.30)

(in the last commutator, the precise normalization will be defined later.) In contrast, operators acting on different momenta always commute:       ap , aq = a†p , a†q = ap , a†q = 0 . (1.31) If we denote by H the Hamiltonian operator of such a system, the property that a†p creates a particle of momentum p (and therefore of energy Ep ) implies that   H, a†p = +Ep a†p . (1.32)

1. BASICS OF Q UANTUM F IELD T HEORY Likewise, since ap destroys a particle with the same energy, we have   H, ap = −Ep ap .

7

(1.33)

(Implicitly in these equations is the fact that particles are non-interacting, so that adding or removing a particle of momentum p does not affect the rest of the system.) In these lectures, we will adopt the following normalization for the free Hamiltonian5 ,

H=

Z


d3 p Ep a†p ap + V Ep , (2π)3 2Ep

(1.34)

where V is the volume of the system. To make contact with the usual treatment6 of a harmonic oscillator in quantum mechanics, it is useful to introduce the occupation number fp defined by, 2Ep V fp ≡ a†p ap .

(1.35)

In terms of fp , the above Hamiltonian reads H=V

Z


d3 p Ep fp + 21 . 3 (2π)

(1.36)

The expectation value of fp has the interpretation of the number of particles par unit of phase-space (i.e. per unit of volume in coordinate space and per unit of volume in momentum space), and the 1/2 in fp + 12 is the ground state occupation of each oscillator7 . Of course, this additive constant is to a large extent irrelevant since only energy differences have a physical meaning. Given eq. (1.34), the commutation relations (1.32) and (1.33) are fulfilled provided that   ap , a†q = (2π)3 2Ep δ(p − q) . (1.37)

5 In a relativistic setting, the measure d3 p/(2π)3 2E has the important benefit of being Lorentz p invariant. Moreover, it results naturally from the 4-dimensional momentum integration d4 p/(2π)4 constrained by the positive energy mass-shell condition 2π θ(p0 ) δ(p2 − m2 ). 6 In relativistic quantum field theory, it is customary to use a system of units in which h ¯ = 1, c = 1 (and also kB = 1 when the Boltzmann constant is needed to relate energies and temperature). In this system of units, the action S is dimensionless. Mass, energy, momentum and temperature have the same dimension, which is the inverse of the dimension of length and duration:             mass = energy = momentum = temperature = length−1 = duration−1 .

Moreover, in four dimensions, the creation and annihilation operators introduced in eq. (1.34) have the dimension of an inverse energy:    †   ap = ap = energy−1

(the occupation number fp is dimensionless.) 7 This is reminiscent of the fact that the energy of the level n in a quantized harmonic oscillator of base energy ω is En = (n + 21 )ω.

8

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.2.2

Scalar field operator, Canonical commutation relations

Note that in quantum mechanics, a particle with a well defined momentum p is not localized at a specific point in space, due to the uncertainty principle. Thus, when we say that a†p creates a particle of momentum p, this production process may happen anywhere in space and at any time since the energy is also well defined. Instead of using the momentum basis, one may introduce an operator that depends on space-time in order to give preeminence to the time and position at which a particle is created or destroyed. It is possible to encapsulate all the ap , a†p into the following Hermitean operator8 φ(x) ≡

Z

 † +ip·x  d3 p ap e + ap e−ip·x , 3 (2π) 2Ep

(1.38)

where p · x ≡ pµ xµ with p0 = +Ep . In the following, we will also need the time derivative of this operator, denoted Π(x), Π(x) ≡ ∂0 φ(x) = i

Z

  d3 p Ep a†p e+ip·x − ap e−ip·x . (2π)3 2Ep

(1.39)

Given the commutation relation (1.37), we obtain the following equal-time commutation relations for φ and Π, 

     φ(x), φ(y) x0 =y0 = Π(x), Π(y) x0 =y0 = 0 , φ(x), Π(y) x0 =y0 = iδ(x−y) . (1.40)

These are called the canonical field commutation relations. In this approach (known as canonical quantization), the quantization of a field theory corresponds to promoting the classical Poisson bracket between a dynamical variable and its conjugate momentum to a commutator: 

Pi , Qj = δij

→



 ^i, Q ^ j = ih ¯ δij . P

(1.41)

In addition to these relations that hold for equal times, one may prove that φ(x) and Π(y) commute for space-like intervals (x − y)2 < 0. Physically, this is related to the absence of causal relation between two measurements performed at space-time points with a space-like separation. It is possible to invert eqs. (1.38) and (1.39) in order to obtain the creation and 8 In

four space-time dimensions, this field has the same dimension as energy:     φ(x) = energy .

1. BASICS OF Q UANTUM F IELD T HEORY

9

annihilation operators given the operators φ and Π. These inversion formulas read Z Z ↔   a†p = −i d3 x e−ip·x Π(x) + iEp φ(x) = −i d3 x e−ip·x ∂0 φ(x) , Z Z ↔   ap = +i d3 x e+ip·x Π(x) − iEp φ(x) = +i d3 x e+ip·x ∂0 φ(x) ,

(1.42)

↔

where the operator ∂0 is defined as ↔

A ∂0 B ≡ A ∂0 B − ∂ 0 A B .

(1.43)

Note that these expressions, although they appear to contain x0 , do not actually depend on time. Using these formulas, we can rewrite the Hamiltonian in terms of φ and Π, Z  H = d3 x 12 Π2 (x) + 21 (∇φ(x))2 + 21 m2 φ2 (x) . (1.44)

From this Hamiltonian, one may obtain equations of motion in the form of HamiltonJacobi equations. Formally, they read δH = Π(x) , δΠ(x)   δH = ∇2 − m2 φ(x) . ∂0 Π(x) = − δφ(x)

∂0 φ(x) =

1.2.3

(1.45)

Lagrangian formulation

One may also obtain a Lagrangian L(φ, ∂0 φ) that leads to the Hamiltonian (1.44) by the usual manipulations. Firstly, the momentum canonically conjugated to φ(x) should be given by Π(x) ≡

δL . δ∂0 φ(x)

(1.46)

For this to be consistent with the first Hamilton-Jacobi equation, the Lagrangian must contain the following kinetic term Z (1.47) L = d3 x 21 (∂0 φ(x))2 + · · · The missing potential term of the Lagrangian is obtained by requesting that we have Z H = d3 x Π(x)∂0 φ(x) − L . (1.48)

10

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

This gives the following Lagrangian, Z  L = d3 x 21 (∂µ φ(x))(∂µ φ(x)) − 21 m2 φ2 (x) . Note that the action. Z S = dx0 L ,

(1.49)

(1.50)

is a Lorentz scalar (this is not true of the Hamiltonian, which may be considered as the time component of a 4-vector from the point of view of Lorentz transformations). The Lagrangian (1.49) leads to the following Euler-Lagrange equation of motion,
x + m2 φ(x) = 0 , (1.51)

which is known as the Klein-Gordon equation. This equation is of course equivalent to the pair of Hamilton-Jacobi equations derived earlier. c sileG siocnarF

1.2.4

Noether’s theorem

Conservation laws in a physical theory are intimately related to the continuous symmetry of the system. This is well known in Lagrangian mechanics, and can be extended to quantum field theory. Consider a generic Lagrangian L(φ, ∂µ φ) that depends on fields and their derivatives with respect to the spacetime coordinates, and assume that the theory is invariant under the following variation of the field, φ(x)

→

φ(x) + ε Ψ(x) .

(1.52)

Such an invariance is said to be continuous when it is valid for any value of the infinitesimal parameter ε. If the Lagrangian is unchanged by this transformation, we can write 0 = = =

∂L ∂L εΨ + ε∂µ Ψ δL = ∂φ ∂(∂µ φ)   ∂L ∂L ∂µ εΨ + ε∂µ Ψ µ ∂(∂ φ) ∂(∂µ φ)   ∂L Ψ . ε ∂µ ∂(∂µ φ) | {z }

(1.53)

Jµ

In the second line, we have used the Euler-Lagrange equation obeyed by the field. The 4-vector Jµ is known as the Noether current associated to this symmetry. The fact

1. BASICS OF Q UANTUM F IELD T HEORY

11

that the variation of the Lagrangian is zero implies the following continuity equation for this current ∂ µ Jµ = 0 .

(1.54)

This is the simplest case of Noether’s theorem, where the Lagrangian itself is invariant. But for the theory to be unmodified by the transformation of eq. (1.52), it is only necessary that the action be invariant, which is also realized if the Lagrangian is modified by a total derivative, i.e. δL = ε Kµ .

(1.55)

(The proportionality to ε follows from the fact that the variation must vanish when ǫ → 0.) When the variation of the Lagrangian is a total derivative instead of zero, the continuity equation is modified into:
∂µ Jµ − Kµ = 0 , (1.56)

where Jµ is the same current as before. As we shall see later, there are situations where a conservation equation such as (1.54) is violated by quantum effects, due to a delicate interplay between the symmetry responsible for the conservation law and the ultraviolet structure of the theory. c sileG siocnarF

1.3 Interacting scalar fields 1.3.1

Interaction term

Until now, we have only considered non-interacting particles, which is of course of very limited use in practice. That the Hamiltonian (1.34) does not contain interactions follows from the fact that the only non-trivial term it contains is of the form a†p ap , that destroys a particle of momentum p and then creates a particle of momentum p (hence nothing changes in the state of the system under consideration). By momentum conservation, this is the only allowed Hermitean operator which is quadratic in the creation and annihilation operators. Therefore, in order to include interactions, we must include in the Hamiltonian terms of higher degree in the creation and annihilation operators. The additional term must be Hermitean, since H generates the time evolution, which must be unitary. The simplest Hermitean addition to the Hamiltonian is a term of the form Z λ HI = d3 x φn (x) , (1.57) n! where n is a power larger than 2. The real constant λ is called a coupling constant and controls the strength of the interactions, while the denominator n! is a symmetry

12

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

factor that will prove convenient later on. At this point, it seems that any degree n may provide a reasonable interaction term. However, theories with an odd n have an unstable vacuum, and theories with n > 4 are non-renormalizable in four space-time dimensions, as we shall see later. For these reasons, n = 4 is the only case which is widely studied in practice, and we will stick to this value in the rest of this chapter. With this choice, the Hamiltonian and Lagrangian read H= L=

Z

Z

d3 x d3 x

1

2 2 Π (x)

1

+ 12 (∇φ(x))2 + 21 m2 φ2 (x) +

2 (∂µ φ(x))(∂

µ

φ(x)) − 21 m2 φ2 (x) −



λ 4 4! φ (x)



λ 4 4! φ (x)

,

, (1.58)

and the Klein-Gordon equation is modified into
λ x + m2 φ(x) + φ3 (x) = 0 . 6

1.3.2

(1.59)

Interaction representation

A field operator that obeys this non-linear equation of motion can no longer be represented as a linear superposition of plane waves such as (1.38). Let us assume that the coupling constant is very slowly time-dependent, in such a way that lim

x0 →±∞

λ=0.

(1.60)

What we have in mind here is that λ goes to zero adiabatically at asymptotic times, i.e. much slower than all the physically relevant timescales of the theory under consideration. Therefore, at x0 = ±∞, the theory is a free theory whose spectrum is made of the eigenstates of the free Hamiltonian. Likewise, the field φ(x) should be in a certain sense “close to a free field” in these limits. In the case of the x0 → −∞ limit, let us denote this by9 lim

x0 →−∞

φ(x) = φin (x) ,

(1.61)

where φin is a free field operator that admits a Fourier decomposition similar to eq. (1.38), φin (x) ≡

Z

h i d3 p † +ip·x −ip·x a e + a e . p,in p,in (2π)3 2Ep

(1.62)

9 In this equation, we ignore for now the issue of field renormalization, onto which we shall come back later (see the section 1.9).

1. BASICS OF Q UANTUM F IELD T HEORY

13

Eq. (1.61) can be made more explicit by writing φ(x) = U(−∞, x0 ) φin (x) U(x0 , −∞) ,

(1.63)

where U is a unitary time evolution operator defined as a time ordered exponential of the interaction term in the Lagrangian, evaluated with the φin field: Z t2 dx0 d3 x LI (φin (x)) , (1.64) U(t2 , t1 ) ≡ T exp i t1

where λ φ4 (x) . LI (φ(x)) ≡ − 4!

(1.65)

This time evolution operator satisfies the following properties U(t, t) = U(t3 , t1 ) = U(t1 , t2 ) =

1 U(t3 , t2 ) U(t2 , t1 ) (for all t2 ) −1 † U (t2 , t1 ) = U (t2 , t1 ) .

(1.66)

One can then prove that h i λ (x +m2 )φ(x)+ φ3 (x) = U(−∞, x0 ) (x +m2 )φin (x) U(x0 , −∞) . (1.67) 6

This equation shows that φin obeys the free Klein-Gordon equation if φ obeys the non-linear interacting one, and justifies a posteriori our choice of the unitary operator U that connects φ and φin .

1.3.3

In and Out states

The in creation and annihilation operators can be used to define a space of eigenstates  of the free Hamiltonian, starting from a ground state (vacuum) denoted 0in . For instance, one particle states would be defined as   pin = a† 0in . (1.68) p,in The physical interpretation of these states is that they are states with a definite particle content at x0 = −∞, before the interactions are turned on10 .

In the same way as we have constructed in field operators, creation and annihilation operators and states, we may construct out ones such that the field φout (x) is a 10 For an interacting system, it is not possible to enumerate the particle content of states, because of quantum fluctuations that may temporarily create additional virtual particles.

14

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

free field that coincides with the interacting field φ(x) in the limit x0 → +∞ (with  0out , we the same caveat about field renormalization). Starting from a vacuum state  may also define a full set of states, such as pout , that have a definite particle content at x0 = +∞. It is crucial to observe that the in and out states are not identical:  

   0out 6= 0in (they differ by the phase 0out 0in ) , pout 6= pin , · · · (1.69)

Taking the limit x0 → +∞ in eq. (1.63), we first see that11 ap,out = U(−∞, +∞) ap,in U(+∞, −∞) ,

a†p,out = U(−∞, +∞) a†p,in U(+∞, −∞) ,

(1.70)

from which we deduce that the in and out states must be related by   αout = U(−∞, +∞) αin .

(1.71)

The two sets of states are identical for a free theory, since the evolution operator reduces to the identity in this case. c sileG siocnarF

1.4 LSZ reduction formulas Among the most interesting physical quantities are the transition amplitudes

 q1 q2 · · · out p1 p2 · · · in , (1.72)

whose squared modulus enters in cross-sections that are measurable in scattering experiments. Up to a normalization factor, the square of this amplitude gives the probability that particles with momenta p1 p2 · · · in the initial state evolve into particles with momenta q1 q2 · · · in the final state.

A first step in view of calculating transition amplitudes is to relate them to expectation values involving the field operator φ(x). In order to illustrate the main steps in deriving such a relationship, let us consider the simple case of the transition amplitude between two 1-particle states, 

qout pin . (1.73)  Firstly, we write the state |pin as the action of a creation operator on the corresponding vacuum state, and we replace the creation operation by its expression in terms of φin ,  



= qout a†p,in 0in qout pin Z 

(1.74) = −i d3 x e−ip·x qout Πin (x) + iEp φin (x) 0in . 11 The evolution operator from x0 = −∞ to x0 = +∞ is sometimes called the S-matrix: S ≡ U(+∞, −∞).

15

1. BASICS OF Q UANTUM F IELD T HEORY

Next, we use the fact that φin , Πin are the limits when x0 → −∞ of the interacting fields φ, Π, and we express this limit by means of the following trick: lim

x0 →−∞

0

F(x ) =

lim

x0 →+∞

0

F(x ) −

Z +∞

dx0 ∂x0 F(x0 ) .

(1.75)

−∞

The term with the limit x0 → +∞ produces a term identical to the r.h.s. of the first line of eq. (1.74), but with an a†p,out instead of a†p,in . At this stage we have

 qout pin

=



 0out aq,out a†p,out 0in Z 

+i d4 x ∂x0 e−ip·x qout Π(x) + iEp φ(x) 0in . (1.76)

In the first line, we use the commutation relation between creation and annihilation operators to obtain 

(1.77) 0out aq,out a†p,out 0in = (2π)3 2Ep δ(p − q) .

This term does not involve any interaction, since the initial state particle simply goes through to the final state (in other words, this particle just acts as a spectator in the process). Such trivial terms always appear when expressing transition amplitudes in terms of the field operator, and they are usually dropped since they do not carry any interesting physical information. We can then perform explicitly the time derivative in the second line to obtain12 Z  . 



qout pin = i d4 x e−ip·x (x + m2 ) qout φ(x) 0in , (1.78)

. where we use the symbol = to indicate that the trivial non-interacting terms have been dropped.

Next, we repeat the same procedure for the final state particle: (i) replace the annihilation operator aq,out by its expression in terms of φout , (ii) write φout as a limit of φ when x0 → +∞, (iii) write this limit as an integral of a time derivative plus a term at x0 → −∞, that we rewrite as the annihilation operator aq,in :

 qout pin

. =

Z

 i d4 x e−ip·x (x + m2 ) 0out aq,in φ(x) 0in Z 


+i d4 y ∂y0 eiq·y 0out Π(y) − iEq φ(y) φ(x) 0in . (1.79)

12 We use here the dispersion relation p2 − p2 = m2 of the incoming particle to arrive at this expression. 0 The mass that should enter in this formula is the physical mass of the particles. This remark will become important when we discuss renormalization.

16

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

However, at this point we are stuck because we would like to bring the aq,in to the right where it would annihilate 0in , but we do not know the commutator between aq,in and the interacting field operator φ(x). The remedy is to go one step back, and note that we are free to insert a T-product in
Πout (y) − iEq φout (y) φ(x)

=

y0 →+∞

T



Π(y) − iEq φ(y) φ(x)

(1.80)

since the time y0 → +∞ is obviously larger than x0 . Then the boundary term at y0 → −∞ will automatically lead to the desired ordering φ(x) aq,in ,

 qout pin

. =

Z

i d4 x e−ip·x (x + m2 ) Z

+i d4 y ∂y0 eiq·y





 0out φ(x) aq,in 0in | {z } 0


0out T Π(y) − iEq φ(y) φ(x) 0in . (1.81)

Performing the derivative with respect to y0 , we finally arrive at

Z  . 

qout pin = i2 d4 xd4 y ei(q·y−p·x) (x+m2 )(y+m2 ) 0out T φ(x)φ(y) 0in . (1.82)

Such a formula is known as a (Lehmann-Symanzik-Zimmermann) reduction formula. The method that we have exposed above on a simple case can easily be applied to the most general transition amplitude, with the following result for the part of the amplitude that does not involve any spectator particle:

Z m  . m+n Y q1 · · · qn out p1 · · · pm in = i d4 xj e−ipi ·xi (xi + m2 ) i=1

×

ZY n

d4 yj eiqj ·xj (yj + m2 )

j=1

 × 0out T φ(x1 ) · · · φ(xm )φ(y1 ) · · · φ(yn ) 0in .

(1.83)

The bottom line is that an amplitude with m + n particles is related to the vacuum expectation value of a time-ordered product of m + n interacting field operators (a slight but important modification to this formula will be introduced in the section 1.9, in order to account for field renormalization). Note that the vacuum states on the left and on the right of the expectation value are respectively the out and the in vacua. c sileG siocnarF

1. BASICS OF Q UANTUM F IELD T HEORY

17

1.5 From transition amplitudes to reaction rates All experiments in particle physics amount to a measurement that answers the following question: given a certain setup that defines an initial state, how many reactions of a certain type occur per unit time? The concept of “reaction of a certain type” may vary widely depending on the number of criteria that are imposed on the final state for the reaction to be worth counting. For instance, one may consider the reaction e+ e− → anything, the reaction e+ e− → µ+ µ− , or even a reaction with the same particles in the initial and final states, but where in addition the final muons are required to have momenta in a certain range. As we have seen in the previous section, the LSZ reduction formulas express transition amplitudes between states with a definite particle content in terms of correlation functions of the field operators that are calculable in quantum field theory. The missing link to connect this to experimental measurements is an explicit formula relating reaction rates to these transition amplitudes.

1.5.1

Invariant cross-sections

Definition of a cross-section : In a scattering experiment such as those performed in a particle collider, the observed reaction rate results from a combination of some factors that depend on the accelerator design (the fluxes of particles in the colliding beams), and a factor that contains the genuine microscopic information about the reaction. In general, this microscopic input is given in terms of a quantity called a cross-section, that has the dimension of an area. Consider two colliding beams, containing particles of type 1 and 2, respectively. For simplicity, assume that the two beams have a uniform particle density, and let us denote S their common transverse area. If during the experiment, N1 particles of the first beam and N2 particles of the second beam fly by the interaction zone, the cross-section for the process 1 + 2 → F, where F is some final state, is the quantity σ12→F defined by  Number of times F is  N N 1 2 σ12→F . = S seen in the experiment

(1.84)

In this formula, the left hand side is measured experimentally, while in the right hand side the ratio N1 N2 /S depends only on the setup of the collider13 . Therefore, the cross-section can be obtained as the ratio of two known quantities. Note that the cross-section in general depends on the momenta p1,2 of the particles participating in the collision (and on the momenta of the particles in the final state F), but in a Lorentz covariant way, i.e. only through Lorentz scalars such as (p1 + p2 )2 . 13 In practice, the beam conditions are monitored by measuring in parallel the event rate of another reaction, whose cross-section is already accurately known.

18

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Normalization of 1-particle states : An important point in the determination of the cross-section is the normalization of the 1-particle states. We have Z  2

 (1.85) pin pin = d3 x x pin = 2Ep (2π)3 δ(0) . | {z } V

In the first equality, we have inserted set of position eigenstates in order to

a complete  highlight the interpretation of pin pin as the integral of the square of a wavefunction. The second equality follows from the canonical commutation relation between creation and annihilation operators. This equation means that our convention of normalization of the states corresponds to “2E particles per unit volume”. We are using quotes here because 2E does not have the correct dimension to be a proper density of particles. This is mostly an aesthetic problem: this convention of normalization will cancel out eventually, since cross-sections are defined in such a way that they do not depend on the incoming fluxes of particles. Example of a non-interacting theory : These normalization issues can be clarified by considering the trivial example of a non-interacting theory. In this case, the exact result for the transition amplitude between two 1-particle states is 

(1.86) qout pin = 2Ep (2π)3 δ(q − p) . By squaring this amplitude, we obtain 

qout pin 2 = 4Ep Eq V (2π)3 δ(q − p) .

(1.87)

and integrating over q with the Lorentz invariant measure d3 q/((2π)3 2Eq ) gives Z 

 d3 q qout pin 2 = pin pin = initial number of particles . (1.88) 3 (2π) 2Eq

Since we are considering a non-interacting theory, we know without any calculation that every particle in the initial state should be present in the final state with the same momentum. Therefore, the integral in the left hand side of the previous equation is the number of particles in the final state, and the quantity 

qout pin 2

d3 q (2π)3 2Eq

(1.89)

counts those that have their momentum in a volume d3 q centered around q. More generally, for an n-particle final state, n

 Y q1 · · · qn out · · · in 2 j=1

d3 qj (2π)3 2Eqj

(1.90)

is the number of events where the final state particles have their momenta in the volume d3 q1 · · · d2 qn centered on (q1 , · · · , qn ).

19

1. BASICS OF Q UANTUM F IELD T HEORY

General squared amplitude : Consider now a transition from a 2 amplitude

 particle state to a final state with n particles, q1 · · · qn out p1 p2 in . By momentum conservation, all the contributions to this amplitude are proportional to a delta function,

Xn  q1 · · · qn out p1 p2 in ≡ (2π)4 δ p1 +p2 −

j=1

and its squared modulus reads



 2 q1 · · · qn out p1 p2 in

=


qj T(q1,··· ,n |p1,2 ) , (1.91)

Xn
(2π)4 δ p1 + p2 − qj j=1 2 × (2π)4 δ(0) T(q1,··· ,n |p1,2 ) . | {z }

(1.92)

VT

This expression contains the square of the delta function. One of these factors becomes a delta of zero, which has the interpretation of space-time volume VT in which the process takes place. Since the initial state contains a fixed number of particles of each kind (1 and 2) per unit volume in all space, we expect the total number of events to be extensive, because interactions may happen in all the volume at any time. This is the meaning of the factor VT that appears in this square. From the insight gained by studying the non-interacting theory, this square weighted by the Lorentz invariant phase-space measure of the final state counts the number of events in which the final state particles have momenta in the volume d3 q1 · · · d2 qn centered on (q1 , · · · , qn ): Number of events n

 2 Y = q1 · · · qn out p1 p2 in j=1

d3 qj (2π)3 2Eqj

2 Xn = VT T(q1,··· ,n |p1,2 ) (2π)4 δ p1 + p2 −

j=1

|

{z

qj

dΓn (p1,2 )

n
Y j=1

d3 qj . (2π)3 2Eqj } (1.93)

(dΓn (p1,2 ) is the invariant final state measure subject to the constraint of momentum conservation.) Cross-section in the target frame : At this point, the relationship with the crosssection of this transition is most easily established in the rest frame of one of the initial state particles, e.g. the particle 2 (this frame is called the target frame). Consider a thin

20

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

v1T

ℓ

S v1

2

Figure 1.1: Geometry of a two-body cross-section in the target frame. The two volumes represent the particles that can take part in the reaction in the duration T .

1

slice of this target, of transverse section S and infinitesimal thickness ℓ, as shown in the figure 1.1. The interaction volume is the volume of this target, i.e. V = Sℓ. Given the normalization of the 1-particle states, it contains N2 = 2m2 Sℓ particles of type 2. In the target frame, the particles 1 have a velocity v1 = p1 /Ep1 . Therefore, within a time interval T , N1 = 2Ep1 Sv1 T = 2p1 ST of them travel through the interaction zone. Using eqs. (1.84) and (1.93), we thus obtain the following expression for the cross-section in the target frame14 Z 2 1 σ12→1···n target = dΓn (p1,2 ) T(q1,··· ,n |p1,2 ) . (1.94) 4m2 p1 frame

Note that this is the total cross-section for a final state with n particles, since we have integrated over all the final state momenta. By undoing some of these integrations, we may obtain a cross-section which is differential with respect to some of the kinematical variables that characterize the final state (e.g., the angle at which a final state particle is scattered). Cross-section in the center of momentum frame : Another important frame in view of the setup of many experiments in particle physics is the center of momentum frame. This is the observer’s frame in experiments where two beam of same-mass particles and equal energies are collided head on. In this frame, we have p1 + p2 = 0 ,

E1 = E2 ,

s ≡ (p1 + p2 )2 = 4E21 .

(1.95)

Since s is Lorentz invariant, its expression in the rest frame of the particle 2 is ′ s = (m2 + E1′ )2 − p12 (in this paragraph, the primes indicate kinematical variables in the target frame). Moreover, simple kinematics show that the combination m2 p1′ in the target frame becomes √ (1.96) m2 p1′ = s p1

 dimensions, q1 · · · qn out p1 p2 in ∼ (mass)−(2+n) , T(q1,··· ,n |p1,2 ) ∼ (mass)2−n and dΓn ∼ (mass)2n−4 , and therefore this formula indeed gives an area. 14 Regarding

21

1. BASICS OF Q UANTUM F IELD T HEORY

in the center of momentum frame. Therefore, the expression of the cross-section in this frame reads σ12→1···n

center of momentum

1 = √ 4 s p1

Z

2 dΓn (p1,2 ) T(q1,··· ,n |p1,2 ) .

(1.97)

Likewise, obtaining the expression of a cross-section in a frame where the two beams have different momenta is a simple matter of relativistic kinematics (this is useful when the detector apparatus is neither the rest frame of one of the particles, nor the center of momentum frame, and one counts events in terms of some kinematical variable measured in this frame – alternatively, one may boost all the measured final state momenta in order to convert them to momenta in one of the above two frames). c sileG siocnarF

1.5.2

Decay rates

Another very common type of observable is the decay rate Γ of a particle, defined so that Γ dt is the decay probability of a particle at rest in the time interval dt. The decay rate can be obtained from matrix elements with a 1-particle initial state,

Xn  q1 · · · qn out p1 in ≡ (2π)4 δ p1 −

j=1


qj T(q1,··· ,n |p1 ) .

(1.98)

Squaring this matrix element again produces a space-time volume factor VT , and integrating over the invariant phase space of the final state particles gives ZY n

|

j=1

Z 2  2 d3 qj

= VT dΓ (p ) |p ) p q · · · q T(q . n in out 1 1,··· ,n 1 1 1 n (2π)3 2Eqj {z } {z } | Total number of decays

Decays per unit of time and volume

(1.99)

Given the normalization of the 1-particles states, a sample of volume V contains N1 = 2Ep1 V particles, and the average number of decays in the time interval T is therefore 2Ep1 Γ VT . From this, we get the following expression for the decay rate 1 Γ= 2Ep1

Z

2 dΓn (p1 ) T(q1,··· ,n |p1 ) .

(1.100)

A differential decay rate can be obtained by leaving some of the final state kinematical variables unintegrated. c sileG siocnarF

22

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.6 Generating functional 1.6.1

Definition

To facilitate the bookkeeping, it is useful to introduce a generating functional that encapsulates all the expectation values, by defining Z[j] ≡ = Note that

Z ∞ X 

1 d4 x1 · · · d4 xn ij(x1 ) · · · ij(xn ) 0out T φ(x1 ) · · · φ(xn ) 0in n! n=0 Z 

0out T exp i d4 x j(x)φ(x) 0in . (1.101)

 Z[0] = 0out 0in 6= 1

(1.102)

in an interacting theory (but if the vacuum state is stable, then this vacuum to vacuum transition amplitude must be a pure phase whose squared modulus is one). From this functional, the relevant expectation values are obtained by functional differentiation 

δn Z[j] 0out T φ(x1 ) · · · φ(xn ) 0in = . (1.103) iδj(x1 ) · · · iδj(xn ) j=0

The knowledge of Z[j] would therefore give access to all the transition amplitudes. However, it is in general not possible to derive Z[j] in closed form, and we need to resort to perturbation theory, in which the answer is obtained as an expansion in powers of the coupling constant. c sileG siocnarF

1.6.2

Relation to the free generating functional

The generating functional can be brought to a more useful form by first writing φ(x1 ) · · · φ(xn ) = U(−∞, x01 ) φin (x1 ) U(x01 , x02 ) φin (x2 ) · · · φin (xn ) U(x0n , ∞) . (1.104)

For convenience, we split the leftmost evolution operator as U(−∞, x01 ) = U(−∞, +∞) U(+∞, x01 ) .

(1.105)

Noticing that the formula (1.104) is true for any ordering of the times x0i and using the expression of the U’s as a time-ordered exponential, we have Z T φ(x1 ) · · · φ(xn ) = U(−∞, +∞) T φin (x1 ) · · · φin (xn ) exp i d4 x LI (φin (x)) ,

23

1. BASICS OF Q UANTUM F IELD T HEORY

(1.106) where the time-ordering in the right-hand side applies to all the operators on its right. This leads to the following representation of the generating functional

Z[j] =

=

Z h i  0out U(−∞, +∞) T exp i d4 x j(x)φin (x) + LI (φin (x)) 0in | {z }

0in   Z Z 

δ 4 exp i d x LI 0in T exp i d4 x j(x)φin (x) 0in . iδj(x) | {z }



Z0 [j]

(1.107)

This expression of Z[j] is the most useful, since it factorizes the interactions into a (functional) differential operator acting on Z0 [j], the generating functional for the non-interacting theory. c sileG siocnarF

1.6.3

Free generating functional

It turns out that the latter is calculable analytically. The main difficulty in evaluating Z0 [j] is to deal with the non-commuting objects contained in the exponential. A central mathematical result that we shall need is a particular case of the BakerCampbell-Hausdorff formula (see the section 4.1.5 for a derivation), if [A, [A, B]] = [B, [A, B]] = 0 ,

1

eA eB = eA+B e 2 [A,B] .

(1.108)

This formula is applicable here because commutators [a, a† ] are c-numbers that commute with everything else. In order to apply it, let us slice the time axis into an infinite number of small intervals, by writing

T exp

Z +∞ −∞

d4 x O(x) =

+∞ Y

i=−∞

T exp

Z x0i+1

d4 x O(x) ,

(1.109)

x0 i

where the intermediate times are ordered according to · · · x0i < x0i+1 < · · · . The product in the right hand side should be understood with the convention that the factors are ordered from left to right when the index i decreases. When the size ∆ ≡ x0i+1 − x0i of these intervals goes to zero, the time-ordering can be removed in

24

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

the individual factors15 : T exp

Z +∞ −∞

d4 x O(x) = lim+ ∆→0

+∞ Y

i=−∞

exp

Z x0i+1

d4 x O(x) .

(1.110)

x0 i

A first application of the Baker-Campbell-Hausdorff formula leads to Z

Z  T exp i d4 x j(x)φin (x) = exp i d4 x j(x)φin (x)

1Z   × exp − d4 xd4 y θ(x0 − y0 ) j(x)j(y) φin (x), φin (y) . 2 (1.111) Note that the exponential in the second line is a c-number.  In the end, we will need to evaluate the expectation value of this operator in the 0in vacuum state. Therefore, it is desirable to transform it in such a way that the annihilation operators are on the right and the annihilation operators are on the left. This can be achieved by writing φin (x)

=

(+)

φin (x) ≡ (−)

φin (x) ≡

(+)

(−)

φin (x) + φin (x) , Z d3 p a† e+ip·x , (2π)3 2Ep p,in Z d3 p ap,in e−ip·x , (2π)3 2Ep

(1.112)

and by using once again the Baker-Campbell-Hausdorff formula. We obtain Z

T exp i d4 x j(x)φin (x)

Z 

Z  (+) (−) = exp i d4 x j(x)φin (x) exp i d4 x j(x)φin (x)

1 Z  (+)  (−) × exp d4 xd4 y j(x)j(y) φin (x), φin (y) 2

1Z   × exp − d4 xd4 y θ(x0 − y0 ) j(x)j(y) φin (x), φin (y) . 2 (1.113) 15 Field



operators commute for space-like intervals,  O(x), O(y) = 0 if (x − y)2 < 0 .

Moreover, when ∆ → 0, the separation between any pair of points x, y with x0i < x0 , y0 < x0i+1 is always space-like.

1. BASICS OF Q UANTUM F IELD T HEORY

25

The operator that appears in the right hand side of the first line is called a normalordered exponential, and is denoted by bracketing the exponential between a pair of colons (: · · · :): Z

Z 

Z  (+) (−) 4 : exp i d x j(x)φin (x) : ≡ exp i d x j(x)φin (x) exp i d4 x j(x)φin (x) . 4

(1.114)

A crucial property of the normal ordered exponential is that its in-vacuum expectation value is equal to unity: Z 

0in : exp i d4 x j(x)φin (x) : 0in = 1 . (1.115) Therefore, we have proven that the generating functional of the free theory is a Gaussian in j(x),

1Z  Z0 [j] = exp − d4 xd4 y j(x)j(y) G0F (x, y) , (1.116) 2

where G0F (x, y) is a 2-point function called the free Feynman propagator and defined as    (+)  (−) G0F (x, y) = θ(x0 − y0 ) φin (x), φin (y) − φin (x), φin (y) .

1.6.4

(1.117)

Feynman propagator

Since the commutators in the right hand side of eq. (1.117) are c-numbers, we can also write G0F (x, y) = =

   (+)   (−) 0in θ(x0 − y0 ) φin (x), φin (y) − φin (x), φin (y) 0in 

0in T φin (x)φin (y) 0in . (1.118)

In other words, the free Feynman propagator is the in-vacuum expectation value of the time-ordered product of two free fields. Using the Fourier mode decomposition of φin and the commutation relation between creation and annihilation operators, the Feynman propagator can be rewritten as follows G0F (x, y) =

Z

 d3 p θ(x0 − y0 ) e−ip·(x−y) + θ(y0 − x0 ) e+ip·(x−y) . 3 (2π) 2Ep (1.119)

26

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

In the following, we will also make an extensive use of the Fourier transform of this propagator (with respect to the difference of coordinates xµ − yµ , since it is translation invariant): e 0 (k) ≡ G F =

Z

d4 (x − y) eik·(x−y) G0F (x, y) Z0 Z  0 0 1 +∞ 0 i(k0 −Ek )z0 dz0 ei(k +Ek )z .(1.120) dz e + 2Ek 0 −∞

The remaining Fourier integrals over z0 are not defined as ordinary functions. Instead, they are distributions, that can also be viewed as the limiting value of a family of ordinary functions. In order to see this, let use write Z +∞ 0

Likewise Z0

−∞

0

dz0 eiaz = lim+ ǫ→0

0

dz0 eiaz = lim+ ǫ→0

Z +∞

0

dz0 ei(a+iǫ)z =

0

Z0

i . a + i0+

0

dz0 ei(a−iǫ)z = −

+∞

i . a − i0+

(1.121)

(1.122)

Therefore, the Fourier space Feynman propagator reads e 0 (k) = G F

k2

i . − m2 + i0+

(1.123)

e 0 (k) is Lorentz invariant. Henceforth, G0 (x, y) is also Lorentz invariant16 . Note that G F F It is sometimes useful to have a representation of eq. (1.123) in terms of distributions. This is provided by the following identity:   i 1 + πδ(z) , (1.124) = i P z + i0+ z where P(1/z) is the principal value of 1/z (i.e. the distribution obtained by cutting out –symmetrically– an infinitesimal interval around z = 0). As far as integration over the variable z is concerned, this prescription amounts to shifting the pole slightly below the real axis, or equivalently to going around the pole at z = 0 from above (the 16 This is somewhat obfuscated by the fact that the step functions θ(±(x0 − y0 )) that enter in the definition of the time-ordered product are not Lorentz invariant. The Lorentz invariance of time-ordered products follows from the following properties:

• if (x − y)2 < 0, then the two fields commute and the time ordering is irrelevant,

• if (x − y)2 ≥ 0, then the sign of x0 − y0 is Lorentz invariant.

27

1. BASICS OF Q UANTUM F IELD T HEORY

term in πδ(z) can be viewed as the result of the integral on the infinitesimally small half-circle around the pole):

z

z

i0 +

0

From eq. (1.123), it is trivial to check that G0F (x, y) is a Green’s function of the operator x + m2 (up to a normalization factor −i): (x + m2 ) G0F (x, y) = −iδ(x − y) .

(1.125)

Strictly speaking, the operator x +m2 is not invertible, since it admits as zero modes all the plane waves exp(±ik · x) with an on-shell momentum k20 = k2 + m2 . The i0+ prescription in the denominator of eq. (1.123) amounts to shifting infinitesimally the zeroes of k20 = k2 + m2 in the complex k0 plane, in order to have a well defined inverse. The regularization of eq. (1.123) is specific to the time-ordered propagator. Other regularizations would provide different propagators; for instance the free retarded propagator is given by e 0 (k) = G R

i (k0 + i0+ )2 − (k2 + m2 )

.

(1.126)

One can easily check that its inverse Fourier transform is a function G0R (x, y) that satisfies (x + m2 ) G0R (x, y) = −iδ(x − y) , G0R (x, y) = 0 if x0 < y0 .

(1.127)

In other words, G0R is also a Green’s function of the operator x + m2 , but with boundary conditions that differ from those of G0F . c sileG siocnarF

1.7 Perturbative expansion and Feynman rules The generating functional Z[j] is usually not known analytically in closed form, but is given indirectly by eq. (1.107) as the action of a functional differential operator

28

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

that acts on the generating functional of the free theory. The latter is a Gaussian in j, whose variance is given by the free Feynman propagator G0F . Although not explicit, this formula provides a straightforward method for obtaining vacuum expectation values of T-products of fields to a given order in the coupling constant λ. c sileG siocnarF

1.7.1

Examples

Let to order λ1 the following two functions: first 

us  illustrate

this by computing 0out 0in and 0out T φ(x)φ(y) 0in . In order to make the notations a bit lighter, we denote G0xy ≡ G0F (x, y). At order one in λ, we have

 0out 0in

= =

"

#  4 Z λ δ 4 2 d z + O(λ ) Z0 [j]|j=0 Z[0] = 1 − i 4! iδj(z) Z λ 1−i d4 z G0zz2 + O(λ2 ) , (1.128) 8

and

 0out T φ(x)φ(y) 0in # "  4 Z δ δ2 Z0 [j] λ 2 4 + O(λ ) 2 d z = 1−i 4! iδj(z) i δj(x)δj(y) j=0 Z Z λ λ d4 z G0zz2 − i d4 z G0xz G0zz G0zy + O(λ2 ) = G0xy − i G0xy 8 2 Z i h λ d4 z G0zz2 + O(λ2 ) = 1−i 8 | {z } Z[0]

×

h

G0xy

λ −i 2

Z

i d4 z G0xz G0zz G0zy + O(λ2 ) .

(1.129)

Although the final expressions at order one are rather simple, the intermediate steps are quite cumbersome due to the necessity of taking a large number of functional  derivatives. Moreover, the expression of the 2-point function 0out T φ(x)φ(y) 0in becomes simpler after we notice that one can factor out Z[0]. This property is in fact completely general; all transition amplitudes contain a factor Z[0]. From the remark made after eq. (1.102), this factor is a pure phase and its squared modulus is one and will have no effect in transition probabilities. Therefore, it would be desirable to identify from the start the terms that lead to this prefactor, to avoid unnecessary calculations. c sileG siocnarF

29

1. BASICS OF Q UANTUM F IELD T HEORY

1.7.2

Diagrammatic representation

This simplification follows a quite transparent rule if we represent the above expressions diagrammatically, by introducing the following notation G0xy ≡ x

y .

(1.130)

The functions considered above can be represented as follows: Z[0] = 1 +

1 8

+ O(λ2 )

z

 0out T φ(x)φ(y) 0in = x + 81 x

y

z

y

+

1 x 2

z

y

+ O(λ2 ) .

(1.131)

The graphs that appear in the right hand side of these equations are called Feynman diagrams. By adding to eq. (1.130) the rule that each vertex should have a factor −iλ and an integration over the entire space-time, then these graphs are in one-to-one correspondence with the expressions of eqs. (1.128) and (1.129). For now, we have recalled explicitly the numerical prefactors (1/8, 1/2,...) but they can in fact be recovered simply from the symmetries of the graphs. In the second of eqs. (1.131), the second term of the right hand side contains a factor which is not connected to any of the points x and y. These disconnected graphs are precisely the ones responsible for the factor Z[0] that appears in all transition amplitudes. We can therefore disregard these type of graphs altogether. c sileG siocnarF

1.7.3

Feynman rules

The diagrammatic representation of eqs. (1.131) can in fact be used to completely bypass the explicit calculation of the functional derivatives of Z0 [j]. The rules that p govern this construction are called Feynman  of order λ to a

rules. The contributions n-point time-ordered product of fields 0out Tφ(x1 ) · · · φ(xn ) 0in can be obtained as follows: 1. Draw all the graphs (with only vertices of valence 4) that connect the n points x1 to xn and have exactly p vertices. Graphs that contain a subgraph which is not connected to any of the xi ’s should be ignored. 2. Each line of a graph represents a free Feynman propagator G0F . 3. Each vertex represents a factor −iλ and an integral over the space-time coordinate assigned to this vertex.

30

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS 4. The numerical prefactor for a given graph is the inverse of the order of its discrete symmetry group. As an illustration, we indicate below the generators of these symmetry groups and their order for the graphs that appear in eqs. (1.131):

z

x

z

y

−→

order 8

−→

1 , 8

−→

order 2

−→

1 . 2

(1.132)

Note that this rule for obtaining the symmetry factor associated to a given graph is correct only if the corresponding term in the Lagrangian has been properly symmetrized. For instance, the operator φ4 should appear in the Lagrangian with a prefactor 1/4!.

1.7.4

Connected graphs

At the step 1, graphs made of several disconnected subgraphs can usually appear in certain functions, provided that each subgraph is connected to at least one of the points xi . For instance, a 4-point function contains a piece which is simply made of the product of two 2-point functions. In addition, it contains terms that correspond to a genuine 4-point function, not factorizable in a product of 2-point functions. The factorizable pieces are usually less interesting because they can be recovered from already calculated simpler building blocks17 . For this reason, it is sometimes useful to introduce the generating function of the connected graphs, denoted W[j]. This functional is very simply related to Z[j] by W[j] = log Z[j] .

(1.133)

To give a glimpse of this identity, let us write W[j] =

Z ∞ X 1 d4 x1 · · · d4 xn Cn (x1 , · · · , xn ) j(x1 ) · · · j(xn ) , n!

(1.134)

n=1

17 Moreover, in scattering amplitudes, these disconnected contributions are not physically interesting. For instance, if an m + n → p + q amplitude factorizes into the product of two sub-amplitudes (m → p and n → q, respectively), then the corresponding sub-processes can happen at very distant locations in space-time, which is usually not what one wants.

31

1. BASICS OF Q UANTUM F IELD T HEORY

where the Cn (x1 , · · · , xn ) are n-point functions whose diagrammatic representation contain only connected graphs. If we expand Z[j] = exp W[j], we obtain Z Z[j] = 1 + d4 x C1 (x) j(x) Z i h 1 d4 xd4 y C2 (x, y) + C1 (x)C1 (y) j(x)j(y) + | {z 2!  }

0out T φ(x)φ(y) 0in Z h 1 d4 xd4 yd4 z C3 (x, y, z) + C2 (x, y)C1 (z) + 3! +C2 (y, z)C1 (x) + C2 (z, x)C1 (y) i +C1 (x)C1 (y)C1 (z) j(x)j(y)j(z) {z |  }

0out T φ(x)φ(y)φ(z) 0in +···

(1.135)

This expansion highlights how the vacuum expectation values of time-ordered products of fields can be factorized into products of connected contributions. c sileG siocnarF

1.7.5

Feynman rules in momentum space

Until now, we have obtained Feynman rules in terms of objects that depend on spacetime coordinates, leading to expressions for the perturbative expansion of the vacuum expectation value of time-ordered products of fields. However, in most practical applications, we need subsequently to use the LSZ reduction formula (1.83) to turn these expectation values into transition amplitudes. This involves the application of the operator i( + m2 ) to each external point, and a Fourier transform. Firstly, note that thanks to eq. (1.125), the application of i( + m2 ) simply removes the external line to which it is applied: # " (x + m2 )

z

x

=

x

.

(1.136)

Thus, these operators just produce Feynman graphs that are amputated of all their external lines. Then, the Fourier transform can be propagated to all the internal lines of the graph, leading to an expression that involves propagators and vertices that depend only on momenta. The Feynman rules for obtaining directly these momentum space expressions are: 1 ′ . The graph topologies that must be considered is of course unchanged. The momenta of the initial state particles are entering into the graph, and the momenta of the final state particles are going out of the graph

32

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

2 ′ . Each line of a graph represents a free Feynman propagator in momentum space e 0 (k) G F 3 ′ . Each vertex represents a factor −iλ(2π)4 δ(k1 + · · · + k4 ), where the ki are the four momenta entering into this vertex

3 ′′ . All the internal momenta that are not constrained by these delta functions should be integrated over with a measure d4 k/(2π)4 4 ′ . Symmetry factors are computed as before. For instance, these rules lead to:

=

P p2

q2

k

= p1

q1

Z

i d4 k 4 2 (2π) k − m2 + i0+ Z (−iλ)2 d4 k i i = . 4 2 2 + 2 (2π) k −m +i0 (p1 +p2 −k)2 −m2 +i0+ −i

λ 2

(1.137)

1.7.6

Counting the powers of λ and h ¯

The order in λ of a (connected) graph G is of course related to the number of vertices nV in the graph, G ∼ λnV .

(1.138)

This can also be related to the number of loops of the graph, which is a better measure of its complexity since it determines how many momentum integrals it contains. Let us denote nE the number of external lines, nI the number of internal lines and nL the number of loops. These parameters are related by the following two identities: 4nV

=

2nI + nE

nL

=

nI − nV + 1 .

(1.139)

The first of these equations equates the number of “handles” carried by the vertices, and the number of propagator endpoints that must attached to them. The right hand side of the second equation counts the number of internal momenta that are not constrained by the delta functions of momentum conservation carried the vertices (the +1 comes from the fact that not all these delta functions are independent - a linear combination of them must simply tell that the sum of the external momenta must be

1. BASICS OF Q UANTUM F IELD T HEORY

33

zero, and therefore does not constrain the internal ones in any way). From these two identities, one obtains nV = nL − 1 +

nE , 2

(1.140)

and the order in λ of the graph is also G ∼ λnL −1+nE /2 .

(1.141)

According to this formula, the order of a graph depends only on the number of external lines nE (i.e. on the number of particles involved in the transition amplitude under consideration), and on the number of loops. Thus, the perturbative expansion is also a loop expansion, with the leading order being given by tree diagrams, the first correction in λ by one-loop graphs, etc... It turns out that the number of loops also counts the order in the Planck constant h ¯ of a graph. Although we have been using a system of units in which h ¯ = 1, it is easy to reinstate h ¯ by the substitution Z

1 S x + m2 λ 4  S → = − d4 x φ(x) φ(x) + φ (x) . (1.142) h ¯ 2 h ¯ 4!¯h From this, we see that h ¯ enters in the Feynman rules as follows Propagator : Vertex :

ih ¯ , p2 − m2 + i0+ λ −i , h ¯

(1.143)

and the order in h ¯ of a graph is given by G ∼h ¯ nI −nV ∼ h ¯ nL −1 .

(1.144)

Therefore, each additional loop brings a power of h, ¯ and the loop expansion can also be viewed as an expansion in powers of h. ¯ c sileG siocnarF

1.8 Calculation of loop integrals 1.8.1

Wick’s rotation

Let us consider the first of the examples given in eq. (1.137) and define Z λ d4 k i −iΣ(P) ≡ −i . 2 (2π)4 k2 − m2 + i0+

(1.145)

34

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

k0

-Ep+i0 +

Figure 1.2: Wick rotation in the complex k0 plane.

-i0 +

Ep

In order to calculate the momentum integral, it is useful to perform a Wick rotation, in which we rotate the k0 integration axis by 90 degrees to bring it along the imaginary axis, as illustrated in the figure 1.2. The integrals along the horizontal and vertical axis are opposite because the shaded domain does not contain any of the poles of the Feynman propagator, and because the propagator vanishes as k−2 0 when |k0 | → ∞. The integral along the vertical axis amounts to writing k0 = −iκ with κ varying from −∞ to +∞. After this transformation, the integral of eq. (1.145) becomes λ Σ(P) = 2

Z

1 d4 kE , (2π)4 k2E + m2

(1.146)

where kE is the Euclidean 4-vector defined by kiE = k (i = 1, 2, 3) and k4E = κ, with squared norm k2E = k2 + κ2 . c sileG siocnarF

1.8.2

Volume element in D dimensions

When the integrand depends only on the norm |kE |, we can separate the radial integration on |kE | from the angular integration over the orientation of the vector in 4dimensional Euclidean space. In D dimensions, the volume measure for a rotationally invariant integrand reads dD kE = D VD (1) kED−1 dkE ,

(1.147)

where VD (kE ) is the volume of the D-dimensional ball of radius kE . These volumes can be determined recursively by Zπ V1 (kE ) = 2kE , VD (kE ) = kE dθ sin θ VD−1 (kE sin θ) . (1.148) 0

35

1. BASICS OF Q UANTUM F IELD T HEORY Therefore, we have V2 (kE ) = πk2E ,

4π 3 k , 3 E

V3 (kE ) =

V4 (kE ) =

π2 4 k . 2 E

(1.149)

Although knowing V4 (kE ) is sufficient for performing a radial momentum integral in four dimensions, it is interesting to have the formula for an arbitrary dimension, in view of applications to dimensional regularization. More generally, we have VD+1 (1) = VD (1) π1/2

1.8.3

Γ(D 2 + 1) Γ(D 2

+

3 2)

and VD (1) =

2 πD/2 . D Γ(D 2)

(1.150)

Feynman parameterization of denominators

Let us now consider the second diagram of eq. (1.137) (with the notation P ≡ p1 +p2 ), −iΓ4 (P) ≡

Z (−iλ)2 d4 k i i . 2 (2π)4 k2 −m2 +i0+ (P − k)2 −m2 +i0+

(1.151)

In this more complicated example, an extra difficulty is that the integrand is not rotationally invariant. The following trick, known as Feynman parameterization can be used to rearrange the denominators18 : 1 = AB

Z1 0

dx . [xA + (1 − x)B]2

(1.152)

The denominator resulting from this transformation is x(k2 −m2 +i0+ )+(1−x)((P−k)2 −m2 +i0+ ) = l2 −m2 −∆(x, P)+i0+ , (1.153) where we denote l ≡ k − (1 − x)P and ∆(x, P) ≡ −x(1 − x)P2 . At this point, we can apply a Wick rotation19 to the shifted integration variable l, in order to obtain Γ4 (P) = −

λ2 2

Z1 0

dx

Z

1 d4 lE , (2π)4 [l2E + m2 + ∆(x, P)]2

(1.154)

where the integrand is again invariant by rotation in 4-dimensional Euclidean space. c sileG siocnarF

18 For

n denominators, this formula can be generalized into Z1 X 1 1 = Γ (n) dx1 · · · dxn δ(1 − . xi ) A1 A2 · · · An [x A + · · · + xn An ]n 1 1 0 i

19 It is allowed because the integration axis can be rotated counterclockwise without passing through the poles in the variable l0 .

36

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.9 K¨allen-Lehmann spectral representation As we shall see now, the limit in eq. (1.61) that relates the interacting field φ and the free field of the interaction picture φin is too naive. One of the consequences is that we will have to make a slight modification to the reduction formula (1.83). Consider the time-ordered 2-point function,

 0out T φ(x)φ(y) 0in

=



θ(x0 − y0 ) 0out φ(x)φ(y) 0in 

+θ(y0 − x0 ) 0out φ(y)φ(x) 0in .

(1.155)

For each of the expectation values in the right hand side, let us insert an identity operator between the two field operators, written in the form of a sum over all the possible physical states, X  λ λ . (1.156) 1= states λ

The states λ can be arranged into classes inside which the states differ only by a boost. A class of states, that we will denote α, is characterized by its particle content and by the relative momenta of these particles. Within a class, the total momentum of can be varied by applying a Lorentz boost. For a class α, we will denote the state  αp the state of total momentum p. Each class of states has an invariant mass mα , such that the total energy p0 and total momentum p of the states in this class obey p20 − p2 = m2α . In addition, it is useful to isolate the vacuum in the sum over the states. Therefore, the identity operator can be rewritten as X Z   d3 p αp α p , p 1 = 0 0 + (1.157) (2π)3 2 p2 + m2α classes α where we have written the integral over the total momentum of the states in a Lorentz invariant fashion. (We need not specify if we are using in or out states here.)

When we insert this identity operator between the two field operators, the vacuum does not contribute. For instance 

0out φ(x) 0 = 0 . (1.158)

(φ creates or destroys a particle, and therefore has a vanishing matrix element between ^ we can write vacuum states.) Using the momentum operator P,

 0out φ(x) αp

= = =

^  ^ 0out eiP·x φ(0)e−iP·x αp 

0out φ(0) αp e−ip·x 

0out φ(0) α0 e−ip·x .

(1.159)

37

1. BASICS OF Q UANTUM F IELD T HEORY

The second line uses the fact that the total momentum in the vacuum state is zero, and is p for the state αp . In the last equality, we have applied a boost that cancels the total momentum p, and used the fact that the vacuum is invariant, as well as the scalar field φ(0). Therefore, we obtain the following representation for the time-ordered 2-point function

 0out T φ(x)φ(y) 0in =

X



  0out φ(0) α0 α0 φ(0) 0in

classes α 

d3 p p θ(x0 − y0 )e−ip·(x−y) + θ(y0 − x0 )eip·(x−y) , × (2π)3 2 p2 + m2α {z } | Z

G0 (x,y;m2 α) F

(1.160)

where the underlined integral, G0F (x, y; m2α ), is the Feynman propagator for a hypothetical scalar field of mass mα (compare this integral with eq. (1.119)). It is customary to rewrite the above representation as Z∞ 

dM2 0out T φ(x)φ(y) 0in = ρ(M2 ) G0F (x, y; M2 ) , (1.161) 2π 0 where ρ(m2 ) is the spectral function defined as X  

δ(M2 − m2α ) 0out φ(0) α0 α0 φ(0) 0in . (1.162) ρ(M2 ) ≡ 2π classes α This function describes the invariant mass distribution of the non-empty states of the theory under consideration, and the exact Feynman propagator is a sum of free Feynman propagators with varying masses, weighted by this mass distribution.

In a theory of massive particles, the spectral function has a delta function corresponding to states containing a single particle of mass m, and a continuum distribution20 that starts at the minimal invariant mass (2m) of a 2-particle state: ρ(M2 ) = 2π Z δ(M2 − m2 ) + continuum for M2 ≥ 4m2 ,

(1.163)

where Z is the product of matrix elements that appear in eq. (1.162), in the case of 1-particle states. In a theory with interactions, Z in general differs from unity (in fact, it may be infinite). Note that in this equation, m must be the physical mass of the particles, as it would be inferred from the simultaneous measurement of their energy and momentum. As we shall see shortly, this is not the same as the parameter we denoted m in the Lagrangian. 20 Between the 1-particle delta function and the 2-particle continuum, there may be additional delta functions corresponding to multi-particle bound states (to have a stable bound state, the binding energy should decrease the mass of the state compared to the mass 2m of two free particles at rest).

38

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Taking the Fourier transform of eq. (1.161) and using eq. (1.163) for the spectral function, we obtain the following pole structure for the exact Feynman propagator: e (p) = G F

iZ + terms without poles . p2 − m2 + i0+

(1.164)

Therefore, the parameter Z that appears in the spectral function has also the interpretation of the residue of the single particle pole in the exact Feynman propagator. The fact that Z 6= 1 calls for√a slight modification of the LSZ reduction formulas.  Eq. (1.163) implies that a factor Z appears in the overlap between the state φ(x) 0in  and the 1-particle state pin . In other words, φ(x) creates a particle with probability Z rather than 1. Therefore, there should be a factor Z−1/2 for each incoming and outgoing particle in the LSZ reduction formulas that relate transition amplitudes to products of fields φ:

 m+n  . i q1 · · · qn out p1 · · · pm in = √ Z ZY m n Y 4 −ipi ·xi 2 × d xj e (xi + m ) d4 yj eiqj ·xj (yj + m2 ) i=1

j=1

 × 0out T φ(x1 ) · · · φ(xm )φ(y1 ) · · · φ(yn ) 0in .

(1.165)

In practical calculations, the factor Z at a given order of perturbation theory is obtained by studying the 1-particle pole of the dressed propagator, as the residue of this pole. It is common to introduce a renormalized field φr defined as a rescaling of φ, φ≡

√

Z φr .

(1.166)

By construction, the Feynman propagator defined from the 2-point time-ordered product of φr has a single-particle pole of residue 1. In other words, we may replace in the right hand side of the LSZ reduction formula (1.165) all the fields by renormalized fields, and at the same time remove all the factors Z−1/2 . c sileG siocnarF

1.10 Ultraviolet divergences and renormalization Until now, we have not attempted to calculate explicitly the integrals over the Euclidean momentum kE in eqs. (1.146) and (1.154). In fact, these integrals do not converge when |kE | → ∞, and as such they are therefore infinite. These infinities are called ultraviolet divergences. c sileG siocnarF

39

1. BASICS OF Q UANTUM F IELD T HEORY

1.10.1

Regularization of divergent integrals

As we shall see shortly, this has very deep implications on how we should interpret the theory. However, before we can discuss this, it is crucial to make the integrals temporarily finite in order to secure the subsequent manipulations. This procedure, called regularization, amounts to altering the theory to make all the integrals finite. There is no unique method for achieving this, and the most common ones are the following: • Pauli-Villars method : modify the Feynman propagator according to i k2 − m2 + i0+

i i − . (1.167) k2 − m2 + i0+ k2 − M2 + i0+

→

When |kE | ≫ M, this modified propagator decreases as |kE |−4 instead of |kE |−2 for the unmodified propagator, which is usually sufficient to render the integrals convergent. The original theory (and its ultraviolet divergences) are recovered in the limit M → ∞.

• Lattice regularization : replace continuous space-time by a regular lattice of points, for instance a cubic lattice with a spacing a between the nearest neighbor sites. On such a lattice, the momenta are themselves discrete, with a maximal momentum of order a−1 . Therefore, the momentum integrals are replaced by discrete sums that are all finite. The original theory is recovered in the limit a → 0. A shortcoming of lattice regularization is that the discrete momentum sums are usually much more difficult to evaluate than continuum integrals, and that it breaks the usual space-time symmetries such as translation and rotation invariance. This is nevertheless the basis of numerical Monte-Carlo methods (lattice field theory). • Cutoff regularization : cut the integration over the norm of the Euclidean momentum by |kE | ≥ Λ. The underlying theory is recovered in the limit Λ → ∞. This is a commonly used regularization in scalar theories, due to its simplicity and because it preserves all the symmetries of the theory.

• Dimensional regularization : this method is based on the observation that the integral Z∞ 0

dkE

kED−1 2 [kE + ∆]n

= =

1 2

Z∞

D

du

0 D ∆ 2 −n

2

u 2 −1 [u + ∆]n

Z1

D

D

dx xn− 2 −1 (1−x) 2 −1 (1.168) 0 {z! } | ! Γ

n−

D 2

Γ

Γ (n)

D 2

40

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS is well defined for almost any D except for D = 2n, 2n + 2, 2n + 4, · · · and D = 0, −2, −4, · · · thanks to the analytical properties of the Gamma function21 . Dimensional regularization keeps the number of space-time dimensions D arbitrary in all the intermediate calculations, and at the end one usually writes D = 4 − 2ǫ with ǫ ≪ 1. This regularization does not break any of the symmetries of the theory, including gauge invariance (which is not the case of cutoff regularization). There is an extra complication: the coupling constant λ is a priori dimensionless only when D = 4. In order to keep the dimension of λ unchanged, we must introduce a parameter µ that has the dimension of a mass, and replace λ by λµ4−D . Note that the field φ(x) has the dimension of a mass to the power (D − 2)/2. Setting D = 4 − 2ǫ, the singular part of the integrals Σ(P) and Γ4 (P) introduced above as examples is λ m2 1 + O(1) , 2 (4π)2 ǫ λ2 1 1 Γ4 (P) = − + O(1) . 2 (4π)2 ǫ

Σ(P) = −

1.10.2

(1.169)

Mass renormalization

Let us now make a few observations: • The above divergent terms are momentum independent22 , • They appear in 2-point and 4-point functions only. Moreover, it is important to realize that the parameters (m2 and λ) in the Lagrangian are not directly observable quantities by themselves23 . For instance, the mass of a particle is a measurable property of the particle (e.g. by measuring both its energy and its momentum, via p20 − p2 ). In quantum field theory, this definition of the mass corresponds to the location of the poles of the propagator in the complex p0 plane. However, as we shall see, loop corrections modify substantially the propagator, and it turns out that the parameter m in the free propagator has in fact little to do with this physical mass. If we dress the propagator by summing the multiple insertions of the 1-loop correction −iΣ,

e (P) ≡ P G F

P +

P +

P +

+ ... ,

21 Γ (z) is analytic in the complex plane, at the exception of a discrete series of simple poles, located at zn = −n for n ∈ , with residues (−1)n /n!. 22 These examples are not completely general. As we shall see later, divergent terms proportional to P 2 may also appear in the 2-point function. 23 In this regard, it is important to realize that the renormalization of the parameters of the Lagrangian would be necessary even in a theory that has no divergent loop integrals.

◆

1. BASICS OF Q UANTUM F IELD T HEORY

41 (1.170)

we obtain e (P) = G F

i , p20 − p2 − m2 − Σ + i0+

(1.171)

from which it is immediate to see that this loop correction alters the location of the pole, now given by p20 − p2 =

2 |m {z+ Σ}

.

(1.172)

new squared mass

Since the propagator given in eq. (1.171) includes loop corrections, its poles ought to give a value of the mass closer to the physical one. Therefore, it is tempting to write: m2phys = m2 + Σ + O(λ2 ) .

(1.173)

Of course, since Σ is infinite, the only way this can be satisfied is that the parameter m2 that appears in the Lagrangian be itself infinite, with an opposite sign in order to cancel the infinity from Σ. To further distinguish it from the physical mass, the parameter m in the Lagrangian is usually called the bare mass, while mphys is the physical –or renormalized– mass. c sileG siocnarF

1.10.3

Field renormalization

Note that the 1-loop function Σ in a theory with a φ4 interaction is somewhat special, because at this order it is independent of the momentum P. Being a constant, the infinity it contains can be absorbed entirely into a redefinition of the bare mass, but the residue of the pole remains equal to 1. However, starting at two loops, the 2-point functions that correct the propagator are usually momentum dependent, as is the case for instance with this graph:

It is convenient to expand Σ(P2 ) around the physical mass: Σ(P2 ) = Σ(m2phys ) + (P2 − m2phys ) Σ ′ (m2phys ) + 21 (P2 − m2phys ) Σ ′′ (m2phys ) + · · · (1.174) e to have a pole at P2 = m2 , we need to impose For the resummed propagator G F phys m2phys = m2 + Σ(m2phys ) ,

(1.175)

42

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

that generalizes eq. (1.173) to a momentum-dependent Σ. Then, in the vicinity of the pole, the dressed propagator behaves as e (P) G F

≈

P 2 →m2 phys

i (1 −

Σ ′ (m2phys )) (P2

− m2phys ) + i0+

.

(1.176)

This indicates that the field renormalization factor Z cannot be equal to 1 when the propagator is corrected by a momentum-dependent loop. Instead, we have Z=

1 1−

Σ ′ (m2phys )

.

(1.177)

Moreover, Weinbergs’s theorem implies that the ultraviolet divergences of the 2-point function Σ(P2 ) arise only in Σ(m2phys ) and in the first derivative Σ ′ (m2phys ), while higher derivatives are all finite. Eqs. (1.175) and (1.177) therefore indicate that these infinities can be “hidden” in the bare mass m2 and in a multiplicative field renormalization factor Z. c sileG siocnarF

1.10.4

Ultraviolet power counting

From the above considerations, it appears crucial that Σ has divergences only in its 0th and 1st order Taylor coefficients and Γ4 only in the 0th order, in order to be able to absorb the divergences by a proper definition of m2 , Z and λ. A simple dimensional argument gives plausibility to this assertion (of which Weinberg’s theorem provides a more rigorous justification). Let us assume that we scale up all the internal momenta of a graph by some factor ξ. In doing this, a graph G with nV vertices and nI internal lines will scale as G ∼ ξD nL −2nI ,

(1.178)

assuming D space-time dimensions for more generality. The exponent ω(G) ≡ D nL − 2nI is called the superficial degree of divergence of the graph. This exponent characterizes how the graph diverges when all its internal momenta are rescaled uniformly: • ω(G) ≥ 0 : The graph has an intrinsic divergence. • ω(G) < 0 : The graph may be finite, or may contain a divergent subgraph. More precisely, the convergence theorem states that a graph G is finite if ω(G) < 0, and the degrees of divergence of all its subgraphs are negative as well. Of course, subgraphs do not always satisfy this condition. But in the renormalization process, the divergent subgraphs will have been dealt with at an earlier stage since they occur at a lower order of the perturbative expansion.

43

1. BASICS OF Q UANTUM F IELD T HEORY

The superficial degree of divergence signals all the n-point functions that may have ultraviolet divergences of their own (as opposed to being divergent because of a divergent subgraph). Using eqs. (1.139), ω(G) can be rewritten in the following way ω(G) = 4 − nE + (D − 4) nL .

(1.179)

An important consequence of this formula is that in 4 dimensions the superficial degree of divergence of a graph does not depend on the number of loops, but only on the number of external lines. When D = 4, the only functions that have a non-negative ω are the 2-point function and the 4-point function24 . It is important to realize that this does not mean that a 6-point cannot be divergent. However, it can diverge only if it contains a divergent 2-point or 4-point subgraph. Moreover, the value of the superficial degree of divergence indicates the maximal power of the ultraviolet cutoff that may appear in these functions: • 2-point: up to Λ2 • 4-point: up to log(Λ) Note also that if we differentiate a graph with respect to the invariant norm P2 of one of its external momenta, we get   ∂G = 2 − nE + (D − 4) nL . (1.180) ω ∂P2 (ω further decreases by two units with each additional derivative with respect to P2 .) Therefore, the momentum derivative Σ ′ (P2 ) of the 2-point function has ω = 0 in D = 4, and its higher derivatives all have ω < 0. The fact that only Γ4 (m2phys ), Σ(m2phys ) and Σ ′ (m2phys ) have ω ≥ 0 is the reason why it is possible to get rid of all the divergences of this theory (in 4 dimensions) by a redefinition of the parameters of the Lagrangian. This theory is said to be renormalizable. c sileG siocnarF

1.10.5

Ultraviolet classification of quantum field theories

In dimensions lower than 4, ω(G) is a strictly decreasing function of the number of loops, which indicates that graphs with a given nE do not develop new divergences beyond a certain loop order. Such theories are said super renormalizable because they only have a finite number of divergent graphs. Conversely, in dimensions higher than 4, ω(G) increases with the number of loops, and any function will eventually 24 Functions with an odd number of external lines vanish in the theory under consideration. Note also that 0-point functions (vacuum graphs) have a superficial degree of divergence equal to 4, indicating that they may contain up to quartic divergences ∼ Λ4 .

44

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

become divergent at some loop order. These theories are usually25 non renormalizable. One may think of introducing, as they become necessary, additional operators in the Lagrangian with a coupling constant adjusted to cancel the new divergences that arise at a given loop order. However, an infinite number of such parameters would need to be introduced, thereby reducing drastically the predictive power of this type of theory26 . As we have seen, the renormalizability of a field theory depends both on the interaction terms it contains, and on the dimensionality of space-time. In fact, a simpler equivalent criterion is the mass dimension of the coupling constant in front of the interaction term: • dim > 0 : super-renormalizable, • dim = 0 : renormalizable, • dim < 0 : non-renormalizable. For instance, the “coupling constant” m2 in front of the mass term has always a mass dimension equal to two, and this term is therefore super-renormalizable. In contrast, the coupling constant λ in front of a φ4 interaction has a mass dimension 4 − D, and is (super)renormalizable in dimensions less than or equal to four. c sileG siocnarF

1.10.6

Renormalization in perturbation theory, Counterterms

A convenient setup for casting the renormalization procedure within perturbation theory is to write the bare Lagrangian, L=



1 λb 1 ∂µ φb ∂µ φb − m2b φ2b − φ4b , 2 2 4!

(1.181)

(here we denote φb , mb and λb the bare field, mass and coupling, to stress that they are not the physical ones) as the sum of a renormalized Lagrangian and a correction: L

=

Lr

≡

∆L

≡

Lr + ∆L

1 λr 1 ∂µ φr ∂µ φr − m2r φ2r − φ4r 2 2 4!
µ
1 1 1 2 ∆ ∂µ φr ∂ φr − ∆m φr − ∆λ φ4r . 2 Z 2 4!

(1.182)

25 It may happen that an internal symmetry, such as a gauge symmetry, renders a function finite while its superficial degree of divergence is non negative. 26 Non-renormalizable field theories may nevertheless be used as low energy effective field theories, where they approximate below a certain cutoff a more fundamental –possibly unknown– theory supposedly valid above the cutoff.

45

1. BASICS OF Q UANTUM F IELD T HEORY

Lr contains the renormalized (i.e. physical) mass mr and coupling constant λr (the latter may be defined from the measurement of some cross-section chosen as reference).√In ∆L, the coefficients ∆Z , ∆m , ∆λ are called counterterms. Recalling that φb = Z φr , the bare and physical parameters and the counterterms must be related by ∆Z = Z − 1 ∆m = Zm2b − m2r ∆λ = Z2 λb − λr .

(1.183)

The terms in ∆L are treated as a perturbation to Lr , and one may introduce extra Feynman rules for the various terms it contains:

1 1 ∆ ∂µ φr ∂µ φr − ∆m φ2r 2 Z 2 −

1 ∆λ φ4r 4!

P

→

→

=

−i ∆Z P2 + ∆m

=

−i ∆λ




(1.184)

At tree level, only the term Lr is used, and by construction the physical quantities computed at this order will depend only on physical parameters. Higher orders involve divergent loop corrections. The counterterms ∆Z , ∆m , ∆λ should be adjusted at every order to cancel the new divergences that arise at this order. In particular, after having included the contribution of the counterterms, the self-energy Σ(P2 ) are usually required to satisfy the following conditions27 : Σ(m2r ) = 0 ,

Σ ′ (m2r ) = 0 .

(1.185)

With this choice, it is not necessary to dress the external lines with the self-energy in the LSZ reduction formulas for transition amplitudes. Indeed, the renormalization conditions (1.185) imply that i( + m2r ) GF = 1 ,

lim (−iΣ)GF = 0 .

p2 →m2 r

(1.186)

For each external line, the reduction formula contains an operator i(x + m2r ) acting on the corresponding external propagator. If this propagator is dressed, this gives 

i( + m2r ) GF + GF (−iΣ)GF + GF (−iΣ)GF (−iΣ)GF + · · · = 1 . (1.187) | {z } dressed propagator

Therefore, all the terms are zero except the first one, and we can ignore self-energy corrections on the external lines. c sileG siocnarF

27 Strictly

speaking, the only requirement is that the counterterms cancel the infinities, which does not fix uniquely their finite part. Various renormalization schemes are possible, that differ in how these finite parts are chosen.

46

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.10.7

BPHZ renormalization

The actual proof of renormalizability is more complicated than what this superficial discussion based on power counting may suggest. Indeed, a crucial aspect is to show that the divergences can be removed via the subtraction of local terms only, i.e. that the divergences are polynomial in the external momenta. While this is trivial in all the one-loop graphs we have considered, it is not obviously true beyond one-loop. As an illustration, let us consider the following example of a two-loop contribution to the 4-point function: k

1

3 l

2

. 4

A

B

C

In the graph A, the loop we have represented with a thicker line is divergent, and is multiplied by a non-polynomial function of P2 ≡ (p1 + p2 )2 coming from the rest of the graph. The Feynman rules give the following integrand for this graph: IA =

(−iλ)3 0 GF (k)G0F (k − P) G0F (l)G0F (l + k + p3 ) . 2

(1.188)

The superficial degree of divergence of the integration over l is ω(A; l) = 0 (at fixed k), and therefore the boldface loop is logarithmically divergent. The diagram B consists in subtracting from this loop a polynomial in its external momenta, whose degree is precisely equal to its superficial degree of divergence. Since ω(A; l) = 0, the subtraction is the zeroth order of the Taylor expansion of that loop (underlined in the following equation): IA+B =

i h (−iλ)3 0 GF (k)G0F (k−P) G0F (l)G0F (l+k+p3 )−G0F (l)G0F (l) . (1.189) 2

Now, the degree of divergence in l of the combination inside the square brackets is ω(A + B; l) = −1, and the integration over l is therefore convergent in four dimensions. After the momentum l has been integrated out, we are left with a function of k whose behaviour is k0 , up to logarithms, whose integral is thus divergent. Since the degree of divergence in k is ω(A + B; k) = 0, this overall divergence can again be removed by subtracting the zeroth order of the Taylor expansion with respect to the external momenta, i.e. IA+B+C

=

h i (−iλ)3 0 GF (k)G0F (k − P) G0F (l)G0F (l + k + p3 ) − G0F (l)G0F (l) 2 h i −G0F (k)G0F (k) G0F (l)G0F (l + k) − G0F (l)G0F (l)

. (1.190)

47

1. BASICS OF Q UANTUM F IELD T HEORY

After these two successive subtractions, we have obtained a function whose integral on both k and l is completely finite. Moreover, at each step, we have subtracted only quantities that are polynomial in the external momenta of the corresponding loop (with a degree equal to the superficial degree of divergence of the loop). This recursive procedure for constructing a subtracted integrand is known as BogoliubovParasiuk-Hepp-Zimmermann renormalization. c sileG siocnarF

1.11 Spin 1/2 fields 1.11.1

Dimension-2 representation of the rotation group

In ordinary quantum mechanics, the spin s is related to the dimension n of representations of the rotation group by n = 2s + 1 .

(1.191)

Thus, spin 1/2 corresponds to representations of dimension 2. Such a representation is based on the (Hermitean) Pauli matrices: ! ! ! 0 1 0 −i 1 0 1 2 3 , (1.192) σ = , σ = , σ = 1 0 i 0 0 −1 from which we can construct the following unitary 2 × 2 matrices
U ≡ exp − 2i θi σi .

(1.193)

That the Pauli matrices (up to a factor 2) are generators of the Lie algebra of rotations can be seen from 

 Ji , Jj = i ǫijk Jk

1.11.2

with Ji ≡

σi . 2

(1.194)

Spinor representation of the Lorentz group

This idea can be extended to quantum field theory in order to encompass all the Lorentz transformations rather than just the spatial rotations. We are therefore seeking a dimension 2 representation of the commutation relations (1.10). Firstly, let us assume that we know a set of four n × n matrices γµ that satisfy the following anti-commutation relation:  µ ν γ , γ = 2 gµν 1n×n . (1.195)

48

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Such matrices are called Dirac matrices. From these matrices, it is easy to check that the matrices Mµν ≡

i  µ ν γ ,γ 4

(1.196)

form an n-dimensional representation of the Lorentz algebra. However, an exhaustive search indicates that the smallest matrices that fulfill eqs. (1.195) (in four space-time dimensions, i.e. for µ, ν = 0, · · · , 3) are 4 × 4. Several unitarily equivalent choices exist for these matrices. A possible representation (known as the Weyl or chiral representation) is the following28 ! ! 0 1 0 σi 0 i . (1.197) γ ≡ , γ ≡ −σi 0 1 0 In this representation, the generators for the boosts and for the rotations are ! ! 0 1 ijk σk 0 i σi 0i ij , M = ǫ M =− . (1.198) 2 0 −σi 2 0 σk Given a Lorentz transformation Λ defined by the parameters ωµν , let us define
i U1/2 (Λ) ≡ exp − ωµν Mµν . 2

(1.199)

A Dirac spinor is a 4-component field ψ(x) that transforms as follows: ψ(x)

→

U1/2 (Λ) ψ(Λ−1 x) .

(1.200)

In other words, the matrix U1/2 defines how the four components of this field transform under a Lorentz transformation (since these four components mix, ψ(x) is not the juxtaposition of four scalar fields). The fact that the lowest dimension for the Dirac matrices is 4 indicates that the spinor ψ(x) describes two spin-1/2 particles: a particle and its antiparticle, that are distinct from each other. c sileG siocnarF

1.11.3

Dirac equation and Lagrangian

Let us now determine an equation of motion obeyed by this field, such that it is invariant under Lorentz transformations. Since the Mµν ’s act only on the Dirac indices, a trivial answer could be the Klein-Gordon equation,
x + m2 ψ(x) = 0 . (1.201) 28 Although it is sometimes convenient to have an explicit representation of the Dirac matrices, most manipulations only rely on the fact that the obey the anti-commutation relations (1.195).

1. BASICS OF Q UANTUM F IELD T HEORY

49

But there is in fact a stronger equation that remains invariant when ψ is transformed according to eq. (1.200). Notice first that µ µ ν U−1 1/2 (Λ)γ U1/2 (Λ) = Λ ν γ .

(1.202)

This equation indicates that rotating the Dirac indices of γµ with U1/2 is equivalent to transforming the µ index as one would do for a normal 4-vector. Using this identity, we can check that under the same Lorentz transformation we have
iγµ ∂µ − m ψ(x)

→

Therefore, the Dirac equation,
iγµ ∂µ − m ψ(x) = 0 ,


U1/2 (Λ) iγµ ∂µ − m ψ(Λ−1 x) .

(1.203)

(1.204)

is Lorentz invariant. This equation implies the Klein-Gordon equation (to see it, apply the operator iγµ ∂µ + m on the left), and is therefore stronger. The Dirac matrices are not Hermitean. Instead, they satisfy
† γµ = γ0 γµ γ0 .

(1.205)

Therefore, the Hermitean conjugate of U1/2 (Λ) is U†1/2 (Λ) = exp



i i 0 ωµν (Mµν )† = γ0 exp ωµν Mµν γ0 = γ0 U−1 1/2 (Λ) γ . 2 2 (1.206)

Because of this, the simplest Lorentz scalar bilinear combination of ψ’s is ψ† γ0 ψ (instead of the naive ψ† ψ). It is common to denote ψ ≡ ψ† γ0 . From this, we conclude that the Lorentz scalar Lagrangian density that leads to the Dirac equation reads
L = ψ iγµ ∂µ − m ψ(x) . (1.207)

1.11.4

Basis of free spinors

Before quantizing the spinor field in a similar fashion as the scalar field, we need to find plane wave solutions of the Dirac equation. There are two types of solutions: ψ(x) = u(p) e−ip·x with (pµ γµ − m) u(p) = 0 , ψ(x) = v(p) e+ip·x with (pµ γµ + m) v(p) = 0 .

(1.208)

The solutions u(p) and v(p) each form a 2-dimensional linear space, and it is customary to denote a basis by us (p) and vs (p) (the index s, that takes two values

50

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

s = ±, is interpreted as the two spin states for a spin 1/2 particle). A convenient normalization of the base vectors is ur (p)us (p) = 2mδrs , vr (p)vs (p) = −2mδrs ,

u†r (p)us (p) = 2Ep δrs , v†r (p)vs (p) = 2Ep δrs ,

ur (p)vs (p) = vr (p)us (p) = 0 .

(1.209)

When summing over the spin states, we have: X X /+m, /−m, us (p)us (p) = p vs (p)vs (p) = p

(1.210)

s=±

s=±

/ ≡ pµ γµ . where we have introduced the notation p c sileG siocnarF

1.11.5

Canonical quantization

From the Lagrangian (1.207), the momentum canonically conjugated to ψ(x) is Π(x) = iψ† (x) .

(1.211)

Trying to generalize the canonical commutation relation of scalar field operators (1.40) would lead to   ψa (x), ψ†b (y) x0 =y0 = δ(x − y)δab , (1.212)

where we have written explicitly the Dirac indices a, b. However, by decomposing ψ(x) on a basis of plane waves by introducing creation and annihilation operators, 

XZ d3 p † +ip·x −ip·x ψ(x) ≡ , (1.213) a v (p)e + b u (p)e sp s sp s (2π)3 2Ep s=± one would find a Hamiltonian which is not bounded from below. The resolution of this paradox is that the commutation relation (1.212) is incorrect, and should be replaced by an anti-commutation relation,  ψa (x), ψ†b (y) x0 =y0 = δ(x − y)δab , (1.214)

which leads to anti-commutation relations for the creation and annihilation operators   arp , a†sq = brp , b†sq = (2π)3 2Ep δ(p − q)δrs . (1.215)

(All other combinations are zero.) These anti-commutation relations imply that the square of creation operators is zero, which means that it is not possible to have two particles with the same momentum and spin in a quantum state. This is nothing but the Pauli exclusion principle. This is the simplest example of the spin-statistics theorem, which states that half-integer spin particles must obey Fermi statistics. c sileG siocnarF

51

1. BASICS OF Q UANTUM F IELD T HEORY

1.11.6

Free spin-1/2 propagator

From eq. (1.213), we obtain the following expression for the free Feynman propagator of the Dirac field29 S0F (x, y) ≡



0 θ(x0 − y0 )ψa (x)ψb (y) − θ(y0 − x0 )ψb (y)ψa (x) 0 {z } | T (ψa (x)ψb (y))

=

Z

4

/ + m) i(p d p −ip·(x−y) . e 4 2 (2π) p − m2 + i0+ {z } |

(1.216)

S0 (p) F

The diagrammatic representation of this propagator is a line with an arrow: p

S0F (p)

1.11.7

=

.

(1.217)

LSZ reduction formula for spin-1/2

The LSZ reduction formula for transition amplitudes with fermions and/or antifermions in the initial and final states reads:

 m+n Z Z  . i 4 −ip·x qσ qσ · · · out ps ps · · · in = d x e d4 x e−ip·x · · · | {z } | {z } Z1/2 n particles

m particles

Z

Z → → / x −m) uσ (q)(−i ∂ / y +m) × d4 y e+iq·y d4 y e+iq·y · · · vs (p)(i ∂ 

× 0out T ψ(x)ψ(y)ψ(x)ψ(y) · · · 0in ←

←

/ x +m)us (p) (−i ∂ / y −m)vσ (q) , ×(i ∂

(1.218)

where we give examples for fermions and anti-fermions (indicated by a bar over the momentum and spin), both for the initial and final states. Besides the requirement that the external lines of the Feynman graphs should be amputated, this formula leads 29 We have introduced a minus sign in the definition of the time-ordered product of Dirac fields. One would have to mimic the derivation of the section 1.6 in order to see that this is the propagator that naturally appears in the generating functional for the amplitudes with fermions.

52

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

to the following prescriptions for the open ends of fermionic lines: Incoming fermion :

p

=

u(p)

Incoming anti-fermion :

p

=

v(p)

Outgoing fermion :

p

=

u(p)

Outgoing anti-fermion :

p

=

v(p) .

Note that when writing the expression corresponding to a given Feynman graph, the fermion lines it contains must be read in the direction opposite to the arrow carried by the lines. c sileG siocnarF

1.12 Spin 1 fields 1.12.1

Classical electrodynamics

The best known spin-1 particle is the photon. In classical electrodynamics, the electric field E and magnetic field B obey Maxwell’s equations, ∇·E=ρ

∇ × B − ∂t E = J

∇ × E + ∂t B = 0 ∇·B=0,

(1.219)

written here in terms of charge density ρ and current J. The local conservation of electrical charge implies the following continuity equation ∂t ρ + ∇ · J = 0 .

(1.220)

The last two Maxwell’s equations are automatically satisfied if we write the E, B fields in terms of potentials V and A, E ≡ ∂t A + ∇V

,

B ≡ −∇ × A .

(1.221)

53

1. BASICS OF Q UANTUM F IELD T HEORY

This representation is not unique, since E and B are unchanged if we transform the potentials as follows: V → V + ∂t χ ,

A → A − ∇χ ,

(1.222)

where χ is an arbitrary function of space and time. Eq. (1.222) is called a (Abelian) gauge transformation. Quantities that do not change under (1.222) are said to be gauge invariant. For instance, the electrical and magnetic fields are invariant. c sileG siocnarF

1.12.2

Classical electrodynamics in Lorentz covariant form

In order to make manifest the properties of Maxwell’s equations under Lorentz transformations, let us firstly rewrite them in covariant form. Introduce a 4-vector Aµ and a rank-2 tensor Fµν , Aµ ≡ (V, A) ,

Fµν ≡ ∂µ Aν − ∂ν Aµ .

(1.223)

(Fµν is called the field strength.) Recalling that ∂µ = (∂t , −∇), gauge transformations take the following form Aµ → Aµ + ∂µ χ ,

(1.224)

and Fµν is gauge invariant. Moreover, we see that Ei = F0i

Bi =

,

1 2

ǫijk Fjk .

(1.225)

If we also encapsulate ρ and J in a 4-vector, Jµ ≡ (ρ, J) ,

(1.226)

the first two Maxwell’s equations and the continuity equation read ∂µ Fµν = −Jν

,

∂µ J µ = 0 .

(1.227)

The last two Maxwell’s equations become ǫµνρσ ∂ν Fρσ = 0 .

(1.228)

(It is automatically satisfied thanks to the antisymmetric structure of Fµν .) A Lorentz scalar Lagrangian density whose Euler-Lagrange equations of motion are the Maxwell’s equations is 1 L ≡ − Fµν Fµν + Jµ Aµ . 4

(1.229)

54

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Because of the term Jµ Aµ that couples the potential to the sources, this Lagrangian density is not gauge invariant, but the action (integral of L over all space-time) is, provided that the current is conserved (i.e. satisfies the continuity equation). Indeed, we have Z Z d4 x Jµ Aµ → d4 x Jµ (Aµ + ∂µ χ) =

Z

4

µ

Z

d x J Aµ − d4 x χ ∂µ Jµ + | {z }

boundary . term

(1.230)

0

(The boundary term is zero if we assume that there are no sources at infinity.)

1.12.3

c sileG siocnarF

Canonical quantization in Coulomb gauge

Although it leads to Maxwell’s equations, the above Lagrangian has an unusual property, related to gauge invariance: the conjugate momentum of the potential A0 is identically zero, Π0 (x) ≡

δL =0. δ∂0 A0 (x)

(1.231)

Therefore, we cannot quantize electrodynamics simply by promoting the Poisson bracket between A0 and its conjugate momentum to a commutator. However, this problem is not intrinsic to quantum mechanics: the very same issue arises when trying to formulate classical electrodynamics in Hamilton form. The resolution of this problem is to fix the gauge, i.e. to impose an extra condition on the potential Aµ such that a unique Aµ corresponds to given E and B fields. Possible gauge conditions are: nµ Aµ = 0

Axial gauge :

(nµ is a fixed 4-vector) ,

µ

∂ Aµ = 0 , ∇·A=0.

Lorenz gauge : Coulomb gauge :

(1.232)

Let us illustrate this procedure in Coulomb gauge30 . Firstly, let us decompose the vector potential Ai into longitudinal and transverse components: Ai = Aik + Ai⊥ ,

(1.233)

with Aik ≡

∂i ∂j ∂2

Aj

,

 ∂i ∂j  Ai⊥ ≡ δij − 2 Aj . ∂

(1.234)

30 One may start from another gauge condition, and follow a similar line of reasoning in order to derive a quantized theory of the photon field in another gauge. However, as we shall see later, we can make the gauge fixing much more transparent by using functional quantization.

55

1. BASICS OF Q UANTUM F IELD T HEORY

The Coulomb gauge condition is equivalent to Aik = 0. The remaining components of Aµ are therefore A0 and the two components of Ai⊥ , in terms of which the Lagrangian reads: L

=

1 1 1 (∂t Ai⊥ )(∂t Ai⊥ ) − (∂j Ai⊥ )(∂j Ai⊥ ) + (∂i A0 )(∂i A0 ) 2 2 2 1 +(∂t Ai⊥ )(∂i A0 ) + (∂i Aj⊥ )(∂j Ai⊥ ) + J0 A0 − Ji Ai⊥ . 2

(1.235)

Note that the two underlined terms will vanish in the action, after an integration by parts (thanks to the transversality of Ai⊥ ). The Euler-Lagrange equation for the field A0 is ∂2 A0 = J0 ,

(1.236)

i.e. the Poisson equation with source term J0 . Note that this equation has no time derivative. Therefore, A0 reflects instantaneously the changes of the charge density J0 (this does not contradict special relativity, since A0 is not an observable – only E and B are). Ignoring all the terms that would vanish in the action upon integration by parts, we may thus rewrite the Lagrangian as L=

1 1 1 1 (∂t Ai⊥ )(∂t Ai⊥ ) − (∂j Ai⊥ )(∂j Ai⊥ ) − Ji Ai⊥ + J0 2 J0 , 2 2 2 ∂

(1.237)

and obtain the following Euler-Lagrange equation of motion for the field Ai⊥ :  ∂i ∂j   Ai⊥ = − δij − 2 Jj , ∂

(1.238)

i.e. a massless Klein-Gordon equation with the transverse projection of the charge current as source term. In this form, electrodynamics has no redundant degrees of freedom, and can now be quantized in the vacuum (J0 = Ji = 0) in the canonical way. Firstly, we define the momentum conjugated to Ai⊥ , Πi⊥ (x) ≡

δL δ ∂t Ai⊥ (x)

= ∂t Ai⊥ (x) .

(1.239)

Then, we promote Ai⊥ and Πi⊥ to quantum operators, and we impose on them the following canonical equal-time commutation relations,   ∂i ∂j  Ai⊥ (x), Πj⊥ (y) x0 =y0 = i δij − 2 δ(x − y) , ∂  i    A⊥ (x), Aj⊥ (y) x0 =y0 = Πi⊥ (x), Πj⊥ (y) x0 =y0 = 0 .



(1.240)

56

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

(In the first of these relations, the transverse projector in the right hand side follows from the fact that Ai⊥ and Πj⊥ are both transverse.) These commutation relations can be realized by decomposing Ai⊥ on a basis of solutions of the Klein-Gordon equation, i.e. plane waves: i X Z d3 p h i † +ip·x i∗ −ip·x ǫ (p) a e + ǫ (p) a e , (1.241) Ai⊥ (x) ≡ λp λ λp (2π)3 2|p| λ λ=1,2

where the two vectors ǫi1,2 (p) are polarization vectors orthogonal to p, p · ǫλ (p) = 0 .

(1.242)

In 3 spatial dimensions, a basis of such vectors has two elements, that we have labeled with λ = 1, 2. In addition, it is convenient to normalize the polarization vectors as follows ǫλ (p) · ǫ∗λ ′ (p) = δλλ ′

,

X

j ij ǫi∗ λ (p)ǫλ (p) = δ −

λ=1,2

pi pj . p2

(1.243)

With this choice, the commutation relations of eqs. (1.240) are equivalent to the following commutation relations between creation and annihilation operators: 

   aλp , aλ ′ q = a†λp , a†λ ′ q = 0 ,   aλp , a†λ ′ q = (2π)3 2|p| δλλ ′ δ(p − q) .

1.12.4

(1.244)

Feynman rules for photons

Eq. (1.241) can be inverted to obtain the creation and annihilation operators as a†λp

=

−i ǫi∗ λ (p)

Z

↔

d3 x e−ip·x ∂0 Ai⊥ (x) ,

Z ↔ aλp = +i ǫiλ (p) d3 x e+ip·x ∂0 Ai⊥ (x) ,

(1.245)

With these formulas, it is easy to derive the LSZ reduction formulas for photons in the initial and final states,

qλ ′ · · · out pλ · · · | {z } | {z }

n photons

Z

m photons

 . in =



i

Z1/2

m+n Z

d4 x e−ip·x ǫi∗ λ (p) x · · ·


 × d4 y e+iq·y ǫjλ ′ (q) y · · · 0out T Ai⊥ (x)Aj⊥ (y) · · · 0in .

(1.246)

57

1. BASICS OF Q UANTUM F IELD T HEORY

The free Feynman propagator of the photon (in Coulomb gauge) can be read off the quadratic part of the Lagrangian (1.237). In momentum space, it reads   i j p i δij − ppp2 j = . (1.247) G0F ij (p) = i p2 + i0+ The operator ǫiλ (p) x in the reduction formula simply amputates the external photon line to which it is applied31 . Transition amplitudes with incoming and outgoing photons are therefore given by amputated graphs, with a polarization vector contracted to the Lorentz index of each external photon. c sileG siocnarF

1.13 Abelian gauge invariance, QED So far, we have derived a quantized field theory for spin 1/2 fermions and a quantized field theory of photons (in the absence of charged sources), but they appear as unrelated constructions. The next step is to combine the two into a quantum theory of charged fermions that interact electromagnetically via photon exchanges. c sileG siocnarF

Global U(1) symmetry of the Dirac Lagrangian

1.13.1

Firstly, note that the fermion Lagrangian is invariant under the following transformation of the fermion field ψ

→

Ω† ψ ,

(1.248)

where Ω is a phase (i.e. an element of the group U(1)), provided that we consider only rigid transformations (i.e. independent of the space-time point x). By Noether’s theorem (see the section 1.2.4), this continuous symmetry corresponds to the existence of a conserved current, Jµ = ψ γµ ψ .

(1.249)

It is indeed straightforward to check from Dirac’s equation that ∂ µ Jµ = 0 . 31 Note



that δij −

pi pj  j ǫλ (p) = ǫiλ (p) . p2

Therefore, the transverse projectors attached to the external photon lines can be dropped.

(1.250)

58

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The physical interpretation of this current emerges from the spatial integral of the time component J0 , Z Q ≡ d3 x J0 (x) . (1.251) Using the Fourier mode decomposition (1.213) of the spinor ψ(x), we obtain the following expression:

 XZ d3 p asp a†sp + b†sp bsp Q = 3 (2π) 2Ep s=± Z 

X d3 p = b†sp bsp − a†sp asp + (infinite) constant . 3 (2π) 2Ep s=±

(1.252)

Thus, the operator Q counts the number of particles created by b† minus the number of particles created by a† . If we assign a charge +1 to the former and −1 to the latter, we can interpret Q as the operator that measures the total charge in the system. c sileG siocnarF

1.13.2

Minimal coupling to a spin-1 field

Secondly, the gauge transformation of the potential Aµ given in eq. (1.224) can also be written in the following form32 : Aµ with

→

Ω† Aµ Ω + i Ω† ∂µ Ω ,

Ω(x) ≡ e−i χ(x) .

(1.253)

(1.254)

When written in this form, the gauge transformation of the photon field appears to be also generated by the group U(1). Unlike the quantum field theory for fermions, the photon Lagrangian is invariant under local gauge transformations, i.e. where Ω depends on x in an arbitrary fashion. Therefore, at this point we have two disjoint quantum field theories: a theory of non-interacting charged fermions that has a global U(1) invariance, and a theory of non-interacting photons that has a local U(1) invariance, but no coupling between the two. Let us see what minimal modification would be necessary in order to promote the U(1) symmetry of the fermion sector into a local symmetry. An immediate obstacle is that Ω(x) ∂µ Ω† (x) 6= ∂µ .

(1.255)

32 Naturally, Ω† Aµ Ω = Aµ . We have used this somewhat more complicated form to highlight the analogy with the non-Abelian gauge theories that we will study later.

59

1. BASICS OF Q UANTUM F IELD T HEORY

Equivalently, the problem comes from the fact that the derivative ∂µ ψ does not transform in the same way as ψ itself when Ω depends on x. Instead, we have ∂µ ψ

→

∂µ Ω† ψ = Ω† ∂µ ψ + (∂µ Ω† ) ψ .

(1.256)

But we see that the second term can be connected to the variation of a photon field under the same transformation. This suggests that the combination (∂µ − iAµ )ψ has a simpler transformation law:
∂µ − iAµ ψ

→




 ∂µ − i Ω† Aµ Ω + iΩ† ∂µ Ω Ω† ψ

= Ω† ∂µ − iAµ ψ + Ω† Ω(∂µ Ω† ) + (∂µ Ω)Ω† ψ . {z } | ∂µ (ΩΩ† )=0

(1.257)

The operator Dµ ≡ ∂µ − iAµ is called a covariant derivative. The above calculation shows that ψDµ ψ is invariant under local gauge transformations. c sileG siocnarF

1.13.3

Abelian gauge theories

This observation is the basis of (Abelian) gauge theories: the minimal change to the Dirac Lagrangian that makes it locally gauge invariant introduces a coupling ψAµ ψ between two fermion fields and a spin-1 field such as the photon. The complete Lagrangian of this theory therefore reads: 1 / − m) ψ . L = − Fµν Fµν + ψ iD 4

(1.258)

We already know the Feynman rules for the photon and fermion propagators, and the prescription for external photon and fermion lines. The only additional Feynman rule is the following interaction vertex,

µ

= −iγµ ,

(1.259)

that can be read off directly from the Lagrangian. Quantum Electrodynamics (QED) is a quantum field theory that describes the interactions between electromagnetic radiation (photons) and charged particles (electrons and positrons for instance), whose Lagrangian is of the form (1.258). The only necessary generalization compared to the previous discussion is to introduce a

60

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

parameter e that represents the (bare) electrical charge of the electron, which leads to the following changes: Covariant derivative :

Dµ ≡ ∂µ − i e Aµ

Gauge transformation of the photon : Aµ → Ω† Aµ Ω + Electrical current : Photon-electron vertex :

i † µ Ω ∂ Ω e

e ψγµ ψ − i e γµ

. (1.260)

1.14 Charge conservation, Ward-Takahashi identities 1.14.1

Charge of 1-particle states

The charge operator Q defined in eq. (1.251) is invariant by translation in time (because Jµ is a conserved current) and in space (because it is integrated over all space). Since the current Jµ is a 4-vector, Q is also invariant under Lorentz transformations. Therefore Q conserves the energy and momentum of the states on which it acts. When acting on the vacuum state, one has  Q 0 = 0 . (1.261)  When acting on a 1-particle state αp , Q gives another state with the same 4momentum, and therefore the same invariant mass. But since single particle states are separated  in the spectral function of the theory,  from states with a higher occupancy Q |αp must in fact be proportional to αp itself,   Q |αp = qα,p αp . (1.262) In other words, 1-particle states are eigenvectors of the charge operator. Since Q is Lorentz invariant, the eigenvalue qα,p cannot depend on the momentum p (nor on the spin state of the particle), and it can only depend on the species of particle α. We will thus denote it qα , and call it the electrical charge of the particle of type α.

In theories with 1-particle states that do not correspond to the fundamental fields of the Lagrangian (e.g. composite bound states made of several elementary particles), one may go a bit further. The canonical anti-commutation relations imply  0    J (x), ψ(y) x0 =y0 = −e ψ(x) δ(x − y) , Q, ψ(y) = −e ψ(y) . (1.263) More generally, for any local function F(ψ(x), ψ† (x)), we have   Q, F(ψ(y), ψ† (y)) x0 =y0 = −e (n+ − n− ) F(ψ(y), ψ† (y))

(1.264)

61

1. BASICS OF Q UANTUM F IELD T HEORY

† where n+ is the number of ψ’s in F and n− the number  of ψ ’s. If we evaluate this identity between the vacuum and a 1-particle state αp , we obtain 

0 F(ψ(y), ψ† (y)) αp (qα − (n+ − n− )e) = 0 . (1.265)

Therefore, if the operator F(ψ, ψ† ) can create the particle α from the vacuum (i.e. the matrix element in the left hand side is non-zero), then we must have qα = (n+ − n− ) e .

(1.266)

In other words, the charge of the particle α is the number of ψ’s it contains, minus the number of ψ† ’s, times the electrical charge e of the field ψ (as it appears in the Lagrangian). The non-trivial aspect of this assertion comes from the fact that it does not depend on the (usually complicated and non-perturbative) interactions that produce the binding. So far, we have not discussed the renormalization of the parameter e. Its renormalized value er should be such that the covariant derivative retains its form33 when expressed in terms of the renormalized photon field Aµ r , i.e. ∂µ − i er Aµ r . Since the field

Aµ r 1/2

Aµ b = Z3

(1.267)

is related to the bare photon field

Aµ b

Aµ r ,

by (1.268)

the bare and renormalized charges must be related by −1/2

eb = Z3

er .

(1.269)

In combination with eq. (1.266), this means that the charges of all 1-particle states are −1/2 , regardless of the species of particle contained renormalized by the same factor Z3 in the state. For this to work, cancellations between various Feynman graphs are necessary. These cancellations are a consequence of the local gauge invariance of the theory, and in their simplest form they can be encapsulated in the Ward-Takahashi identities, that we shall derive now. c sileG siocnarF

1.14.2

Ward-Takahashi identities

Amplitudes with amputated external photon lines can be obtained as follows: Z Mµ1 µ2 ··· (q1 , q2 , · · · ) = d4 x1 d4 x2 · · · e−iq1 ·x1 e−iq2 ·x2 · · ·  

× βout T Jµ1 (x1 )Jµ2 (x2 ) · · · αin , (1.270)

33 The implicit assumption of this sentence is that the renormalization of QED preserves its local gauge invariance.

62

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where only electromagnetic currents appear inside the T-product, and all the external charged particles are kept in the initial and final states α and β (and are therefore on-shell). 1 Let us contract the Lorentz index µ1 with the momentum qµ 1 of the first photon. After an integration by parts, this reads

q1,µ1 M

µ1 µ2 ···

(q1 , q2 , · · · ) =

Z

−i d4 x1 d4 x2 · · · e−iq1 ·x1 e−iq2 ·x2 · · ·

  × 0out ∂µ1 T Jµ1 (x1 )Jµ2 (x2 ) · · · 0in . (1.271)

The derivative of the T-product involves two types of terms: (i) terms where the derivative acts directly on the current Jµ1 (x1 ), that are zero thanks to current conservation, and (ii) terms where it acts on the theta functions that order the times inside the T-product. With two currents, the latter term reads34    ∂ T Jµ (x)Jν (y) = δ(x0 − y0 ) J0 (x), Jν (y) = 0 . ∂xµ

(1.272)

This generalizes to more than two currents, and we therefore have quite generally q1,µ1 Mµ1 µ2 ··· (q1 , q2 , · · · ) = 0 .

(1.273)

The same property would hold for all the external photon lines of the amplitude. This equation is known as the Ward-Takahashi identity. A consequence of eq. (1.273) is that QED transition amplitudes are unchanged if the photon propagators or polarization vectors are modified by terms proportional to the momentum pµ , G0F µν (p) ǫµ λ (p)

→

→

G0F µν (p) + aµ pν + bν pµ µ ǫµ λ (p) + c p .

(1.274)

This is precisely the modification of the Feynman rules one would encounter by using a different gauge fixing in the quantization of the theory. Thus, the Ward-Takahashi identities imply the gauge invariance of the transitions amplitudes in QED. c sileG siocnarF

34 This step of the argument would fail if we had kept charged field operators inside the T-product, because their equal-time commutator with J0 is non-zero. Therefore, the Ward-Takahashi identities are valid provided all the external charged particles are on-shell, but there is no such requirement for the neutral external particles (e.g. the photons).

63

1. BASICS OF Q UANTUM F IELD T HEORY

1.15 Spontaneous symmetry breaking 1.15.1

Introduction

Until now, our discussion of the symmetry of a theory has been limited to a study of its Lagrangian or Hamiltonian, and we have tacitly assumed that the symmetry of the Lagrangian implies that the physics of this system exhibits the symmetry under consideration to its full extent. However, strictly speaking, a symmetric Lagrangian only implies that the corresponding equations of motion are symmetric, i.e. that a symmetry transformation applied to a solution of the equations of motion gives another solution. In other words, the symmetry of the Lagrangian implies that the set of the solutions of the equations of motion is symmetric, not that every individual solution is symmetric. A spontaneously broken symmetry is a symmetry of the Lagrangian which is not realized by the ground state. c sileG siocnarF

Let us first recall a standard result of quantum mechanics, that on the surface seems to forbid the possibility of non-symmetric ground states. Consider a quantum system of Hamiltonian H, which is also invariant under a discrete symmetry R such that R2 = 1 (such as a mirror symmetry). The Hamiltonian commutes with the symmetry generator,   R, H = 0 , (1.275)

which implies that H and R are diagonalizable simultaneously. Since R2 = 1, the eigenvalues of R are ±1, and the eigenstates of H are all either symmetric or antisymmetric under R.

φ

φ

φ

Figure 1.3: Left to right: potential for the Hamiltonians H0 , H0 + V and H0 + e. V +V

In order to see how this result is circumvented in quantum field theory, let us consider a simple explicit realization of this situation by a potential made of two infinite wells centered at φ = ±φ∗ , mirror symmetric with respect to φ = 0. Let us

64

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

denote H0 the Hamiltonian  of this system. Each of the wells has its own ground state, that we denote 0+ and 0− , respectively. They are degenerate in energy, transform into one another by the action of R, and have a vanishing overlap,  

 R 0+ = 0− , 0+ 0− = 0 . (1.276)

Then, we introduce a perturbation V, also mirror symmetric, such that the energy barrier between the two wells becomes finite (this interactions acts as a kind of coupling between the two wells). With this perturbation, we have h0+ |H0 + V|0+ i = h0− |H0 + V|0− i = a , h0+ |H0 + V|0− i = h0+ |V|0− i = b 6= 0 , h0− |H0 + V|0+ i = h0− |V|0+ i = b ,

(1.277)

with a, b real. With b 6= 0, the eigenstates of the full Hamiltonian   H ≡ H0 + V are no longer the states 0+ and 0− , but the combinations35 0+ ± 0− , whose eigenvalues are a ± b. These eigenstates are symmetric or anti-symmetric under R.

Note now that b is proportional  to the time derivative of the tunneling amplitude  between the states 0+ and 0= . Therefore, it is exponentially suppressed when the system has a large spatial volume, since the field must go from +φ∗ to −φ∗ in the entire volume during this transition. In fact, b is identically zero if the volume is infinite, and the two eigenstates |0+ i ± |0− i are degenerate. If this system is then

 e the diagonal matrix elements36 0± V e 0± perturbed by an anti-symmetric term V, of the perturbation are much larger than the splitting 2b between the eigenvalues of H. Therefore, under the effect of this perturbation, the ground state (now unique since e breaks the symmetry R) of the Hamiltonian is very close to one of the perturbation V  the original states 0± . c sileG siocnarF

1.15.2

Degenerate vacua with a continuous symmetry

Let us now return  to the case of a continuous symmetry. When the volume is infinite, a ground state v is characterized by the fact that it is en eigenstate of the momentum Pi with a null eigenvalue37 P |vi = 0 .

(1.278)

 this conclusion to hold, the matrix elements of H between 0± and the excited states should be negligible. Otherwise, the ground state of the perturbed Hamiltonian will be a more general linear combination of the eigenstates of H0 . 36 The non-diagonal matrix elements of V e are zero per our assumption that V e is odd under R. 37 Multiparticle states whose total momentum is zero can be excluded by the fact that they are separated from ground states by a finite threshold. 35 For

65

1. BASICS OF Q UANTUM F IELD T HEORY There is in general a whole set of such states, that we may choose as orthogonal, hu|vi = δuv .

(1.279) 

For any matrix element u A(x)B(0) v of the equal-time product of two local operators, we may insert a complete basis of states in order to get 

u A(x)B(0) v

=

X   u A(0) w w B(0) v

vacua w

+

Z

 

d3 p X u A(0) N, p N, p B(0) v e−ip·x , 3 (2π) N

(1.280)

 where we have  separated the ground states w from the continuum of populated states N, p (the label N – possibly continuous– distinguishes all those states that have the same total momentum p). To obtain this relationship, we have used the translation invariance of the ground states, and the fact that P is the generator of spatial translations. Since the states N, p belong to a continuum of states, the integral on the second line is smooth enough and vanishes when |x| → ∞ by Riemann’s lemma. Therefore, we have: X   

lim u A(x)B(0) v = u A(0) w w B(0) v . (1.281) |x|→+∞

vacua w

Likewise, we may prove: lim

|x|→+∞

X   

u B(0)A(x) v = u B(0) w w A(0) v .

(1.282)

vacua w

Causality implies that A(x)B(0) = B(0)A(x) since

the separation 

between the two points is space-like, so that the matrix elements u|A(0)|v and u|B(0)|v may be viewed as commuting Hermitean matrices, that we can diagonalize simultaneously. Moreover, since A and B are arbitrary local Hermitean operators, this property is in fact true for all such operators. By choosing properly the basis of the vacua when the volume is infinite, all the local Hermitean operators have vanishing matrix elements between distinct vacua:

 u|A(0)|v = δuv av . (1.283) Consequently, any local interaction term that breaks the symmetry responsible for the degeneracy of these vacua is diagonal  in this basis. Therefore, it lifts the degeneracy and promotes one of the states v to the status of true ground state of the system (instead of a symmetric linear combination of the v ’s). c sileG siocnarF

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.15.3

Conserved currents and charges

The fact that the Lagrangian is invariant under a continuous symmetry implies the existence of conserved currents Jµ a (x) such that ∂ µ Jµ a (x) = 0 .

(1.284)

Let us assume that the symmetry transformation is of the form φi (x)

→

φi (x) + i ǫa ta ij φj (x) ,

(1.285)

where the ta ij are the generators of the Lie algebra of the group or transformations, in the representation where the fields φi live. For the fields φi to be Hermitean, the numbers ta ij must be purely imaginary (this would be the case if the φi are in the adjoint representation of the Lie algebra). From Noether’s theorem, the conserved currents read: X δφi (x) δL . (1.286) Jµ a (x) = δ∂µ φi (x) δǫa i

By integrating over all space, we obtain conserved charges Z Qa (x0 ) ≡ d3 xJ0a (x0 , x) .

(1.287)

The time component of the currents has the form J0a (x) = i πi (x) ta ij φj (x) ,

(1.288)

where πi (x) is the canonical momentum associated with φi (x). Since the matrices ta have imaginary components, these currents are Hermitean, as well as the charges Qa . Using the canonical commutation relations,   φi (x), φj (y) x0 =y0 = 0 ,   πi (x), πj (y) x0 =y0 = 0 ,   φi (x), πj (y) x0 =y0 = i δij δ(x − y) , (1.289) we find the following equal-time commutator between components of the conserved currents:  0    b Ja (x), J0b (y) x0 =y0 = ta ij tkl πi (x)φj (x), πk (y)φl (y)   = i δ(x − y) ta , tb ij πi (x)φj (x) . (1.290) Using also the commutation relation that defines the Lie bracket,  a b t , t = i fabc tc ,

(1.291)

67

1. BASICS OF Q UANTUM F IELD T HEORY (the fabc are real numbers) we get  0  Ja (x), J0b (y) x0 =y0 = δ(x − y) fabc J0c (x) .

(1.292)

By integrating over the positions x and y, this becomes a commutator between the conserved charges,   Qa (x0 ), Qb (x0 ) = fabc Qc (x0 ) . (1.293) In other words, the charges Qa (x0 ) form a real representation of the Lie algebra. In addition, the commutator between the conserved charges and the field operators is given by38 : 

 Qa (x0 ), φi (x)

= = =

Z   i d3 y πk (y)ta kl φl (y), φi (x) x0 =y0 Z i d3 y(−i)δ(x − y)δki ta kl φl (x) ta ij φj (x) .

(1.294)

Note that the above commutation relations are not affected by the spontaneous breaking of symmetry, since they follow from the properties of the field operators, regardless of the nature of the ground state of system. c sileG siocnarF

1.15.4

Ground state

The ground state of the system is characterized by the expectation values of the field operators:



 φi ≡ 0|φi (x)|0 . (1.295)

In order to see whether the ground state is invariant under the of the symmetry

action  transformations, let us study the variation of the quantities φi :

 δ φi

=

= = =



 0|δφi (x)|0

 i ǫa ta ij 0|φj (x)|0  

 i ǫa 0 Qa (x0 ), φi (x) 0 

i ǫa 0 Qa φi (x) − φi (x)Qa 0 .

(1.296)

Thus, it is clear that these expectation values are invariant if the ground state is annihilated by the all the generators of the Lie algebra (i.e. if Qa 0 = 0 for all a). c sileG siocnarF

38 Since

the charges are conserved, we are free to evaluate them at the same time as the field φi .

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.15.5

Spectral properties

Consider now the expectation value in the ground state of the commutator between the conserved currents and the field operators:     

 µ 0 Ja (x), φi (y) 0 = 0 Jµ a (x − y), φi (0) 0 h XZ  

= d4 p δ(p − pN ) 0 Jµ a (x − y) N N φi (0) 0 N

 i

− 0 φi (0) N N Jµ a (x − y) 0 Z h X   ip·(x−y) d4 p δ(p − pN ) 0 Jµ = a (0) N N φi (0) 0 e N

  ip·(y−x) i

− 0 φi (0) N N Jµ . a (0) 0 e

(1.297)

 In the second line, we have summed over a complete set of states N , arranged according to their 4-momentum pN . We have also used the translation invariance of the ground state, and the properties of states with a definite momentum under translations. If we define µ i Fa,i (p) ≡ (2π)3

e µ (p) ≡ (2π)3 iF a,i

X N

X N

 

δ(p − pN ) 0 Jµ a (0) N N φi (0) 0

 

δ(p − pN ) 0 φi (0) N N Jµ a (0) 0 ,

(1.298)

we have
µ e µ (p) ∗ , Fa,i (p) = − F a,i

(1.299)

since Jµ a and φi are Hermitean. Moreover, Lorentz invariance implies that these objects have the following form: µ Fa,i (p) = pµ θ(p0 ) ρa,i (p2 ) , e µ (p) = pµ θ(p0 ) ρ ea,i (p2 ) , F a,i

(1.300)

ea,i are functions (so far unspecified) depending only where ρa,i and ρ on the invariant p2 . The factor θ(p0 ) follows form the fact that the physical states N have a positive energy. Then, by inserting unit factor given by Z 1 = ds δ(p2 − s) , (1.301)

1. BASICS OF Q UANTUM F IELD T HEORY

69

we obtain Z   

µ ea,i (s) ∆(y − x; s) , 0 [Ja (x), φi (y)] 0 = −∂µ x ds ρa,i (s) ∆(x − y; s) + ρ (1.302)

where we denote ∆(x − y; s) ≡

Z

d4 p 2πθ(p0 ) δ(p2 − s) eip·(x−y) . (2π)4

(1.303)

This function obeys the Klein-Gordon equation with the “mass” s: (x + s)∆(x − y; s) = 0 ,

(1.304)

and is Lorentz invariant. When the interval x − y is space-like, it cannot depend separately on x0 − y0 since the sign of x0 − y0 is not invariant for a space-like separation. Therefore, for such an interval, it can depend only on (x − y)2 and s, and we have ∆(x − y; s) = ∆(y − x; s) if (x − y)2 < 0 .

(1.305)

Therefore, Z   

µ 0 [Ja (x), φi (y)] 0 = −∂µ ρa,i (s) ∆(y−x; s) if (x−y)2 < 0 . x ds ρa,i (s)+e

(1.306)

Since the commutator in the left hand side vanishes for local operators with a spacelike separation, we get39 : ea,i (s) = 0 . ρa,i (s) + ρ

(1.307)

Returning to the case of a generic interval x − y, we thus have: Z 

µ   0 [Ja (x), φi (y)] 0 = −∂µ x ds ρa,i (s) ∆(x − y; s) − ∆(y − x; s) . (1.308)

By applying the derivative ∂xµ to both sides of this equation, and using the KleinGordon equation and the fact that the current Jµ a (x) is conserved, we get Z   0 = ds s ρa,i (s) ∆(x − y; s) − ∆(y − x; s) , (1.309) 39 This

property, combined with eq. (1.299), implies that ρa,i (p2 ) is real.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

which implies s ρa,i (s) = 0 .

(1.310)

Therefore, ρa,i (s) = 0 for all s 6= 0, and the only possible support of ρa,i (s) is localized at s = 0 (in the form of a delta function so that integrals over s are non-zero). Let us now show that it is not possible that ρa,i (s) be identically zero everywhere (including at s = 0) when the symmetry is spontaneously broken. By setting µ = 0 and x0 = y0 , and using eq. (1.303), we obtain 

0 0 [Ja (x), φi (y)] 0

=

x0 =y0

=

Z

p 2i dsd4 p p2 +s ρa,i (s) eip·(x−y) δ(p2 − s) Z i δ(x − y) ds ρa,i (s) . (1.311)

Then, we can integrate over x and use the commutation relation (1.294) in order to get Z

 ta φ = i ds ρa,i (s) . j ij

(1.312)

Thus, the functions ρa,i (s) that have  a non-zero integral are in one-to-one correspondence with the non-zero ta φ j , i.e. with the fact that the ground state is non ij invariant under the action of some of the symmetry generators. When this happens, we must have

 ρa,i (s) = −i δ(s) ta ij φj .

(1.313)

This equation is the essence of Goldstone’s theorem. Note now that ρa,i is a spectral function similar to the one defined in the section 1.9. Therefore, the presence of a δ(s) in this function signals the existence of a one-particle state with zero mass in the sum of eq. (1.298) (multiparticle states with a null total momentum would produce a continuum extending down to s = 0 rather than a delta function). Moreover, this results indicates that there are as many such massless particles (called NambuGoldstone modes) as there are broken symmetries by the ground state.  0 is invariant under rotations, which Finally, let us note that the state φ (0) i

 implies that the matrix element N φi (0) 0 is zero unless the state N has a vanishing helicity. Thus, only spin 0 particles can  contribute to the δ(s) in the nonzero spectral functions. Moreover, 0 J0a (0) N vanishes for any state N whose quantum numbers differ from those of J0a . Thus, the Nambu-Goldstone modes are spin-0 particles that have the same internal quantum numbers as J0a . c sileG siocnarF

1. BASICS OF Q UANTUM F IELD T HEORY

71

1.16 Perturbative unitarity Unitarity is one of the pillars of quantum mechanics, since it is tightly related to the conservation of probability. A completely general consequence of unitarity is the optical theorem, whose perturbative translation becomes manifest in the so-called Cutkosky’s cutting rules. c sileG siocnarF

1.16.1

Optical theorem

The “S-matrix” is the name given to the evolution operator that relates the in and out states:

αout ≡ αin S . (1.314)

In a unitary field theory, the S matrix is a unitary operator on the space of physical states: SS† = S† S = 1 .

(1.315)  This property means that for a properly normalized initial physical state αin , we have X 2 |hβout |αin i| = 1 , (1.316) states β

where the sum includes only physical states. In other words, in any interaction process, the state α must evolve with probability one into other physical states. In general, one subtracts from the S-matrix the identity operator, that corresponds to the absence of interactions, and one writes: S ≡ 1 + iT .

(1.317)

Therefore, one has 1 = (1 + iT )(1 − iT † ) = 1 + iT − iT † + TT † ,

(1.318)

or equivalently −i(T − T † ) = TT † .

(1.319)  Let us now take the expectation value of this identity in the state αin , and insert the identity operator written as a complete sum over physical states between T and T † in the right hand side. This leads to: X

 hαin |T |βin i 2 . −i αin |T − T † |αin = (1.320) states β

72

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Equivalently, this identity reads Im hαin |T |αin i =

2 1 X hαin |T |βin i . 2

(1.321)

states β

This identity is known as the optical theorem. It implies that the total probability to scatter from the state α to any state β equals twice the imaginary part of the forward transition amplitude α → α. c sileG siocnarF

1.16.2

Cutkosky’s cutting rules

Eq. (1.321) is valid to all orders in the interactions. But as we shall see it also manifests itself in some properties of the perturbative expansion. Let us first consider i as an example a scalar field theory, with a cubic interaction in − 3! λφ3 (x). Firstly, decompose the free Feynman propagator in two terms, depending on the ordering between the times at the two endpoints: G0F (x, y) ≡ θ(x0 − y0 )G0−+ (x, y) + θ(y0 − x0 )G0+− (x, y) .

(1.322)

The 2-point functions G0−+ and G0+− are therefore defined as 

G0−+ (x, y) ≡ 0in φin (x)φin (y) 0in ,



G0+− (x, y) ≡ 0in φin (y)φin (x) 0in . (1.323)

In order to streamline the notations, it is convenient to rename G0F by G0++ , and to introduce another propagator with a reversed time ordering: G0−− (x, y) ≡ θ(x0 − y0 )G0+− (x, y) + θ(y0 − x0 )G0−+ (x, y) .

(1.324)

The usual Feynman rules in coordinate space amount to connect a vertex at x and a vertex at y by the propagator G0++ (x, y). The coordinate x of each vertex is integrated out over all space-time, and a factor −iλ is attached to each vertex. We will call + this type of vertex. Thus, the Feynman rules for calculating transition amplitudes involve only the + vertex and the G0++ propagator. Let us then introduce a vertex of type −, to which a factor +iλ is assigned (instead of −iλ for the vertex of type +). The integrand of a Feynman graph G is a function G(x1 , x2 , · · · ) of the coordinates xi of its vertices. We will generalize this function by assigning + or − indices to all the vertices, G(x1 , x2 , · · · )

→

Gǫ1 ǫ2 ··· (x1 , x2 , · · · ) ,

(1.325)

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1. BASICS OF Q UANTUM F IELD T HEORY

where the indices ǫi = ± indicate which is the type of the i-th vertex. The usual Feynman rules thus correspond to the function G++··· . These generalized integrands are constructed according to the following rules: + vertex :

−iλ

,

− vertex :

+iλ

,

Propagator from ǫ to ǫ ′ :

G0ǫǫ ′ (x, y)

.

(1.326)

Let us assume that the i-th vertex carries the largest time among all the vertices of the graph. Since x0i is largest than all the other times, then the propagator that connects this vertex to an adjacent vertex of type ǫ at the position x is given by G0±ǫ (xi , x) = G0−ǫ ǫ (xi , x) .

(1.327)

In other words, this propagator depends only on the type ǫ of the neighboring vertex, but not on the type of the i-th vertex. Therefore, we have G···[+i ]··· (x1 , x2 , · · · ) + G···[−i ]··· (x1 , x2 , · · · ) = 0 ,

(1.328)

where the notation [±i ] indicates that the i-th vertex has type + or − (the types of the vertices not written explicitly are the same in the two terms, but otherwise arbitrary). This identity, known as the largest time equation, follows from eq. (1.327) and from the sign change when a vertex changes from + to −. A similar identity also applies to the sum extended to all the possible assignments of the + and − indices: X Gǫ1 ǫ2 ··· (x1 , x2 , · · · ) = 0 . (1.329) {ǫi =±}

This is obtained by pairing the terms and using eq. (1.328). It is crucial to observe that this identity is now valid for any ordering of the times at the vertices of the graph. Therefore, it is also valid in momentum space after a Fourier transform. If we isolate the two terms where all the vertices are of type + or all of type −, this also reads X G++··· + G−−··· = − Gǫ1 ǫ2 ··· , (1.330) {ǫi =±} ′

where the symbol {ǫi = ±} ′ indicates the set of all the vertex assignments, except + + · · · and − − · · · . Using eq. (1.119), G0++ (x, y) =

Z

 d3 p θ(x0 − y0 ) e−ip·(x−y) + θ(y0 − x0 ) e+ip·(x−y) , 3 (2π) 2Ep

74

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS (1.331)

and comparing with eq. (1.324), we can read off the following representations for G0−+ and G0+− , G0−+ (x, y)

=

G0+− (x, y)

=

Z

Z

d3 p e−ip·(x−y) (2π)3 2Ep d3 p e+ip·(x−y) . (2π)3 2Ep

(1.332)

Likewise, we obtain G0−− (x, y)

=

Z

 d3 p 0 0 +ip·(x−y) 0 0 −ip·(x−y) θ(x − y ) e + θ(y − x ) e , (2π)3 2Ep (1.333)

Note that G0++ + G0−− = G0−+ + G0+− . Using the following representation for step functions: Z

0 0 dp0 −i eip0 (x −y ) , 2π p0 − i0+ Z 0 0 i dp0 eip0 (x −y ) , θ(y0 − x0 ) = + 2π p0 + i0

θ(x0 − y0 ) =

(1.334)

we can derive the momentum space expressions of these propagators: i , − m2 + i0+  ∗ −i = G0++ (p) , G0−− (p) = 2 p − m2 − i0+ G0−+ (p) = 2π θ(+p0 )δ(p2 − m2 ) , G0++ (p) =

p2

G+− (p) = 2π θ(−p0 )δ(p2 − m2 ) .

(1.335)

Therefore, the momentum space Feynman rules for the − sector are the complex conjugate of those for the + sector, since we have also +iλ = (−iλ)∗ . Note that for this assertion to be true, it is crucial that the coupling constant λ be real, which is a condition for unitarity. The Fourier transform of an amputated Feynman graph G gives a contribution to a transition amplitude (recall the LSZ reduction formula), i.e. a matrix element of the S

75

1. BASICS OF Q UANTUM F IELD T HEORY

operator. Therefore, Γ ≡ iG gives a matrix element of the T operator. Therefore, after Fourier transform, eq. (1.330) becomes Im Γ++··· =

1 2

X   iΓ ǫ

1 ǫ2 ···

{ǫi =±} ′

.

(1.336)

If the graph contains N vertices, there are a priori 2N − 2 terms in the right hand side of this equation. However, this number is considerably reduced if we notice that the +− and −+ propagators can carry energy only in one direction (from the − vertex to the + vertex), because of the factors θ(±p0 ). This constraint on energy flow forbids “islands” of vertices of type + surrounded by only type − vertices, or the reverse. From the LSZ reduction formula (1.82) and the definition (1.120) of the Fourier transformed propagators, we see that the notation G−+ (p) implies a momentum p defined as flowing from the + endpoint to the − endpoint: G−+ (p) =

p +

-

.

(1.337)

Thus, the proportionality G−+ (p) ∝ θ(p0 ) indicates that the energy flows from the + endpoint to the − endpoint. Let us consider the example of a very simple 1-loop two-point function40 Γ (p),

p −iΓ (p) =

.

(1.338)

Because of the constrained energy flow direction in the propagators G−+ , G+− , if the momentum p is entering into the graph from the left with p0 > 0, the only assignments that mix + and − vertices must divide the graph into two connected subgraphs: a connected part made only of + vertices that comprises the vertex where p0 > 0 enters in the graph, and a connected part containing only − vertices comprising the vertex where the energy leaves the graph. For the topology shown in eq. (1.338), there is only one possibility,

p −iΓ+− (p) =

,

(1.339)

where the vertex of type − is circled in the diagrammatic representation. The division of the graph into these two subgraphs may be materialized by drawing a line (shown in gray above) through the graph. This line is called a cut, and the rules for calculating 40 Momentum

conservation implies that it depends on a single momentum p.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

the value of a graph with a given assignment of + and − vertices are called Cutkosky’s cutting rules. For instance, in the case of the above example, they lead immediately to the following expression41 for the imaginary part of Γ++ , Z λ2 1 d4 k Im Γ++ (p) = G−+ (k)G−+ (p − k) , (1.340) 2 2 (2π)4 that can be rewritten as Im Γ++ (p) =

Z λ2 d4 k1 2πθ(k01 )δ(k21 − m2 ) 4 (2π)4 Z 4 d k2 2πθ(k02 )δ(k22 − m2 )(2π)4 δ(p − k1 − k2 ) . × (2π)4 (1.341)

In the right hand side of  this equation, we recognize the square of the transition amplitude k1 k2out pin (whose value at tree level is simply λ), integrated over the (symmetrized) accessible phase-space for a 2-particle final state. We can therefore view this equation as a perturbative realization of the optical theorem at order λ2 . Indeed, at this order, the only states β that may be included in the sum over final states are 2-particle states42 . The considerations developed on this example can be generalized to the 2-point function at any loop order. We can write 1 X (iΓγ (p)) , (1.342) Im Γ++ (p) = 2 cuts γ where the sum is now limited reduced to a sum over all the possible cuts (with the + vertices left of the cut and the − vertices right of the cut). As an illustration, let us consider the following 2-loop example, for which three cuts are possible:

Im Γ++ (p) =

.

(1.343)

At this order start to appear various contributions to the right hand side of eq. (1.321): the central cut corresponds to a 3-body final state, while the other two cuts correspond to an interference between the tree level and the 1-loop correction to a 2-body decay. c sileG siocnarF

41 The

first factor 1/2 comes from eq. (1.336), and the second 1/2 is the symmetry factor of the graph for a scalar loop. In the formula for Im Γ++ , it has the interpretation of the factor that symmetrizes a 2-particle final state. 42 This result is consistent with the formula (1.100) for a decay rate, if we note that the decay rate Γ of a
particle is related to the imaginary part of the corresponding self-energy by Γ = Im Γ++ (p) /Ep . This can be seen as follows: after resumming the self-energy Γ++ (p) on the propagator, the imaginary part makes it decay as G++ (x, y) ∼ exp(−(Im Γ++ )|x0 − y0 |/2Ep ), and the particle density, quadratic in the field operator, decays as the square of the propagator.

77

1. BASICS OF Q UANTUM F IELD T HEORY

1.16.3

Fermions

In the case of spin 1/2 fermions, the propagators connecting the various types of vertices are given by

/ + m) i(p , − m2 + i0+ / + m) −i(p , S0−− (p) = 2 p − m2 − i0+ / + m)θ(−p0 )δ(p2 − m2 ) , S0−+ (p) = 2π (p

S0++ (p) =

p2

/ + m)θ(+p0 )δ(p2 − m2 ) . S0+− (p) = 2π (p

(1.344)

The cutting rules for fermions are therefore similar to those for scalar particles. The possibility to interpret the cut fermion propagators in terms of on-shell final state fermions is a consequence of the following identities:

/+m= p /−m= p

X

us (p)us (p) ,

spin s X

vs (p)vs (p) ,

(1.345)

spin s

p that are valid when p0 = p2 + m2 > 0. In the case of the propagator S0−+ (p), we may attach the spinor us (p) to the amplitude on the right of the cut, and the spinor us (p) to the amplitude on the left, which are precisely the spinors required by the LSZ formula for a fermion of momentum p in the final state. In the case of S0+− (p), for which p0 < 0, we should first write

S0+− (p) = =

/ − m)θ(−p0 )δ(p2 − m2 ) −2π(−p X vs (−p)vs (−p)θ(−p0 )δ(p2 − m2 ) , −2π

(1.346)

spin s

in order to see that it corresponds to an anti-fermion in the final state. c sileG siocnarF

78

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

1.16.4

Photons

Coulomb gauge : For photons in Coulomb gauge, the reasoning is very similar to the case of fermions. Firstly, the four different types of propagators read   i j i δij − ppp2 ij , G0++ (p) = p2 + i0+   i j −i δij − ppp2 ij , G0−− (p) = p2 − i0+   i j ij G0−+ (p) = 2π θ(+p0 ) δij − ppp2 δ(p2 ) ,   i j ij (1.347) G0+− (p) = 2π θ(−p0 ) δij − ppp2 δ(p2 ) .

Recalling also that X

j ij ǫi∗ λ (p)ǫλ (p) = δ −

λ=±

pi pj , p2

(1.348)

we see that the projector that appears in the cut propagators can be interpreted as the polarization vectors that should attached to amplitudes for each final state photon. Therefore, the cutting rules in Coulomb gauge have a direct interpretation in terms of the optical theorem. This simplicity follows from the fact that the only propagating modes are physical modes in Coulomb gauge. c sileG siocnarF

Feynman gauge : This interpretation is not so direct in covariant gauges, such as the Feynman gauge. In this gauge, the free photon propagator is given by: µν G0++ (p) = −gµν

p2

i . + i0+

(1.349)

The factor −gµν does not change anything to the cutting rules, and simply appears as a prefactor in all propagators: µν G0−− (p) = −gµν

−i , p2 − i0+

µν G0−+ (p) = −2π gµν θ(+p0 )δ(p2 ) , µν G0+− (p) = −2π gµν θ(−p0 )δ(p2 ) .

(1.350)

^ direction, i.e. Let us assume for definiteness that the photon momentum p is in the z p = (0, 0, p). Therefore, the two physical polarizations vectors, orthogonal to p, can be chosen as follows ǫµ 1 (p) ≡ (0, 1, 0, 0) , µ ǫ2 (p) ≡ (0, 0, 1, 0) .

(1.351)

79

1. BASICS OF Q UANTUM F IELD T HEORY They are orthonormal ǫλ (p) · ǫλ ′ (p) = −δλλ ′ ,

(1.352)

µν and transverse: pµ ǫµ that appears in the cut 1,2 (p) = 0. However, the tensor −g photon propagators cannot be written as a sum over physical polarizations: X µ ∗ −gµν 6= ǫλ (p)ǫν (1.353) λ (p) . λ=1,2

Only the µ = 1, 2 components of these tensors are equal. As a consequence, it seems that Cutkosky’s cutting rules may lead to terms that we cannot interpret as physical final photon states, which would violate the optical theorem. If this was the case, then perturbation theory would not be consistent with unitarity. To see how this paradox is resolved, let us introduce two more (unphysical) polarization vectors 43 : 1 ǫµ + (p) ≡ √ (1, 0, 0, 1) , 2 1 µ ǫ− (p) ≡ √ (1, 0, 0, −1) , 2

(1.355)

thanks to which we may now write µ ∗ ν ∗ ν gµν = ǫµ + (p)ǫ− (p) + ǫ− (p)ǫ+ (p) −

X

ν ∗ ǫµ λ (p)ǫλ (p) .

(1.356)

λ=1,2

In other words, the physical polarization sum in the right hand side of eq. (1.353) is equal to −gµν , plus some extra terms that are proportional to pµ of pν . When we use Cutkosky’s cutting rules in order to calculate the imaginary part of graph, a cut photon line carrying the momentum pµ leads to an expression that has the following structure: ∗

µν ] (iMν iMµ 2 (p)) , 1 (p) [−g

(1.357)

ν where iMµ 1 and iM2 are the amplitudes on the left and on the right of the cut, respectively. Here, we have highlighted only one of the cut photons, and the other cut lines have not been written explicitly since they do not play any role in the 43 For

an arbitrary momentum p, these polarization vectors read: ǫµ + (p) ≡ √ ǫµ − (p) ≡ √

1 2|p| 1 2|p|

(p0 , p) , (p0 , −p) .

(1.354)

80

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

argument. Moreover, only the tensor structure of the cut propagator matters, and we have therefore only written the factor −gµν . The above quantity can be rewritten as " # X µ ∗ µ µ µ ν ∗ ν ∗ ν ∗ iM1 (p) ǫλ (p)ǫλ (p) −ǫ+ (p)ǫ− (p) −ǫ− (p)ǫ+ (p) (iMν 2 (p)) λ=1,2

=

iMµ 1 (p)

"

X

ν ∗ ǫµ λ (p)ǫλ (p)

λ=1,2

#

∗

(iMν 2 (p)) .

(1.358)

Indeed, the last two terms are zero thanks to the Ward identity satisfied44 by the ν amplitudes iMµ 1 and iM2 : ν pµ Mµ 1 (p) = pν M2 (p) = 0 ,

(1.359)

ǫµ + (p)

and because is proportional to pµ . Therefore, the non-physical photon degrees of freedom, that may appear in the cutting rules in covariant gauges, are in fact canceled by the Ward identities satisfied by QED amplitudes. c sileG siocnarF

1.16.5

Schwinger-Keldysh formalism

Perturbation theory

provides a way of computing order by order transition amplitudes like p′ q′ out pqin . The calculation of these matrix elements is amenable via the LSZ reduction formulas to the expectation value of time-ordered products of field operators, between the in- and out- vacuum states, for instance 

0out T φ(x1 )φ(x2 )φ(x3 )φ(x4 ) 0in , the calculation of which can be performed with the usual Feynman rules.

However, there is a class of more general problems that cannot be addressed by this standard perturbation theory. One of the simplest problems 

of that kind is the evaluation of the expectation value of the number operator αin a†out (p)aout (p) αin , that counts the particles of momentum p in the final state, given that the initial state was this matrix element, one needs to calculate the amplitude state α. To evaluate

the  αin φ(x)φ(y) αin , that has no time ordering, and where one has in states on both sides. More generally, one sometimes needs the amplitudes



 0in T φ(x1 ) · · · φ(xn ) T φ(y1 ) · · · φ(yp ) 0in , where T denotes the anti-time ordering. The Schwinger-Keldysh formalism is tailored for addressing these more general questions. Moreover, as we shall see, it is formally identical to Cutkosky’s cutting rules. c sileG siocnarF

44 When an amplitude has external charged particles, the Ward identity is satisfied only if these particles are on-shell. This is indeed the case here, because all the cut lines are on-shell, as well as all the incoming particles.

81

1. BASICS OF Q UANTUM F IELD T HEORY Schwinger-Keldysh perturbation theory :

Consider the expectation value



 0in T φ(x1 ) · · · φ(xn ) T φ(y1 ) · · · φ(yp ) 0in .

(1.360)

As we did in the derivation of ordinary perturbation theory, let us first replace each Heisenberg field operator by its counterpart in the interaction representation, using eq. (1.63). After some rearrangement of the evolution operators, we get :

 0in T φ(x1 ) · · · φ(xn ) Tφ(y1 ) · · · φ(yp ) 0in = Z +∞ i

h = 0in T φin (x1 ) · · · φin (xn ) exp i d4 x LI (φin (x)) −∞

h

×T φin (y1 ) · · · φin (yp ) exp i

Z +∞ −∞

i  d4 x LI (φin (x)) 0in . (1.361)

Here, we have exploited the fact that the factor U(−∞, +∞) that appears in these manipulations is the anti-time ordered exponential of the interaction term, in order to write this formula in a more symmetric way. To go further, it is useful to imagine that the time axis is in fact a contour C made of two branches labeled + and − running parallel to the real axis, as illustrated in figure 1.4. This contour is oriented, with

C

+

x0

− Figure 1.4: Time contour in the Schwinger-Keldysh formalism.

the + branch running in the direction of increasing time, followed by the − branch running in the direction of decreasing time. Then, it is convenient to introduce a path ordering, denoted by P and defined as a standard ordering along the contour C. In more detail, one has  T A(x)B(y)      T A(x)B(y) P A(x)B(y) =  A(x)B(y)     B(y)A(x)

if if if if

x0 , y0 ∈ C+ ,

x0 , y0 ∈ C− ,

x0 ∈ C− , y0 ∈ C+ ,

x0 ∈ C+ , y0 ∈ C− .

(1.362)

82

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

One can use this contour ordering to write the previous equations in a much more compact way. In particular, eq. (1.361) can be generalized into : 

− 0in P φ (x1 ) · · · φ− (xn )φ+ (y1 ) · · · φ+ (yp ) 0in = Z 

− + + d4 x LI (φin (x)) 0in . = 0in P φ− (x ) · · · φ (x )φ (y ) · · · φ (y ) exp i 1 n 1 p in in in in C

(1.363)

The differences compared to eq. (1.361) are threefold : i. A single overall path ordering takes care automatically of both the time ordering and the anti-time ordering contained in the original formula, ii. For this trick to work, one must (temporarily) assume that the fields on the upper and lower branch of the contour C are distinct: φ+ and φ− respectively, iii. The time integration in the exponential is now running over both branches of the contour C. The advantage of having introduced this more complicated time contour is that it leads to a expressions that are formally identical to those of ordinary perturbation theory, provided one replaces the time ordering by the path ordering and provided one extends the time integration from to C. In particular, one can first define a generating functional, Z 

SK (1.364) Z [j] ≡ 0in T exp i d4 x j(x)φ(x) 0in ,

❘

C

that encodes all the correlators considered in this section, provided the external source j has distinct values j+ and j− on the two branches of the contour (the superscript SK is used to distinguish this generating functional from the standard one). As in the case of Feynman perturbation theory, one can write this generating functional as:   Z Z 

δ 0in T exp i d4 x j(x)φin (x) 0in , (1.365) ZSK [j] = exp i d4 x LI iδj(x) C C | {z } ZSK 0 [j]

with

ZSK 0 [j]

=

G0C (x, y) ≡



1Z exp − d4 xd4 y j(x)j(y) G0C (x, y) 2 C 

0in P φin (x)φin (y) 0in .

(1.366)

The free propagator G0C , defined on the contour C, is a natural extension of the Feynman propagator (in particular, it coincides with the Feynman propagator if the

83

1. BASICS OF Q UANTUM F IELD T HEORY

two time arguments are on the + branch of the contour). Besides the propagator, the other change to the perturbative expansion in the Schwinger-Keldysh formalism is that the time integration at the vertices of a diagram must run over the contour C instead of the real axis. The connection with Cutkosky’s cutting rules appears when we break down the propagator into 4 components G0±± (x, y), depending on whether the times x0 , y0 are on the upper or lower branch of the contour. An explicit calculation of these free propagators leads to G0++ (x, y) = G0−− (x, y) = G0+− (x, y) = G0−+ (x, y) =

Z

d4 p e−ip·(x−y) , (2π)4 p2 − m2 + iǫ Z 4 e−ip·(x−y) d p , −i 4 2 (2π) p − m2 − iǫ Z 4 d p −ip·(x−y) e 2πθ(−p0 )δ(p2 − m2 ) , (2π)4 Z 4 d p −ip·(x−y) e 2πθ(+p0 )δ(p2 − m2 ) . (2π)4 i

(1.367)

The time integration on the contour C is also split into two terms, the upper branch corresponding to a vertex + (−iλ) and the lower branch to a vertex − (+iλ, because of the minus sign due to integrating from +∞ to −∞). In the Schwinger-Keldysh formalism, the vacuum-vacuum diagrams are simpler than in conventional perturbation theory. Here, one has

 ZSK [0] = 0in 0in = 1 , (1.368)

which means that all the connected vacuum-vacuum diagrams are zero. This is due to the fact that in this formalism one is calculating correlators that have the in- vacuum on both sides. This cancellation works individually for each diagram topology, and results from a cancellation between the various ways of assigning the + and − indices to the vertices of a diagram (a vacuum-vacuum diagram with a fixed assignment of + and − vertices is not zero in general). This cancellation can be viewed as a consequence of eq. (1.329). c sileG siocnarF

Relation between the functionals Z[j] and ZSK [j] : There is a useful functional relation between the generating functional of conventional perturbation theory Z[j], and that of the Schwinger-Keldysh formalism : Z [j+ , j− ] = exp SK

Z

4

4

d xd y

G0+− (x, y) x y

 δ2 Z[j+ ] Z∗ [j− ] . δj+ (x)δj− (y) (1.369)

84

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

(Here, in order to avoid any confusion, we write explicitly the two components + and − of the source j in the Schwinger-Keldysh generating functional.) Thanks to this formula, one can construct diagrams in the Schwinger-Keldysh formalism by stitching an ordinary Feynman diagram and the complex conjugate of another Feynman diagram. In order to prove this relation, it is sufficient to establish it for the free theory, since the interactions are always trivially factorizable (see eqs. (1.107) and (1.365)). c sileG siocnarF

Chapter 2

Functional quantization 2.1 Path integral in quantum mechanics Let us consider a quantum mechanical system with a single degree of freedom, whose Hamiltonian is H≡

P2 + V(Q) . 2m

(2.1)

The position and   momentum operators Q and P obey the following commutation relation Q, P = i. We would like to calculate the probability for the system to start at the position qi at a time ti and end at the position qf at the time tf . The answer 2 may be obtained as ψ(qf , tf ) by solving Schr¨odinger’s equation with an initial wavefunction localized at qi , i∂t ψ(q, t) = H ψ(q, t) ,

ψ(q, ti ) ≡ δ(q − qi ) .

(2.2)

More formally, in the Schr¨odinger picture, it is given by the squared modulus of the following transition amplitude

−iH(t −t )  f i qf e qi , (2.3)  where q denote the eigenstate of the position operator with eigenvalue q. Let us subdivide the time interval [ti , tf ] into N equal sub-intervals, by introducing: ∆≡

tf − ti N

,

tn ≡ ti + n ∆ .

(2.4)

(Therefore, we have t0 = ti and tN = tf .) The time evolution operator can be factorized as e−iH(tf −ti ) = e−iH(tN −tN−1 ) ×e−iH(tN−1 −tN−2 ) ×· · ·×e−iH(t1 −t0 ) . (2.5)

85

86

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Between the successive factors in the right hand side, we can insert the identity operator written as a complete sum over the position eigenstates: Z +∞  dq q q , (2.6) 1= −∞

and the transition amplitude (2.3) becomes

 qf e−iH(tf −ti ) qi =

Z N−1 Y j=1





 dqj qf e−i∆H qN−1 qN−1 e−i∆H qN−2 · · · 

· · · q1 e−i∆H qi .

(2.7)

Note that this formula, illustrated in the figure 2.1, is exact for any value of N. In the

q

t q

t Figure 2.1: Illustration of eq. (2.7) with 10 and 200 intermediate points. The endpoints are fixed, while the intermediate points are integrated over. The line segments connecting the points are just a help to guide the eye, but there is no “path” at this stage.

87

2. F UNCTIONAL QUANTIZATION

Hamiltonian (2.1), the kinetic energy and potential energy terms do not commute, which complicates the evaluation of its exponential. We can remedy this situation by using the Baker-Campbell-Hausdorff formula, that we shall write here as follows e∆(A+B) = e∆A e∆B e−

∆2 3 2 [A,B]+O(∆ )

.

(2.8)

In the limit ∆ → 0 (i.e. N → ∞), we may neglect the last factor since the product of all such factors goes to unity1 when N → ∞. Therefore, each elementary factor of eq. (2.7) is rewritten as

 qi+1 e−i∆H qi

≈ =

 P2 qi+1 e−i∆ 2m e−i∆V(Q) qi Z  P2  dpi

qi+1 e−i∆ 2m pi pi e−i∆V(Q) qi , 2π (2.9)

where we have introduced the identity operator, written this time as a complete sum over momentum eigenstates: Z dp  p p . (2.10) 1≡ 2π

In the two factors, the exponential operator depends only on P or Q, and the matrix elements are trivial to evaluate by using the fact that the operators are enclosed between momentum and position eigenstates: p2  P2  i

qi+1 e−i∆ 2m pi = e−i∆ 2m qi+1 pi ,



−i∆V(Q)  qi = e−i∆V(qi ) pi qi . pi e



Using now

 q p = eipq ,

(2.12)

we arrive at the formula2 


qi+1 e−i∆H qi = e−i∆H(pi ,qi ) ei pi (qi+1 −qi ) 1 + O(∆2 ) . 1 We

(2.11)

(2.13)

use

2

2

2

lim eα1 /N eα2 /N · · · eαN /N = 1 ,

N→∞

P provided that the sum i αi ’s does not diverge too quickly. 2 A bit more care is necessary for Hamiltonians that are not separable into a sum of a P-dependent term and a Q-dependent term. A proper treatment should use Weyl’s prescription for defining the quantum Hamiltonian operator from the classical Hamiltonian. In eq. (2.13), one would obtain H(pi , 21 (qi +qi+1 )) instead of H(pi , qi ).

88

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

If we define q˙ i ≡ (qi+1 − qi )/∆ the slope of the line segments in the figure 2.1, and we take the limit N → ∞, we may write the transition amplitude as a path integral:

 qf e−iH(tf −ti ) qi =

Z



q(ti )=qi q(tf )=qf

 Dp(t)Dq(t)

Z tf
 ˙ dt p(t)q(t) − H(p(t), q(t)) . × exp i

(2.14)

ti

  One should be aware of the fact that the functional measure Dq(t)Dp(t) in general lacks solid mathematical foundations, although it allows for some powerful manipulations that would be extremely cumbersome to perform at the level of quantum operators. Note that at the boundaries ti,f the position is well defined, and therefore the momentum is not constrained (by the uncertainty principle). A crucial aspect of eq. (2.14) is that all the objects that appear in the right hand side are ordinary c-numbers that commute, while the left hand side is made of quantum operators and states. In this section, we have started from the conventional formulation of transition amplitudes in quantum mechanics, in order to arrive at the formula (2.14). However, one may now “forget” the canonical formalism and view the path integral expression of transition amplitudes as another way of going from a classical Hamiltonian H to a quantized theory. For a Hamiltonian where the P dependence has no powers higher than quadratic, as in the example of eq. (2.1), it is possible to perform exactly the integral over p(t). This type of integral is called a Gaussian path integral. Gaussian path integrals can be evaluated in the same way as their ordinary counterparts, using the following formulas, Z +∞

−x2 /(2σ)

dx e

=

√

2πσ ,

−∞

Z +∞

2

dx e±ix

/(2σ)

π√

= e±i 4

2πσ , (2.15)

−∞

and treating each p(t) as an independent variable. In the present case, we need the integral Z

p2 ˙ i∆(pq− 2m )

dp e

π −i 4

=e |

r

2

2πm i∆ mq˙ 2 e . {z ∆ }

(2.16)

prefactor independent of q,q˙

  The (infinite in the limit ∆ → 0) prefactors can be hidden in the measure Dq(t) since they do not depend on the path, and we are therefore led to the following

89

2. F UNCTIONAL QUANTIZATION formula:

−iH(t −t )  f i qi qf e

=

Z



Z



q(ti )=qi q(tf )=qf

=

 Dq(t) exp i

Z tf ti

 dt L(q(t))

 Dq(t) eiS[q(t)] ,

(2.17)

q(ti )=qi q(tf )=qf

where L(q) is the classical Lagrangian: m q˙ 2 − V(q) 2 and S[q] the corresponding classical action. L(q) ≡

(2.18)

c sileG siocnarF

2.2 Classical limit, Least action principle In the previous section, we have written all the formulas with h ¯ = 1. Had we kept the Planck constant, the final formula would have been: Z

−iH(t −t )    i f i qf e qi = Dq(t) eh¯ S[q(t)] . (2.19) q(ti )=qi q(tf )=qf

(This can be guessed a posteriori based on the fact that h ¯ has the dimension of an action.) Because of the factor i inside the exponential, this integral is wildly oscillating, except in the immediate vicinity of the function qc (t) that realizes the extremum of the action. Note that this function is precisely the solution of the classical EulerLagrange equations of motion. Roughly speaking, the phase oscillations become significant when S[q(t)] − S[qc (t)] ≥ 2π h ¯ , (2.20) and paths that fulfill this inequality do not contribute to the path integral. Therefore, in the limit h ¯ → 0, the path integral is dominated by the unique path qc (t), i.e. by the classical trajectory of the system. The path integral formalism thus provides a very intuitive way of connecting smoothly quantum and classical mechanics. c sileG siocnarF

2.3 More functional machinery 2.3.1

Time-ordered products

Consider the matrix element 

−iH(t −t ) f 1 qf e Q e−iH(t1 −ti ) qi ,

(2.21)

90

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

q

t Figure 2.2: Illustration of eq. (2.19). The paths whose action is far apart from the classical extremum are plotted in fainter colours. The solid black line is the classical trajectory.

that measures the expectation value of the position at the time t1 . In order to evaluate this object, we need to insert on either side of the position operator Q an identity operator written as a complete sum over position eigenstates, i.e. Q

→

Z

  dqdq ′ q q Q q ′ q ′ = | {z } q δ(q−q ′ )

Z

 dq q q q .

(2.22)

This leads immediately to the following path integral representation: Z   

−iH(t −t ) −iH(t1 −ti ) f 1 qi = Dq(t) q(t1 ) eiS[q(t)] . (2.23) qf e Qe q(ti )=qi q(tf )=qf

Likewise, if t2 > t1 , we have:

 qf e−iH(tf −t2 ) Q e−iH(t2 −t1 ) Q e−iH(t1 −ti ) qi = Z   = Dq(t) q(t1 ) q(t2 ) eiS[q(t)] .

(2.24)

q(ti )=qi q(tf )=qf

If we introduce a time-dependent position operator Q(t) ≡ eiHt Q e−iHt ,

(2.25)

91

2. F UNCTIONAL QUANTIZATION and its eigenstates   q, t ≡ eiHt q ,

(2.26)

the previous equation takes a much more compact form

 qf , tf Q(t2 )Q(t1 ) qi , ti

=

t2 >t1

Z



q(ti )=qi q(tf )=qf

 Dq(t) q(t1 ) q(t2 ) eiS[q(t)] . (2.27)

The condition t2 > t1 is crucial here, because the left hand side would be quite different if the times are ordered differently. In contrast, the objects q(t1 ) and q(t2 ) in the right hand side are ordinary numbers that commute. One may render this formula true for any ordering between t1 and t2 by introducing a T-product, that ensures that the operator with the largest time is always on the left:


 qf , tf T Q(t1 )Q(t2 ) qi , ti

=

Z



q(ti )=qi q(tf )=qf

 Dq(t) q(t1 ) q(t2 ) eiS[q(t)] . (2.28)

This formula generalizes to n factors:


 qf , tf T Q(t1 ) · · · Q(tn ) qi , ti

=

Z



 Dq(t) q(t1 ) · · · q(tn ) eiS[q(t)] .

q(ti )=qi q(tf )=qf

(2.29) This result is extremely important in applications to quantum field theory, since time-ordered products of field operators are the central objects that appear in the LSZ reduction formulas. One may also apply differential operators containing time derivatives on this equation, for instance:
 ∂

qf , tf T Q(t1 ) · · · Q(tn ) qi , ti ∂t1 Z   ˙ 1 ) · · · q(tn ) eiS[q(t)] . Dq(t) q(t =

(2.30)

q(ti )=qi q(tf )=qf

In other words, a time derivative in the integrand of the path integral also applies to the step functions that enforce the time ordering in the left hand side. c sileG siocnarF

92

2.3.2

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Functional sources and derivatives

The amplitudes of the form (2.29) can all be encapsulated into the following generating functional: Z tf 

(2.31) dt j(t) Q(t) qi , ti , Zfi [j(t)] ≡ qf , tf T exp i ti

where j(t) is some arbitrary function of time. From Zfi [j], the amplitudes can be recovered by functional differentiation: 


δn Zfi [j] qf , tf T Q(t1 ) · · · Q(tn ) qi , ti = n . (2.32) i δj(t1 ) · · · δj(tn ) j≡0

Functional derivatives obey the usual rules of differentiation, with the additional property that the values of the function j(t) at different times should be viewed as independent variables, i.e. δj(t) = δ(t − t ′ ) . δj(t ′ )

(2.33)

From this formula, one may also read the dimension of a functional derivative: h δ i     dim = −dim j(t) − dim t . (2.34) δj(t)

From eq. (2.29), we can derive an expression of the generating functional Zfi as a path integral, Z R tf   Dq(t) eiS[q(t)]+i ti dt j(t)q(t) , (2.35) Zfi [j(t)] = q(ti )=qi q(tf )=qf

that involves only the commuting c-number q(t) and no time-ordering. Note also that there is an Hamiltonian version of this path integral: Z   Dp(t)Dq(t) Zfi [j(t)] = q(ti )=qi q(tf )=qf

Z tf
 ˙ × exp i dt p(t)q(t) − H(p(t), q(t)) + j(t)q(t) . (2.36) ti

2.3.3

Projection on the ground state at asymptotic times

So far in this section, we have considered amplitudes where the initial and final states are position eigenstates. However, the path integral formalism is not limited to this

2. F UNCTIONAL QUANTIZATION

93

 situation. Let us assume for instance that the system is in a state ψi at the time ti  and in the state ψf at the time tf . For any operator O, the expectation value between these two states can be related to transitions between position eigenstates by writing Z  



(2.37) ψf , tf O ψi , ti = dqi dqf ψ∗f (qf ) ψi (qi ) qf , tf O qi , ti ,

where

 ψ(q) ≡ q ψ

(2.38)  is the position representation of the wavefunction of the state ψ . However, the use of this formula is cumbersome in practice, because of the integrations over qi,f .

In the special  case where the initial and final states are the ground state of the Hamiltonian, 0 , and the initial and final times are −∞ and +∞, there is trick to n of the Hamiltonian, circumvent this difficulty. Let us introduce the eigenstates

 with eigenvalue En and eigenfunction ψn (q) ≡ q n , and write  qi , ti

=

= =

 eiHti qi ∞ X   eiHti n n qi

n=0 ∞ X

n=0

 ψ∗n (qi ) eiEn ti n .

(2.39)

We will assume that the Hamiltonian is shifted by a constant so that the energy of the ground state 0 is E0 = 0. Now, we multiply the Hamiltonian by 1 − i0+ , where 0+ denotes some positive infinitesimal number. All the factors exp(i(1 − i0+ )En ti ) go to zero when ti → −∞, except for n = 0. Therefore, after this alteration of the Hamiltonian, we have:   lim qi , ti = ψ∗0 (qi ) 0 . (2.40) ti →−∞

We can then weight this equation by a function ϕ(qi ), Z Z   dqi ϕ(qi ) qi , ti = dqi ϕ(qi )ψ∗0 (qi ) 0 , lim ti →−∞ | {z }

 0 ϕ

(2.41)

i.e.

 0 =

1

 ti →−∞ 0 ϕ lim

Z

 dqi ϕ(qi ) qi , ti .

(2.42)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

 Any function ϕ(q) such that the state ϕ has a non-zero overlap with the ground

state 0 is appropriate in this role, but the simplest expressions are obtained with the constant function ϕ(q) = 1, corresponding to the momentum eigenstate p = 0. Likewise, changing H → (1 − i0+ )H has a similar effect on the final state in the limit tf → +∞,

(2.43) lim qf , tf = ψ0 (qf ) 0 . tf →+∞

From these considerations, when the initial and final states at ±∞ are the ground state, we can write the generating functional in the following simple path integral form: Z   Z[j(t)] = Dp(t)Dq(t)

Z
 ˙ × exp i dt p(t)q(t) − (1 − i0+ )H(p(t), q(t)) + j(t)q(t) . (2.44)

From the discussion after eq. (2.42), we see that the boundary conditions on the paths are not important. They only affect an overall prefactor, that can be adjusted by hand in such a way that Z[0] = 1. After performing the Gaussian functional integral over p(t), we can rewrite this expression in Lagrangian form: Z   Z[j(t)] = Dq(t)

Z
 mq˙ 2 (t) × exp i dt (1 + i0+ ) − (1 − i0+ )V(q(t)) + j(t)q(t) . 2 (2.45) The term in (i0+ )q˙ 2 may be viewed as contributing to the convergence of the integral at for a confining potential such that V(q) → +∞ when large velocities. Likewise, q → ∞, the term in (i0+ )V(q) contributes to the convergence at large coordinates. c sileG siocnarF

2.3.4

Functional Fourier transform

Given a functional F[q(t)], its functional Fourier transform is defined by Z

Z    e F[p(t)] ≡ Dq(t) F[q(t)] exp i dt p(t)q(t) .

(2.46)

In other words, the Fourier conjugate of the “variable” q(t) is another function of time, p(t). Eq. (2.46) may be inverted by Z Z

   F[p(t)] exp − i dt p(t)q(t) . (2.47) F[q(t)] ≡ Dp(t) e

The usual properties of ordinary Fourier transforms extend to the functional case, e.g.:

95

2. F UNCTIONAL QUANTIZATION • The Fourier transform of a constant is a delta function, • The Fourier transform of a Gaussian is another Gaussian,

• The Fourier transform of a product is the convolution product of the Fourier transforms. c sileG siocnarF

2.3.5

Functional translation operator

The functional derivative δ/δj(t) may be viewed as the generator of translations in the space of the functions j(t). Its exponential provides a translation operator:

Z δ  dt a(t) exp F[j(t)] = F[j(t) + a(t)] , (2.48) δj(t) for any functional F[j(t)]. Another extremely important formula is

Z 

Z  δ n  exp dt j(t)q(t) exp λ dt δj(t) {z } | A[j,q;λ]

= exp |

Z


 . dt j(t)q(t) + λ qn (t) {z }

(2.49)

B[j,q;λ]

The proof of this formula consists in noticing that A[j, q; λ = 0] = B[j, q; λ = 0], and in comparing their (ordinary) derivatives with respect to λ: Z Z  δ n ∂λ A[j, q; λ] = λ dt A[j, q; λ] = λ dt qn (t) A[j, q; λ] , δj(t) Z ∂λ B[j, q; λ] = λ dt qn (t) B[j, q; λ] . (2.50) Therefore A[j, q; λ] and B[j, q; λ] are equal at λ = 0 and obey the same differential equation. c sileG siocnarF

2.3.6

Functional diffusion operator

It is sometimes useful to evaluate the action of an operator which is quadratic in functional derivatives. The result is given by Z

Z

Z   σ(t)  δ 2  a2 (t)  dt exp F[j] = Da(t) exp − dt F[j+a] . (2.51) 2 δj(t) 2σ(t)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

In order to establish this formula, consider the following differential equation,

Z σ(t)  δ 2  dt ∂z F[j(t); z] = F[j; z] , (2.52) 2 δj(t) where z is an ordinary real-valued variable. One may view this equation as a diffusion equation in the space of the functions j(t), and F[j; z] as a density functional on this space. The left hand side of eq. (2.51) is the formal expression of the solution of this equation at z = 1, if we interpret F[j] as its initial condition at z = 0. In order to show that it is equal to the right hand side, one should first transform the diffusion equation (2.52) by performing a functional Fourier transform, e F[k(t); z] ≡

∂z e F[k(t); z]

=

Z

Z   Dj(t) exp i dt j(t)k(t) F[j(t)]

Z σ(t) 2  e − dt k (t) F[k(t); z] . 2 

The solution of the latter equation is simply

Z σ(t) 2  e e k (t) F[k(t); z = 0] , F[k(t); z = 1] = exp − dt 2

(2.53)

(2.54)

and the inverse Fourier transform of this solution leads to the right hand side of eq. (2.51). c sileG siocnarF

2.4 Path integral in scalar field theory The functional formalism that we have exposed in the context of quantum mechanics can now be extended to quantum field theory. The main change is that the functions over which one integrates are functions of time and space (as opposed to functions of time only in quantum mechanics). All the result of the previous section can be translated into analogous formulas in quantum field theory, thanks to the following correspondence: q(t) p(t) j(t)

←→

←→ ←→

φ(x) Π(x) j(x) (2.55)

The main results of the previous section, namely that time-ordered products of operators in the canonical formalism become simple products of ordinary functions in the path integral representation, and that the ground state at ±∞ can be obtained

97

2. F UNCTIONAL QUANTIZATION

by relaxing the boundary conditions and multiplying the Hamiltonian by 1 − i0+ , remain true in this new context. Thus, the analogue of eq. (2.44) in a real scalar field theory is: Z

 DΠ(x)Dφ(x)

Z
 + ˙ × exp i d4 x Π(x)φ(x)−(1−i0 )H(Π, φ) + j(x)φ(x) .

Z[j] =



(2.56)

Since the Hamiltonian is quadratic in Π, H=

1 2 1 1 Π + (∇φ) · (∇φ) + m2 φ2 + V(φ) , 2 2 2

it is easy to perform the (Gaussian) functional integration on Π, to obtain: Z

Z
   Z[j] = Dφ(x) exp i d4 x L(φ) + j(x)φ(x) ,

(2.57)

(2.58)

where

L(φ) ≡


1 ˙ 2 − 1 (1 − i0+ ) (∇φ) · (∇φ) + m2 φ2 − (1 − i0+ )V(φ) . (1 + i0+ )φ 2 2 (2.59)

Note that the 1 − i0+ in front of the interaction potential plays no role if we turn off adiabatically the coupling constant when |x0 | → ∞. Using the analogue of eq. (2.49), we can separate the interactions as follows Z

 δ  Z[j] = exp − i d4 x V Z0 [j] , (2.60) iδj(x) with Z0 [j] ≡ L0 (φ)

=

Z



Z 
 Dφ(x) exp i d4 x L0 (φ) + j(x)φ(x) ,


1 ˙ 2 − 1 (1 − i0+ ) (∇φ) · (∇φ) + m2 φ2 . (2.61) (1 + i0+ )φ 2 2

The functional integral that gives Z0 [j] in eq. (2.61) is Gaussian in φ and can be performed in a straightforward manner, giving 

1Z d4 xd4 y j(x)j(y) G0F (x, y) , (2.62) Z0 [j] = exp − 2

98

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where G0F (x, y) is the inverse of the operator h i i (1 + i0+ )∂20 − (1 − i0+ )(∇2 + m2 ) .

(2.63)

Note that the terms in i0+ ensure the existence of this inverse. Going to momentum space, we see that the Fourier transform of this inverse is i (1 + i0+ )k20 − (1 − i0+ )(k2 + m2 )

,

(2.64)

which after some rearrangement of the i0+ ’s appears to be nothing but eq. (1.123). Although the canonical quantization of a scalar field theory was tractable, we see on this example that the path integral approach provides a much quicker way of obtaining the expression of the free generating functional, with the correct pole prescription for the free Feynman propagator. c sileG siocnarF

2.5 Functional determinants In the earlier sections of this chapter, we have been a bit cavalier with Gaussian integrations, since we have disregarded the constant prefactors they produce. This was legitimate in the problems we were considering, since the normalization of the generating functional can be fixed by hand. However, in certain situations, these prefactors depend crucially on quantities that have a physical significance, e.g. on a background field. In order to compute this prefactor, let us start from a simple 1-dimensional Gaussian integral, r Z +∞ 1 2π − 2 ax2 dx e = . (2.65) a −∞ The first stage of generalization is to replace x by an n-component vector x ≡ (x1 , · · · , xn ), and the positive number a by a positive definite symmetric matrix A, and to consider the integral ZY n 1 T I(A) ≡ dxi e− 2 x Ax . (2.66) i=1

This integral can be calculated by representing the vector x in the orthonormal basis made of Q the eigenvectors of A (such a basis exists, since A is symmetric). The measure i dxi is unchanged, because the diagonalization of the matrix can be done by an orthogonal transformation. Therefore, the above integral also reads ZY n r n Y 1P 2π a i y2 −2 i i = , (2.67) I(A) = dyi e ai i=1

i=1

99

2. F UNCTIONAL QUANTIZATION

where the numbers ai are the eigenvalues of A. This result can be written in a much more compact form: (2π)n/2 I(A) = √ . det A

(2.68)

This reasoning can be generalized to the functional case by writing: Z

1Z  h i−1/2   d4 xd4 y φ(x)A(x, y)φ(y) = det (A) Dφ(x) exp − , (2.69) 2

where A(x, y) is a symmetric operator. In this formula, we have still disregarded some truly constant (and infinite) prefactors, made of powers of 2π. One can also generalize this Gaussian integral to the case where the vector x is complex, J(A) ≡

ZY n

†

dxi dx∗i e−x

Ax

i=1

=

(2π)n , det A

(2.70)

where A is a Hermitean matrix. The functional analogue of this integral is Z



Z  h i−1  , Dφ(x)Dφ∗ (x) exp − d4 xd4 y φ∗ (x)A(x, y)φ(y) = det (A) (2.71)

Zeta function regularization : Despite the elegance of this formula, one should keep in mind that the functional determinant det A is most often infinite, because the spectrum of the operator extends to infinity. A common regularization technique for functional determinants is based on a generalization of Riemann’s ζ function. Let the λn be the eigenvalues of A, and define:
X 1 ζA (s) ≡ tr A−s = . λsn n

(2.72)

(The function ζA is called the zeta function of the operator A.) The determinant of A is related to this function by

det A = exp − ζA′ (0) . (2.73)

The sum over n in the definition of ζA usually converges only if Re (s) is large enough (how large depends on the distribution of eigenvalues at large n), but not for s = 0. However, like in the case of Riemann’s zeta function, ζA (s) can be analytically continued to most of the complex s-plane, which provides a regularized definition of the determinant. c sileG siocnarF

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Diagrammatic interpretation : Let us consider as an example the operator Aϕ ≡  + λ2 ϕ2 , where ϕ(x) is a background field. The inverse of this operator is the propagator of a scalar particle (with a φ4 interaction) over the background field ϕ. We can skip the regularization step if we make a ratio with the determinant of the similar operator with no background field:
det 
. R≡ (2.74) det  + λ2 ϕ2 A very useful formula relates the determinant of an operator to the trace of its logarithm,


det A = exp Tr log A . (2.75)

This formula can be proven (heuristically, since the objects we are manipulating may not be finite) by writing both sides of the equation in terms of the eigenvalues of A: X
Y

det A = λn = exp log λn = exp Tr log A . (2.76) n

n

Therefore, the ratio defined in eq. (2.74) can be rewritten as 
 2 −1 . R = exp −Tr log 1 + λϕ 2 

(2.77)

Writing −1 = iG0F , and expanding the logarithm gives R = exp

∞

X  n
  2 1 Tr − i λϕ G0F . 2 n

(2.78)

n=1

The argument of the exponential has a simple interpretation as a 1-loop diagram made of a line dressed with insertions of the background field, the index n being the number of such insertions:

n
 2 1 G0F = Tr − i λϕ 2 n

.

|

{z

n insertions

}

(2.79)

Each of the insertions of the background field (shown by lines terminated by a dot in the above diagram) corresponds to a factor −i λ2 ϕ2 . The prefactor 1/n is the symmetry factor for the cyclic permutations of the n insertions. The argument of the exponential is a sum of connected 1-loop diagrams. Taking the exponential to obtain the ratio R simply produces all the multiply connected graphs made of products of such 1-loop diagrams. c sileG siocnarF

101

2. F UNCTIONAL QUANTIZATION

2.6 Quantum effective action 2.6.1

Definition

The action S[φ] that enters in the path integral representation of the generating functional Z[j] is the classical action. Its parameters reflect the interactions among the constituents of the system at tree level, but in order to express higher order corrections loop corrections are necessary. The quantum effective action, denoted Γ [φ], is defined as the functional that would produce the all-orders value of the interactions solely from tree-level contributions. Γ [φ] should coincide with the classical action at lowest order of perturbation theory, but also encapsulates all the higher order corrections. One may write Γ [φ] formally as Z ∞ X 1 d4 x1 · · · d4 xn φ(x1 ) · · · φ(xn ) Γn (x1 , · · · , xn ) . (2.80) Γ [φ] ≡ n! n=2

Γ2 (x1 , x2 ) is therefore the inverse of the exact propagator, Γ4 (x1 , · · · , x4 ) is the exact 4-point function (in coordinate space), etc... c sileG siocnarF

2.6.2

Relation between Γ [φ] and W[j]

Until now, we have introduced the generating functional of the vacuum expectation value of time-ordered products of fields, Z[j], as well as the functional W[j] ≡ log Z[j] that generates the subset made of connected Feynman graphs. Recall that in term of path integrals, Z Z h i   W[j] (2.81) Z[j] = e = Dφ(x) exp iS[φ(x)] + i d4 x j(x)φ(x) . Let us replace the classical action S[φ] by the quantum effective action Γ [φ] in the previous formula, to define Z Z h i   WΓ [j] ZΓ [j] = e = Dφ(x) exp iΓ [φ(x)] + i d4 x j(x)φ(x) . (2.82)

This functional generates graphs whose building blocks are the exact propagator (Γ2−1 ), and the exact vertices (Γ3 , Γ4 . · · · ). From the definition of Γ [φ] as the “action” that would generate the exact theory at tree level, we conclude that WΓ [j]|tree = W[j] .

(2.83)

In other words, the tree diagrams of WΓ [j] should be equal to the all-orders W[j]. The tree diagrams may be isolated by reintroducing Planck’s constant in the definition of ZΓ [j] as follows Z Z hi i   WΓ [j;h] ¯ = Dφ(x) exp Γ [φ(x)] + d4 x j(x)φ(x) . (2.84) ZΓ [j; h] ¯ =e h ¯

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

As we have discussed in the section 1.7.6, the order in h ¯ of a connected graph is h ¯ nL −1 ,

(2.85)

where nL is the number of loops of the graph. Therefore, the functional WΓ [j; h] ¯ has the following loop expansion: WΓ [j; h] ¯ =

∞ X

h ¯ nL −1 WΓ ,nL [j] , | {z } nL =0 nL loops

(2.86)

and the tree level contributions in WΓ [j] are the terms that survive in the formal limit h ¯ → 0: ¯ WΓ [j; h] ¯ . WΓ [j]|tree = lim h h→0 ¯

(2.87)

But from our discussion of the classical limit of path integrals in section 2.2, we know that the limit h ¯ → 0 corresponds to the extremum of the argument of the exponential, i.e. δΓ [φ] + j(x) = 0 . δφ(x)

(2.88)

Note that this equation is the analogue of the usual Euler-Lagrange equation of motion, with the quantum effective action in place of the classical action. This equation implicitly defines φ as a function of j, that we will denote φj , in terms of which we can write Z i hi WΓ [j;h] ¯ Γ [φj (x)] + d4 x j(x)φj (x) , (2.89) e ≈ exp h→0 ¯ h ¯ which leads to the following relationship between the quantum effective action and the generating functional of connected graphs: Z Γ [φj ] = −i W[j] − d4 x j(x)φj (x) . (2.90) Therefore, Γ [φ] can be obtained as the Legendre transform of the generating functional W[j] of the connected graphs. Note that the “quantum equation of motion” (2.88) may also be viewed as defining j in terms of φ, that we shall denote jφ . Eq. (2.90) may therefore also be written as Z

Γ [φ] = −i W[jφ ] − d4 x jφ (x)φ(x) .

(2.91)

103

2. F UNCTIONAL QUANTIZATION

Taking a functional derivative of this equation with respect to φ(y) and using the chain rule, we obtain Z Z δjφ (x) δj (x) δW[j] δΓ [φ] = −i d4 x − j (y) − d4 x φ φ(x) . (2.92) φ δφ(y) δj(x) j=j δφ(y) δφ(y) φ | {z } −jφ (y)

This leads to

φ(x) = −i

δW[j] δj(x) j=j

,

or equivalently

φj (x) = −i

φ

 δW[j]

= φ(x) j . δj(x) (2.93)

In other words, φj is the connected 1-point function (i.e. the vacuum expectation value of the field) in the presence of the source j. c sileG siocnarF

2.6.3

Second derivative of the effective action

Differentiating eq. (2.88) with respect to j(y) gives: δ(x − y) = = =

δ δΓ [φj ] δj(y) δφj (x) Z δ2 Γ [φj ] δφj (z) − d4 z δj(y) δφj (x)δφj (z) Z δ2 Γ [φj ] δ2 W[j] . i d4 z δj(y)δj(z) δφ (z)δφ (x) | {z } | j {z j } −

G(y,z)connected

(2.94)

Γ2 (z,x)

This formula shows a posteriori that (up to a factor i) the coefficient Γ2 in the expansion (2.80) is indeed the inverse of the exact connected 2-point function, as was expected from our request that the effective action Γ [φ] reproduces the full content of the theory. By parameterizing the inverse propagator in terms of a self-energy Σ as follows, G−1 = G−1 0 + iΣ ,

(2.95)

we see that the second derivative of the quantum effective action is nothing but the self-energy. An important class of diagrams in this discussion are the one-particle irreducible (1PI) diagrams, that are those that remain connected if one cuts any one

104

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

of their internal propagators. For instance, the first of these diagrams is 1PI while the second one is not: 1PI diagram :

Non-1PI diagram : The concept of 1PI diagrams is crucial in the summation of a self-energy to all orders. Indeed, repeated insertions of a non-1PI self-energy would lead to the erroneous multiple countings of identical graphs. To avoid this, a self-energy should only contain 1PI graphs, and we conclude that the second derivative of the quantum effective action is one-particle irreducible. c sileG siocnarF

2.6.4

One-particle irreducibility

The quantum effective action Γ [φ] is in fact one-particle irreducible at all orders in φ, not just at quadratic order in φ as the above argument suggests. By exponentiating eq. (2.91) and using the path integral definition of exp(W[j]), we first obtain

ei Γ [φ]

= =

Z    Dϕ exp i S[ϕ] + d4 x j(x)(ϕ(x) − φ(x)) j=jφ Z Z     . Dϕ exp i S[ϕ + φ] + d4 x j(x)ϕ(x)

Z



(2.96)

j=jφ

(In the second line, we have shifted by φ the integration variable ϕ.) Thus, the quantum effective action can be obtained from a shifted classical action, to which is added a source jφ that implicitly depends on φ via the quantum equation of motion (2.88). The expansion of the shifted classical action S[ϕ + φ] leads to a number of vertices, some of which are φ-dependent. Thus, Γ [φ] is the sum of the connected (because we must take the logarithm in order to extract Γ [φ]) vacuum graphs build with these φ-dependent vertices and the φ-dependent source jφ . To every line of such a graph is associated a free propagator G0 , determined from the quadratic term in the action. A very important property is the fact that the expectation value of ϕ(x) with this

105

2. F UNCTIONAL QUANTIZATION shifted action vanishes: Z Z  

   ϕ(x) ≡ Dϕ ϕ(x) exp i S[ϕ + φ] + jϕ =

Z



=

−φ(x) ei Γ [φ] + e−i

=

ei Γ [φ]

 δW[j]

R



j=jφ

Z

 Dϕ (ϕ(x) − φ(x)) exp i S[ϕ] + d4 x j(ϕ − φ) 

jφ

δ eW[j] iδj(x) j=jφ

 =0. − φ(x) iδj(x) j=jφ {z } |

j=jφ

(2.97)

0

Note that in order to obtain the final zero, it is crucial that j be set to jφ at the end. Let us now consider a one-particle reducible vacuum graph G that may possibly contribute to Γ [φ]. Because it is reducible, this graph contains at least one bare propagator that connects two subgraphs A and B, such that the two subgraphs become disconnected when removing this propagator, GAB ≡

Z

A

x

y

B

.

(2.98)

x,y

When summing over all graphs that may enter in B, we get Z X

 GAB = d4 xd4 y A(x) G0 (x, y) ϕ(y) = 0 ,

(2.99)

B

thanks to the previous result on the expectation value of ϕ. Therefore, the one-particle reducible graphs do not contribute to Γ [φ], which generalizes to all orders in φ what we had already seen for the quadratic terms. c sileG siocnarF

2.6.5

One-loop effective action

At one loop, one may obtain a closed expression for the quantum effective action. For this, write the Lagrangian as a renormalized Lagrangian plus counter-terms: L ≡ Lr (φr ) + ∆L(φr ) ,

(2.100)

both depending on the renormalized field φr . We will denote Sr and ∆S the corresponding actions. Likewise, we write the external source j = jr + δj, where jr is the current that solves the following equation: δSr [φr ] + jr (x) = 0 , (2.101) δφr (x) ϕ

106

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

i.e. the current that solves at lowest order the defining equation of the effective action. The correction ∆j is then adjusted order by order so that the expectation value of the field remains equal to ϕ at all orders,

 ϕ(x) = φr (x) j +∆j . (2.102) r

In the path integral representation of the generating functional Z[j], we write the field as φr = ϕ + η: Z[j] =

Z



 iSr [ϕ+η]+∆S[ϕ+η]+R d4 x (jr +∆j)(ϕ+η) Dη(x) e ,

(2.103)

and we expand the argument of the exponential in powers of η up to quadratic order: Z Sr [ϕ + η] + d4 x jr (ϕ + η) =

Z Sr [ϕ] + d4 x jr (x)ϕ(x) Z   δS [φ ] r r 4 + j η(x) + d x r δφr (x) ϕ Z  1 δ2 Sr [φr ]  + d4 xd4 y η(x) η(y) 2 δφr (x)δφr (y) ϕ +··· (2.104)

Note that the term linear in η is zero by virtue of eq. (2.101). Therefore, we may rewrite Z[j] as follows  Z R 4   ei Sr [ϕ]+∆S[ϕ]+ d x jϕ Dη(x) ei Sϕ [η]+∆Sϕ [η] , (2.105)

Z[j] = where we denote 1 Sϕ [η] ≡ 2

Z

 d xd y η(x) 4

4

δ2 Sr [φr ]  η(y) + · · · δφr (x)δφr (y) ϕ

(2.106)

(Likewise, ∆Sϕ [η] results from the expansion in powers of η of the counter-terms.) At one loop, it is sufficient to keep only the quadratic terms in η, and the path integral gives a determinant: 

−1/2 δ2 Sr [φr ]  i det − 2 δφr (x)δφr (y) ϕ    i 1 δ2 Sr [φr ]  = exp − tr ln − . 2 2 δφr (x)δφr (y) ϕ 

(2.107)

2. F UNCTIONAL QUANTIZATION

107

At this order, the generating functional of connected graphs reads3 Z   1 δ2 Sr [φr ]  W[j] = i Sr [ϕ]+∆S[ϕ]+ d4 x jϕ − tr ln +· · · (2.108) 2 δφr (x)δφr (y) ϕ from which we obtain the following quantum effective action  δ2 Sr [φr ]  i + ··· Γ [ϕ] = Sr [ϕ] + ∆S[ϕ] + tr ln 2 δφr (x)δφr (y) ϕ

(2.109)

Note that the object inside the logarithm is the inverse of the propagator dressed by the background field ϕ. c sileG siocnarF

2.7 Two-particle irreducible effective action 2.7.1

Definition and equations of motion

The quantum effective action Γ [φ] studied in the previous section can be extended into a functional Γ [φ, G] that depends on a field φ and a propagator G. The starting point of this derivation is to introduce a second source k(x, y) that couples to a pair of fields ϕ(x)ϕ(y). The corresponding generating functional W[j, k] for connected graphs is given by Z Z Z     1 W[j,k] k(x, y) ϕ(x)ϕ(y) . (2.110) e = Dϕ exp i S[ϕ]+ j(x)ϕ(x)+ 2 x,y x In terms of graphs, W[j, k] is the sum of the connected vacuum graphs built with the bare propagator and the vertices defined by the classical action S[ϕ], with the external source j, and with a kind of non-local mass term k(x, y). Let us denote δW[j, k] ≡ φj,k (x) , iδj(x)  1 δW[j, k] ≡ φj,k (x)φj,k (y) + Gj,k (x, y) . iδk(x, y) 2

(2.111)

In the second equation, we have separated a disconnected part φj,k (x)φj,k (y) and a connected two-point function Gj,k (x, y). Both the field φj,k and the propagator Gj,k depend on the sources, which we indicate by the subscript j, k. Conversely, we may formally invert these equations to define φ, G dependent sources, jφ,G and kφ,G . 3 We

have dropped a factor − 2i inside the tr ln, since it only produces an additive constant.

108

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS Then, the Legendre transform that defined Γ [φ] from W[j] can be generalized into Γ [φ, G] =

Z

−i W[jφ,G , kφ,G ] − d4 x jφ,G (x)φ(x) Z   1 d4 xd4 y kφ,G (x, y) φ(x)φ(y) + G(x, y) . − 2

(2.112)

By taking derivatives with respect to φ(x) or G(x, y), we obtain the following equations: Z Γ [φ, G] + jφ,G (x) + d4 y kφ,G (x, y) φ(y) = 0 , δφ(x) Γ [φ, G] 1 + kφ,G (x, y) = 0 . δG(x, y) 2

(2.113)

Note that the first of these equations generalizes the quantum equation of motion (2.88) with the adjunction of a self-energy kφ,G (x, y). c sileG siocnarF

2.7.2

Two-particle irreducibility

From the Legrendre transform of eq. (2.112), we obtain the following path integral representation of the functional Γ [φ, G]: ei Γ [φ,G]

Z   Dϕ exp i S[ϕ] + j(x)(ϕ(x) − φ(x)) x Z   1 + k(x, y) ϕ(x)ϕ(y) − φ(x)φ(y) − G(x, y) 2 x,y Z Z    Dϕ exp i S[ϕ+φ]+ j(x)ϕ(x) x Z   1 k(x, y) ϕ(x)ϕ(y)+2ϕ(x)φ(y)−G(x, y) , + 2 x,y (2.114) Z

=

=



where we have omitted the subscript φ, G on the sources j, k for the sake of brevity. From the second equation, we first obtain

 ϕ(x)

=

Z

=

ei Γ [φ,G]



Z   Dϕ ϕ(x) exp i S[ϕ + φ] + jϕ +  δW[j, k] iδj(x)

k 2

 − φ(x) j=jφ,G = 0 . k=kφ,G



ϕϕ + 2ϕφ − G



(2.115)

109

2. F UNCTIONAL QUANTIZATION

Like in the case of the 1PI functional Γ [φ], this identity ensures that the one-particle reducible graphs do not contribute to Γ [φ, G]. But as we shall see now, the functional Γ [φ, G] is limited to a much more restricted set of graphs, since only the two-particle irreducible graphs contribute, i.e. the graphs that cannot be made disconnected by removing two arbitrary propagators. Consider a 2-particle reducible graph,

GAB ≡

Z

x

A

x,y

,

B

(2.116)

y

in which we have exhibited the two bare propagators that would disconnect the graph if removed. Summing over the graphs that can contribute to B, we may write this as Z X

 (2.117) GAB = d4 xd4 y A(x, y) ϕ(x)ϕ(y) c , B

(the subscript c indicates that we keep only the connected part of the two point function) with

 ϕ(x)ϕ(y) c

≡ e−i Γ [φ,G]

Z



 Dϕ ϕ(x)ϕ(y) ei




R k S[ϕ+φ]+ jϕ+ 2 [ϕϕ+2ϕφ−G]

R k δeW[j,k] = −φ(x)φ(y) + 2 e| −iΓ [φ,G] e−i{zjφ+ 2 (G−φφ)} iδk(x, y) e−W[j,k]

= G(x, y) .

(2.118)

In the second equality, we have  ignored some terms that have already been shown to vanish when studying ϕ(x) , and we have extracted the combination ϕ(x)ϕ(y) by differentiating with respect to k(x, y). From this identity, we obtain XZ B

x

A x,y

B y

=

Z

.

A x,y

(2.119)

G(x,y)

In other words, when summing over all the possible graphs contributing to B, the 2-particle reducible block is replaced by a single propagator G(x, y), and the resulting graph is two-particle irreducible. Thus, the functional Γ [φ, G], when expressed in terms of the 2-point function G, is made only of 2-particle irreducible graphs, and its derivatives with respect to the field φ are the 2-particle irreducible n-point functions. Thus, Γ [φ, G] is the generating functional in φ of the 2PI correlation functions. c sileG siocnarF

110

2.7.3

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Loop expansion of Γ [φ, G]

2PI functional at null field : Consider first the 2PI effective action at null field, Γ [0, G]. At φ ≡ 0, we have −2

δΓ [0, G] = k0,G . δG

(2.120)

Using the path integral representation (2.114), and replacing k0,G by the above equality, we get4 ei Γ [0,G]

= =


δΓ [0,G]
R  Dϕ ei S[ϕ]+tr [G−ϕϕ] δG + j0,G ϕ X 2PI vacuum diagrams built with the . propagator G and the vertices from S[ϕ] Z



(2.121)

The first equality is as an implicit identity obeyed by Γ [0, G]. The diagrammatic interpretation in the second line follows from the discussion at the end of the previous subsection. In the right hand side, the term in j0,G ϕ cancels the 1-particle reducible tadpole contributions, while the term in G − ϕϕ is the one that eliminates the 2particle reducible ones (by replacing chains like G0 ΣG0 Σ · · · ΣG0 by G). Note that the bare propagator G0 defined by the quadratic part of the classical action S[ϕ] does not appear in the final result for Γ [0, G], since it is replaced systematically by G. Only the interaction terms of the classical action matter, since they define the vertices that connect the G’s in the diagrammatic representation of Γ [0, G]. c sileG siocnarF

Legrendre transform at fixed k : Then, it is useful to introduce the first Legrendre transform of W[j, k], i.e. the 1PI effective action at fixed k, Z Γk [φ] ≡ −i W[jφ,k , k] − d4 x jφ,k (x)φ(x) , (2.122) where the source jφ,k now has an implicit dependence on the field φ and on the second source k. This functional obeys: δΓk [φ] δk(x, y)

=

Z Z δjφ,k (z) δW δW δjφ,k (z) + d4 z − d4 z φ(z) iδk(x, y) iδj(z) j=jφ,k δk(x, y) δk(x, y) {z } | φ(z)

= 4 We

δW . iδk(x, y)

use the compact notation

(2.123)

R

x,y

A(x, y)B(x, y) = tr (AB).

111

2. F UNCTIONAL QUANTIZATION Given this identity, it is natural to define G(x, y) as follows,
1 δΓk [φ] φ(x)φ(y) + G(x, y) . ≡ δk(x, y) 2

(2.124)

In this equation, we may either view G as a function of k, φ or k as a function (that we denote kφ,G ) of φ, G. Adopting the latter point of view, we may perform a second Legendre transform to obtain a functional of φ and G: Γ [φ, G] ≡ Γkφ,G [φ] −


1 tr kφ,G [G + φφ] . 2

(2.125)

(We use the same notation Γ [φ, G] because we shall prove shortly that this functional is identical to the one we have defined earlier by a double Legendre transform.) This definition leads to  δΓ  δΓ [φ, G] 1 δkφ,G  1  δkφ,G k = − k(x, y) + tr [G + φφ] , − 2 tr δG(x, y) 2 δk kφ,G δG δG {z } | δΓ [φ, G] δφ(x)

=

−jφ,k − + tr |

Z

0

kφ,G (x, y)φ(y)

y

  δΓ δkφ,G  1  δkφ,G k − 2 tr [G + φφ] , δk kφ,G δφ δG {z }

(2.126)

0

that are identical to eqs. (2.113). This proves that the definition (2.125) of the 2PI effective action is equivalent to the original definition (2.112). c sileG siocnarF

Path integral representation of Γk [φ] : Since we have Z
R 1R   W[j,k] e = Dϕ ei S[ϕ]+ jϕ+ 2 ϕkϕ ,

Γk [φ] is the 1PI effective action of the modified classical action Z 1 Sk [ϕ] ≡ S[ϕ] + d4 xd4 y ϕ(x) k(x, y) ϕ(y) , 2 and it admits the following path integral representation Z R δΓk [φ]
  . eiΓk [φ] = Dϕ ei Sk [φ+ϕ]− ϕ δφ

(2.127)

(2.128)

(2.129)

Thus, if we denote Γk,1 [φ] ≡ Γk [φ] − Sk [φ] the terms at 1-loop and higher orders, we have Z R δ(Sk [φ]+Γk,1 [φ])
  iΓk,1 [φ] δφ . (2.130) e = Dϕ ei Sk [φ+ϕ]−Sk [φ]− ϕ

112

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Let us now expand the shifted action Sk [φ + ϕ] Z Z δ2 Sk [φ] δSk [φ] 1 Sk [φ + ϕ] ≡ Sk [φ] + ϕ ϕ + ϕ + Sint [φ; ϕ] , (2.131) δφ 2 δφδφ | {z } k+iG−1 φ

where Sint [φ; ϕ] denotes the terms of degree at least three in ϕ in the Taylor expansion of S[φ + ϕ], and G−1 φ is the inverse of the tree-level propagator in the background field φ. Therefore, Γk,1 [φ] can also be written as eiΓk,1 [φ] =

Z



 Dϕ ei

R δΓk,1 [φ]
1R −1 δφ 2 ϕ[k+iGφ ]ϕ+Sint [φ;ϕ]− ϕ

Diagrammatic interpretation of Γ [φ, G] : Γ [φ, G] = const + S[φ] +

i 2

.

(2.132)

Let us now write Γ [φ, G] as follows:

tr ln(G−1 ) +

i 2

ln(G−1 φ G) + Γ2 [φ, G] . (2.133)

This equation defines Γ2 [φ, G] so that the left hand side is indeed Γ [φ, G]. The Combining eqs. (2.125), (2.132) and (2.133), we must have Γ2 [φ, G] = const −

1 2

tr ([kφ,G + i G−1 φ ]G) −

i 2

tr ln(G−1 ) + Γk,1 [φ] . (2.134)

In order to eliminate kφ,G , we may use
δΓ2 [φ, G] 1 i −1 + kφ,G + Gφ − G−1 = 0 , δG 2 2

that follows from eqs. (2.113) and (2.133). We thus obtain Z
R δΓ [φ]
δΓ2   k,1 iΓ2 [φ,G] e = Dϕ ei Sφ,G [ϕ]+tr [G−ϕϕ] δG − ϕ δφ ,

(2.135)

(2.136)

with

Sφ,G [ϕ] ≡

i 2

Z

ϕ G−1 ϕ + Sint [φ; ϕ] .

(2.137)

Finally, using the fact that −jφ,G = δΓk,1 /δφ and comparing with eq. (2.121), we see that Γ2 [φ, G] is the sum of the 2PI vacuum graphs built with the propagator G and the vertices obtained from the expansion of S[φ + ϕ]. The first four terms of the expansion of Γ2 [φ, G] in a scalar field theory with φ4 interaction are shown in the figure 2.3.

113

2. F UNCTIONAL QUANTIZATION

Figure 2.3: Beginning of the diagrammatic expansion of Γ2 [φ, G] in a scalar field theory with quartic interaction. The lines terminated by a cross are the field φ and the black lines represent the propagator G.

2.7.4

Dyson equation

After setting the source k = 0, the equation of motion for the propagator (2.135) becomes −i G−1 = −i G−1 φ −2 |

δΓ2 [φ, G] , δG } {z

(2.138)

−Σ

which is known as the Dyson equation, that resums the self-energy Σ on the propagator. Convoluting this equation by G on the right gives −iG−1 φ G + Σ G = −i .

(2.139)

Recall that Gφ is the tree-level propagator in a background field φ, i.e. i G−1 φ =

δ2 S[φ] = −( + m2 ) − V ′′ (φ) , δφδφ

(2.140)

where V(φ) is the interaction potential in the Lagrangian. Therefore, the equation of motion has the following more explicit form    + m2 + V ′′ (φ) + Σ G = −i . (2.141)

A closed system of equations is obtained by adding the equation of motion of φ, obtained as δΓ/δφ = 0 (in a system with symmetry φ → −φ, and assuming that there is no spontaneous breaking of this symmetry, the field expectation value is zero and we may simply set φ = 0 in the above equation for G). Note that the self-energy Σ is itself a functional of φ and G. In the 2PI framework, the self-energy is the derivative of Γ2 [φ, G] with respect to the propagator, Σ = −2

δΓ2 [φ, G] . δG

(2.142)

114

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Diagrammatically, this derivative amounts to opening one internal line of the graphs that contribute to Γ2 . For instance the graphs of the figure 2.3 give the following topologies in Σ: Σ∼

.

Note that the 2-particle irreducibility of Γ2 is equivalent to the 1-particle irreducibility of Σ, which is crucial in order to avoid including multiple times the same contributions when summing the self-energy to all orders. In practice, a truncation is necessary in order to obtain equations that have a finite number of terms. For instance, keeping only the first diagram in Γ2 gives the tadpole diagram in Σ (but bear in mind that since this tadpole contains the full propagator G, the solution of the corresponding Dyson equation resums an infinite set of Feynman graphs). c sileG siocnarF

2.8 Euclidean path integral and Statistical mechanics 2.8.1

Statistical mechanics in path integral form

A path integral formalism also exists for statistical mechanics. In order to illustrate this, let us consider again the quantum mechanical system described by the Hamiltonian of eq. (2.1). Our goal is to calculate the partition function in the canonical ensemble5 ,
Zβ ≡ Tr e−βH , (2.143) where β is the inverse temperature (it is customary to use a system of units in which Boltzmann’s constant kB is equal to unity – therefore temperature has the same dimension as energy.) More generally, one may want to calculate the following canonical ensemble expectation values,


 −βH O . (2.144) O β ≡ Z−1 β Tr e The cyclicity of the trace leads to an important identity for expectation values of products of operators:


−βH O1 (t)O2 (t ′ ) β ≡ Z−1 O1 (t)O2 (t ′ ) β Tr e
−βH = Z−1 O1 (t) |e+βH{ze−βH} O2 (t ′ ) β Tr e 1

=

=

−βH Z−1 O2 (t ′ ) e−βH O1 (t)e+βH β Tr e {z } |



 O2 (t )O1 (t + iβ) β , ′

O1 (t+iβ)




(2.145)

5 In theories with a conserved quantity, it is also possible to study the grand canonical ensemble. One needs to substitute H → H − µQ in the definition of the partition function, where Q is the operator of the conserved charge and µ the associated chemical potential.

115

2. F UNCTIONAL QUANTIZATION

where we have formally identified the density operator exp(−βH) with a time evolution operator for an imaginary time iβ. This relationship is called the Kubo-MartinSchwinger (KMS) identity. Although we have established it for an expectation value of a product of two operators, it is completely general. The identification of the density operator with an imaginary time evolution operator is at the heart of the formalism to evaluate canonical ensemble expectation values. If we represent the trace that appears in the partition function in the coordinate basis, Z 

(2.146) Zβ = dq q e−βH q ,

the integrand in the right hand side is a transition amplitude similar to eq. (2.3), except that initial and final coordinates are identical, and the time interval is imaginary. We can nevertheless formally reproduce all the manipulations of the section 2.1, with an initial time ti ≡ 0 and a final time tf ≡ −iβ. It is common to introduce the Euclidean time τ ≡ it, with τ varying from 0 to β. The only changes to our original derivation of the path integral is that the path q(t) must be replaced by a path q(τ) whose time derivative is the Euclidean velocity q˙ E , related to the usual velocity q˙ by q˙ ≡

dq dq =i . dt dτ |{z}

(2.147)

˙E q

We obtain the following path integral representation of the partition function: Zβ

=

Z

=

dq

Z



 Dp(τ)Dq(τ) exp

q(0)=q q(β)=q

Z



 Dp(τ)Dq(τ) exp

q(0)=q(β)

Zβ 0

Zβ 0


 dτ i p(τ)q˙ E (τ) − H(p(τ), q(τ))


 dτ i p(τ)q˙ E (τ) − H(p(τ), q(τ)) . (2.148)

In the second line, we have simplified the boundary conditions of the path q(τ), since the only constraint it must obey is to be β-periodic in imaginary time. The integration over the momentum p(τ) is again Gaussian, and after performing it we obtain the following expression Zβ =

Z



 Dq(τ) exp −

q(0)=q(β)

Zβ 0

|

dτ


 m 2 . q˙ E (τ) + V(q(τ)) 2 {z }

The quantity SE [q] is called the Euclidean action.

SE [q(τ)]

(2.149)

116

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Then, we can generalize this formalism to calculate ensemble averages of timeordered (in imaginary time) products of position operators. For instance, the analogue of eq. (2.28) is

Tr e−βH Tτ Q(τ1 )Q(τ2 ) =

Z



 Dq(τ) e−SE [q(τ)] q(τ1 )q(τ2 ) ,

q(0)=q(β)

(2.150) where the symbol Tτ denotes the time-ordering in the imaginary time τ. Likewise, we may define a generating functional for these expectation values −βH

Tr e

Tτ exp

Zβ 0


dτ j(τ)Q(τ) =

Z



Rβ  Dq(τ) e−SE [q(τ)]+ 0 dτ j(τ)q(τ) .

q(0)=q(β)

(2.151)

2.8.2

Statistical field theory

This formalism can be extended readily to a quantum field theory. In this context, it can be used to calculate canonical ensemble expectation values of operators for a system of relativistic particles. One can write directly the following generalization of eq. (2.151), Zβ
d4 xE j(x)φ(x) Tr e−βH Tτ exp | {z0 } Z[j;β]

=

Z



Rβ 4  Dφ(x) e−SE [φ(x)]+ 0 d xE

j(x)φ(x)

,

(2.152)

φ(0,x)=φ(β,x)

where the measure d4 xE stands for dτ d3 x. Like in the case of ordinary QFT in Minkowski space-time, we can isolate the interactions by writing:  

Z δ 4 Z0 [j; β] , (2.153) Z[j; β] = exp − d xE LE,I δj(x) where LE,I is the interaction term in the Euclidean Lagrangian density, and Z0 [j; β] is the generating functional of the non-interacting theory: Z0 [j; β] =

Z



h  Dφ(x) exp −

φ(0,x)=φ(β,x)

Zβ 0

d4 xE

1

2

i
(∂τ φ)2 +(∇φ)2 +m2 φ2 −jφ .

117

2. F UNCTIONAL QUANTIZATION

(2.154) The Gaussian path integral in this expression leads to: Z0 [j; β] = exp

1 Zβ 2

0

 d4 xE d4 yE j(x) G0E (x, y) j(y) ,

(2.155)

where the free Euclidean propagator G0E (x, y) is the inverse of the operator m2 − ∂2τ −∇2 over the space of functions that are β-periodic in the imaginary time variable. Because of this periodicity, the “energy” variable, conjugate to the Euclidean time, is discrete: ωn ≡

2πn β

(n ∈ ❩) .

(2.156)

In terms of these energies, called Matsubara frequencies, the free Euclidean propagator in momentum space reads e 0 (ωn , p) = G E

ω2n

1 . + p2 + m2

(2.157)

Note that the denominator cannot vanish, and therefore this propagator does not need an i0+ prescription for being fully defined. Eqs. (2.153) and (2.155) lead to a perturbative expansion that can be cast into an expansion in terms of Feynman diagrams. The Feynman rules associated to these graphs are very similar to those already encountered when calculating scattering amplitudes, with only a few modifications: Propagators : Vertices :

1 , ω2n + p2 + m2 X X

− λ 2π δ ωni (2π)3 δ pi , i 3

Loops :

Z 1 X d p . β (2π)3

(2.158) (2.159)

i

(2.160)

n∈❩

In other words, the main difference with the usual perturbative expansion is that the energies are replaced by the discrete Matsubara frequencies, and that the loop integration on p0 is replaced by a discrete sum over these frequencies. c sileG siocnarF

118

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 3

Path integrals for fermions and photons In the previous chapter, we have learned that the quantization of a scalar field may be performed by means of the path integral representation. This leads to a much more concise derivation of the generating functional, and of the free propagator, compared to the canonical approach. In this chapter, we will therefore seek a similar path integral formalism for other types of fields, in view of the functional quantization of a gauge theory such as QED (and later, of non-Abelian gauge theories, for which a canonical approach would be extremely difficult to implement). c sileG siocnarF

3.1 Grassmann variables 3.1.1

Definition

In the functional formulation of a scalar field theory, we saw that time-ordered products of field operators correspond to the ordinary product of the integration variable in the integrand of the path integral (see the eq. (2.29)). Ultimately, a path integral representation of the time-ordered product of fermion field operators should allow the same, but with a catch: the T-product for fermions involves a minus sign when two operators are exchanged (see 1.216), that we need to be able to generate in the integrand of a would-be fermionic path integral. This can be achieved with Grassmann numbers1 , that are anti-commuting variables. In a sense, Grassmann 1 Although we call them “numbers”, they are not representable as scalar (e.g. real or complex) variables. A Grassmann number may be represented by a nilpotent 2 × 2 matrix, and the Grassmann algebra with N

119

120

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

numbers are the classical analogue of anti-commuting quantum operators. For a set of Grassmann variables ψi (i = 1 · · · N), we have  ψi , ψj = 0 . (3.1)

The linear space spanned by the ψi ’s is called a Grassmann algebra.

3.1.2

Functions of a single Grassmann variable

Consider first the case N = 1. The square of a Grassmann number ψ is therefore zero, ψ2 = 0, and by induction all higher powers of ψ are also zero. The Taylor expansion of a function of ψ is therefore limited to the first two terms, f(ψ) = a + ψb .

(3.2)

In general, we need to deal with functions f(ψ) that are themselves commuting objects. Therefore, the coefficient a is an ordinary number, while b is another Grassmann number, {b, b} = {b, ψ} = 0. This implies that f(ψ) = a + ψb = a − bψ .

(3.3)

Because of the non-commuting nature of b and ψ, we may define left and right derivatives, denoted by: →

∂ ψ f(ψ) = b ,

←

f(ψ) ∂ ψ = −b .

(3.4)

One may define a linear mapping on functions of a Grassmann variable, that behaves for most purposes as an integration (although it is not an integral in the Lebesgue sense), called the Berezin integral. We require two basic axioms: • Linearity : Z Z dψ α f(ψ) = α dψ f(ψ) ,

(3.5)

generators admits a representation in terms of 2N × 2N matrices, that may be viewed as operators acting on the Hilbert space of N identical fermions of spin 1/2 (of dimension 2N since each spin has two states). For instance, when N = 2, one may represent the Grassmann numbers ψ1,2 as     0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     ψ2 =  ψ1 =   .  , 1 0 0 0 0 0 0 0 0 −1 0 0 0 0 1 0

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS • The integral of a total derivative is zero: Z dψ ∂ψ f(ψ) = 0 . The only definition consistent with these requirements is Z dψ f(ψ) = b ,

121

(3.6)

(3.7)

up to an overall constant that should be the same for all functions. Thus, integration and differentiation of functions of a Grassmann variable are essentially the same thing. In particular, the Berezin integral satisfies: Z Z dψ 1 = 0 , dψ ψ = 1 . (3.8)

Functions of N Grassmann variables

3.1.3

Taylor expansion : We will denote collectively ψ ≡ (ψ1 , · · · , ψN ). The most general function of N Grassmann variables can be written as N X 1 f(ψ) = ψi ψi · · · ψip Ci1 i2 ···ip , p! 1 2

(3.9)

p=0

with implicit summations on the indices in . Terms of degree higher than N cannot exist because they would contain the square of at least one of the ψi ’s, and therefore be zero. We have chosen to write the Grassmann variables on the left of the coefficients → in order to simplify the calculation of the left derivatives ∂ ψ . Note that the last coefficient Ci1 ···iN must be proportional to the Levi-Civita tensor: Ci1 ···iN ≡ γ ǫi1 ···iN .

(3.10)

Note that this last term can also be written as: 1 N!

ψi1 · · · ψiN γ ǫi1 ···iN = ψ1 · · · ψN γ .

(3.11)

Integration : In order to be consistent with eqs. (3.8), the integral of f(ψ) over the N Grassmann variables ψ1 , · · · , ψN , must be given by Z dN ψ f(ψ) = γ . (3.12) The terms of degree 0 through N − 1 in the “Taylor expansion” of f(ψ) cannot contribute to the integral, since at least one of the ψi is absent in these terms, and

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

the integral over this ψi will therefore give zero. A somewhat more explicit formulation of an integral over N Grassmann variables is to write the measure as dN ψ ≡ dψN dψN−1 · · · dψ1 (in this order), and to perform the N integrals successively, starting with the innermost one (i.e. dψ1 ). Therefore Z Z Z Z   N dψ2 dψ1 ψ1 ψ2 · · · ψN = 1 . (3.13) d ψ ψ1 · · · ψN = dψN · · · | {z } | {z1 } 1

Change of variables : Let us now consider a linear change of variables: ψi ≡ Jij θj ,

(3.14)

where θ1 · · · θN are N Grassmann variables. The last term of the expansion of f(ψ), the only one relevant for integration, can be rewritten as ψi1 · · · ψi

N

ǫi1 ···iN γ

= =



Ji1 j1 θj1 · · · JiN jN θjN ǫi1 ···iN γ
det J θj1 · · · θjN ǫj1 ···jN γ .

From this relationship, we conclude that Z Z 
−1 dN ψ f(ψ) = det J dN θ f(ψ(θ)) . | {z } {z } | γ

(3.15)

(3.16)

det (J) γ

Thus, a change of variables in a Grassmann integral involves the inverse of the Jacobian that would normally appear in the same change of variables for a scalar integral. c sileG siocnarF

Gaussian integrals : Let ψ1 , · · · , ψN be N Grassmann variables, and consider the following integral Z
(3.17) I(M) ≡ dN ψ exp 21 ψi Mij ψj ,

where M is an antisymmetric N × N matrix made of commuting numbers (real or complex). Firstly, note that such an integral is non-zero only if N is even. For N = 2, this matrix is of the form ! 0 µ M= , (3.18) −µ 0

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS and the exponential in the integral reads
exp 21 ψi Mij ψj = 1 + µ ψ1 ψ2 .

123

(3.19)

(Recall that functions of two Grassmann variables are in fact polynomials of degree two.) Therefore, in the case N = 2, the Gaussian integral (3.17) reads2  1/2 I(M) = µ = det (M) . (3.20)

In the case of a general even N, the matrix M may be written in the following block diagonal form,   0 µ1  −µ   1 0    QT ,  0 µ 2 (3.21) M=Q     −µ 0 2   .. . | {z } D

where Q is a special3 orthogonal matrix. Defining QT ψ ≡ θ, we have Z
 −1 I(M) = det (Q) dN θ exp 12 θT Dθ . | {z }

(3.22)

µ1 µ2 ···=[det (D)]1/2

But since det (Q) = +1, this becomes  1/2  1/2 I(M) = det (D) = det (M) .

(3.23)

Contrast this with the result of a Gaussian integral in the case of ordinary real variables, eq. (2.68), where the square root of the determinant appeared in the denominator. It is often necessary to perform a Gaussian integral in the presence of a source that shifts the minimum of the quadratic form in the exponential, Z
(3.24) I(M, η) ≡ dN ψ exp 21 ψi Mij ψj + ηi ψi ,

where η is a set of N Grassmann sources. By introducing the new Grassmann variable ψi′ ≡ ψi − M−1 ij ηj , this integral falls back to the previous type, and we obtain:  1/2
I(M, η) = det (M) exp − 21 ηT M−1 η .

2 The

(3.25)

determinant of a real antisymmetric matrix of even size is the square of its Pfaffian and is therefore

positive. 3 Orthogonal matrices have determinant +1 or −1. The special orthogonal matrices are the subgroup of those that have determinant +1.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Gaussian integral with 2N variables : Another useful type of Gaussian integral is Z
J(M) ≡ dN ξdN ψ exp ψi Mij ξj , (3.26)

where M is an N × N matrix of commuting numbers, and ψ and ξ are independent Grassmann variables. The only non-zero contribution to this integral comes from the term of order N in the Taylor expansion of the exponential, Z

1 J(M) = dN ξdN ψ ψi1 Mi1 j1 ξj1 · · · ψiN MiN jN ξjN N! N(N−1) Z

(−1) 2 = dN ξdN ψ ψi1 · · · ψiN ξj1 · · · ξjN N! × Mi1 j1 · · · MiN jN =

(−1)

N(N−1) 2

N!

ǫi1 ···iN ǫj1 ···jN Mi1 j1 · · · MiN jN .

(3.27)

In the second line, we have reordered the Grassmann variables in order to bring all the ψi ’s on the left, and the sign in the prefactor keeps track of the number of permutations that are necessary to achieve this. To give a non-zero result, the indices {in } and {jn } must be permutations of [1 · · · N]: J(M) =

=

(−1)

N(N−1) 2

N! (−1)

ǫ(σ)ǫ(ρ) Mσ(1)ρ(1) · · · Mσ(N)ρ(N)

X

ǫ(σ)ǫ(τσ) M1τ(1) · · · MNτ(N)

σ,ρ∈Sn

N(N−1) 2

N!

X

σ,τ∈Sn

(3.28)

where ǫ(σ) is the signature of the permutation σ, and with τ ≡ ρσ−1 in the second line. Using ǫ(σ)ǫ(τσ) = ǫ(τ), this becomes:  X N(N−1)  1 X J(M) = (−1) 2 ǫ(τ) M1τ(1) · · · MNτ(N) . (3.29) 1 N! τ∈Sn σ∈Sn {z } | {z } | 1

det (M)

Note that this overall sign may be absorbed into a reordering of the measure, since: dN ξdN ψ = (−1)

N(N−1) 2

dξN dψN · · · dξ1 dψ1 .

Therefore, we have Z

dξN dψN · · · dξ1 dψ1 exp ψi Mij ξj = det M .

(3.30)

(3.31)

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3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

3.1.4

Complex Grassmann variables

Now, let us define complex Grassmann variables, from two of the previously defined Grassmann variables ψ and ξ: χ≡

ψ + iξ √ 2

,

ψ − iξ √ . 2

(3.32)

i (χ − χ) √ , 2

(3.33)

χ≡

Conversely, we have χ+χ ψ= √ 2

,

ξ=

and the integrations over these variables are related by dξdψ = i dχdχ , ψξ = −i χχ , Z Z dχdχ χχ = dξdψ ψξ = 1 . From this, we obtain Z
dχdχ exp µ χχ = µ ,

that can be generalized into Z
dχN dχN · · · dχ1 dχ1 exp(χT Mχ) = det M .

(3.34)

(3.35)

(3.36)

In the presence of sources η and η, we obtain the following Gaussian integral: Z




dχN dχN · · · dχ1 dχ1 exp χT Mχ+ηT χ+χT η = det M exp −ηT M−1 η . (3.37)

3.2 Path integral for fermions We now have all the ingredients for building a path integral for spin 1/2 fermions. Let us work our way backwards, starting from a generating functional that generates the free time-ordered products of spinors,

Z  (3.38) Z0 [η, η] ≡ exp − d4 xd4 y η(x)S0F (x, y)η(y) ,

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where S0F (x, y) is the free Dirac time-ordered propagator and η and η are a pair of complex Grassmann-valued sources. Indeed, we have ← δ δ Z0 [η, η] iδη(x) iδη(y) →

= S0F (x, y) .

(3.39)

η=η=0

Taking more than two derivatives (but with an equal number of derivatives with respect to η and with respect to η) will lead to all the contributions in the free time-ordered product of spinors, with the correct signs to account for their anti-commuting nature. Note that using Grassmann-valued sources was necessary in order to get these signs. Then, by comparing eqs. (3.37) and (3.38), we can represent this free generating function as a path integral over Grassmann variables:

Z0 [η, η] =

=

Z



Z  / − m)ψ(x) Dψ(x)Dψ(x) exp i d4 x ψ(x)(i∂

Z



 Dψ(x)Dψ(x) eiS[ψ,ψ] ei


 +η(x)ψ(x) + ψ(x)η(x)
R d4 x η(x)ψ(x)+ψ(x)η(x)

.

(3.40)

We have ignored the determinant, since it is independent of the sources. Instead, one simply adjusts the normalization of the generating functional so that Z0 [0, 0] = 1. The second line shows that the path integral formulation of a field theory of spin 1/2 fermions takes the same form as that of scalar fields, provided we use Grassmann variables instead of commuting c-numbers. c sileG siocnarF

In quantum electrodynamics, fermions interact only by their minimal coupling to the photon fields, LI = −i e ψγµ Aµ ψ .

(3.41)

As in the scalar case, this interaction can be factored out of the generating functional, by writing: Z[η, η] = exp



Z − ie d4 x Aµ (x)

→

←

δ δ γµ iδη(x) iδη(x)



Z0 [η, η] .

(3.42)

Here, we are treating the photon field as a fixed background. When we consider the path integral representation of dynamical photons in the next section, the Aµ (x) inside the exponential will also be replaced by a functional derivative.

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

127

3.3 Path integral for photons 3.3.1

Problems with the naive path integral

In the case of photons, the difficulties encountered in the path integral formulation are of a different nature. Since photons are bosons, we expect that they can be represented by a functional integration over commuting functions Aµ (x). But the gauge invariance of the theory implies that there is an unavoidable redundancy in this representation: the naive path integral over [DAµ (x)] would integrate over infinitely many copies of the same physical configurations. Therefore, we need a way to cut through this redundancy, which is achieved by gauge fixing. In order to better see the nature of this difficulty, let us assume that we can treat Aµ (x) as four scalar fields, and write the following path integral, Z

Z
   µ (3.43) Z0 [j ] ≡ DAµ (x) exp i d4 x − 41 Fµν Fµν + jµ Aµ .

This is a Gaussian integral, since Fµν Fµν is quadratic in the field Aµ , 1 − 4

Z

4

d xF

µν

Fµν

= = =

Z

1 − d4 x ∂µ Aν − ∂ν Aµ ∂µ Aν − ∂ν Aµ 4 Z
1 + d4 x Aµ gµν  − ∂µ ∂ν Aν 2 Z
ν 1 d4 k e µ e (−k) . A (k) gµν k2 − kµ kν A − 4 2 (2π) (3.44)

Performing this Gaussian integral requires the inverse of the object gµν k2 − kµ kν , that one may seek as a linear combination of the metric tensor gµν and kµ kν /k2 , i.e. we are looking for coefficients α and β such that:
ν ρ
gµν k2 − kµ kν α gνρ + β kkk2 = δρµ . (3.45) | {z } ρ α k 2 δρ µ −α kµ k

This equation has clearly no solution, and therefore it is impossible to invert gµν k2 − kµ kν . This means that some eigenvalues
ν of this operator are zero, and that the e µ (k) gµν k2 − kµ kν A e (−k) has flat directions. Along these flat quadratic form A directions, the exponential in the path integral (3.43) does not decrease, which spoils e µ (k) along kµ . its convergence. These flat directions correspond to the projection of A µ Note that they also do not contribute to the linear term j Aµ , for a conserved current that satisfies ∂µ jµ = 0. Therefore, one should not integrate over these components of Aµ in eq. (3.43). c sileG siocnarF

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

3.3.2

Path integral in Landau gauge

A simple way out it to decompose Aµ as follows: Aµ

µ Aµ ⊥ + Ak ,  kµ kν  e gµν − Aν (k) , k2  kµ kν  e ν (k) . A k2

=

e µ (k) ≡ A ⊥

e µ (k) ≡ A k

(3.46)

The functional measure can be factorized as follows 

    DAµ = DAµ DAµ ⊥ k ,

(3.47)

and since nothing depends on Aµ k in the photon kinetic term, we can write Z



Z Z0 [j ] ≡ exp i d4 x jµ Aµ k Z 

Z   µ
4 1 µν F F + j A . (x) exp i d x − × DAµ µν µ ⊥ ⊥ 4 µ



 DAµ k (x)

(3.48)

By integrating by parts the argument of the exponential in the integral on Aµ k , we obtain a delta function of ∂µ jµ . Thus, for external currents that obey the continuity equation ∂µ jµ = 0, this prefactor is an infinite constant that can be ignored. When 2 µ ν restricted to the subspace of Aµ ⊥ , the operator gµν k − k k is invertible, and we can now perform the Gaussian integral, to obtain: Z0 [jµ ] = exp



−

1 2

Z

 d4 xd4 y jµ (x) G0F µν (x, y) jν (y) ,

(3.49)

with the free photon propagator in momentum space given by G0F µν (p) ≡

 pµ pν  −i µν g − . p2 + i0+ p2 c sileG siocnarF

(3.50)

(We have introduced the i0+ prescription that selects the ground state at x0 → ±∞, using the same argument as in the section 2.3.3.) The procedure used here is equivalent to imposing the gauge fixing condition ∂µ Aµ = 0, called Lorenz gauge or Landau gauge. As one can see, the resulting propagator (3.50) differs from the Coulomb gauge propagator given in eq. (1.247). c sileG siocnarF

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

3.3.3

129

General covariant gauges

All gauge fixings amount to constraining in some way the quantity ∂µ Aµ , since it does not appear in the integrand of the photon path integral. Instead of imposing ∂µ Aµ = 0, one may instead impose the more general condition ∂µ Aµ (x) = ω(x) ,

(3.51)

where ω(x) is some arbitrary function of space-time. This can be done by introducing a functional delta function, δ[∂µ Aµ − ω], inside the path integral. However, the introduction of the function ω(x) breaks Lorentz invariance. To mitigate this problem, one integrates over all the functions ω(x), with a Gaussian weight. This amounts to defining the generating functional as follows4 , Z Z 

  ξ d4 x ω2 (x) Z0 [jµ ] ≡ Dω(x) exp − i 2 Z

Z
     µ × DAµ (x) δ ∂µ A − ω exp i d4 x − 14 Fµν Fµν + jµ Aµ , (3.52)

where ξ is an arbitrary constant. Performing the integration on ω(x) thanks to the delta functional, and integrating by parts, this becomes Z

Z  
 Z0 [jµ ] = DAµ (x) exp i d4 x 21 Aµ (gµν −(1−ξ)∂µ ∂ν )Aν +jµ Aµ . (3.53)

From this formula, a standard Gaussian integration tells us that the corresponding photon propagator in momentum space should be the inverse of
i gµν p2 − (1 − ξ)pµ pν . (3.54) ν

ρ

Looking for an inverse of the form α gνρ + β ppp2 , we find   −i gµν 1 pµ pν i 0 µν GF (p) = 2 1− + . p + i0+ p2 + i0+ ξ p2

(3.55)

the argument of the delta function is linear in the variable Aµ that does not appear in the k integrand, we do not need a Jacobian. It is possible to impose non-linear gauge conditions of the form δ[F(∂µ Aµ ) − ω], but this should be done by writing the path integral as follows Z Z

Z       ξ Dω(x) exp − i d4 x ω2 (x) DAµ (x) F ′ (∂µ Aµ ) δ F(∂µ Aµ ) − ω · · · | {z } 2 4 Since

Jacobian

In general, the Jacobian cannot be ignored since it depends on the gauge field, but it can be expressed in terms of ghost fields. Doing this would be an useless complication in QED, but is an essential step in the quantization of non-Abelian gauge theories.

130

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The gauge fixing parameter ξ appears in the propagator, but only in the term proportional to pµ pν . Thanks to the Ward-Takahashi identities, it does not have any incidence on physical results, provided that all the external charged particles are on mass-shell. The Landau gauge of the previous subsection corresponds to ξ → ∞. Another popular choice is the Feynman gauge, obtained for ξ = 1, −i gµν . (3.56) ξ=1 p2 + i0+ Note that one could also introduce a non Lorentz covariant condition inside the delta function, such as δ[∂i Ai − ω], in order to derive the photon propagator in Coulomb gauge via the path integral. G0F µν (p) =

c sileG siocnarF

3.4 Schwinger-Dyson equations 3.4.1

Functional derivation

Consider a Lagrangian density L(φ, ∂µ φ) (φ may be a collection R of fields, but we do not write any index on it to keep the notation light), and S ≡ x L the corresponding action. The generating functional of time-ordered products of fields has the following path integral representation: Z R   (3.57) Z[j] = Dφ(x) eiS[φ]+i jφ .

In the right hand side, φ(x) should be viewed as a dummy integration variable, and the result of the integral should be unmodified if we change φ(x) → φ(x) + δφ(x). This translates into Z Z    iS[φ]+i R jφ 4 δS  . (3.58) 0 = δZ[j] = i Dφ(x) e d x δφ(x) j(x) + δφ(x) Taking n functional derivatives of this identity with respect to ij(x1 ),...,ij(xn ) and setting then j to zero gives: Z Z

  δS Dφ(x) eiS[φ] d4 x δφ(x) i φ(x1 ) · · · φ(xn ) 0 = δφ(x) n  X Y + δ(x − xi ) φ(xj ) . (3.59) j6=i

i=1

Since in this discussion the variation δφ(x) is arbitrary, this implies the following identities Z

  δS 0 = Dφ(x) eiS[φ] i φ(x1 ) · · · φ(xn ) δφ(x) n  X Y + δ(x − xi ) φ(xj ) , (3.60) i=1

j6=i

131

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

known as the Schwinger-Dyson equations (here written in functional form). For instance, in the case of a scalar field theory with a φ4 interaction term, this leads to 
i x + m2 0out T φ(x1 ) · · · φ(xn )φ(x) 0in 

+i λ 0out T φ(x1 ) · · · φ(xn )φ3 (x) 0in 3!

=

n X i=1



Y δ(x − xi ) 0out T φ(xj ) 0in .

(3.61)

j6=i

(We have used the remark following eq. (2.30) in order to let the operator  + m2 act also on the step functions that order the operators in the time-ordered product.) If we convolute this equation with the free Feynman propagator (i.e. the inverse of the operator x + m2 ), the above Schwinger-Dyson equation can be represented diagrammatically as follows: 1

2

1

2

+ x

x

i

=

n X

1 i−1 i+1

x

.

(3.62)

i=1

n

n

|

n

{z contact terms

}

The Schwinger-Dyson equations have several simple consequences. When applied to a free theory (λ = 0) in the case n = 1, we get 
x + m2 0out T φ(x1 )φ(x) 0in = −iδ(x − x1 ) , (3.63)

which is nothing but the equation of motion satisfied by the Feynman propagator. In the general case, if x differs from all the xi ’s, we obtain 
x + m2 0out T φ(x1 ) · · · φ(xn )φ(x) 0in 

+ λ 0out T φ(x1 ) · · · φ(xn )φ3 (x) 0in = 0 . (3.64) 3!

Thus, in a certain sense5 , we can say that time-ordered products of fields satisfy the Euler-Lagrange equation of motion. c sileG siocnarF

3.4.2

Schwinger-Dyson equations and conserved currents

The functional derivative of the action S with respect to φ(x) is given by ∂L ∂L δS = − ∂µ . δφ(x) ∂φ(x) ∂(∂µ φ(x))

(3.65)

5 I.e., up to the terms in δ(x − x ) that may appear in the right hand side, called contact terms. These i contact terms in fact take care of the action of the time derivative on the theta functions of the time ordering operator T.

132

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

When we equate this to zero, we recover the Euler-Lagrange equation of motion. Under an infinitesimal variation δφ(x) of the field, the Lagrangian density varies by δL

= =

∂L ∂L δφ(x) + ∂µ (δφ(x)) ∂φ(x) ∂(∂µ φ(x))   ∂L δS δφ(x) + δφ(x) . ∂µ ∂(∂µ φ(x)) δφ(x)

(3.66)

When the variation δφ(x) corresponds to a symmetry of the Lagrangian, we have δL = 0, and therefore   δS ∂L (3.67) δφ(x) = −∂µ δφ(x) , δφ(x) ∂(∂µ φ(x)) {z } | Jµ (x)

where Jµ is the Noether current associated to this continuous symmetry. In the classical theory, this current is conserved, i.e. ∂µ Jµ = 0, if the fields obey the Euler-Lagrange equation of motion. The Schwinger-Dyson equations provide a quantum analogue of this conservation law, at the level of the expectation values of time-ordered products of fields. In eq. (3.59), we can replace δφ(δS/δφ) by −∂µ Jµ . When the resulting identity is rewritten in terms of operators, the derivative ∂µ should go outside the time-ordering, and we obtain 

∂µ 0out T Jµ (x)φ(x1 ) · · · φ(xn ) 0in n X Y 

+i φ(xj ) 0in = 0 . δ(x − xi ) 0out T δφ(x) j6=i

i=1

(3.68)

Therefore, when a Noether current operator is inserted inside a time-ordered product, it satisfies the continuity equation up to contact terms (coming from the action of ∂0 on the theta functions of the T product). Eq. (3.68) is a generalization of the WardTakahashi identities, already discussed in the context of electric charge conservation. c sileG siocnarF

Note that in some cases, a continuous symmetry does not leave the Lagrangian density invariant, but modifies it by a total derivative, δL = ∂µ Kµ ,

(3.69)

so that only the action is invariant. There is still a conserved current, given by Jµ (x) ≡

∂L δφ(x) − Kµ (x) . ∂(∂µ φ(x))

This however does not modify eqs. (3.68). c sileG siocnarF

(3.70)

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

133

3.5 Quantum anomalies 3.5.1

General considerations

It may happen that some symmetries of the Lagrangian (i.e. symmetries of the classical theory) are broken by quantum corrections. This phenomenon is called a quantum anomaly. One way this may appear is via the introduction of a regularization (e.g. a cutoff), whose effect leaves an imprint on physical results even after the cutoff has been taken to infinity. Here we will adopt a functional point of view on this issue. In the previous section, a crucial point in the derivation of the Schwinger-Dyson equations is that the functional measure must be invariant under the symmetry under consideration. Quantum anomalies may be viewed as an obstruction in defining a functional measure which is invariant under certain symmetries, e.g. axial symmetry. c sileG siocnarF

Let us consider a set of fermion fields ψn (x), that we encapsulate into a multiplet denoted ψ(x), and assume that they interact with a gauge potential Aa µ (x) in a nonchiral way (this is the case of electromagnetic interactions and of strong interactions). Consider now the following transformation of the fermion fields: ψ(x) → U(x)ψ(x) .

(3.71)

The Hermitic conjugate of ψ transforms as: ψ† (x) → ψ† (x)U† (x) ,

(3.72)

so that we have

ψ(x) ≡ ψ† (x)γ0 → ψ† (x)U† (x)γ0 = ψ(x)γ0 U† (x)γ0 .

(3.73)

Since they are Grassmann variables, the measure should be transformed with the inverse of the determinant of the transformation. Since the transformation under consideration is local in x, it reads 

 DψDψ →

  1 DψDψ , det (U) det (U)

(3.74)

where the matrices U and U carry both indices for the fermion species and space-time indices: Uxm,yn ≡ Umn (x) δ(x − y) , Uxm,yn ≡ (γ0 U† (x)γ0 )mn δ(x − y) .

3.5.2

(3.75)

Non-chiral transformations

Let us consider the following transformation: U(x) = eiα(x)t ,

(3.76)

134

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

❘

where α(x) ∈ and where t is a Hermitean matrix that does not contain γ5 ≡ 0 1 2 3 i γ γ γ γ . Therefore: U† (x) = e−iα(x)t ,

(3.77)

and Z

=

(UU)xm,yn

Z

= =

d4 z

X

Uxm,zp Uzp,yn

p

d4 z δ(x − z)δ(z − y)

X

e−iα(z)t

p

δmn δ(x − y) .



mp



eiα(z)t



pn

(3.78)

Thus UU = 1, which implies det U det U = 1, and the fermion measure is invariant under this kind of transformations. This means that this symmetry does not exhibit quantum anomalies. c sileG siocnarF

3.5.3

Chiral transformations

Let us now define the right-handed and left-handed projections of a spinor,     1 − γ5 1 + γ5 ψ , ψL ≡ ψ, ψR ≡ 2 2

(3.79)

and consider a transformation that acts differently on these two components: 5

U(x) = eiα(x)γ

t

,

(3.80)

where t is again a Hermitean matrix. Such transformations are called chiral transformations. The matrix γ5 ≡ iγ0 γ1 γ2 γ3 satisfies γ5 5†


2

=1,

γ = γ5 , {γ5 , γ0 } = 0 ,

(3.81)

which implies: 5

5

γ0 U† (x)γ0 = γ0 e−iα(x)γ t γ0 = eiα(x)γ

t

= U(x) .

(3.82)

Thus U = U, and det U = det U. Unless this determinant is equal to one, the measure is not invariant and transforms according to: 

 DψDψ →

  1 DψDψ . 2 (det U)

(3.83)

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3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

Consider an infinitesimal transformation of the form given in eq. (3.80). We can write: (U − 1)xm,yn = i α(x)(γ5 t)mn δ(x − y) .

(3.84)


−2

(3.85)

In order to calculate det U −2

(det U)

, we use the formula6 :

= e−2 tr ln U .

In the present case, we have: −2

(det U)

= ≈

α≪1

=


 exp −2 tr ln 1 + iα(x) γ5 t δ(x − y) 
 exp −2 i tr α(x) γ5 t δ(x − y)  Z  exp i d4 x α(x)A(x) ,

(3.86)

with a function A(x) whose formal expression is A(x) ≡ −2 tr (γ5 t) δ(x − x) .

(3.87)

In this equation, the trace symbol tr denotes both a trace on the indices carried by the Dirac matrices and a trace on the fermion species. In terms of this function, the measure transforms as R 4     (3.88) DψDψ → ei d x α(x)A(x) DψDψ .

The fact that this measure is not invariant under the transformation (3.80) implies that there exists fermion loop corrections that break the invariance under chiral transformations, even if the Dirac Lagrangian itself is invariant (this is the case when one considers a global transformation, i.e. a constant α(x), and the fermions are massless). The prefactor that alters the measure can be absorbed into a redefinition of the Lagrangian, L(x) → L(x) + α(x)A(x) .

(3.89)

All happens as if the Lagrangian itself was not invariant under this transformation. If one integrates out the fermion fields in order to obtain an effective theory for the other fields, the term in α(x)A(x) must be included in the Lagrangian of this effective theory in order to correctly account for the quantum anomalies. c sileG siocnarF

6 If

the λi are the eigenvalues of U, we have: X Y
ln λi = etr λi = exp det U = i

i

ln U

.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

3.5.4

Calculation of A(x)

At first sight, the expression (3.87) of the anomaly function A(x) is very poorly defined: the trace is zero, but it is multiplied by an infinite δ(0). In order to manipulate finite expressions, we must first regularize the delta function. This can be done by writing: !  / 2x D 5 δ(x − y) , (3.90) A(x) = −2 lim tr γ t F − 2 y→x,M→+∞ M / x is the Dirac operator7 where D
/ x ≡ γµ ∂µ − i g ta Aa D µ (x) ,

(3.91)

and where F(s) is a function such that F(0) = 1 , F(+∞) = 0 , s F′ (s) = 0 at s = 0 and at s = +∞ .

(3.92)

A covariant derivative is mandatory in eq. (3.90), since an ordinary derivative would break gauge invariance. Then, we replace the delta function by its Fourier representation: Z 4 d k ik(x−y) δ(x − y) = e , (3.93) (2π)4 which leads to A(x) = =

! / 2x D eik(x−y) lim tr γ t F − 2 −2 y→x,M→+∞ M   Z 4 / +D / x )2 (ik d k 5 . (3.94) −2 lim tr γ t F − (2π)4 M→+∞ M2 Z

d4 k (2π)4



5

The second equality follows from lim F(∂x ) eik·(x−y) = F(ik + ∂x ) .

y→x

(3.95)

7 We are considering here the case where the fermions are coupled to a non-Abelian gauge field. The a index a carried by Aa µ is a “colour” index, and the t ’s are the generators of the Lie algebra representation where the fermions live. g is the coupling of the fermions to the gauge fields. See the next chapter for more details.

137

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS The function A(x) can then be rewritten as follows:  2 !  Z 4 /x d k D 4 5 /+ A(x) = −2 lim M tr γ t F − ik , M→+∞ (2π)4 M

(3.96)

by redefining the integration variable, k → Mk. Then, we can write: 2  2  / /x k · Dx D D / + x = k2 − 2i , − − ik M M M

(3.97)

and expand the function F(·) in powers of 1/M. The only terms that give a non-zero contribution to A(x) should not go to zero too quickly when M → +∞: only the terms decreasing at most as 1/M4 should be kept. Moreover, the Dirac trace should be non-zero, which implies that the matrix γ5 must be accompanied by at least four / 2x in eq. (3.97), that ordinary γµ matrices. The matrices γµ come from the term D brings two of them8 , and we therefore need to go to the second order in the Taylor expansion of the function F(·). In fact, a single term fulfills all these constraints: Z 4   d k ′′ 2 5 /4 . (3.98) F (k ) tr γ t D A(x) = − x (2π)4 By a Wick’s rotation (k → iκ, k2 → κ2 ), we obtain9 : Z

′′

4

2

2

d k F (k ) = 2iπ

+∞ Z

dκ κ3 F′′ (κ2 ) = iπ2 .

(3.99)

0

The last equality is obtained by two successive integrations by parts. We also have: / 2x D

= = = =

ν Dµ x Dx γ µ γ ν 1 µ ν D D ({γµ , γν } + [γµ , γν ]) 2 x x 1 D2x + [Dµ , Dν ] [γµ , γν ] 4 x x ig D2x − ta Fµν a [γµ , γν ] . 4

(3.100)

Using tr (γ5 γµ γν γρ γσ ) = −4 i ǫµνρσ ,

(3.101)

we obtain A(x) = − 8 In

g2 ρσ a b ǫµνρσ Fµν a (x) Fb (x) tr (t t t) , 16π2

(3.102)

this counting, we assume that the matrix t does not contain Dirac matrices. that the rotationally invariant measure in 4-dimensional Euclidean space is 2π2 κ3 dκ.

9 Recall

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where the trace is now only on the fermion species. When t is the identity matrix, the integral of A(x) depends only on topological properties of the gauge field configuration and takes discrete values. In the context of anomalies, it is called the Chern-Pontryagin index. Moreover, the Atiyah-Singer theorem relates this invariant to the zero modes of the Euclidean Dirac operator in this gauge field (see the section 3.5.7). c sileG siocnarF

3.5.5

Anomaly of the axial current

When the action is invariant under global chiral transformations, its variation under local chiral transformation may be written as Z δS = d4 x Jµ (3.103) 5 (x) ∂µ α(x) , where Jµ 5 (x) is the axial current. Integrating by parts, and identifying this variation with the term obtained in the previous section, we should have g2 ρσ a b ǫµνρσ Fµν (3.104) a (x) Fb (x) tr (t t t) , 16π2 where h·iA is an average over the fermion fields, in a fixed gauge field configuration. h∂µ Jµ 5 (x)iA = −

c sileG siocnarF

3.5.6

Axial anomaly in the u and d quarks sector

Consider the sector of the two lightest quarks flavours, u et d. If one neglects their mass, the corresponding action is invariant under the following chiral transformations: δu = i αγ5 u ,

δd = −i αγ5 d .

(3.105)

The matrix t in quark flavour space that corresponds to this transformation is ! 1 0 t= . (3.106) 0 −1 Strong interaction : Through the strong interactions, all quark flavours couple identically with the gluons (i.e. all quarks belong to the same representation of the SU(3) algebra). In other words, the matrices ta that describe this coupling do not depend on the quark flavour (equivalently, one may say that they are proportional to the identity in quark flavour space). The trace that appears in the anomaly function can be factored into separate flavour and colour factors tr (ta tb t) = trcolour (ta tb ) × trflavour (t) = 0 . | {z }

(3.107)

1−1=0

This means that the anomalies that may occur in the gluon-gluon term cancel between the u and d flavours of quarks. c sileG siocnarF

3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

139

Electromagnetic interaction : The situation is different with electromagnetic interactions, because the u and d quarks have different electrical charges. Now the matrices ta are the direct product of a charge matrix in flavour space, ! 2 0 Q≡ 3 , (3.108) 0 − 13 and the identity 1colour in colour space, since all the quark colours couple identically to photons. Therefore, the trace in the anomaly function is trflavour (Q2 t) × trcolour (1colour ) =

Nc , 3

(3.109)

where Nc = 3 is the number of colours. This leads to A(x) = −

e2 Nc ǫµνρσ Fµν (x) Fρσ (x) , 48π2

(3.110)

where Fµν is the electromagnetic field strength. Decay of the neutral pion in two photons : At low energy, the strong interactions may be described by an effective theory that couples a doublet of fermions ψ (the u and d quarks), the three pions π and a field σ. The interaction term in this model is LI ≡ λ ψ(σ + iπ · σγ5 )ψ ,

(3.111)

where σi (i = 1, 2, 3) are the Pauli matrices. Note that π3 must be the neutral pion, since it couples diagonally to the two components of the doublet (σ3 is a diagonal matrix). This interaction term is invariant under the transformation (3.105) provided that the fields σ and π transform as σ → σ − α π3

,

π1,2 → π1,2

,

π3 → π3 + ασ .

(3.112)

Moreover, the masses of nucleons are due to a spontaneous breaking of this symmetry,

 in which the σ field has a non-zero expectation value in the ground state: σ = fπ . Thus the variation of the field π3 is δπ3 = fπ α. When photons are added to this model, there is no direct coupling between the neutral pion and the photon. Let us now consider the theory that would result from integrating out the quark fields. The anomaly (3.110) would produce a term Lanom (x) = −

e2 Nc ǫµνρσ Fµν (x) Fρσ (x) α(x) . 48π2

(3.113)

in the Lagrangian. This term should be canceled somehow, because we are now talking about an effective theory of pions and photons, that should be chiral invariant.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The resolution of this issue is that this effective theory contains a coupling between the neutral pion and two photons, of the form: Lπ0 γγ = −

e2 Nc ǫµνρσ Fµν (x) Fρσ (x) π3 (x) . 48π2 fπ

(3.114)

The decay rate of a neutral pion into two photons can be easily determined from the effective coupling (3.114): Γ (π0 → 2γ) =

N2c α2em m3π . 144π3 f2π

(3.115)

This result could also be obtained by computing the transition amplitude at one loop from a neutral pion to two photons in the effective model we started from. The present considerations show that this decay is in fact controlled to a large extent by a quantum anomaly.

3.5.7

Atiyah-Singer index theorem

a Covariant derivatives Dµ = ∂µ − i g Aa µ t are anti-Hermitean, because the gauge a potential Aµ is real and the colour matrices ta are Hermitean (recall that an ordinary derivative is anti-Hermitean). However, γ0 is Hermitean, while γ1,2,3 are antiHermitean. Therefore, the Dirac operator Dµ γµ in Minkowski space is neither Hermitean not anti-Hermitean.

Let us introduce an Euclidean time via x4 ≡ ix0 . Likewise, we also have: ∂4 = i∂0

,

A4 = iA0

,

γ4 = iγ0 ,

(3.116)

and the measure over space-time becomes d4 x = i d4 xE where d4 xE is the measure over 4-dimensional Euclidean space (d4 xE = dx1 dx2 dx3 dx4 ). The Dirac operator becomes: / = D

4 X

a i (∂i − i g Aa i t )γ ,

(3.117)

i=1

where the index i runs from 1 to 4. Now, the Dirac matrices γi are all anti-Hermitean, which implies that the Euclidean Dirac operator is Hermitean. It can therefore be diagonalized in an orthonormal basis of eigenfunctions φk : / x φk (x) = λk φk (x) , D Z d4 xE φ†k (x)φk′ (x) = δkk′ ,

(3.118)

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3. PATH INTEGRALS FOR FERMIONS AND PHOTONS

with real eigenvalues λk . Note also that these eigenfunctions must obey the following completeness relation: X φk (x)φ†k (y) = δ(x − y) . (3.119) k

Consider now the case where t is the identity, t φk (x) = φk (x) ,

(3.120)

and use the completeness identity in order to express the delta function in the anomaly function A(x) in eq. (3.90): A(x) =

= =

−2

−2 −2

lim

y→x,M→+∞

lim

y→x,M→+∞

lim

M→+∞

X k

tr



X k



/2 D γ F − x2 M  5

tr

!

φ†k (y) γ5

λ2 F − k2 M



X



φk (x)φ†k (y)

k

/2 D F − x2 M

φ†k (x)γ5 φk (x) .

!



φk (x)

(3.121)

Thus, we obtain the following relationship, g2 32π2

Z

b a b d4 xE ǫijkl Fa ij (x) Fkl (x) tr(t t ) Z X  λ2  Z 1 =− d4 xE φ†k (x)γ5 φk (x) , d4 xE A(x) = lim F − k2 M→+∞ 2 M k

(3.122)

between an integral that involves the field strength of a gauge field configuration and a sum over the spectrum of the Euclidean Dirac operator (in the same gauge field). Since γ5 anticommutes with the Dirac operator,  5 / =0, γ ,D (3.123) c sileG siocnarF

/ x with the eigenvalue −λk : the state φk′ ≡ γ5 φk (x) is also an eigenfunction of D / x (γ5 φk (x)) = −λk (γ5 φk (x)) . D

(3.124)

/ x is Hermitean, When λk 6= 0, the state φk′ is distinct from the state φk (x). Since D they are in fact orthogonal: Z Z † 4 5 (3.125) d xE φk (x)γ φk (x) = d4 xE φ†k (x)φk′ (x) = 0 .

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

This implies that none of the eigenfunctions φk with a non-zero eigenvalue can contribute to the right hand side of eq. (3.122). The only contributions to eq. (3.122) come from the eigenfunctions for which λk = 0, i.e. the zero modes of the Euclidean Dirac operator. Since we have assumed that f(0) = 1, we have: g2 32π2

Z

b a b d4 xE ǫijkl Fa ij (x) Fkl (x) tr(t t ) =

X Z

d4 xE φ†k (x)γ5 φk (x) .

k|λk =0

(3.126)

 / x = 0, we can choose these zero modes in such a way that they are also Since γ5 ,D eigenmodes of γ5 , with eigenvalues +1 or −1. We can thus divide the zero modes in two families, the right-handed and the left-handed zero modes: / x φR (x) = 0 , D / x φL (x) = 0 , D

γ5 φR (x) = +φR (x) , γ5 φL (x) = −φL (x) .

(3.127)

Using also the fact that the eigenfunctions are normalized as follows, Z

Z

d4 xE φ†R (x)φR (x) = 1 , d4 xE φ†L (x)φL (x) = 1 ,

we obtain the following identity Z g2 b a b d4 xE ǫijkl Fa ij (x) Fkl (x) tr(t t ) = nR − nL , 32π2

(3.128)

(3.129)

where nR and nL are the numbers of right-handed and left-handed zero modes, respectively. This formula is the Atiyah-Singer index theorem. It tells us that the integral in the left hand side is an integer, despite being the integral of a quantity that changes continuously when one deforms the gauge field. Different considerations, from the study of Euclidean gauge field configurations known as instantons, provide another insight on this integral by relating it to the third homotopy group of the gauge group, π3 (SU(Nc )) = ❩. c sileG siocnarF

Chapter 4

Non-Abelian gauge symmetry Gauge theories are quantum field theories with matter fields (usually spin 1/2 fermions, but also possibly scalars) and gauges potentials in such a way that the Lagrangian is invariant under the action of a local continuous transformation. Quantum Electrodynamics is the simplest such theory, with a local U(1) invariance. Given Ω(x) ∈ U(1), the various objects that enter in the theory transform as follows: ψ Aµ Fµν D

→

→

→

µ

→

Ω−1 ψ , i Aµ + Ω−1 ∂µ Ω , e Fµν ,

  i Ω−1 Dµ Ω = ∂µ − ie Aµ + Ω−1 ∂µ Ω . e

(4.1)

In this construction, the gauge transformation of Aµ could have been found by requesting that Dµ ψ transforms as ψ itself, Dµ ψ

→

Ω−1 (x) Dµ ψ ,

(4.2)

with Dµ ≡ ∂µ − ieAµ the covariant derivative. The field strength Fµν would then be defined as ∂µ Aν − ∂ν Aµ . Our goal is now to generalize the concept of gauge theory to more general groups of transformations, in view of applications to the electroweak and to the strong interactions. In these two cases, the internal group of transformations is SU(2) and SU(3), respectively, but we will consider in most of this chapter a general Lie group. Our goal is to construct a consistent field theory that generalizes eqs. (4.1) to the case where Ω(x) belongs to some general Lie group G. c sileG siocnarF

143

144

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

4.1 Non-abelian Lie groups and algebras 4.1.1

Lie groups

Let us start by recalling that a Lie group is a group which is also a smooth manifold. The group operation will be denoted multiplicatively, as in Ω2 Ω1 , and we will denote the identical element by 1 and the inverse of a group element Ω by Ω−1 . The fact that a Lie group G is also a manifold allows to use concepts of differential geometry in their study. Matrix Lie groups, that will be our main concern in view of applications to quantum field theory, are closed subsets of GL(n, ), the general linear group of n × n matrices on the field of complex numbers. Here is a list of some classical examples of matrix Lie groups, along with their definition: Special linear groups : SL(n, ) det (Ω) = 1

❈

❈❘

Special orthogonal group :

SO(n)

ΩT Ω = 1 , det Ω = 1

Unitary group :

U(n)

Ω† Ω = 1

Special unitary group :

SU(n)

Ω† Ω = 1 , det Ω = 1 (4.3)

4.1.2

Lie algebras

Geometrically, the Lie algebra g is a vector space that may be viewed as tangent to the group at the identity Ω = 1. Therefore, its dimension is the same as that of the group manifold. The group multiplication induces on the tangent space a non-associative multiplication, the Lie bracket, thereby turning it into an algebra. The Lie algebra completely encapsulates the local properties of the underlying Lie group, and if the group is simply connected its Lie algebra defines it globally. Because they are linear spaces, Lie algebras are usually easier to study than their group counterpart, although they provide most of the information. In the specific case of matrix Lie groups, the corresponding Lie algebra can be defined as the following set of matrices1  (4.4) g ≡ X eit X ∈ G, for all real t .

The matrix exponential eX is defined from the Taylor series of the exponential by eX ≡

∞ X Xn , n!

(4.5)

n=0

1 The prefactor i inside the exponential is common in the quantum physics literature, but seldom used in mathematics. Its main benefit is to make X a Hermitean matrix when the group elements are unitary.

145

4. N ON -A BELIAN GAUGE SYMMETRY

that converges for all finite size matrices X since this series has an infinite radius of convergence. A crucial property of the matrix exponential is that eX+Y 6= eX eY

if [X, Y] 6= 0 .

(4.6)

Instead, one may use Trotter’s formula (also known as the Lie product formula)2 :  n eX+Y = lim eX/n eY/n . (4.7) n→∞

(See the figure 4.2 for a geometrical illustration of this formula.) Note that for X to be in the algebra, it is sufficient that exp(tX) ∈ G for t in a neighbourhood of t = 0. Then, since the group G is closed under multiplication, this property extends to all t’s on the real axis. In fact, the mapping t → exp(tX) is a group homomorphism (from the additive group to G) that spans a one-dimensional subgroup of G. From the definition (4.4) of the Lie algebra, and using Trotter’s formula, one can check that any real linear combination of elements of g is in g, i.e. that g is a real vector space. Therefore, every element of g can be written as a linear combination of some basis elements ta ,

❘

X = Xa ta

(Xa ∈

❘) ,

(4.8)

with an implicit sum on the index a. The ta ’s are called the generators of the algebra. c sileG siocnarF

Thanks to the exponential mapping (4.4), the properties of the Lie groups listed in eqs. (4.3) translate into specific properties of the matrices X in the corresponding algebras: Special linear groups : sl(n, ) tr (X) = 0 Special orthogonal group : so(n) XT = −X

❈❘

Unitary group :

u(n)

X† = X

Special unitary group :

su(n)

X† = X , tr (X) = 0 (4.9)

Note that the conditions imposed on Ω in eqs. (4.3) are non-linear, in contrast with the linear conditions obeyed by the matrices X in eqs. (4.9). This is why a Lie group is a curved manifold, while a Lie algebra is a linear space. 2A

sketch of the proof is the following:   Y +n + O(n−2 ) = exp X+Y + O(n−2 ) , eX/n eY/n = 1 + X n n  n   eX/n eY/n = exp X + Y + O(n−1 ) .

146

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Lie Group manifold

G

e it X iX

Figure 4.1: Lie group and Lie algebra.

1

Lie Algebra

4.1.3

g

Geometrical interpretation

First note that we have d itX e iX = dt

.

(4.10)

t=0

The group elements exp(itX) form a smooth curve on the group manifold (t = 0 corresponds to the identity), and iX may be viewed as the vector tangent to this curve at the identity, as illustrated in the figure 4.1. The non-commutativity of the group is related to the curvature of the corresponding manifold3 . Because of this curvature, a displacement eiX followed by a displacement eiY does not lead to the same point as the two displacements performed in reverse order. This geometrical representation also provides an interpretation of Trotter’s formula for the exponential of a sum, as shown in the figure 4.2. The dimension of the Lie algebra equals the number of independent directions on the group manifold. From the conditions listed in (4.9) on the matrices X ∈ g, it is easy to determine the dimension of these algebras (viewed as algebras over the field ). The dimensions are listed in the table 4.1 for some common cases.

❘

Moreover, as we shall see in the section 4.1.5, the group multiplication can be inferred from that on the Lie algebra, via the Baker-Campbell-Hausdorff formula. Despite these correspondences, the Lie algebra may not reflect the global properties of the group (e.g. whether it is connected), and distinct Lie groups may have isomorphic Lie algebras. This is for instance the case of U(1) and SO(2), SO(3) and SU(2), or SU(2) × SU(2) and SO(4). c sileG siocnarF

3 This assertion could be made more precise as follows: it is possible to define a metric tensor on the group manifold, and the corresponding Ricci curvature tensor. This curvature may then be expressed in terms of the constants that define the commutators between the generators of the algebra (see eq. (4.14)).

147

4. N ON -A BELIAN GAUGE SYMMETRY

Table 4.1: Dimensions of a few common Lie algebras.

Lie algebra sl(n, ) so(n) u(n) su(n)

❘

Dimension n2 − 1 n(n − 1)/2 n2 2 n −1

eitX e i t (X+Y)

Figure 4.2: Geometrical interpretation of Trotter’s formula: the broken path, made of a succession of elementary steps eitX/n and eitY/n , approximates better and better the curve eit(X+Y) on the group manifold as n → ∞.

4.1.4

iX

i(X+Y)

iY

eitY

Lie bracket and structure constants

Consider an element Ω of the Lie group and an element X of the Lie algebra. For any real number t, we have
itX exp i t Ω−1 XΩ = Ω−1 e (4.11) |{z} Ω , |

∈G

{z

∈G

}

where the equality follows from the Taylor series of the exponential. From the definition of the Lie algebra, this implies that Ω−1 X Ω ∈ g. Therefore, if X, Y ∈ g we also have e−itX Y eitX ∈ g ,

(4.12)

and the derivative with respect to t at t = 0 is also an element of the algebra,   −i X, Y ∈ g . (4.13)

In other words, −i times the commutator of two elements of a Lie algebra is another element of the algebra. Thus −i[·, ·] is the multiplication law4 in g (it is also called 4 In

contrast, the ordinary product of two elements of the algebra is in general not in the algebra.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

the Lie bracket). Therefore, the commutators between its generators can be written as  a b t , t = i fabc tc , (4.14)

where the fabc are real numbers called the structure constants. The antisymmetry of the commutator implies that fabc = −fbac . Given three elements X, Y, Z ∈ g of the algebra, their commutator satisfies the Jacobi identity          X, Y, Z + Y, Z, X + Z, X, Y = 0 , (4.15) which implies the following relationship among the structure constants: fade fbcd + fbde fcad + fcde fabd = 0 .

4.1.5

(4.16)

Baker-Campbell-Hausdorff formula

Given an element X ∈ g, we may define a function from g to g as follows:   adX (Y) ≡ −i X, Y .

(4.17)

The function adX is called the adjoint mapping at the point X. The exponential of the adjoint mapping plays an important role, thanks to the following formula5 eadX Y = e−iX Y eiX .

(4.18)

This allows to write the derivative of the exponential of a (matrix-valued) function as follows: ad

e X(t) − 1 dX(t) d iX(t) e = i eiX(t) . dt adX(t) dt

(4.19)

(This is known as Duhamel’s formula6 .) The non-trivial aspect of this formula is that it is true even when X(t) does not commute with its derivative. Then, given X, Y ∈ g, 5 This

may be proven by considering a one-parameter family of such equalities: t adX

e

Y = e−itX Y eitX ,

and by noting that the left and right hand sides coincide at t = 0, and obey identical differential equations with respect to the parameter t. 6 Note that this formula is equivalent to: Z1 d iX(t) dX(t) isX(t) e =i e . ds ei(1−s)X(t) dt dt 0 This latter form can be proven by writing eiX(t+ε) = ei




X(t)+εX ′ (t)+O(ε2 )

= lim

n→∞



ei

X(t) n

ε

ei n X

′

 (t)+O(ε2 ) n

,

(we use Trotter’s formula to obtain the second equality) and by expanding the right hand side to first order in ε.

149

4. N ON -A BELIAN GAUGE SYMMETRY let us define a matrix Z(t) by eiZ(t) ≡ eiX eitY .

(4.20)

Differentiating both sides with respect to t (using eq. (4.19) for the left hand side), we obtain #−1 " ad dZ(t) e Z(t) − 1 = Y. dt adZ(t)

(4.21)

From eq. (4.18), we can also see that adZ(t)

e

= et adY eadX .

(4.22)

Integrating eq. (4.21) from t = 0 to t = 1, we obtain the following identity: 

iX iY

ln e e



= iX + i

Z1

dt F et adY eadX

0




Y,

(4.23)

where the function F(·) is defined by F(z) ≡

ln(z) . z−1

(4.24)

Eq. (4.23) is the integral form of the Baker-Campbell-Hausdorff formula. In order to recover the more familiar expansion in nested commutators, note that et adY eadX = 1 + t adY + adX + 21 (t2 ad2Y + ad2X ) + t adY adX + · · · 1 (4.25) F(z) = 1 − (z − 1) + 31 (z − 1)2 · · · . 2 This leads to        i   1 ln eiX eiY = i(X + Y) − X, Y − X, X, Y − Y, X, Y + · · · (4.26) 2 12

(Explicit expressions for all the coefficients of this series are given by Dynkin’s formula.) In applications to quantum field theory, we usually need only the first two terms of this expansion because the commutators we encounter are commuting numbers and all the subsequent terms are zero. Besides being an intermediate step in the derivation of eq. (4.26), the integral form (4.23) shows that the group product can be reconstructed from Lie algebra manipulations (since the right hand side of this equation contains only objects that belong to the algebra). c sileG siocnarF

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4.1.6

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Representations

A real representation of a Lie group G is a group homomorphism from elements of G to elements of GL(n, ), i.e. a mapping π from G to GL(n, ) that preserves the group structure:

❘

π(1) = 1

,

❘

π(Ω2 Ω1 ) = π(Ω2 )π(Ω1 ) .

(4.27)

A representation is said to be faithful if it is a one-to-one mapping. Likewise, one may define representations of a Lie algebra as homomorphisms from g to gl(n, ), i.e. mappings π that preserve the Lie algebra structure:   π(αX + βY) = α π(X) + β π(Y) , π([X, Y]) = π(X), π(Y) . (4.28)

❘

a Note that if we define ta π ≡ π(t ) the images of the generators, then they obey a b abc c [tπ , tπ ] = i f tπ with the same structure constants as in the original Lie algebra. Since we are focusing on matrix Lie groups, their elements are already matrices, and one may wonder what representations are good for. In fact, it is often important to know how a given group (e.g. the rotation group SO(3)) acts on a more general linear space. In the example of SO(3), even though the “defining” action is on 3 in terms of 3 × 3 matrices, the group has many other matrix representations made of objects that act on spaces other than 3 .

❘

❘

Singlet representation : The singlet representation, or trivial representation, is the representation for which the mapping is π(Ω) = 1 for all Ω’s. The objects that belong to this representation space are invariant under the transformations of the group G. In quantum field theory, one says that these objects are “neutral” (under the group G). Fundamental representation : The fundamental representation, or standard representation, is the smallest faithful representation. It is also the representation obtained when π is the identical map. In other words, in the fundamental representation, the elements of a matrix Lie group are simply represented by themselves. In the case of compact simple Lie algebras (that give consistent non-Abelian gauge theories, as we shall see in the next section), it is possible to choose the generators ta f (the subscript f b denotes the fundamental representation) is such a way that tr (ta f tf ) is proportional to the identity. A customary choice of their normalization is to impose:
δab b . (4.29) tr ta = f tf 2 This sets the normalization of the structure constants, through eq. (4.14). Then, one usually normalizes the generators of other representations in such a way that they fulfill the commutation relation (4.14) with the same structure constants (but the trace formula (4.29) with a prefactor 1/2 is in general only valid in the fundamental representation).

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4. N ON -A BELIAN GAUGE SYMMETRY

Adjoint representation : The adjoint representation of a Lie group G is a representation as linear operators that act on the Lie algebra g, defined by the following mapping: Ω ∈ G → AdΩ ∈ GL(g)

such that AdΩ (X) = Ω−1 XΩ .

(4.30)

If the dimension of the Lie algebra is d, then AdΩ may be viewed as a d × d matrix. We may also define the adjoint representation of the algebra g, as follows: X ∈ g → adX ∈ GL(g)

such that adX (Y) = −i[X, Y] .

(4.31)

It is sufficient to know the adjoint representation of the generators ta , for which one often uses the following notation a . i adta ≡ Tadj

(4.32)

a Note that Tadj can be represented by a d × d matrix. Using Jacobi’s identity, one may   a check that adta , adtb = −adi[ta ,tb ] = fabc adtc . Therefore, the Tadj ’s fulfill the a same commutation relations as the t ’s themselves:  a b c Tadj , Tadj = i fabc Tadj . (4.33)

Using eq. (4.14), we find that the components of these matrices are given by
a Tadj = −i fcab . (4.34) bc

In other words, the adjoint representation is a representation by matrices whose size is the dimension of the algebra, and in which the components of the generators are the structure constants. That eqs. (4.33) and (4.34) are consistent is a consequence of the Jacobi identity (4.16) satisfied by the structure constants. c sileG siocnarF

A common use of the adjoint representation is to rearrange expressions such as e−iX Y eiX = eadX Y ,

(4.35)

where X and Y are in some representation r of the Lie algebra. Using X = Xa ta r and Y = Ya ta , we can rewrite this as follows r h i a iXa Tadj c Yb . (4.36) e eadX Y = eXa adta Yb tb = t r r cb

Thus, we have i h i h e−iX Y eiX = eiXadj c

cb

Yb ,

(4.37)

where the left hand side may be in any representation r. In other words, the right and left multiplication by a group element and its inverse can be rewritten as a left multiplication by the adjoint of this group element.

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4.2 Yang-Mills Lagrangian 4.2.1

Covariant derivative and gauge transformations

When trying to extend the concept of gauge theory to a non-Abelian symmetry, it is useful to first construct a covariant derivative. This object, denoted Dµ , is a deformation of the ordinary derivative that transforms as follows: Dµ

→

Ω−1 (x) Dµ Ω(x) ,

(4.38)

where Ω(x) is a spacetime dependent element of a Lie group G. Let us look for a covariant derivative of the form Dµ ≡ ∂µ − ig Aµ (x) ,

(4.39)

where g is a coupling constant similar to the constant e in QED and Aµ (x) a 4-vector (in quantum field theory, this field is called a gauge field). The transformation law (4.38) is satisfied provided that Aµ (x) transforms in a very specific way. Note first that the ordinary derivative ∂µ is invariant (i.e. it belongs to the singlet representation). If we denote AΩ µ (x) the transformed Aµ (x), then we must have: ∂µ − ig AΩ µ (x) = =

  Ω−1 (x) ∂µ − ig Aµ (x) Ω(x)
∂µ + Ω−1 (x) ∂µ Ω(x) − ig Ω−1 (x) Aµ (x) Ω(x) ,

(4.40)

from which we obtain the transformation law7 of Aµ (x): Aµ (x)

→

−1 AΩ (x) Aµ (x) Ω(x) + µ (x) ≡ Ω


i −1 Ω (x) ∂µ Ω(x) . (4.41) g

From eqs. (4.17), (4.18) and (4.19), we see that if Ω is an element of a Lie group G, then Ω−1 ∂µ Ω belongs to the Lie algebra g. Thus, if the second term in the right hand side of eq. (4.41) belongs to the representation r of the Lie algebra, the first term should also be in this representation for consistency. The same applies to Aµ , that we can decompose as follows: a Aµ (x) ≡ Aa µ (x) tr ,

(4.42)

where the ta r are the generators of the algebra in the representation r. 7 From this transformation law, we see that field configurations of the form ig−1 Ω−1 ∂ Ω may be µ transformed into the null field Aµ ≡ 0. Such configurations are called pure gauge fields.

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4. N ON -A BELIAN GAUGE SYMMETRY

Infinitesimal transformations : Eq. (4.41) specifies how the field Aµ changes under any transformation of G. However, it is sometimes useful to consider infinitesimal transformations, i.e. Ω close to 1. This is done by writing Ω = exp(ig θa ta r ), with |θa | ≪ 1, and by expanding eq. (4.41) to order one in θa . The variation of Aµ is given by:     δAµ = −∂µ θr (x) + i g Aµ (x), θr (x) = − Dµ , θr (x) , (4.43)

where we have defined θr ≡ θa ta r . This can also be written more explicitly as
abc (4.44) δAa θb (x) Acµ (x) = − Dadj µ ab θb (x) , µ = −∂µ θa (x) + g f where Dadj µ is the covariant derivative in the adjoint representation.

4.2.2

Non-Abelian field strength

In the previous section, we have introduced a vector field Aµ in order to define a covariant derivative. To interpret Aµ as describing a spin-1 particle, we should construct a kinetic term for this field, with the constraint that it is invariant under the transformations (4.41). In the case of quantum electrodynamics, this Lagrangian was − 41 Fµν Fµν , where the field strength was defined as Fµν ≡ ∂µ Aν − ∂ν Aµ . However, a direct verification indicates that this expression of the field strength cannot lead to an invariant Lagrangian in the case of a theory with non-Abelian symmetry. In order to mimic QED, we aim at constructing a Lagrangian with second order derivatives. Indeed, since the field Aµ (x) has the dimension of a mass, two derivatives and two powers of the field would provide the required dimension 4 for a Lagrangian in four space-time dimensions. A useful intermediate step is the construction of a field that depends only on Aµ (x) and has a simple transformation law. From the transformation law of the covariant derivative, we find that the commutator [Dµ , Dν ] transforms as     D µ , Dν → Ω−1 (x) Dµ , Dν Ω(x) . (4.45) More explicitly, this commutator reads      Dµ , Dν = −ig ∂µ Aν − ∂ν Aµ − ig Aµ , Aν . | {z }

(4.46)

Fµν

This generalizes the field strength Fµν to an arbitrary gauge group G. Note the commutator between gauge fields, that did not exist in QED. By construction, the field strength is an element of algebra, in the same representation as Aµ , a Fµν (x) ≡ Fa µν (x) tr ,

(4.47)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

and its transformation law8 is Fµν (x)

→

Ω−1 (x) Fµν (x) Ω(x) .

(4.48)

Like in QED, one may define non-Abelian electrical and magnetic fields by Eia = F0i a

,

Bia =

1 2

ǫijk Fjk a ,

(4.49)

but with an important difference: these E and B fields are not gauge invariant (they belong to the representation r of the Lie algebra). Instead, they transform covariantly as follows: Ei (x)

→

4.2.3

Ω−1 (x) Ei (x) Ω(x) ,

Bi (x)

Lagrangian

→

Ω−1 (x) Bi (x) Ω(x) . (4.50)

In order to build a kinetic term for Aµ from Fµν , we must contract all the Lorentz indices to have a Lorentz invariant Lagrangian. This forces us to have at least two F’s, since gµν Fµν = 0. Therefore, if we restrict to objects of mass dimension 4, this kinetic term should be quadratic in F, with a dimensionless prefactor. The most general9 term of this kind is LA ≡ −hab Fa µν Fb µν ,

(4.51)

where hab is a constant real symmetric matrix in the group indices. In addition, for this Lagrangian to define a consistent field theory, the matrix hab should be positive definite (otherwise some parts of the kinetic term would have the wrong sign and the energy of the system would not be bounded from below). Under an infinitesimal gauge transformation, the variation of this Lagrangian is δ LA = −2 hab Fa µν θc fdcb Fd µν ,

(4.52)

and for the kinetic term to be gauge invariant we must have hab fdcb Fa µν Fd µν = 0 . | {z }

(4.53)

sym. in a,d

This condition is satisfied for any gauge field configuration provided that fdcb hba + facb hbd = 0 .

(4.54)
Note that ❤ab ≡ tr (ta tb ) is a solution of this constraint, since tr Fµν Fµν is obviously gauge invariant given the transformation law (4.48) for the field strength, but the positivity condition imposes some restrictions on the kind of Lie algebra we may use. c sileG siocnarF

8 The field strength associated to a pure gauge field is zero, since there exists a transformation Ω for which Aµ becomes the null field. µν ρσ 9 We ignore for now the operator ǫ µνρσ hab Fa Fb . This term will be discussed in the section 4.5.

4. N ON -A BELIAN GAUGE SYMMETRY

4.2.4

155

Constraints on the Lie algebra

Eq. (4.54), combined with the fact that hab is a real symmetric positive-definite matrix, strongly constrains the Lie algebras that lead to consistent (gauge invariant, with a positive definite kinetic energy) non-Abelian gauge theories. Complete antisymmetry of the structure constants : Let us start from a diagonalization of the matrix hab , hab ≡ Otac λc Ocb = Oca Ocb λc ,

(4.55)

where Oac is a real orthogonal matrix. Since the matrix hab is positive definite, all the eigenvalues λc are positive, and we can define a square root of the matrix by Ωab ≡ Oca Ocb λ1/2 , c

hab = Ωac Ωcb .

(4.56)

Note that Ωab is a real symmetric matrix. Now, let us introduce a new basis for the algebra, defined by a

b t ≡ Ω−1 ab t ,

b

ta = Ωab t .

(4.57)

This is a legitimate change of basis for a real algebra since the matrix Ω is real and has no vanishing eigenvalue (all the eigenvalues λc are strictly positive since hab is positive definite). The commutator of two of these new generators is  a b c −1 a ′ b ′ c ′ t , t = i Ω−1 (4.58) Ωc ′ c t . ′Ω ′ f } | aa bb {z fabc

By rewriting eq. (4.54) in terms of the new structure constants fabc and by using the fact that Ω is invertible, we get fdca + facd = 0 .

(4.59)

In other words, eq. (4.54) implies that there exists a basis in which the structure constants are also antisymmetric under the exchange of the first and third indices. From this, we conclude that they are in fact completely antisymmetric10 (and not just in the first two indices, as implied by their definition in terms of the Lie bracket). Allowed sub-algebras : Consider now the generators in the adjoint representation, whose components are given by
a Tadj = −i fcab = −i fabc . (4.60) bc 10 Assuming they are non-zero, this requires an algebra with at least 3 generators, since it is not possible to construct an antisymmetric rank-3 tensor with indices that take less than 3 values.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Since the structure constants are real and antisymmetric, these generators are Hermitean matrices. Thanks to this property, there exists a basis in which all the adjoint generators have a common block diagonal structure:

a Tadj

 a D(1)  0  =  0  .. .

0

 ... . . .   , . . .  .. .

0

Da (2)

0

0 .. .

Da (3) .. .

(4.61)

a where the sizes of the blocks are the same for all the Tadj ’s. This block decomposition can be obtained recursively, until one gets blocks that are not further reducible. If d is the dimension of the adjoint representation (i.e. also the dimension of the Lie group), it corresponds to a decomposition of d into orthogonal subspaces that are invariant under the action of all the generators. Regarding the Lie algebra, this indicates that it is a direct sum of simple sub-algebras11 , and u(1) sub-algebras (if some diagonal blocks Da (n) are zero for all a’s). In addition, these simple sub-algebras are compact a b because Kab ≡ tr (Tadj Tadj ), restricted to the corresponding subspace, is positive 12 definite . Indeed, P when the structure constants are totally antisymmetric, we have Kab Xa Xb = c,d Xa facd )2 ≥ 0 for any vector X. Moreover, there is no non-zero vector X for which this quadratic form is zero, because otherwise we would have c Tadj X = 0 for all c, which means that this vector X would define a u(1) sub-algebra and cannot be part of the subspace associated with a simple sub-algebra.

❘

Standard form of the Lagrangian : be written as 

 c Tadj ,h = 0 ,

for all c .

Note now that the constraint (4.54) can also

(4.62)

This implies that hab has the same block decomposition as the adjoint generators (see eq. (4.61)), with diagonal blocks that are proportional to the identity (with positive 11 A Lie algebra is not simple if there exists a set of generators T α (the number of which is strictly smaller than the dimension of the algebra) which is closed under commutation with the algebra, i.e. a , T α ] = gaαβ T β for all a, α. If we write these new generators as linear combinations T α ≡ V α T a , [Tadj a adj the closure of the sub-algebra under commutation implies that the set of vectors {V α } is the basis of a a ’s. Conversely, if we have an invariant subspace that subspace invariant under the action of all the Tadj cannot be reduced to a smaller one, then it corresponds to a simple sub-algebra. a b 12 The coefficients K ab define the Killing form, K(X, Y) ≡ tr (adX adY ) = −Kab X Y . Since Kab is positive definite in compact Lie algebras, it naturally defines a distance d2 (X, Y) ≡ Kab (X − Y)a (X − Y)b . This distance in g gives rise locally to a distance in the underlying Lie group G, which can then be extended to the entire group into a metric invariant under the group action.

157

4. N ON -A BELIAN GAUGE SYMMETRY prefactors)  2 α(1) 1  0  h=  0  .. .

0

0

α2(2) 1

0

0 .. .

α2(3) 1 .. .

...



. . .   . . .  .. .

(4.63)

The prefactors α2(i) can be absorbed into the normalization of the gauge field and the coupling constant of the corresponding sub-algebra, by writing α(i) Fµν = ∂µ Aν′ − ∂ν Aµ′ − ig ′ [Aµ′ , Aν′ ]

(4.64)

with Aµ′ ≡ α(i) Aµ and g ′ ≡ α−1 (i) g. Therefore, we can always write the Lagrangian as a sum of terms (one for each simple and u(1) sub-algebra) having the following standard form 1 L A = − Fa (x) Fa µν (x) . 4 µν

(4.65)

Despite its resemblance with the photon kinetic term in QED, this Lagrangian has a quite remarkable feature in the case of simple sub-algebras: due to the commutator term in Fµν , LA contains terms that are cubic in Aµ and terms which are quartic in Aµ . These terms are interactions between three and four of the spin-1 particles described by Aµ , respectively. Thus, unlike in QED, the Lagrangian (4.65) has a very rich structure, and defines in itself a very interesting quantum field theory, called Yang-Mills theory. c sileG siocnarF

4.3 Non-Abelian gauge theories A non-Abelian gauge theory is a quantum field theory that has at least a gauge field whose symmetry group is a non-Abelian group G. Thus, the Lagrangian of all non-Abelian gauge theories contains a Yang-Mills term: 1 L A ≡ − Fa (x)Fa µν (x) . 4 µν

(4.66)

If Aµ is the only field of the theory, then it is a plain Yang-Mills theory.

4.3.1

Fermions

However, useful gauge theories in particle physics must also have matter fields, i.e. fermions. Under the action of a Lie group G, a fermion field transforms as ψ(x)

→

Ω−1 (x) ψ(x) ,

ψ(x)

→

ψ(x) Ω−1† (x) ,

(4.67)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where Ω is an element of some representation r of G. By an abuse of language, it is often said that the spinor ψ lives in the representation r (although strictly speaking, the spinor is in the space on which the elements of the group representation are acting). Consider now the Dirac Lagrangian constructed with a covariant derivative,
/ x − m ψ(x) . LD = ψ(x) iD (4.68) Under a gauge transformation, it becomes LD

→


/ x − m ψ(x) . ψ(x) Ω−1† (x)Ω−1 (x) iD {z } |

(4.69)

(ΩΩ† )−1

For this Lagrangian to be gauge invariant, Ω must be a unitary matrix, which restricts to unitary representations of the gauge group (all finite dimensional representations of compact Lie groups are equivalent to a unitary representation). Like in electrodynamics, the necessity of using a covariant derivative in order to have a Dirac Lagrangian invariant under local gauge transformations completely specifies the coupling between the fermions and the gauge field Aµ :
a LI = −ig ψi γµ Aa (4.70) µ tr ij ψj ,

where we have written explicitly the Lie algebra indices i, j of all the objects. These indices, that run from 1 to the size of the representation r, label the “charge” carried by the fermions, while the index a may be viewed as the charge carried by the spin-1 particle associated to the vector field Aa µ (this index runs from 1 to the dimension of the group).

4.3.2

Standard Model

Two important interactions in Nature are described by non-Abelian gauge theories: • Quantum chromodynamics, the quantum field theory of strong interactions, is of this type: the gauge fields are the gluons, and the matter fields are the quarks, of which exist 6 families, or flavours (up, down, strange, charmed, bottom, top). The charge associated to this gauge interaction is called colour. The gauge group of QCD is SU(3), and the quarks live in the fundamental representation (therefore, they can have three different colours). In QCD, the gluons interact equally with the right-handed and left-handed projections of the spinors: it is said to be a non-chiral interaction. • Likewise, the Electroweak theory is a non-Abelian gauge theory with the gauge group SU(2)×U(1), but with the peculiarity that the SU(2) acts only on the lefthanded projection of the fermions. In other words, the right-handed fermions belong to the singlet representation of SU(2) (while the left-handed fermions are arranged in doublets, corresponding to the fundamental representation of SU(2)).

4. N ON -A BELIAN GAUGE SYMMETRY

159

It is also possible to couple a (charged) scalar field φ(x) to a gauge potential Aµ . Under a local gauge transformation, φ transforms as follows φ(x)

→

Ω† (x) φ(x) ,

(4.71)

and therefore the following Lagrangian density is invariant under local gauge transformations:
†

Lscalar = Dµ φ(x) Dµ φ(x) − m2 φ† (x)φ(x) − V φ† (x)φ(x) . (4.72)

(The potential should depend on the scalar field via the combination φ† φ in order to be gauge invariant). The most important example of such a scalar in particle physics is the Higgs boson. In the Standard Model, the potential of the Higgs field is symmetric under the gauge transformations, but has minima at non-zero value of the field φ, leading to spontaneous symmetry breaking. Because of its coupling to the gauge potentials and to the fermions, the Higgs field vacuum expectation value turns them into massive particles (see the next section for a discussion of this phenomenon).

4.3.3

Classical equations of motion

From the Lagrangians (4.66), (4.68) and (4.72), it is straightforward to obtain the classical Euler-Lagrange equations of motion. For the fermions, we simply obtain the Dirac equation
/ − m ψ(x) = 0 . iD

(4.73)

For scalar fields, the classical equation of motion is a deformation of the Klein-Gordon equation, in which the ordinary derivatives are replaced by covariant derivatives: h i Dµ Dµ + m2 + V ′ φ† (x)φ(x) φ(x) = 0 . (4.74) For the gauge field Aµ , the derivatives of the various pieces of the Lagrangian read: ∂LA = −Fa µν , ∂(∂µ Aa ν) ∂LA c µν = g fabc Ab , µF ∂Aa ν ∂LD = g ψ γν ta ψ , ∂Aa ν  

† ∂Lscalar = ig φ† ta Dν φ − Dν φ ta φ . a ∂Aν

∂µ

(4.75)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

This leads to the following equation of motion 

Dµ , Fµν



a

= −Jν a ,



† a  ν a † a Jν a = g ψ γ t ψ + ig φ t Dν φ − Dν φ t φ ,

(4.76)

known as Yang-Mills equation. From the Dirac and Klein-Gordon equations, one may check that the colour current Jν a is covariantly conserved:   Dν , Jν = 0 . (4.77)

The field strength also obeys another equation, known as the Bianchi identity,       Dµ , Fνρ + Dν , Fρµ + Dρ , Fµν = 0 , (4.78) that follows from the Jacobi identity between covariant derivatives. c sileG siocnarF

4.3.4

Useful su(N) identities

Feynman graphs relevant for the Standard Model involve manipulations of the su(N) generators (for N = 2, 3), mostly in the fundamental representation (since all matter fields are in this representation). In this section, we derive some useful formulas that help in these calculations. Fierz identity : In the case of su(N), there are N2 − 1 generators ta f , while the linear space of all N × N Hermitean matrices has a dimension N2 . A basis of the latter can be obtained by adding the identity matrix to the ta f ’s. Thus, any N × N Hermitean matrix M can be written as M = m0 1 + ma ta f .

(4.79)

Since the ta f ’s are traceless, we have m0 =

1 tr (M) , N

ma = 2 tr (M ta ) .

(4.80)

Considering the entry ij of the matrix M, we can write Mij

= =



1 a Mkk δij + 2 Mlk ta f kl tf ij N h1

i a Mlk δkl δij + 2 ta t f kl f ij . N

(4.81)

161

4. N ON -A BELIAN GAUGE SYMMETRY Since this is true for any Hermitean matrix M, we must have

1 a δkl δij + 2 ta f kl tf ij = δil δjk , N

(4.82)

which is usually written as follows i

1 1h a δ δ − δ δ . ta t = il jk ij kl f ij f kl 2 N

(4.83)

This formula is called a Fierz identity. It has a convenient diagrammatic representation, j

i

a (ta f )ij (tf )kl =

= k

l

1 2

−

1 2N

, (4.84)

in which the solid blobs represent the ta f matrices, and the wavy line indicates that the indices a are contracted. In the right hand side, the solid lines indicate how the indices ijkl are connected by the delta symbols. By contracting the indices jk in the Fierz identity (4.83), we obtain: a ta f tf




il

=

N2 − 1 δil . 2N

(4.85)

a The quadratic combination ta f tf , called the fundamental Casimir operator, is proportional to the identity (and therefore commutes with everything). The prefactor is sometimes denoted Cf ≡ (N2 − 1)/(2N).

The diagrammatic representation (4.84) provides a very convenient way of obtaining certain identities involving the generators of the fundamental representation. As an illustration, let us consider the following example:

b a ta f tf tf =

b a

a

=

1 2

=

1 b 1 b 1 tr (tb t =− t . f )1 − 2 2N f 2N f

−

1 2N (4.86)

For the first term, we have used the fact that a closed loop in this diagrammatic representation corresponds to a trace over the colour indices, and the tracelessness the generators. Likewise, one would obtain

b c a b ta f tf tf tf tf =

c a

b

a

b

=

1 1  1 + 2 tcf . 4 N

(4.87)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

More su(N) formulas : Contrary to the commutator, the anti-commutator of two matrices of the algebra does not belong to the algebra. However, in the case of su(N), it can be decomposed as a linear combination of the identity and the generators of the algebra:  a b δab tf , tf = N 1 + dabc tcf . (4.88)

The first term is obtained by taking the trace of the equation, using eq. (4.29) and the fact that the generators are traceless. The constants dabc are sometimes called the symmetric structure constants. Therefore, the product of two generators of the fundamental representation can be written as  1  δab abc abc c b . (4.89) 1 + (d + i f ) t ta t = f f f 2 N From this, we deduce the following identities b c tr ta f tf tf




=




facd fbcd

=

dacd dbcd

=

facd dbcd

=

ade bef cfd

=

b a c tr ta f tf tf tf

f

f

f

=


1 abc d + i fabc , 4 1 − δbc , 4N N δab ,  4 − N δab , N 0, N abc f . 2

(4.90)

Note that the third of these equations provides the trace of the product of two generators in the adjoint representation:
a b tr Tadj Tadj = N δab . (4.91)

4.4 Spontaneous gauge symmetry breaking 4.4.1

Dirac fermion masses and chiral symmetry

Chiral gauge theories, i.e. gauge theories in which the left and right handed fermions belong to distinct representations, are rather special regarding the masses of these fermions. Let us recall that the left and right spinors are defined by 1 + γ5 1 − γ5 ψ , ψL ≡ ψ, 2 2 1 + γ5 0 1 − γ5 0 ψ R = ψ† γ , ψL = ψ† γ . 2 2 ψR ≡

(4.92)

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4. N ON -A BELIAN GAUGE SYMMETRY

Consider first the term of the Dirac Lagrangian that does not depend on the mass. It can be decomposed as follows in terms of the left and right handed spinors: /ψ ψD

X

=

ψ†

ǫ,ǫ ′ =±

X

=

1 + ǫγ5 0 1 + ǫ ′ γ5 / γ D ψ 2 2

ψ†

ǫ=ǫ ′ =±

1 + ǫγ5 0 / ψR + ψ L D / ψL . / ψ = ψR D γ D 2

(4.93)

Therefore, this terms does not mix the left and right spinors, and is invariant under independent gauge transformations of the two spinor helicities. In particular, it is perfectly possible that they belong to different representations of the Lie algebra. c sileG siocnarF

In contrast, a Dirac mass term m ψψ has the following decomposition in terms of the left and right handed spinors: m ψψ

=

m

X

ψ†

1 + ǫγ5 0 1 + ǫ ′ γ5 γ ψ 2 2

ψ†

1 + ǫγ5 0 γ ψ = m ψR ψL + m ψL ψR . 2

ǫ,ǫ ′ =±

X

=

ǫ=−ǫ ′ =±

(4.94)

If ψR and ψL belong to different representations and transform independently under the gauge transformations, such a term is not gauge invariant and is therefore not allowed. Therefore, generically, fermions must be massless in a chiral gauge theory. The most prominent example of this situation is the Standard Model, where the gauge group is SU(3) × SU(2) × U(1), and where the left and right handed fermions transform differently under the SU(2) × U(1) part. More precisely, the two chiral components have different charges (called hypercharge in this context) under U(1), and the left handed fermions form SU(2) doublets while the right handed ones are singlet under SU(2). Thus, in such a gauge theory, fermions should naively be massless, while experimental evidence shows that they are massive.

4.4.2

Coupling to a scalar field, Yukawa terms

Let us focus on the case where the right handed fermions are singlet under the gauge group, while the left handed ones belong to a non trivial representation. This means that they transform as follows ψR ψL

→

→

ψR ,

ψR

Ω−1 ψL ,

ψL

→

→

ψR , ψL Ω .

(4.95)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Thus, a way out to construct an operator which is bilinear in the fermions, mixes the left and right components, and does not contain derivatives is to introduce a scalar field Φ that transforms in the same way as ψL , Ω−1 Φ .

(4.96)

This operator, called a Yukawa term, reads   λ ψL ri Φi ψR r ,

(4.97)

Φ

→

where λ is a coupling constant, r is a Dirac index, and i is the index that labels the components of the Lie algebra representation to which ψL and Φ both belong. From the way the indices are contracted, this term is both gauge and Lorentz invariant. Note that since the contraction of the Dirac indices between the two spinors already produces a Lorentz invariant object, the field Φ must be Lorentz invariant on its own, and thus must be a scalar. At this point, the term of eq. (4.97) is not yet a mass term, but simply a tri-linear interaction term between fermions and the newly introduced

 scalar field. However, a mass term is generated if the vacuum expectation value Φi is non-zero (as we shall see, this is related to the spontaneous breaking of the gauge symmetry). Therefore, we may redefine the scalar field by writing (for the sake of this example, we choose this expectation value to point in the direction i = 1)   v   0  (4.98) Φ ≡ Φv + ϕ , Φv ≡   ..  , . 0 and the term (4.97) becomes     λ v ψL r1 ψR r + λ ψL ri ϕi ψR r . λ ψL ri Φi ψR r = |{z}

(4.99)

m

If the fermion in the right handed singlet matches the first component of the left handed multiplet, then the first term in the right hand side is a Dirac mass term for this fermion (the fermions corresponding to the other components of the multiplet remain massless13 ). The second term in the right hand side is a genuine interaction term between the fermions and the fluctuating part of the scalar. Interestingly, with 13 In the Standard Model, where the left handed fermions belong to the fundamental representation of SU(2), it is possible to give a mass to the second component of the doublet. Indeed, by noting that Ωt2 ΩT
= t2 for any matrix Ω in the fundamental representation of SU(2), we see that the term iλ ψL t2 Φ∗ ψR is gauge invariant and gives a mass λv to the second component of the left handed doublet when the vacuum expectation value of the scalar field has a non-zero first component.

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4. N ON -A BELIAN GAUGE SYMMETRY

this mechanism, the strength of this interaction is proportional to the mass of the fermion. In a theory where several fermions acquire their masses by coupling to the expectation value of the same scalar field, this leads to a definite prediction: the ratios of the couplings must equal the ratios of the masses (but the masses themselves are not predicted, since the Yukawa couplings λ are free parameters). Family mixing : When there are several families of fermions (that we label by an extra index f in this paragraph), the Yukawa term of eq. (4.97) can be generalized into
λff ′ ψL fri Φi ψR f ′ r ,

(4.100)

without spoiling Lorentz or gauge invariance. Thus, with a non-zero vacuum expectation value of the scalar field, we get a fermion mass matrix which is in general not diagonal in the fermion families. Note that here, we are implicitly choosing a basis of fermion fields in which the couplings to the gauge bosons are diagonal, i.e. for which the vertex with one gauge boson and two fermions does not mix the fermion families. Conversely, we could choose a basis of fermion fields in which the mass matrix is diagonal. In this alternate basis, the interactions with the gauge bosons are no longer diagonal, i.e. the coupling to a gauge boson may change the type of fermion. These non-diagonal interactions are described by a matrix known as the Cabbibo-Kobayashi-Maskawa in the sector of quarks.

4.4.3

Higgs mechanism

Until now, we have not explicited the mechanism by which the scalar field Φ may have a non-zero vacuum expectation value. The simplest gauge invariant Lagrangian that exhibits this phenomenon is
2
†
λ Lscalar = Dµ Φ(x) Dµ Φ(x) + m2 Φ† (x)Φ(x) − Φ† (x)Φ(x) . (4.101) 4 Note the unusual sign of the mass term. Because of this feature, the value Φ = 0 is a local maximum of the potential, and cannot be a stable field configuration. Instead, this potential has minima for Φ† Φ = 2m2 /λ, which corresponds to a gauge invariant shell of non-trivial minima. The general arguments developed in the section 1.15 also apply here: in an infinite volume, the system chooses as its ground state one of these minima (as opposed to a symmetric linear combination of all the minima). Let us denote Φv the ground state on which the system settles. The gauge group G contains a subgroup H that leaves Φv invariant, called the stabilizer of Φv , and the set of the minima of the potential can be identified with the coset space14 G/H. Then, c sileG siocnarF

14 Given a group G and H one of its subgroups, two elements Ω and Ω ′ are said to be H-equivalent if Ω−1 Ω ′ ∈ H. The quotient of the group G by this equivalence relationship, also called coset space and denoted G/H, is the set of the resulting equivalence classes. When H is a normal subgroup (i.e. ΩHΩ−1 = H for all Ω ∈ G), the coset space is itself a group.

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Φv

H

G/H

Figure 4.3: Illustration of the symmetry breaking pattern in the case of a potential with G = O(3) symmetry. The set of minima of the potential is a 2-dimensional sphere. The stabilizer of the minimum Φv is H = O(2). The dark circular arrow shows the action of the generator of h, while the lighter arrows show the action of the generators of the complementary set.

the generators ta of the Lie algebra g can be divided in two sets: a basis of h (for a > n), and a complementary set (for 1 ≤ a ≤ n): 1≤a≤n : a>n :

ta 6 0, ij Φvj = a tij Φvj = 0 .

(4.102)

In the cases of interest in quantum field theory, the Lie algebra g is a direct sum   g = h ⊕ m , with h, m ⊂ m , (4.103)

and the complementary set of generators is a basis of m. (This is called a reductive decomposition of g). In this case, the tangent space to G/H at the origin can be identified with m, via the following mapping X ∈ m → eitX H ∈ G/H .

(4.104)

Thus, G/H is obtained by exponentiation of the elements of m, and any configuration of the scalar field may be parameterized as follows: n  X 
Φ(x) ≡ exp i ϑa (x)ta Φv + r(x) .

|

a=1

{z

Ω−1 (x)

}

(4.105)

In this representation, r(x) denotes the “radial” field variables, while the ϑa (x) are the “angular” ones. From eqs. (4.102), the latter correspond to the generators broken by the spontaneous symmetry breaking. Therefore, if it were not for the coupling of Φ to the gauge fields through the covariant derivatives, we would conclude from

167

4. N ON -A BELIAN GAUGE SYMMETRY

Goldstone’s theorem that the modes r(x) are massive while the modes ϑa (x) are the massless Nambu-Goldstone bosons. However, this conclusion is altered by the minimal coupling to a gauge field because it is possible to absorb the matrix Ω−1 (x), that contains the would-be Nambu-Goldstone modes, into a gauge transformation of that field. Indeed, we may write Dµ Φ

=

Dµ Ω−1 Φv + r)

=

Ω−1 ΩDµ Ω−1 (Φv + r) , | {z }

(4.106)

′ Dµ

with Dµ′ ≡ ∂µ − igAµ′ ,

Aµ′ ≡ ΩAµ Ω−1 +

i Ω∂µ Ω−1 . g

(4.107)

We see that after this gauge transformation of Aµ (this choice of gauge is known as the unitary gauge), only the modes r(x) can still be considered as physical dynamical modes of the scalar field, and the kinetic term of the scalar Lagrangian can thus be rewritten as Dµ Φ


†

Dµ Φ





†
′ = D µ (Φv + r) Dµ′ (Φv + r)
†
′ = D µ r Dµ′ r
†

†
′ ′ + − igA µ Φv Dµ′ r + D µ r − igA ′ µ Φv
†
′ + − igA µ Φv − igA ′ µ Φv . (4.108) {z } | 1 ′ bµ ′a 2 Mab Aµ A

In this expression, the last term is particularly interesting, since it provides a mass for some of the gauge bosons. More explicitly, the mass matrix is given by b Mab ≡ 2 g2 Φ†vi ta ik tkj Φvj .

(4.109)

Note that since 1 2

Mab Xa Xb = g2

2 X Xb tb kj Φvj ≥ 0 ,

(4.110)

k

this mass matrix is positive and has a number of flat directions equal to the number of generators ta that annihilate Φv . From this, we conclude that the gauge bosons that become massive via this mechanism are those that couple to the generators of the broken symmetries.

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4.5 θ-term and strong-CP problem 4.5.1

CP-odd gauge invariant operator

In the construction of the Lagrangian of Yang-Mills theory, we have argued that the only dimension four gauge invariant local operator is an operator quadratic in the field strength Fµν a . All the Lorentz indices should be contracted in order to obtain a a Lorentz invariant Lagrangian density. An obvious possibility is Fµν a Fµν , which is the combination that appears in the Yang-Mills action. However, there exists another Lorentz invariant contraction, obtained by introducing the Levi-Civita tensor,

Lθ ≡

g2 θ ǫµνρσ tr (Fµν Fρσ ) . 32π2

(4.111)

The prefactor 1/32π2 will appear convenient later, and the coupling constant in front of this term is usually denoted θ. Consequently, this term is referred to as the θ-term.

4.5.2

Expression as a total derivative

Firstly, we should clarify why we have not considered this term right away when we listed the possible gauge invariant operators that may enter in a non-Abelian gauge theory. As we shall prove now, the θ-term is a total derivative. Therefore, it does not enter in the field equations of motion, and has also no influence on perturbation theory. Since our discussion has been so far centered on the perturbative expansion, this term was irrelevant. However, the θ-term –that we cannot exclude on the grounds of symmetries– may lead to non-perturbative effects that we shall discuss in this section. c sileG siocnarF

Let us consider the following vector15 : h g abc a b c i a f Aν Aρ Aσ . Kµ ≡ ǫµνρσ Aa F − ν ρσ 3 15 Note

that this vector can also be expressed as a trace:

  2ig Aν Aρ Aσ . Kµ ≡ 2ǫµνρσ tr Aν Fρσ + 3

(4.112)

4. N ON -A BELIAN GAUGE SYMMETRY

169

The divergence of this vector is given by ∂µ Kµ

=

h

a a abc b c ǫµνρσ ∂µ Aa Aρ Aσ ν ∂ρ Aσ − ∂σ Aρ + gf a a +Aa ν ∂µ ∂ρ Aσ − ∂µ ∂σ Aρ

=


c abc b +g fabc (∂µ Ab Aρ (∂µ Acσ ) ρ )Aσ + g f
b c g abc a
c g Aν ∂µ Ab − fabc ∂µ Aa ν Aρ Aσ − f ρ Aσ 3 3i
g b c − fabc Aa ν Aρ ∂µ Aσ 3 1 µνρσ h a a c d e ǫ Fµν Fρσ − g2 fabc fade Ab µ Aν Aρ Aσ 2 i  g c a a b c a a . + fabc Ab µ Aν (∂ρ Aσ − ∂σ Aρ ) − Aρ Aσ (∂µ Aν − ∂ν Aµ ) 3 (4.113)

The two terms of the third line are antisymmetric under the exchange (µν) ↔ (ρσ), while the prefactor ǫµνρσ is symmetric under this exchange. These terms are therefore zero after summing over the indices νρσµ. Then, the term on the second line can be written as follows: g2 ǫµνρσ tr ([Aµ , Aν ][Aρ , Aσ ])  = g2 ǫµνρσ tr Aµ Aν Aρ Aσ + Aν Aµ Aσ Aρ

 −Aν Aµ Aρ Aσ − Aµ Aν Aσ Aρ .

(4.114)

Each term is a trace of four factors, and is invariant under cyclic permutations of the indices. Since cyclic permutations are odd in four dimensions, the ǫµνρσ tensor changes sign under such a permutation, and the contraction with the trace is zero. Therefore, we obtain: ∂µ Kµ =

1 µνρσ a a ǫ Fµν Fρσ , 2

(4.115)

which is proportional to the θ-term. More precisely, we have Lθ =

4.5.3

g2 θ ∂µ Kµ . 32π2

(4.116)

Proof in terms of differential forms

The elementary proof of this result that we have presented in the previous subsection is arguably rather cumbersome. This could have been made much more compact by

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

using the language of differential forms. The simplest differential forms are 1-forms, that one may think of as the contraction of a spacetime dependent vector aµ (x) and of the differential element dxµ , as in A ≡ aµ (x) dxµ .

(4.117)

1-forms measure the variation of a function along an infinitesimal one-dimensional path (thus, 1-forms may be integrated along a path γ, yielding a number). Higher degree forms may be constructed thanks to the exterior product, denoted ∧. A basis of 2-forms is provided by the products dxµ ∧ dxν , that are the areas of the infinitesimal quadrangles of edges (dxµ , dxν , −dxµ , −dxν ). The exterior product is defined to be antisymmetric, dxµ ∧ dxν = −dxν ∧ dxµ ,

(4.118)

which corresponds to a definition of oriented areas, depending on the order in which the edges of the quadrangle are traveled:

dxµ dxν

=

− dxν

dxµ

This also naturally implies that dxµ ∧ dxµ = 0, in accordance with the fact that a quadrangle of edges (dxµ , dxµ , −dxµ , −dxµ ) is reduced to a line segment and thus has zero area. The most general 2-form can be written as F ≡ fµν (x) dxµ ∧ dxν ,

(4.119)

where fµν is antisymmetric under the exchange of the µ, ν indices. 2-forms can be integrated over a two-dimensional manifold, to give a number. In d-dimensional space, one may iteratively construct p-forms for any p ≤ d (higher degree forms are zero by antisymmetry of the exterior product). In particular, in four dimensions, the volume element weighted by the fully antisymmetric tensor ǫµνρσ can be written as d4 x ǫµνρσ = dxµ ∧ dxν ∧ dxρ ∧ dxσ , which allows the following compact notation Z Z 4 µνρσ d xǫ Aµ Aν Aρ Aσ = A ∧ A ∧ A ∧ A .

(4.120)

(4.121)

Here, it is important to note that A ∧ A 6= 0 when Aµ belongs to a non-Abelian Lie algebra, despite the antisymmetry of the exterior product.

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4. N ON -A BELIAN GAUGE SYMMETRY

Another important operation on differential forms is the exterior derivative d, defined as d ω ≡ dxµ ∧ ∂µ ω .

(4.122)

Thus, the exterior derivative of a p-form is a (p + 1)-form. For instance, given a 1-form A = aµ dxµ , we have

d A = dxµ ∧ ∂µ aν dxν = 12 ∂µ aν − ∂ν aµ dxµ ∧ dxν . (4.123) Note that since ordinary derivatives commute, we have16 d2 ω = 0 .

(4.124)

When applying the exterior derivative to the exterior product of two forms, one should distribute the partial derivative on the two factors, and account for the fact that the exterior derivative contains an anticommuting dxµ . Thus, if A is a p-form, we have
d A ∧ B = dA ∧ B + (−1)p A ∧ dB . (4.125)

Differential forms also provide a unified version of various formulas of vector calculus (e.g., Kelvin–Stokes and Ostrogradsky–Gauss theorems), known as Stokes theorem. Given a form ω and a manifold M, Stokes theorem states that Z Z ω= dω , (4.126) M

∂M

where ∂M is the boundary of M. c sileG siocnarF

In order to cast the θ-term in the language of differential forms, let us firstly introduce the gauge potential 1-form: A ≡ ig Aµ dxµ .

(4.127)

Then, we have
ig ∂µ Aν − ∂ν Aµ dxµ ∧ dxν 2  g2  Aµ , Aν dxµ ∧ dxν , A∧A=− 2

dA =

(4.128)

and we see that the field strength Fµν appears in the coefficients of the following 2-form, F ≡ dA − A ∧ A =


ig ∂µ Aν − ∂ν Aµ − ig [Aµ , Aν ] dxµ ∧ dxν . (4.129) {z } 2 | Fµν

16 A differential form ω whose exterior derivative is zero (dω = 0) is said to be closed. A differential form χ which is the exterior derivative of another form (χ = dω) is said to be exact.

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Therefore, the integrand of the θ-term may be written compactly as    4  d4 x ǫµνρσ tr Fµν Fρσ = − 2 tr F ∧ F . g

(4.130)

Likewise, we have

d4 x Kµ = −

  1 4 µ dx ∧ tr A ∧ F + A ∧ A ∧ A . g2 3

(4.131)

Then, note that 

  1 d tr A ∧ F + A ∧ A ∧ A = tr dA ∧ F − A ∧ dF + (dA) ∧ A ∧ A 3

 = tr dA ∧ dA − 2 (dA) ∧ A ∧ A   = tr F ∧ F , (4.132)

where we have used the cyclicity of the trace and the fact that commuting dA with other forms does not bring any sign since it is a 2-form. In order to obtain the last line, we have used
tr A ∧ A ∧ A ∧ A = 0 , (4.133)

which is a consequence of the fact that a cyclic permutation of four objects is odd. Eq. (4.132) is the translation in terms of forms of the fact that the θ-term is the derivative of the vector Kµ . Thanks to Stokes theorem, the integral of the θ-term over a four-dimensional manifold M can be rewritten as an integral over its boundary (located at infinity if M is the entire 4 ), Z     Z 1 (4.134) tr A ∧ F + A ∧ A ∧ A . tr F ∧ F = 3 ∂M M

❘

4.5.4

Effect of the θ-term on the Euclidean path integral

We have already encountered the integral of the θ-term over Euclidean spacetime in the context of anomalies and the Atiyah-Singer index theorem (see eq. (3.129)): Z d4 xE Lθ = nθ , n∈ , (4.135)

❩

where the integer n is related to the chirality of the zero modes of the Dirac operator in the gauge field configuration. When added to the Yang-Mills action, the integral of the θ-term modifies the Euclidean path integral as follows Z Z R 4     DAµ · · · e−S[A,··· ] → DAµ · · · e−S[A,··· ]− d xE Lθ Z X   DAµ · · · n e−S[A,··· ] , (4.136) e−nθ = n∈❩

4. N ON -A BELIAN GAUGE SYMMETRY

173

  where the measure DAµ n is restricted to the gauge fields of index n. Thus, the effect of the θ-term is to reweight the gauge field configurations by a factor (e−θ )n that depends only on θ and on the index n. Note that since n is an integer, the path integral is periodic in θ, with a period 2iπ.

4.5.5

Strong CP-problem

As we have seen in the section 3.5.6, an effective description of the interactions of nucleons with pions is provided by the linear σ model, whose interaction term is LI ≡ λ ψ(σ + iπ · σγ5 )ψ .

(4.137)

However, this does not include any CP-violating interactions, such as those that may result by the θ-term. Its effects may be included in the effective theory by generalizing the interaction term into LI ≡ ψ(λσ + π · σ(iλγ5 + λ))ψ .

(4.138)

By a matching with the underlying theory, the new coupling λ can be related to the parameter θ by the following estimate |λ| ≈ 0.038 |θ| .

(4.139)

Then, the effective theory (4.138) can be used to estimate the neutron electric dipole moment DN (in the chiral limit where the pion mass mπ is much smaller than the nucleon mass mN ). This leads to   m ln mN π ≈ 5 × 10−16 θ e · cm . (4.140) DN ≈ λ λ e 4π2 mN Current experimental limits on the neutron electric dipole moment indicate that D ≤ 3 × 10−26 e · cm , (4.141) N implying that

|θ| . 10−10 .

(4.142)

We thus face a paradoxical situation. The gauge symmetry of quantum chromodynamics allows the addition of the θ-term to the Yang-Mills action, and without any prior knowledge of the coupling θ, one may expect that natural values are of order unity. This constitutes the strong-CP problem: lacking a symmetry principle that would force θ to be zero, why is it nevertheless extremely small?

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4.5.6 θ-term and quark masses There is an interesting interplay between the θ-term and chiral transformations of quark fields: ψf −→ eiγ5 αf ψf ,

(4.143)

where f is an index labeling the quark flavours and the αf are real phases. Under this transformation, the functional measure for the quarks is not invariant, but transforms as follows Z  X h    i a DψDψ −→ exp − αf DψDψ . (4.144) d4 x ǫµνρσ Fa µν Fρσ 2 32π f

The same effect would have been obtained by a change of the angle θ: X θ→θ−2 αf .

(4.145)

f

For the quarks, we can write generically the following mass term17 X

Mf ψ f

X 1 + γ5 1 − γ5 ψf + ψf , M∗f ψf 2 2

(4.146)

f

f

that transforms into the following under the above chiral transformation X

e2iαf Mf ψf

X 1 + γ5 1 − γ5 ψf + ψf . e−2iαf M∗f ψf 2 2

(4.147)

f

f

This is equivalent to transforming the quark masses as follows: Mf → e2iαf Mf .

(4.148)

Since any change of θ can be absorbed by a chiral transformation of the quarks, whose effect is to multiply the quark masses by phases, physical quantities cannot depend separately on θ and on the quark masses. Instead, they can depend only on the following combination Y eiθ Mf , (4.149) f

which is invariant. This discussion indicates that the θ-term has no effect if at least one of the quarks is massless. Unfortunately, a massless up quark (the lightest quark) does not seem consistent with existing experimental and lattice evidence. c sileG siocnarF

17 If

the masses are complex, then the symmetries P and CP are explicitly broken.

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4. N ON -A BELIAN GAUGE SYMMETRY

4.5.7

Link with the topology of gauge fields

Using Stokes’ theorem, the integral of the θ-term over Euclidean spacetime may be rewritten as an integral over a surface localized at infinity: Z Z Z g2 θ g2 θ 4 µ d4 xE Lθ = lim dSµ Kµ , (4.150) d x ∂ K = µ E 32π2 32π2 R→∞ S3,R

where S3,R is a 3-dimensional sphere of radius R and dSµ the measure on this surface. Let us now assume that the coloured objects of the problem are comprised in a finite region of space-time, so that the gauge field configuration goes to a pure gauge at infinity. Such a field can be written as Aµ (x) = aµ (x) +

i † Ω (b x) ∂µ Ω(b x) , g

(4.151)

where Ω(b x) is an element of the gauge group that depends only on the direction of the vector xµ , and aµ (x) is the deviation from the asymptotic pure gauge. For the total field to be a pure gauge at infinity, this deviation must decrease faster than |x|−1 . When |x| → +∞, Aν (x) goes to 0 as |x|−1 , while Fρσ (x) goes to 0 faster than |x|−2 (since Aν (x) goes to a pure gauge), and we have: Kµ

−→

|x|→+∞

4ig µνρσ ǫ tr (Aν Aρ Aσ ) ∼ |x|−3 , 3

(4.152)

and Z

d4 xE Lθ

θ lim 24π2 R→∞

=

Z

S3,R

dS b xµ ǫµνρσ


×tr Ω† (∂ν Ω)Ω† (∂ρ Ω)Ω† (∂σ Ω) ,

(4.153)

where we have used dSµ = b xµ dS, with dS the element of area on the 3-sphere. Note that the integrand decreases as R−3 because of the three derivatives, while dS ∼ R3 . Therefore, the integral is in fact independent of the radius R and we can drop the limit: Z

θ d xE L θ = 24π2 4

Z

S3


dS b xµ ǫµνρσ tr Ω† (∂ν Ω)Ω† (∂ρ Ω)Ω† (∂σ Ω) . (4.154)

Thus, the integral of the θ-term depends only on the function Ω(b x), that maps the 3-dimensional sphere S3 onto the gauge group: Ω :

S3

7−→

G.

(4.155)

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It turns out that these mappings can be grouped in equivalence classes of Ω’s that can be deformed continuously into one another. On the contrary, Ω’s that belong to distinct classes cannot be related by a continuous deformation. The set of these classes possesses a group structure, and is called the third homotopy group of G, denoted π3 (G). For all SU(N) groups with N ≥ 2, the third homotopy group is isomorphic to (❩, +). The interpretation of eq. (4.154) is that the integral of the θ-term depends only on the class to which Ω belongs, and is therefore a topological quantity that can change only in discrete amounts. This discussion provides another point of view on the Atiyah-Singer index theorem, where the same integral was related to the chirality imbalance between the zero modes of the Euclidean Dirac operator in a background gauge field.

4.6 Non-local gauge invariant operators 4.6.1

Two-fermion non-local operator

The discussion in the previous sections exhausts the local gauge invariant objects of dimension less than or equal to 4. However, it is sometimes useful to construct gauge invariant non-local operators, for instance in the definition of parton distributions. The simplest operator of this type is an operator with two spinor fields at different space-time positions, ψ(y) W(y, x) ψ(x). Since the transformation laws of the two spinors involve different Ω’s, such an operator is gauge invariant only if the object W(y, x) between the spinors transforms as follows: W(y, x)

4.6.2

→

Ω† (y) W(y, x) Ω(x) .

(4.156)

Wilson lines

In order to construct such an object, let us define a path γµ (s) that goes from x to y, γµ (0) = xµ ,

γµ (1) = yµ ,

(4.157)

and consider the following differential equation  dγµ  dW Dµ (γ(s)) W = 0, with initial condition W(0) = 1. ≡ ds ds

(4.158)

where the notation Dµ (γ(s)) indicates that the gauge field in the covariant derivative must be evaluated at the point γµ (s). In other words, the covariant derivative of W, projected along the tangent vector to the path γµ (s), is zero. From this definition, it follows that W(s) is an element of the representation r of the gauge group if Aµ is in the representation r of the algebra.

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4. N ON -A BELIAN GAUGE SYMMETRY

Note that when the gauge field Aµ is zero everywhere, then the solution is trivially W(s) = 1. For a generic gauge field, the value of the solution18 at s = 1 is a property of the path γµ and of the gauge potential Aµ . This object, that we will denote as Wyx [A; γ] ≡ W(1) ,

(4.159)

is called a Wilson line. Let us now study how it changes under a gauge transformation Ω. From the transformation law of the covariant derivative, the differential equation that defines the transformed WΩ (s) is dγµ † Ω (γ(s))Dµ (γ(s))Ω(γ(s))WΩ (s) = 0, with initial condition WΩ (0) = 1. ds (4.160) If we define Z(s) ≡ Ω(γ(s))WΩ (s), this equation is equivalent to dγµ Dµ (γ(s)) Z(s) = 0 , ds

with initial condition Z(0) = Ω(x) .

(4.161)

Comparing this equation with the original equation (4.158), we obtain Z(s) = W(s) Ω(x) ,

i.e. WΩ (s) = Ω† (γ(s)) W(s) Ω(x) .

(4.162)

Looking now at the point s = 1, we see that the Wilson line transforms as Wyx [A; γ]

→

Ω† (y) Wyx [A; γ] Ω(x) .

(4.163)

Thus, the Wilson line transforms precisely as we wanted in eq. (4.156), and we conclude that the operator ψ(y)Wyx [A; γ]ψ(x) is gauge invariant. Note that the Wilson line Wyx [A; γ], solution of eq. (4.158) at s = 1, can also be written as a path-ordered exponential,   Z (4.164) Wyx [A; γ] = P exp ig dxµ Aµ (x) . c sileG siocnarF

γ

Although this compact notation is suggestive, it is often useful to return to the defining differential equation (4.158).

4.6.3

Path dependence

By inserting a Wilson line between the points x and y, we can construct a gauge invariant non-local operator ψ(y) · · · ψ(x). However, in doing so, we have introduced 18 Note that if the initial condition is W(0) = Ω instead of 1, then the solution would be changed as 0 follows W(s) → W(s)Ω0 .

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a path γ, for which there are infinitely many possible choices since only its endpoints are fixed. It turns out that in general, the Wilson line depends on the path γ, i.e. Wyx [A; γ] 6= Wyx [A; γ ′ ] .

(4.165)

This implies that, although we may define gauge invariant non-local bilinear operators, their definition is not unique and each choice of the path connecting the two points leads to a different operator.

4.6.4

Case of pure gauge fields

When the gauge potential is a pure gauge field, there exists a function Ω(x) such that AΩ µ (x) =

i † Ω (x) ∂µ Ω(x) . g

(4.166)

Since this field is a gauge transformation of the null field Aµ ≡ 0, Wilson lines in this pure gauge field are given by Wyx [AΩ ; γ] = Ω† (y) Ω(x) .

(4.167)

In other words, in a pure gauge field, the Wilson lines depend only on their endpoints, but not on the path chosen to connect them. This is the only exception to the remark of the previous paragraph. Conversely, a gauge potential Aµ (x) in which the Wilson lines depend only on the endpoints is a pure gauge. A function Ω(x) that gives this gauge potential through eq. (4.166) can be constructed as a Wilson line from x to some arbitrary base point x0 : Ω(x) = Wx0 x [A; γ] .

(4.168)

(The path γ can be chosen arbitrarily.)

4.6.5

Wilson loops

A Wilson loop is a special kind of Wilson line, where the initial point and endpoint are identical, x = y, and therefore the path γ is a closed loop:   I (4.169) W[A; γ] = P exp ig dxµ Aµ (x) . γ

Note that they are a property of the closed loop γ, and do not depend on the choice of the starting point x. Because they have identical endpoints, the trace of a Wilson loop

4. N ON -A BELIAN GAUGE SYMMETRY

179

is gauge invariant. From the result of the previous paragraph, they are equal to the identity in a pure gauge field, but they depend non-trivially on the path in a generic gauge field19 . In Abelian gauge theories, the Wilson loop can be rewritten in terms of the integral of the field strength Fµν over a surface Σ of boundary γ, by using Stokes theorem:  gZ   I  µ = exp i exp ig dx Aµ (x) dxµ ∧ dxν Fµν (x) . (4.170) Abelian 2 Σ γ

Generalizations of this formula to the non-Abelian case exist, that involve a pathordering in the left hand side (thus giving a Wilson loop) and a surface-ordering in the right hand side. For infinitesimally small closed loops, a more direct connection to the field strength may be established. Consider for instance a small square closed path in the (12) plane, γ=

a

x a

The Wilson loop along this path may be approximated by

W[A; γ] ≈ exp − iga A2 (x + a2 ^) exp − iga A1 x + a2 ^ı + a^)

× exp iga A2 (x + a^ı + a2 ^) exp iga A1 (x + a2 ^ı) ,

(4.171)

where we make an error of order a3 on each of the Wilson lines at the edges of the square. By expanding the exponentials, we obtain
W[A; γ] = 1 + iga2 ∂1 A2 (x) − ∂2 A1 (x)
−g2 a2 A2 (x)A1 (x) − A1 (x)A2 (x) + O(a3 ) =

1 + ig a2 F12 (x) + O(a3 ) .

(4.172)

Thus, the first non-trivial correction to a small Wilson loop is the area of the loop times the field strength projected on the plane of the loop. Since W[A; γ] is an element of the representation r of the group, in the vicinity of the identity, it may be represented as

2 a a 3 W[A; γ] = exp i ǫ αa ta r + ǫ β tr + O(ǫ )
ǫ2 a b a b 2 a a α α tr tr + O(ǫ3 ) , = 1r + i ǫ αa ta r + ǫ β tr − 2 (4.173) 19 Wilson loops are extensively used in lattice gauge theories. Moreover, Giles’ theorem states that all the gauge invariant information contained in a gauge potential Aµ can be reconstructed from the trace of Wilson loops (assuming we know Wilson loops for arbitrary loops).

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where ǫ is an infinitesimal parameter quantifying how close W[A; γ] is from the identity, and 1r is the identity matrix in the representation r. Comparing eqs. (4.172) and (4.173), we see that we must identity ǫ ≡ ga2 and αa ≡ Fa 12 (x). The formula (4.172) is insufficient in order to determine βa , since this term gives a contribution of order a4 in the Wilson loop. But we can nevertheless use eq. (4.173) in order to determine the lowest order correction to the trace of the Wilson loop,
g2 a4 a
a b 6 tr (W[A; γ]) = tr 1r − F12 (x)Fb 12 (x) tr tr tr + O(a ) , 2

(4.174)

where we have used the fact that the generators ta r are traceless for the su(N) algebra. Eq. (4.174) is the basis of the discretization of the Yang-Mills action, the first step in the formulation of lattice gauge theories.

4.6.6

Wilson lines and eikonal scattering

Wilson lines also appear in the high energy limit of scattering by an external potential, known as the eikonal limit. Consider the following S-matrix element,  

Sβα ≡ βout αin = βin U(+∞, −∞) αin , (4.175) for the transition between two arbitrary states made of quarks, antiquarks and gluons, α and β. In the second equality, U(+∞, −∞) is the evolution operator from the initial to the final state. It can be expressed as the time ordered exponential of the interaction part of the Lagrangian, h Z i U(+∞, −∞) = T exp i d4 x LI (φin (x)) , (4.176)

where φin denotes generically the fields in the interaction picture. In this discussion, LI contains both the self-interactions of the fields, and their interaction with the external field. Consider now the high energy limit of this scattering amplitude, (∞)

Sβα ≡

lim

ω→+∞



 3 3 βin e−iωK U(+∞, −∞) e+iωK αin {z } |

(4.177)

boosted state

where K3 is the generator of Lorentz boosts in the +z direction. Before doing any calculation, a simple argument can help understand what happens in this limit. Quite generally, scattering amplitudes are proportional to the overlap in space-time between the wavefunctions of the two colliding objects. In the present case, it should scale as the time spent by the incoming state in region occupied by the external field. This duration is inversely proportional to the energy of the incoming state, and goes to zero in the limit ω → +∞. If the interaction between

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4. N ON -A BELIAN GAUGE SYMMETRY

the projectile and the external field was via a scalar exchange, then the conclusion would be that the scattering amplitude vanishes in the high energy limit (in other words, S-matrix elements would go to unity). However, interactions with a colour field involve a vector exchange, i.e. the external field couples to a four-vector Jµ that represents the colour current carried by the projectile, by a term of the form Aµ Jµ . At high energy, the longitudinal component of this four-vector increases proportionally to the energy, and compensates the small time spent in the interaction zone. Thus, for states that interact via a vector exchange20 , we expect that scattering amplitudes have a finite high energy limit (nor zero, nor infinite). This calculation is best done using light-cone coordinates. For any four-vector aµ , one defines a+ ≡

a0 + a3 √ 2

,

a− ≡

a0 − a3 √ . 2

(4.178)

These coordinates satisfy the following formulas, x · y = x+ y− + x− y+ − x⊥ · y⊥ d4 x = dx+ dx− d2 x⊥  = 2∂+ ∂− − ∇2⊥

with

∂+ ≡

∂ ∂ , ∂− ≡ + . ∂x− ∂x

(4.179)

Note also that the non-zero components of the metric tensor are g+− = g−+ = 1 ,

g11 = g22 = −1 .

(4.180)

For a highly boosted projectile in the +z direction, x+ plays the role of the time, and the Hamiltonian is the P− component of the momentum. The generator of longitudinal boosts in light-cone coordinates is K3 = M+− .

(4.181)

Using the commutation relations of the Poincar´e algebra, this leads to the following identities: 3

3

e−iωK P− eiωK 3

−iωK

+

iωK3

e P e −iωK3 j iωK3 e P e

=

e−ω P−

= =

e+ω P+ Pj .

(4.182)

20 By the same reasoning, gravitational interactions, that involve a spin two exchange, would lead to scattering amplitudes that grow linearly with energy.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

They express the fact that, under longitudinal boosts, the components P± of a fourvector are simply rescaled, while the transverse components are left unchanged. Likewise, states, creation operators and field operators are transformed as follows,   3 eiωK p · · · in = (eω p+ , p⊥ ) · · · in 3

3

3

3

eiωK a†in (q) e−iωK = a†in (eω q+ , e−ω q− , q⊥ )

eiωK φin (x) e−iωK = φin (e−ω x+ , eω x− , x⊥ ) .

(4.183)

Note that the last equation is valid only for a scalar field, or for the transverse components of a vector field. In addition, the ± components of a vector field receive an overall rescaling by a factor e±ω . Moreover, since a longitudinal boost does not alter the time ordering, we can also write Z 3 3 −iωK3 iωK3 e U(+∞, −∞) e = T exp i d4 x LI (e−iωK φin (x) eiωK ) . (4.184) The components of the vector current that couples to the target field transform as 3

3

e−iωK Ji (x) eiωK = Ji (e−ω x+ , eω x− , x⊥ ) 3

3

3

3

e−iωK J− (x) eiωK = e−ω J− (e−ω x+ , eω x− , x⊥ ) e−iωK J+ (x) eiωK = eω J+ (e−ω x+ , eω x− , x⊥ ) .

(4.185)

Naturally, the target field Aµ does not change when we boost the projectile. For simplicity, let us assume that Aµ is confined in the region −L ≤ x+ ≤ +L. We can thus split the evolution operator into three factors, U(+∞, −∞) = U(+∞, +L) U(+L, −L) U(−L, −∞) .

(4.186)

The factors U(+∞, +L) and U(−L, −∞) do not contain the external potential. For these two factors, the change of variables e−ω x+ → x+ , eω x− → x− leads to 3

3

3

3

lim e−iωK U(+∞, +L) eiωK

ω→+∞

lim e−iωK U(−L, −∞) eiωK

ω→+∞

=

U0 (+∞, 0)

=

U0 (0, −∞) ,

(4.187)

where U0 is the same as U, but defined with the self-interactions only (since these two factors correspond to the evolution of the projectile while outside of the target field). For the factor U(+L, −L), the change eω x− → x− gives h Z i −iωK3 iωK3 U(+L, −L) e = exp i d2 x⊥ χ(x⊥ ) ρ(x⊥ ) , (4.188) lim e ω→+∞

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4. N ON -A BELIAN GAUGE SYMMETRY

Z   χ(x⊥ ) ≡ dx+ A− (x+ , 0, x⊥ )  Z   ρ(x⊥ ) ≡ dx− J+ (0, x− , x⊥ ) .

with

(4.189)

Thus, the high-energy limit of the scattering amplitude is h Z i 

(∞) Sβα = βin U0 (+∞, 0) exp i d2 x⊥ χ(x⊥ )ρ(x⊥ ) U0 (0, −∞) αin . (4.190)

This formula is an exact result in the limit ω → +∞. One may also note the following important properties: • Only the A− component of the external vector potential, integrated along the trajectory of the projectile, matters. c sileG siocnarF

• The self-interactions and the interactions with the external potential are factorized into three separate factors – this is a generic property of high energy scattering. The role of the longitudinal boost in this factorization is illustrated in the figure 4.4.

Figure 4.4: Illustration of the role of kinematics in the factorization of eq. (4.190). Left: before the boost is applied, quantum fluctuations of the incoming projectile may occur in the region of the external field. Right: after the boost, the region of the external field shrinks due to Lorentz contraction (in the frame of the projectile), and the effect of quantum fluctuations inside this region go to zero.

Eq. (4.190) is an operator formula that still contains the self-interactions of the fields to all orders. In order to evaluate it, one must insert the identity operator written as a sum over a complete set of states on each side of the exponential, (∞)

Sβα

=

 βin U0 (+∞, 0) γin γ,δ h Z i 

× γin exp i d2 x⊥ χ(x⊥ )ρ(x⊥ ) δin 

× δin U0 (0, −∞) αin .

X

(4.191)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The factor X   δin δin U0 (0, −∞) αin

(4.192)

δ

is the Fock expansion of the initial state: it accounts for the fact that the state α prepared at x+ = −∞ may have fluctuated into another state δ before it interacts with the external potential. The matrix elements of U0 that appear in this expansion can be calculated perturbatively to any desired order. There is a similar factor for the final state evolution. 

The interactions with the external field are in the central factor, γin exp ... δin . In order to rewrite it into a more intuitive form, let us first rewrite the operator ρ in terms of creation and annihilation operators. For instance, the fermionic part of the current gives Z dp+ d2 p⊥ d2 q⊥ a
 † ρa (x⊥ ) = g b + bsj p+ q⊥ ei(p⊥ −q⊥ )·x⊥ t 4πp+ (2π)2 (2π)2 f ij si p p⊥  −d†si p+ p⊥ dsj p+ q⊥ e−i(p⊥ −q⊥ )·x⊥ ,

(4.193)

where the ta f are the generators of the fundamental representation of the su(N) algebra and b, d, b† , d† are the annihilation and creation operators for quarks and antiquarks. ρa also receives a contribution from gluons, not written here, obtained with the generators in the adjoint representation and the annihilation and creation operators for gluons instead. This formula captures the essence of eikonal scattering: • Each annihilation operator has a matching creation operator – therefore, the number of quarks and gluons in the state does not change during the scattering, nor their flavour. • The p+ component of the momenta are not affected by the scattering. • The spins are unchanged during the scattering. • The colours and transverse momenta of the constituents of the state may change during the scattering. Scattering amplitudes in the eikonal limit take a very simple form if one trades transverse momentum a transverse

for  position by a Fourier transform. For each , k intermediate state δin ≡ k+ i⊥ , we first define the corresponding light-cone i wave function by : Ψδα ({k+ i , xi⊥ }) ≡

Y Z d2 ki⊥ 

e−iki⊥ ·xi⊥ δin U0 (0, −∞) αin , (4.194) 2 (2π) i∈δ

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4. N ON -A BELIAN GAUGE SYMMETRY

where the index i runs over all the constituents of the state δ. Then, each charged particle going through the external field acquires an SU(N) phase that depends on the representation in which it lives + Ψδα ({k+ i , xi⊥ }) −→ Ψδα ({ki , xi⊥ })

h

Z

Y i∈δ

Ui (x⊥ )

i + a Ui (x⊥ ) ≡ T exp ig dx+ A− a (x , 0, xi⊥ ) tri ,

(4.195)

where ri is the representation corresponding to the constituent i. We recognize in this formula Wilson lines defined on the light-cone direction that corresponds to the boosted projectile. The simplicity of this result is entirely due to kinematics: thanks to the longitudinal boost, the external field is crossed in an infinitesimally short time, during which the transverse positions of the incoming quanta cannot vary.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 5

Quantization of Yang-Mills theory 5.1 Introduction Generically, the Lagrangian density of a non-Abelian gauge theory reads: L

≡


†

Dµ φ(x) Dµ φ(x) − m2 φ† (x)φ(x) − V φ† (x)φ(x)
/ x − m ψ(x) +ψ(x) iD a µν − 41 Fa (x) . µν (x)F

(5.1)

The local non-Abelian gauge invariance of this Lagrangian does not change anything to the quantization of the scalar field φ and of the spinor ψ, for which we may use the standard canonical or path integral approaches, with the result that the usual Feynman rules still apply. The main complication resides in the pure Yang-Mills part (third term) of this Lagrangian, i.e. with the quantization of the gauge potential Aµ . The identification of the degrees of freedom that are made redundant by the gauge symmetry is much more complicated than in QED, and a lot more care is necessary in order to isolate the genuine dynamical variables of the theory. In order to get a sense of the difficulty, let us try to mimic the QED case in order to guess the Feynman rules for non-Abelian gauge fields. Using the explicit form of the field strength, abc b c Fa Aµ Aν , µν = ∂µ Aν − ∂ν Aµ + g f

187

(5.2)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

we can rewrite the Yang-Mills Lagrangian as follows
µν LA = 12 Aa  − ∂µ ∂ν Aa µ g ν
bµ cν abc a −g f ∂µ Aν A A

c dµ eν − 41 g2 fabc fade Ab A , µ Aν A

(5.3)

where we have anticipated an integration by parts in the first (kinetic) term. Note that the kinetic term is formally identical to the kinetic term of a photons, except for the colour index a carried by the gauge potential. Therefore, one may be tempted to generalize the QED Feynman rules to a non-Abelian gauge boson. As in the QED case, the quadratic part of the Lagrangian (5.3) poses a difficulty when trying to determine the free propagator, because the operator between Aµ · · · Aν is not invertible. If we take for granted that a similar gauge fixing procedure (more on this later, as this is in fact the heart of the problem) can be applied here, we may assume that the free gauge boson propagator1 in Feynman gauge is p

G0F µν ab (p) =

=

−i gµν δab , p2 + i0+

(5.4)

and one may read off directly from the Lagrangian (5.3) the following 3-gluon and 4-gluon vertices: aµ k

= p bν

aµ

q

+ gνρ (p − q)µ + gρµ (q − k)ν

cρ

(5.5)

bν

=

cρ

 g fabc gµν (k − p)ρ

dσ

 −i g2 fabe fcde (gµρ gνσ − gµσ gνρ )

+ face fbde (gµν gρσ − gµσ gνρ ) (5.6) + fade fbce (gµν gρσ − gµρ gνσ )

All this seems fine, except for a rather subtle problem that would appear when using this perturbation theory: these Feynman rules lead to amplitudes that do not 1 In this chapter, we use the diagrammatic convention of QCD, where the gauge bosons (gluons) are represented as springs in Feynman diagrams. In the electroweak theory, it is more common to represent them as wavy lines, like the photon in QED.

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5. Q UANTIZATION OF YANG -M ILLS THEORY

fulfill Ward identities, even when all the external coloured particles are on their massshell. From the discussion of perturbative unitarity for amplitudes with external gauge bosons in 1.16.4, the lack of Ward identities seems to imply a violation of unitarity in perturbation theory. Since unitarity is one of the cornerstones of any quantum theory, this is not a conclusion we are ready to accept, and we must conclude that something is missing in the above Feynman rules.

5.2 Gauge fixing In our naive attempt to guess the Feynman rules appropriate for non-Abelian gauge bosons, we have implicitly assumed that the gauge fixing works in the same way as in QED, namely that the gauge fixing trivially leads to the factorization of an infinite factor in the path integral, with no other change to the degrees of freedom that are not constrained by the gauge condition. It turns out that this assumption is incorrect. Let us start from the path integral representation of the expectation value of some gauge invariant operator O(Aµ ): Z 

Z  1

   a µν . (5.7) O ≡ DAa (x) O(A ) exp i d4 x − Fa µ µν F µ 4 | {z } SYM [Aµ ]

Local gauge transformations of the field Aµ , Aµ (x)

→

† AΩ µ (x) ≡ Ω (x) Aµ (x) Ω(x) +

i † Ω (x) ∂µ Ω(x) , g

(5.8)

leave the action and the observable unchanged. Moreover, the functional measure is also invariant, since   Ω    δAa µ (x)  Ω , (5.9) DAa µ (x) = DAa µ (x)] det δAb ν (y)

where the determinant is the Jacobian of the change of coordinates. Using eq. (4.37), this determinant can be rewritten as follows det



 δAΩ a µ (x) δAb ν (y)



=

    det δµ ν δ(x − y) Ωadj (x) ab = 1 ,

(5.10)

since the group element Ωadj is a unitary matrix. Therefore, there is a large amount of redundancy in the above path integral, and it is in fact infinite. By applying a gauge transformation, each field configuration Aµ develops into a gauge orbit (see the figure 5.1), along which the physics is invariant. In order to eliminate this redundancy, we

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Aµ G(Aµ) = 0

gauge fixed Aµ

gauge orbit

Figure 5.1: Illustration of the gauge fixing procedure. The lines represent the gauge field configurations spanned when varying Ω. The shaded surface is the manifold where the gauge condition is satisfied, and the black dots are the gaugefixed field configurations.

5. Q UANTIZATION OF YANG -M ILLS THEORY

191

would like to impose a condition at every space-time point x on the gauge fields,
Ga Aµ (x) = 0 , (5.11) in order to select a unique2 field configuration along each orbit. Geometrically, the gauge condition (5.11) defines a manifold that intersects each orbit, as shown in the figure 5.1, and we choose this intersection as the representative of this field configuration.

5.3 Fadeev-Popov quantization and Ghost fields Thus, we would like to split the integration measure in eq. (5.7) into a physical component in the manifold G(A) = 0, and a component along the gauge orbits that we should factor out. Unfortunately, achieving this in a non-Abelian gauge theory is far more complicated than in QED, because the modification of the gauge potential under a gauge transformation is non-linear. In order to see the difficulty, let us define Z   −1 (5.12) ∆ [Aµ ] ≡ DΩ(x) δ[Ga (AΩ µ )] . ∆[Aµ ] is the determinant of the derivative of the constraint G(Aµ ) with respect to the gauge transformation Ω, at the point where G(Aµ ) = 0,  a δG . (5.13) ∆(Aµ ) = det δΩ Ga (AΩ µ )=0

In QED, for linear gauge fixing conditions, this derivative (and therefore the determinant) is independent of the gauge field, and can be trivially factored out of the path integral. This is not the case in non-Abelian gauge theories, and this determinant is the source of significant complications. One can first prove that the determinant ∆[Aµ ] is gauge invariant. Indeed, changing Aµ → AΘ µ , we have: ∆

−1

[Aµ ] = Θ

= =

Z

Z

Z







Ω′

z}|{  DΩ(x) δ[Ga (AµΘΩ )]

 ′ D(Θ† (x)Ω ′ (x)) δ[Ga (AΩ µ )]

 ′ −1 [Aµ ] . DΩ ′ (x) δ[Ga (AΩ µ )] = ∆

(5.14)

2 It turns out that this is not possible, due to the Gribov ambiguity: all gauge conditions of the form (5.11) have several solutions, called Gribov copies. However, only one of these solutions is a “small field”, while the others are proportional to the inverse coupling g−1 . Since perturbation theory is an expansion around the vacuum (i.e. in the small field regime), these non-perturbatively large copies do not play any role in perturbation theory.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Here, we have used the fact that there exists a group invariant integration measure on a Lie group. By inserting Z   1 = ∆[Aµ ] DΩ(x) δ[Ga (AΩ (5.15) µ )]

inside the path integral (5.7), we obtain Z Z

     a iSYM [Aµ ] Ω O = DΩ(x) DAa . (5.16) µ (x) ∆[Aµ ] δ[G (Aµ )] O(Aµ ) e

Now, we change the integration variable of the second integral according to Aµ → † AΩ µ . In this transformation, the measure [DAµ ], the Yang-Mills action SYM [Aµ ], the observable O(Aµ ) and the determinant ∆[Aµ ] are all unchanged (because they are gauge invariant):     † DAΩ = DAµ , µ † SYM [AΩ µ ] = † O[AΩ µ ] =

† ∆[AΩ µ ] =

SYM [Aµ ] , O[Aµ ] , ∆[Aµ ] ,

(5.17)

while the field AΩ µ becomes Aµ . Therefore, we have Z Z

     a iSYM [Aµ ] O = DΩ(x) DAa . (5.18) µ (x) ∆[Aµ ] δ[G (Aµ )] O(Aµ ) e

At this point, the second integral does not contain the gauge transformation Ω anymore, and therefore we have managed   to factorize the “integral along the orbits” in the form of the first integral over DΩ . Dropping this constant factor, we can therefore write an integral free of any redundancy: Z  

 a iSYM [Aµ ] . (5.19) O = DAa µ (x) ∆[Aµ ] δ[G (Aµ )] O(Aµ ) e In the above formula, the determinant ∆[Aµ ] depends on the gauge field and must therefore have an effect on the Feynman rules. The Fadeev-Popov method consists in rewriting this determinant as a path integral. Note that since ∆[Aµ ] appears in the numerator, we need Grassmann variables in order to represent it as a path integral3 , according to eq. (3.36): Z 
 det i M = Dχa (x)Dχa (x)

Z  × exp i d4 xd4 y χa (x) Mab (x, y) χb (y) . (5.20)
3 The factor i in det i M has been included for aesthetic reasons, but does not change anything. In fact any rescaling M → κ M would leave the results unchanged. Indeed, such a change would alter the ghost propagator according to S → κ−1 S, and the ghost-gauge boson vertex by V → κV. Since the ghosts appear only in closed loops, that contain an equal number of propagators and vertices, these factors κ would cancel out.

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5. Q UANTIZATION OF YANG -M ILLS THEORY

An extra generalization, that we have already used in the path integral quantization of the photon (see eq. (3.52)), is to shift the gauge condition from Ga (A) = 0 to Ga (A) = ωa and to perform a Gaussian integration over ωa . The final result takes the following form: Z  

  DAa O = µ (x) Dχa (x)Dχa (x) O(Aµ ) Z   1 ξ a µν − (Ga (Aµ ))2 + χa Mab χb , × exp i d4 x − Fa µν F } | 2 {z } | {z } | 4 {z LYM

LGF

LFPG

(5.21)

where Mab is the derivative of Ga (AΩ ) with respect to the gauge transformation Ω, at the point Ω = 1 (here, we use the fact that the determinant is gauge invariant to choose freely the Ω at which we compute the derivative). The unphysical Grassmann fields χ and χ introduced as a trick to express the determinant are called Fadeev-Popov ghosts, or simply ghosts. Although physical observables do not depend on these fictitious fields, there is in general a coupling between the ghosts and the gauge fields, because the matrix Mab may contain the gauge field. This implies that the ghosts may appear in the form of loop corrections in the perturbative expansion. As we shall see shortly, they are in fact crucial for the consistency of perturbation theory in non-Abelian gauge theories. In particular, the ghosts ensure that the theory is unitary. c sileG siocnarF

5.4 Feynman rules for non-abelian gauge theories Eq. (5.21) contains all the necessary ingredients to complete the Feynman rules that we have started to derive heuristically at the beginning of this chapter. To turn this formula into explicit Feynman rules, we should first choose the gauge fixing function Ga (A), since it enters directly in the term in ξ2 (Ga (A))2 , and implicitly in the matrix Mab that defines the ghost term. In the common situation where this gauge fixing function is linear in Aµ (all our examples will be of this type), then the terms that are quadratic in the gauge field are the same as in QED, and therefore the gauge boson propagator is also the same (except for an extra factor δab that expresses the fact that the free propagation of a gluon does not change its colour). Thus our guess (5.4) for the Feynman gauge propagator was in fact correct. In addition, the gauge fixing term and the ghost term cannot contain terms of degree 3 or 4 in the gauge field, which implies that the vertices given in eqs. (5.5) and (5.6) are also correct. c sileG siocnarF

5.4.1

Covariant gauge

Let us now consider the general covariant gauge, all known as the Rξ -gauge, already introduced in eq. (3.51) for QED. This amounts to choosing the gauge fixing function

194

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

as a Ga (A) ≡ ∂µ Aa µ − ω (x) .

(5.22)

With this gauge fixing, the free gauge boson propagator is p

G0F µν ab (p)

=

−i gµν δab i δab = 2 + 2 p + i0+ p + i0+



1 1− ξ



pµ pν . (5.23) p2

(The simplest form is obtained in the limit ξ → 1, giving the Feynman gauge4 .) The matrix Mab can be calculated by applying an infinitesimal gauge transformation Ω = exp(iθa ta ) to Aµ . The variation of the gauge field is δAa µ (x) = g fabc θb (x) Ac µ (x) − ∂µ θa (x) ,

(5.24)

and the variation of Ga (A) at the point x is

δGa = g fabc ∂µ θb (x) Ac µ (x)+g fabc θb (x) ∂µ Ac µ (x) − θa (x) . (5.25)

Therefore, we have


δGa (A) = g fabc ∂µ Ac µ (x) + g fabc Ac µ (x) ∂µ − δab  , (5.26) b δθ and the terms that depend on the Fadeev-Popov ghosts can be encapsulated in the following effective Lagrangian:  
LFPG = χa − δab  + g fabc ∂µ Ac µ (x) + g fabc Ac µ (x) ∂µ χb (5.27) Mab =

The first term leads to the following propagator for the ghosts:

p 0

GF (p) =

=

i δab . p2 + i0+

(5.28)

Note that it has the form of a scalar propagator, although the ghosts are anti-commuting Grassmann variables. The vertex between ghosts and gauge bosons reads

a

r

q cµ

b

= g fabc (pµ + qµ ) = g fabc rµ .

(5.29)

p

The Feynman rules for non-Abelian gauge theories in covariant gauge are summarized in the figure 5.2, where we have added for completeness the rules relative to fermions. 4 Another popular choice is the Landau gauge, obtained in the limit ξ → +∞, that corresponds to a strict enforcement of the condition ∂µ Aµ = 0. Indeed, in this limit the exponential of i ξ2 (∂µ Aµ )2 in the gauge fixed Lagrangian oscillates wildly –and produces cancellations– unless ∂µ Aµ = 0. Equivalently, the Gaussian distribution for the function ωa (x) has a vanishing width in this limit, which forces the strict equality ∂µ Aa µ = 0.

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5. Q UANTIZATION OF YANG -M ILLS THEORY

p

=

−i gµν δab i δab + 2 p2 + i0+ p + i0+

=

i δij / p − m + i0+

=

i δab p2 + i0+

=

 g fabc gµν (k − p)ρ

p

p

aµ k

p q

bν

aµ

pµ pν p2

+ gνρ (p − q)µ + gρµ (q − k)ν

=

=

−i g γµ ta r

=

g fabc (pµ + qµ ) = g fabc rµ

aµ j

q



+ face fbde (gµν gρσ − gµσ gνρ ) + fade fbce (gµν gρσ − gµρ gνσ )

i

r



 −i g2 fabe fcde (gµρ gνσ − gµσ gνρ )

dσ

a

1 1− ξ

cρ

bν

cρ






ij

cµ b

p

Figure 5.2: Feynman rules of non-Abelian gauge theories in covariant gauge. We also list the rules involving fermions for completeness. Latin characters a, b, c refer to the adjoint representation, while the letters i, j refer to the representation r in which the fermions live.

196

5.4.2

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Axial gauge

µ The axial gauge fixing consists in constraining the value of nµ Aa µ , where n is a fixed 4-vector (when this vector is time-like, this gauge is called the temporal gauge, and when it is light-like, it is called the light-cone gauge). Therefore, the gauge fixing function is a Ga (A) ≡ nµ Aa µ − ω (x) .

(5.30)

After gauge fixing, the quadratic part of the effective Lagrangian reads
1 a µν Aµ g  − ∂µ ∂ν − ξ nµ nν Aa ν , 2

(5.31)

and the free gauge boson propagator is obtained in momentum space by inverting gµν p2 − pµ pν + ξ nµ nν .

(5.32)

The inverse of this matrix must be of the form A gµν + B pµ pν + C nµ nν + D (nµ pν + nν pµ ) .

(5.33)

(This is the most general symmetric tensor that one may construct with gµν , pµ and nµ .) This leads to the following propagator G0F µν ab (p) =


i pµ pν −i δab h µν pµ nν + pν nµ 2 −1 2 . (5.34) + g − n +ξ p p2 + i0+ p·n (p · n)2

Note that this propagator does not vanish as p−2 at large momentum, because of the term proportional to ξ−1 , With this gauge fixing, the variation of the gauge fixing function under an infinitesimal gauge transformation is given by δGa = g fabc θb (x) nµ Ac µ (x) − nµ ∂µ θa (x) .

(5.35)

and the matrix M reads Mab = g fabc nµ Ac µ (x) − δab nµ ∂µ ,

(5.36)

Therefore, the Fadeev-Popov term in the effective Lagrangian is LFPG = χa



 − δab nµ ∂µ + g fabc nµ Ac µ (x) χb ,

(5.37)

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5. Q UANTIZATION OF YANG -M ILLS THEORY

which leads to the following expressions for the ghost propagator and its coupling to the gauge boson:

p 0

GF (p) =

a

r

=−

q cµ

b

δab p · n + i0+

= i g fabc nµ .

(5.38)

p

A significant simplification of these Feynman rules occurs in the limit ξ → ∞ (that one may call the strict axial gauge, since the condition nµ Aa µ = 0 holds exactly in this limit). In this limit, the gauge boson propagator becomes G0F µν ab (p) = and satisfies

−i δab h µν pµ nν + pν nµ pµ pν n2 i + g − , p2 + i0+ p·n (p · n)2

0 µν nµ G0F µν ab (p) = nν GF ab (p) = 0 .

(5.39)

(5.40)

Therefore, the gauge boson propagator gives zero when contracted into the ghostgauge boson vertex, which effectively decouples the ghosts from the gauge bosons. Thus, the limit ξ → ∞ of the axial gauge is ghost-free (but its propagator is arguably much more complicated than the Feynman gauge propagator). c sileG siocnarF

5.5 On-shell non-Abelian Ward identities In Quantum Electrodynamics, the interpretation of Cutkosky’s cutting rules as a perturbative realization of unitarity depends crucially on the Ward-Takahashi identities satisfied by amplitudes with external photons, namely 1 kµ 1 Γµ1 µ2 ··· (k1 , k2 , · · · ) = 0 ,

(5.41)

valid when all the charged external lines are on-shell. Note that this identity does not require to contract the remaining photons with a polarization vector. c sileG siocnarF

In Yang-Mills theory, it turns out that the identity (5.41) is in general not satisfied. Instead, it is replaced by a different on-shell identity, discovered by ’t Hooft. In order to derive it, let us consider a generalized covariant gauge condition of the form ∂µ Aµ a (x) − ζa (x) = ωa (x) .

(5.42)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Like in the Fadeev-Popov quantization, we integrate over the function ωa with a Gaussian weight, which leads to the following gauge fixed Lagrangian: ξ 1 a µν 2 (5.43) − (∂µ Aµ L ≡ − Fa µν F a − ζa ) + χa Mab χb . 4 2 This Lagrangian is the same as the one encountered before (with ζa ≡ 0), with the addition of two terms that depend on the arbitrary function ζa (x) introduced in the gauge fixing: ξ (5.44) − ζa ζa + ξ ζa ∂µ Aµ a . 2 Since ζa is not a dynamical field but merely a parameter, the first term can be factored out of the generating functional for correlation functions, and cancels once it is properly normalized. The second term acts as a source coupled to the divergence of the gauge field. In momentum space, the Feynman rule for the insertion of such a source is k aµ

= i ξ kµ ζea (k) ,

(5.45)

where ζea is the Fourier transform of ζa . This source is always contracted into an external gluon propagator, leading to the combination5

ζeb (k) kν . (5.46) i ξ kµ ζea (k) G0F µν (k) = ab k2 (The propagator is given in eq. (5.23).) In this contraction, the external gluon propagator is replaced by a factor kν directly contracted into the amputated correlation function, independently of the gauge parameter. Since the function ζa has been introduced as part of our choice of gauge fixing condition, gauge invariant quantities should not depend on it. Consequently, the sum of the graphs contributing to gauge invariant quantities with a given non-zero number of insertions of the source ζa must be zero. Consider S-matrix elements, i.e. transition amplitudes between physical states. The graphs contributing to such a matrix element have a number of external gluons corresponding to the in- and outstates of the amplitude, plus possibly some insertions of the source ζa : out

Γ

in

= Γ {µa}i∈[1,n] (k1 · · · kn ; q1 · · · qp ) {z } | {z } {νb}j∈[1,p] | on-shell

×

"

n Y i=1

µi

#

ǫ (k ) × | {z i} physical

"

off-shell

ν p Y qj j ζebj (qj ) j=1

q2j

#

= 0 . (5.47)

5 In axial gauge, the insertion of a source ζ leads to a factor ζ eb (k) kν /(k · n). Thanks to the factor a kν , the identity (5.47) is also valid in axial gauges.

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5. Q UANTIZATION OF YANG -M ILLS THEORY

In the second line, Γ is a (n + p)-gluon amplitude, with amputated external lines. n of these gluons correspond to the in- and out- states, with colour ai . The corresponding momentum ki is on-shell and the Lorentz index µi is contracted with a physical polarization vector. In contrast, the lines to which the sources ζbj are attached are off-shell and contracted with their own momentum qj . When including all the graphs contributing to a given order in the coupling and in ζ, this expression must vanish if p ≥ 1 because it is a contribution to a gauge invariant quantity. Thus, the Ward identity of eq. (5.41) can be adapted to non-Abelian gauge theories, with the restriction ν that all the gluon lines not contracted with qj j must be on-shell and contracted with a physical polarization vector. For instance, the gluon self-energy obeys the following two relations: kµ ǫν (k) Πµν (k) 2 = 0 , kµ kν Πµν (k) = 0 , (5.48) k =0

while in QED it is sufficient to contract the self-energy with a single k (even off-shell) to obtain zero. Note that these identities are insufficient in order to obtain an unitary Smatrix with only internal gluons, because the tensor structure of the internal cut gluon propagators also involves polarizations which are neither physical nor proportional to qν . The Zinn-Justin equation, that we shall derive in the next chapter, may be viewed as a generalization of these Ward identities to off-shell momenta and arbitrary polarizations.

5.6 Ghosts and unitarity 5.6.1

Explicit example

In Abelian gauge theories, we were able to show that cutting rules provide a perturbative realization of the optical theorem, by using the Ward identities obeyed by amplitudes when all the external charged particles are on-shell. These identities were sufficient to conclude that the unphysical polarizations carried by the internal photon lines of a graph cancel when these lines are cut. But in non-Abelian gauge theories, this reasoning faces two difficulties: i. There are no Ward identities similar to those of QED, that could be used to prove unitarity. ii. Higher order graphs in general have ghost loops, whose interpretation is at the moment unclear when such loops are cut. As we shall see, these two issues are in fact related: the cut ghost lines precisely cancel the unphysical polarizations of the cut gluons. Let us first work out an explicit example that illustrates this assertion: the tree level annihilation of a quark and an antiquark into two gluons in QCD. The corresponding diagrams are the following: c sileG siocnarF

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

We denote p et q the momenta of the incoming quark and antiquark, respectively, and k1,2 the momenta of the outgoing gluons (with Lorentz indices µ, ν and colours a, b, respectively). The contribution of the first two graphs is very similar to that of the analogous graphs in QED for the emission of two photons, except for the extra colour matrices at the quark-gluon vertices: 2 | i Mµν (p, q|k , k ) = (i g) v(q) γµ ta 1 2 ab 1+2

i γν tb /1 − q / −m k

i +γν tb γµ ta u(p) . (5.49) / −k /1 − m p

By contracting this amplitude with the photon momentum k1µ , we get:

2 / a k1µ i Mµν ab |1+2 (p, q|k1 , k2 ) = (i g) v(q) k1 t +γν tb

i / −k /1 − m p

i γν tb /1 − q / −m k  /1 ta u(p) . k (5.50)

In the numerator of the first term, we may write /1 = (k /1 − q / − m) + (q / + m) , k

(5.51)

/ + m) = 0. Likewise, we may simplify the second and use the Dirac equation v(q)(q term by using /1 = (p / − m) − (p / −k /1 − m) , k / (p − m)u(p) = 0 ,

(5.52)

which leads to 2 ν a b k1µ i Mµν ab |1+2 (p, q|k1 , k2 ) = i (i g) v(q) γ [t , t ] u(p) .

(5.53)

This is non-zero, because of the non-commutativity of the Lie generators in a nonAbelian gauge theory. However, by using [ta , tb ] = ifabc tc , this result may be

201

5. Q UANTIZATION OF YANG -M ILLS THEORY

related to the third graph, that contains a 3-gluon vertex. If we use the Feynman gauge for the internal gluon propagator, its contribution can be written as c i Mµν ab |3 (p, q|k1 , k2 ) = i g v(q)γρ t u(p)

−i k23

×g fabc [gµν (k2 − k1 )ρ + gνρ (k3 − k2 )µ + gρµ (k1 − k3 )ν ] ,

(5.54)

where we denote k3 ≡ −k1 − k2 . Contracting this amplitude with k1µ gives c k1µ i Mµν ab |3 (p, q|k1 , k2 ) = i g v(q)γρ t u(p)

−i k23

ρ νρ 2 ν ρ ×g fabc [gνρ k22 − kν 2 k2 − g k3 + k3 k3 ] .

(5.55)

ρ In this equation, the term in kν 3 k3 vanishes once contracted with γρ , since we can write

/ − m) + (q / + m)]tc u(p) = 0 . v(q)γρ tc u(p)kρ3 = −v(q)[(p

(5.56)

However, this is not sufficient for (5.55) to fully cancel (5.53). ρ Setting k22 = 0 kills another term in eq. (5.55). The term in kν 2 k2 would be canceled if in addition we contract the amplitudes with a transverse polarization vector ǫ1,2ν (k2 ), since kν 2 ǫ1,2ν (k2 ) = 0. We indeed have:   µν k1µ ǫ1,2ν (k2 ) i Mµν ab |1+2 (p, q|k1 , k2 ) + i Mab |3 (p, q|k1 , k2 ) k2 =0 = 0 . 2

(5.57)

The same cancellation happens if we contract the amplitudes simultaneously with k1µ et k2ν :   µν k1µ k2µ i Mµν (5.58) ab |1+2 (p, q|k1 , k2 ) + i Mab |3 (p, q|k1 , k2 ) = 0 ,

even if the momentum k2 is not on-shell. Thus, we obtain for this process a Ward identity similar to the QED one, provided certain extra conditions are satisfied by the second gluon (both eqs. (5.57) and (5.58) are special cases on the non-Abelian Ward identity (5.47)). These restrictions weaken the resulting identity, and it is not sufficient to eliminate the longitudinal gluon polarizations when we try to recover the amplitude from the imaginary part of the qq¯ → qq¯ forward amplitude at one loop. In particular, some unphysical polarizations will not cancel in the following cut:

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Except for a graph with a quark loop that does not play any role in the present discussion (since it does not give any 2-gluon final state when cut), the complete list of graphs contributing to the qq¯ → qq¯ forward amplitude at one loop is shown in the 5.3. The contribution of the first 5 graphs (i.e. those with gluon internal lines) to the

Figure 5.3: One-loop diagrams contributing to qq¯ → qq¯ .

optical theorem can be calculated easily by noting that it can be expressed in terms of the amplitude we have just calculated: µν µν i Mµν ab (p, q|k1 , k2 ) ≡ i Mab |1+2 (p, q|k1 , k2 )+ i Mab |3 (p, q|k1 , k2 ) , (5.59)

as follows6 1 2

Z

d4 k1 (2π)4

Z

d4 k2 (2π)4 δ(4) (p + q − k1 − k2 ) (2π)4

×2π(−gµρ )θ(k01 )δ(k21 − m2 ) 2π(−gνσ )θ(k02 )δ(k22 − m2 ) ∗ ρσ (5.60) ×i Mµν ab (p, q|k1 , k2 ) (i Mab (p, q|k1 , k2 )) .

For a successful interpretation of this formula as a physical contribution in the optical theorem, only physical polarizations should survive after we have replaced the tensors −gµρ and −gνσ by using (see eq. (1.356)) X µ µ ν ∗ ν ∗ ∗ gµν = ǫµ ǫλ (k)ǫν (5.61) + (k)ǫ− (k) + ǫ− (k)ǫ+ (k) − λ (k) , λ=1,2

ǫµ ± (k)

µ where are unphysical polarizations (with ǫµ + (k) proportional to k ). After this substitution, several terms are not problematic:

• The terms that contain only the polarizations ǫµ 1,2 since they are fully physical. c sileG siocnarF

6 The

factor 1/2 is a symmetry factor due to the presence of two identical gluons in the final state.

203

5. Q UANTIZATION OF YANG -M ILLS THEORY

µ ν ν • The terms containing ǫµ 1,2 ǫ+ or ǫ+ ǫ+ vanish by virtue of eqs. (5.57) and (5.58).

Thus, we need only study the following term 1h ∗ ρσ (i Mµν ab ǫ−µ ǫ+ν ) (i Mab ǫ+ρ ǫ−σ ) 2

∗

ρσ + (i Mµν ab ǫ+µ ǫ−ν ) (i Mab ǫ−ρ ǫ+σ )

i

,

(5.62)

√ µ integrated over the on-shell momenta k1 and k2 . Using ǫµ + (k) = k / 2|k| and eqs. (5.53) and (5.55), we obtain ǫ+µ (k1 ) i Mµν ab = − √

1 g2 abc c /2 kν v(q) k t u(p) . 2 f 2|k1 | k23

(5.63)

Likewise with the other gluon, we have 1 g2 abc c /1 kν v(q) k t u(p) . 1 f 2|k2 | k23 √ Using then ǫµ − (k) = (k0 , −k)/ 2|k|, we get ǫ+ν (k2 ) i Mµν ab = √

|k2 | |k1 | |k1 | = +g2 |k2 |

2 ǫ−ν (k2 ) ǫ+µ (k1 ) i Mµν ab = −g

ǫ+ν (k2 ) ǫ−µ (k1 ) i Mµν ab

(5.64)

1 /2 fabc tc u(p) , v(q) k k23 1 /1 fabc tc u(p) . (5.65) v(q) k k23

Furthermore, notice that /1 + k /2 )u(p) = v(q)(q / +m+p / − m)u(p) = 0 . v(q)(k

(5.66)

Combining these equations, the non-physical contribution to the optical theorem of the diagrams with a gluon loop, (5.62), can be written as follows: g4

   1  /1 fabc tc u(p) v(q) k /1 fabd td u(p) . v(q) k 2 2 (k3 )

(5.67)

If this was all there is, as the naive Feynman rules we tried to guess at the beginning of this chapter would suggest, then we would have to conclude that Yang-Mills theories are inconsistent because they violate unitarity. Fortunately, there is one more graph in figure 5.3, with a ghost loop. Let us first evaluate the annihilation amplitude of the quark-antiquark pair into a ghost-antighost pair: i Mqq→χχ = i g v(q) γρ tc u(p) ¯

i (g fabc kρ1 ) . k23

(5.68)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Squaring this amplitude, and including the − sign7 associated to a ghost loop8 , the contribution of the last graph of fig. 5.3 to the optical theorem becomes −g4

   1  /1 fabc tc u(p) v(q) k /1 fabd td u(p) , v(q) k 2 2 (k3 )

(5.69)

that exactly cancels the unphysical gluon contribution of eq. (5.67). In other words, the optical theorem is satisfied with only physical modes in the final state sum, thanks to a crucial cancellation that involves ghosts.

5.6.2

Becchi-Rouet-Stora-Tyutin symmetry

The cancellation that occurred in the previous example is in fact general: for every gluon loop, there is a graph of identical topology where this loop is replaced by a ghost loop, that cancels the contribution from the unphysical gluon polarizations in the optical theorem. However, it is difficult to turn the calculation of the previous subsection into a general proof. It turns out that this cancellation originates from a residual symmetry of the gauge fixed Lagrangian: although the gauge fixing term explicitly breaks the gauge symmetry, the effective Lagrangian that appears in eq. (5.21) has a remnant of the original gauge symmetry, known as the Becchi-Rouet-Stora-Tyutin symmetry (BRST). Under an infinitesimal gauge transformation parameterized by θa (x), the gauge field and fermion field vary by δAa µ (x) = δψ(x) =

− Dadj µ




ab

θb (x)

−i gθa (x) ta r ψ(x) ,

(5.70)

where r is the representation in which the fermions live. A BRST transformation is similar to the above transformation, but with the substitution θa (x) → −ϑ χa (x), where ϑ is a Grassmann constant9 , δBRST Aa µ (x) =

δBRST ψ(x) =


  Dadj µ ab ϑ χb (x)   i g ϑ χa (x) ta r ψ(x) .

(5.71)

Since the BRST transformation is structurally identical to a local gauge transformation, any gauge invariant combination of gauge fields and fermions is also BRST-invariant. This is therefore the case of the Yang-Mills Lagrangian and the Dirac Lagrangian with 7 There

is no 1/2 symmetry factors for a ghost-antighost final state, because they are not identical. see here how essential it is that ghosts are anti-commuting fields – otherwise, their contribution would not have the proper sign to cancel the unphysical gluon polarizations in the optical theorem. 9 This Grassmann constant makes ϑ χ (x) a commuting object like θ . a a 8 We

205

5. Q UANTIZATION OF YANG -M ILLS THEORY

a minimal coupling of the fermions to the gauge fields. It is customary to introduce a generator QBRST for this transformation, by denoting δBRST = ϑ QBRST . Thus
adj QBRST Aa QBRST ψ(x) = i g χa (x) ta µ (x) = Dµ ab χb (x) , r ψ(x) . (5.72)

Eqs. (5.71) do not tell how ghost and antighost fields transform under BRST. For reasons that will become clear later, we shall impose that the BRST transformation is nilpotent, i.e. that Q2BRST = 0 when applied to any of the fields of the theory. This requirement constrains the BRST transformation of the ghosts. Indeed, a double BRST transformation applied to fermions reads 
a Q2BRST ψ(x) = i g QBRST χa (x) ta r ψ(x) − χa (x) tr QBRST ψ(x)
2 a b = i g QBRST χa (x) ta r ψ(x) + g χa (x)χb (x) tr tr ψ(x) . (5.73) (The BRST generator is an anti-commuting object, which leads to a minus sign in the second term of the first line when we push it through the Grassmann field χa .) Since i abc c 1 a b b tr . We see that χa and χb anti-commute, we can replace ta r tr by 2 [tr , tr ] = 2 f eq. (5.73) will identically vanish provided that 1 QBRST χa (x) = − g fabc χb (x) χc (x) . 2

(5.74)

Then, we can calculate the action of a double BRST transformation on the gauge field,


abc Q2BRST Aa = Dadj QBRST Acµ χb µ µ ab QBRST χb − g f 
 g bcd χc (x)χd (x) = Dadj µ ab − 2 f   −g fabc ∂µ χc −gfcde Aeµ χd χb . (5.75)

The terms linear in the gauge field cancel by using the anti-commuting nature of the χ’s and the Jacobi identity satisfied by the structure constants: − 12 g2 fabe fbcd Aeµ χc (x)χd (x) + g2 fabc fcde Aeµ χd χb   ace cbd cde f + fabc − fadc fcbe} Aeµ χb χd . = 12 g2 −f | {zf 0

(5.76)

The terms with the derivative ∂µ read

− 21 gfacd ∂µ χc χd − gfabc ∂µ χc χb   = 21 g fabc ∂µ (χc χb ) −(∂µ χc )χb + (∂µ χb )χc = 0 . {z } | −(∂µ χc )χb −χc (∂µ χb ) =−∂µ (χc χb )

(5.77)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The double transformation of the ghost field also vanishes Q2BRST χa =

g2 4


abc bde f + facb fbde} χc χd χe |f {z

(5.78)

0

Therefore, the prescription (5.74) for the BRST transformation of a ghost field leads to Q2BRST ψ = 0 ,

Q2BRST Aa µ =0 ,

Q2BRST χa = 0 .

(5.79)

We need now to specify the BRST transformation of the antighost field. Note that in the path integral that gives the Fadeev-Popov determinant, the ghost and antighost fields are treated as independent; therefore the BRST transformation of the antighost does not have to be related to that of the ghost. Let us denote: QBRST χa (x) ≡ Ba (x) ,

(5.80)

where Ba (x) is a commuting field. For QBRST to be nilpotent, we must have in addition: QBRST Ba (x) = 0 .

(5.81)

(And of course Q2BRST Ba (x) = 0.) Consider now a local function Ξ of all the fields (including Ba ), and add its BRST variation to the Yang-Mills and Dirac Lagrangians: L ≡ LYM + LD + QBRST Ξ . | {z }

(5.82)

BRST-invariant

Since QBRST is nilpotent, this Lagrangian is BRST-invariant. Let us choose Ξ ≡ χa (x)

h1 i Ba (x) + Ga (A(x)) , 2ξ

(5.83)

where ξ is a parameter and Ga (A) is the gauge fixing function. We can write10 QBRST Ξ

= =

i h1
∂Ga
i
h 1 a b B + Ga − χa QBRST Ba + Q A µ BRST 2ξ 2ξ ∂Ab µ
1 a a ∂Ga (5.84) − Dadj B B + Ba Ga + χa µ bc χc . b 2ξ ∂Aµ {z } | QBRST χa

LFPG

10 Note

that a minus sign arises when moving QBRST through the anti-commuting field χb .

5. Q UANTIZATION OF YANG -M ILLS THEORY

207

Note that the last term is nothing but the Fadeev-Popov part of the Lagrangian we have derived earlier in this chapter. Moreover, the field Ba enters only quadratically in this Lagrangian. Therefore, the path integral on Ba can be performed trivially11 , Z
ξR 4  a  i R d4 x 1 Ba Ba +Ba Ga a a 2ξ DB (x) e = e−i 2 d x G G . (5.85)

Therefore, after integrating out the auxiliary field Ba , the resulting theory has exactly the same effective Lagrangian as the one resulting from the Fadeev-Popov procedure: Leff = LYM + LD −


ξ a a ∂Ga G G + χa − Dadj µ bc χc . b 2 ∂Aµ

(5.86)

The formal construction we have followed in this section proves that Leff is BRST invariant, but in a somewhat obfuscated manner after the auxiliary field Ba has been integrated out. The BRST invariance of eq. (5.86) is realized if we define the BRST variation of the antighost field as follows QBRST χa = −ξ Ga ,

(5.87)

which is reminiscent of the relationship between Ba and Ga when we do the Gaussian integration on Ba . c sileG siocnarF

5.6.3

BRST current and charge

The Lagrangian (5.82), with the choice (5.83) for the function Ξ, possesses the following symmetries: • Global gauge invariance (because all the colour indices are contracted). • BRST invariance. • Ghost number conservation, if we assign a ghost number +1 to χ’s and −1 to χ’s. The BRST invariance implies the existence of a conserved current: Jµ ≡ BRST

X

Φ∈{Aµ ,ψ,χ,χ,B}


∂L
QBRST Φ . ∂ ∂µ Φ

From the 0-th component of this current, we may obtain the BRST charge Z QBRST ≡ d3 x J0BRST (x0 , x) .

(5.88)

(5.89)

11 Note that this is equivalent to evaluating the argument of the exponential at the stationary point Ba = −ξ Ga , since the stationary phase approximation is exact for Gaussian integrals.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

In fact, this charge generates the BRST transformation in the following sense:   (5.90) i QBRST , Φ ± = QBRST Φ (Φ ∈ {Aµ , ψ, χ, χ, B}) ,

where [·, ·]± is a commutator if Φ is a commuting field and an anti-commutator if Φ is anti-commuting. If we consider free fields (i.e. we set g = 0), and we Fourier decompose all the fields that appear in the (anti)-commutation relations (5.90), Aµ a (x)

=

ψ(x) ≡ χa (x) ≡ χa (x) ≡

Z

 d3 p † +ip·x −ip·x ǫµ + ǫµ∗ λ (p) aaλp e λ (p) aaλp e 3 (2π) 2|p| λ=1,2,+,− Z

 X d3 p † +ip·x +ip·x d v (p)e + b u (p)e s sp s sp (2π)3 2Ep s=± Z 

d3 p † +ip·x +ip·x e + α e α sp ap (2π)3 2|p| Z

 d3 p † +ip·x +ip·x β e + β e , (5.91) sp ap (2π)3 2|p| X

we obtain 

 QBRST , a†aλp ∝ δλ+ α†ap ,  QBRST , αap = 0 ,  QBRST , βap ∝ a†a−p ,     QBRST , b†sp = QBRST , d†sp = 0 .

5.6.4

(5.92)

BRST cohomology, Physical states and Unitarity

The fact that the BRST charge is nilpotent, Q2BRST = 0, has profound implications on the states of the system. The kernel of QBRST is the set of states annihilated by QBRST ,  
(5.93) Ker QBRST ≡ ψ QBRST ψ = 0 .

The set of states that can be obtained by the action of QBRST on another state is called the image of QBRST , 


Im QBRST ≡ QBRST ψ . (5.94) Because QBRST is nilpotent, the image is a subset of the kernel,

Im QBRST ⊂ Ker QBRST .

(5.95)

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5. Q UANTIZATION OF YANG -M ILLS THEORY

Note that states in the image cannot be physical states, because they have a null norm: 

 ψ ψ = φ QBRST QBRST φ = 0 . | {z }

(5.96)

0

Consider now the following equivalence relationship between states in the kernel: two states are considered equivalent if their difference is in the image,  ′  
ψ ∼ ψ if ψ − ψ ′ ∈ Im QBRST . (5.97)

The cohomology of QBRST is the set of classes of equivalent states,


H QBRST ≡ Ker QBRST / Im QBRST .

(5.98)

It turns out that the physical states are the elements of the cohomology  with non-zero norm12 . Indeed, using eqs. (5.92), it is easy to prove that if ψ is a state in the cohomology, then 
a†a{1,2}p ψ ∈ H QBRST 
b†sp ψ ∈ H QBRST 
(5.99) d†sp ψ ∈ H QBRST ,

while

 a†a±p ψ  α†p ψ  β†p ψ




6∈

H QBRST

6∈


H QBRST .

6∈

H QBRST




(5.100)

In other words, adding to the state a physical particle (gluon with a physical polarization, or quark or antiquark) gives another state in the cohomology, while adding to the state a nonphysical quantum (gluon with a non-physical polarization, ghost or antighost) takes the state out of the cohomology. c sileG siocnarF

Furthermore, since the effective Lagrangian is BRST invariant, it corresponds to a Hamiltonian H  that commutes with QBRST . Therefore, a state in the kernel (i.e. for which QBRST ψ = 0) stays in the kernel under the time evolution generated by this Hamiltonian. Furthermore, the time evolution preserves the norm, and therefore states in the cohomology stay in the cohomology at all times. Therefore, starting from a physical states, the time evolution cannot produce unphysical objects in the final state. This explains why unphysical modes cancel in the final states sum in the optical theorem, despite the fact that the internal lines of Feynman graphs may propagate all sorts of unphysical excitations. c sileG siocnarF



restriction is necessary, because one of the classes in H QBRST is Im QBRST itself, that we know has only zero-norm states. 12 This

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 6

Renormalization of gauge theories 6.1 Ultraviolet power counting Before studying in more detail the renormalizability of gauge theories, one may assess the plausibility of this renormalizability by calculating the superficial degree of ultraviolet divergence of graphs in such a theory. Furthermore, this will guide us regarding which classes of graphs may contain divergences. For simplicity, we will consider here a pure Yang-Mills theory, without matter fields (keeping fermions would force us to distinguish the fermion propagators from the gluon and ghost propagators in the counting, because they have different behaviours at large momentum, but would not change the final conclusion). Note that the gluon propagator decreases as (momentum)−2 in the ultraviolet, both in covariant and strict axial gauge. This is also the behaviour of the ghost propagator1 . Moreover, the 3-gluon vertex and the gluon-ghost-antighost vertex have the same scaling with momentum. Therefore, we need not distinguish in the ultraviolet power counting the ghosts and the gluons. Thus, let us consider a generic connected graph G with the following list of propagators and vertices: • nE external lines (gluons or ghosts), • nI internal lines (gluons or ghosts), • n3 trivalent vertices (3-gluon or gluon-ghost-antighost), 1 In strict axial gauge, the ghost propagator behaves differently, but the ghosts decouple completely from the gluons.

211

212

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS • n4 four-gluon vertices, • nL loops.

These quantities are related by the following identities: nE + 2nI = 3n3 + 4n4

(6.1)

nL = nI − (n3 + n4 ) + 1 .

(6.2)

The first equation states that each vertex must have all its “handles” attached to the endpoint of a propagator, and the second equation counts the number of internal momenta that are not determined by energy momentum conservation. In terms of these parameters, the ultraviolet degree of divergence of this graph (in four space-time dimensions) is ω(G) = 4nL − 2nI + n3 .

(6.3)

Note that each trivalent vertex contains one power of momentum and therefore contribute +1 to this counting. Adding eq. (6.1) and four times eq. (6.2), we obtain ω(G) = 4 − nE ,

(6.4)

that does not depend on any of the internal details of the graph. Moreover, the only functions that have intrinsic ultraviolet divergences are the 2-point, 3-point and 4-point functions, which suggests that Yang-Mills theories may indeed be renormalizable. However, a Yang-Mills theory is not simply the addition of gluon and ghost kinetic terms, 3- and 4-gluon vertices, and a ghost-antighost-gluon vertex: all these terms of the Lagrangian are tightly constrained by gauge symmetry. For instance (but this is not the only constraint), all the vertices depend on a unique coupling constant g. Therefore, in order to establish the renormalizability of Yang-Mills theories, one needs to prove that the structure of the divergences in the above listed functions is such that they can be absorbed into a redefinition of the classical Lagrangian that does not upset these tight constraints (up to a renormalization of the fields).

6.2 Symmetries of the quantum effective action 6.2.1

Linearly realized symmetries

After fixing the gauge with the Fadeev-Popov procedure, we have obtained the following effective Lagrangian: Leff = LYM + LD −


ξ a a ∂Ga G G + χa − Dadj µ bc χc . b 2 ∂Aµ

(6.5)

6. R ENORMALIZATION OF GAUGE THEORIES

213

Although the local gauge invariance of the Yang-Mills Lagrangian is now broken (this was precisely the goal of the gauge fixing procedure), this effective Lagrangian has a number of symmetries. One of them is the BRST symmetry, that we have exhibited in the previous chapter. In addition, Leff has the following symmetries: • Ghost number conservation : the effective Lagrangian is invariant under global phase transformations of the ghost and antighost, χ → eiα χ ,

χ → e−iα χ .

(6.6)

Therefore, if we assign a ghost number +1 to the field χ and −1 to the field χ, this quantity is conserved by the Feynman rules of the gauge fixed theory. • Global gauge invariance : since all colour indices are contracted in the effective Lagrangian, it is invariant under gauge transformations that do not depend on spacetime. • Lorentz invariance is of course also present in the effective Lagrangian. For these three symmetries, the infinitesimal variation of the fields is linear in the fields (which is not the case of the BRST symmetry). These linearly realized symmetries of the classical action are inherited directly by the quantum effective action. In order to prove this assertion, let us consider a generic infinitesimal linear transformation of the fields φn (x)

→

φn (x) + ε Fn [x; φ] ,

(6.7)

where φ1 , φ2 , · · · denote the various fields of the theory (gauge fields, ghosts, ...) and Fn [x; φ] is a local function of the fields (for now, we do not assume that it is linear in the fields). We assume that both the classical action and the functional measure are invariant under this symmetry. Consider now the generating functional Z[j], Z
R 4   (6.8) Z[j] ≡ Dφn (x) ei S[φn ]+ d x jn (x)φn (x) ,

where there is one external source jn for each field φn . Since φn (x) is a dummy integration variable in this path integral, we should obtain the same result after performing the change of variable (6.7). Using the fact that this transformation preserves the measure and the classical action, this implies that Z[j]

= ≈


R 4 R 4  Dφn (x) ei S[φn ]+ d x jn (x)φn (x)+ε d x jn (x)Fn [x;φ] Z Z   i S[φn ]+R d4 x jn (x)φn (x)
d4 x jn (x)Fn [x; φ] . Z[j] + iε Dφn (x) e Z



(6.9)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Therefore, for any sources jn , we must have Z

 d4 x jn (x) Fn [x; φ(x)] j = 0 ,

(6.10)

 where · · · j denotes the quantum average in the presence of an external source j, Z
R 4  

 1 Dφn (x) ei S[φn ]+ d x jn (x)φn (x) O[φ] . (6.11) O[φ] j ≡ Z[j]

 (We have normalized it so that 1 j = 1.) Recall now that the sources and field can be related implicitly by using the quantum effective action: jn;φ (x) = −

δΓ [φ] . δφn (x)

(6.12)

Therefore, the condition (6.10) is equivalent to Z

 δΓ [φ] =0, d4 x Fn [x; φ(x)] j φ δφ (x) n

(6.13)

now satisfied for any fields φn . This is known as the Slavnov-Taylor identity. In other words, the functional Γ [φ] is invariant under the transformation

 φn (x) → φn (x) + ε Fn [x; φ] j . (6.14) φ

It is crucial to note that, because the quantum average in the right hand side is performed with the external source jn;φ that depends implicitly on the fields φn , this is a priori not the same transformation as in eq. (6.7). Let us now consider the special case of a transformation of type (6.7) which is linear in the fields. In this case, we may write Z Fn [x; φ] = d4 y fnm (x, y) φm (y) . (6.15) c sileG siocnarF

(In most practical cases, the transformation will be local and the coefficients proportional to δ(x − y), but this restriction is not necessary for the following argument.) For such a linear transformation, we have Z



 (6.16) Fn [x; φ] j = d4 y fnm (x, y) φm (y) j . φ

φ

Recalling

 that jφ is the configuration of the source j such that the quantum average φ(x) j precisely equals φ(x), this in fact reads

 Fn [x; φ] j = Fn [x; φ] . (6.17) φ

It is this last step that fails when Fn is nonlinear in the fields. From eq. (6.17), we see that the transformations (6.14) and (6.7) are identical. We have thus proven that all linearly realized symmetries of the classical action are also symmetries of the quantum effective action.

215

6. R ENORMALIZATION OF GAUGE THEORIES

6.2.2

BRST symmetry and Zinn-Justin equation

Since an infinitesimal BRST variation is not linear in the fields, the BRST symmetry of the classical action is not inherited so simply by the quantum effective action. Instead, it leads to a set of identities that may be viewed as the analogue of Ward identities for the BRST invariance. Their derivation follows the method of the section 3.4.2. Since we need to apply a BRST transformation to the Yang-Mills  path integral,  we should first study how this transformation affects the measure DAµ DχDχ . Under such a transformation, the fields transform into Aa µ χa χa



abc c a adj a → Aa′ Aµ χb µ ≡ Aµ + ϑ Dµ ab χb = Aµ + ϑ ∂µ δab + gf ϑ → χ′a ≡ χa − g fabc χb χc 2 → χ′a ≡ χa + ϑ Ba = χa − ξ ϑ Ga , (6.18)

where ϑ is a Grassmann constant. The Jacobian matrix has the following block structure: ∂ ∂




′ ′ Aa′ µ , χa , χa
Ab ν , χb , χb

 ν δµ (δab − gϑfabc χc ) ∗  0 δab + gϑfabc χc = δ(x−y)  a 0 −ξϑ ∂G ∂Ab ν

 0  0  ,

δab

(6.19)

where the ∗ denotes a non-zero element that we do not need to calculate because it does not contribute to the determinant. From this structure, we see that the determinant is given by the product of the diagonal elements, and is therefore equal to 1 (recall that ϑ2 = 0). In the derivation, it is convenient to introduce sources ja µ , ηa , ηa that couple a respectively to Aa µ , χa , χa , but also two extra sources that couple directly to QBRST Aµ and QBRST χa : Z[j, η, η; ζ, κ] ≡

=

Z

Z   µ DAµ DχDχ exp i d4 x Leff + ja µ Aa + ηa χa + χa ηa

 a +ζµ a QBRST Aµ − κa QBRST χa Z

Z    DAµ DχDχ exp i d4 x Ltot , (6.20) 

where we use the shorthand Ltot for the sum of terms inside the exponential. Note that the coefficients of the new sources ζµ a and κa are BRST invariant since the BRST transformation is nilpotent. Let us now perform a BRST transformation of the

216

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

integration variables inside the path integral. This is just a change of variables, that does not change the value of the path integral. Using the fact that the measure and the Lagrangian Leff are BRST invariant, we obtain

Z[j, η, η; ζ, κ] =

=

Z

Z   DAµ DχDχ exp i d4 x Ltot Z h 
µ × 1 + i d4 x ja µ ϑ QBRST Aa 


i
+ηa ϑ QBRST χa + ϑ QBRST χa ηa Z  δZ δZ + ηa (x) Z[j, η, η; ζ, κ] + i ϑ d4 x ja µ (x) a iδζµ (x) iδκa (x)    δZ ηa (x) . (6.21) −ξ Ga iδj(x)

(Note that ϑ anticommutes with ηa .) Therefore, we conclude that Z

 d4 x ja µ (x)

δZ δZ − ξ Ga + ηa (x) a iδζµ (x) iδκa (x)



δZ iδj(x)



 ηa (x) = 0 . (6.22)

This is one of the forms of the conservation identities. In this derivation, we see that having introduced sources specifically coupled to the BRST variation of the gauge field Aa µ and of the ghost χa avoided the need for terms with higher order derivatives (indeed, these variations are non-linear in the fields, and would have required more derivatives to be expressed as functional derivatives with respect to sources coupled to elementary fields). By writing Z = exp(W), we see that the same identity applies to W, Z

    δW δW δW a − ξ G + η (x) = 0 . (6.23) d4 x ja (x) η (x) a a µ iδζa iδκa (x) iδj(x) µ (x)

(Here, we have assumed that the gauge fixing function is linear in the gauge field.) The next step is to convert this into an identity for the quantum effective action Γ that generates the 1PI graphs. In this transformation, we will keep the auxiliary sources ζa µ and κa unmodified, as parameters. Thus, Γ and W are related by −i W[j, η, η; ζ, κ] = Γ [A, χ, χ; ζ, κ] Z   a a a + d4 x jµ a (x)Aµ (x) + χa (x)η (x) + η (x)χa (x) .

(6.24)

6. R ENORMALIZATION OF GAUGE THEORIES

217

Fields and sources are related by the following quantum equations of motion: δΓ + jµ a (x) = 0 , δAa µ (x) δΓ + ηa (x) = 0 , δχa (x) δΓ + ηa (x) = 0 , δχa (x)

(6.25)

and we also have δW = i Aa µ (x) , δjµ a (x) δΓ δW =i a , δζa δζ (x) µ µ (x) δΓ δW =i a . δκa (x) δκ (x)

(6.26)

Therefore, the conservation identity expressed in terms of the functional Γ reads Z  δΓ δΓ  δΓ δΓ δΓ a (A) d4 x = 0 . (6.27) − ξ G + a δAa δχa (x) δκa (x) δχa (x) µ (x) δζµ (x) This equation can be simplified a bit as follows. By inserting a derivative δ/δχa (x) under the integral in the definition (6.21) of Z, we obtain zero since we now have the integral of a total derivative. Recalling that the Fadeev-Popov term in the effective Lagrangian is LFPG = χa


∂Ga − Dadj µ bc χc , b ∂Aµ

(6.28)

we can perform explicitly this derivative to obtain Z i R 4  h ∂Ga
 adj +η (x) ei d x Ltot . (6.29) − D χ (x) 0 = DAµ DχDχ a µ bc c ∂Ab {z } µ | −Q

BRST

= i

Ab µ (x)

δ δζµ b (x)

This implies the following functional identity h

ηa (x) + i

∂Ga δ i Z=0, µ ∂Ab µ δζb (x)

(6.30)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

or equivalent identities for W or Γ : ηa (x) + i

∂Ga δΓ δΓ + =0. b δχa (x) ∂Aµ δζµ b (x)

∂Ga δW =0 , µ ∂Ab µ δζb (x)

Furthermore, define a slightly modified effective action: Z ξ Γ ≡Γ+ d4 x Ga (A)Ga (A) . 2

(6.31)

(6.32)

Now the BRST conservation identity takes the following more compact form, known as the Zinn-Justin equation: Z

d4 x



δΓ  δΓ δΓ δΓ =0, + µ δAa δχa (x) δκa (x) µ (x) δζa (x)

(6.33)

from which any explicit reference to the gauge fixing function Ga (A) has disappeared, as well as the coupling constant g. Eq. (6.33) applies to the full quantum effective action, that encapsulates the results from all-order perturbation theory. In the next section, we will show that this identity (combined with the other symmetries of the effective action) completely constrains the structure of its local terms of dimension less than or equal to four, forcing them to be identical to those in the classical action (up to a rescaling of the fields and of the coupling constant). c sileG siocnarF

6.3 Renormalizability 6.3.1

Constraints on the counterterms

By taking the h ¯ → 0 limit in eq. (6.33), one immediately concludes that it is also satisfied by the classical action, S, supplemented with ghosts as well as the sources ζµa and κa : S[A, χ, χ; ζ, κ] =

Z

h d4 x −

1 4


adj
µν a µ Fa Fµν + ζµ a + ∂ χa Dµ ab χb i + g2 fabc κa χb χc . (6.34)

By introducing the following compact notation, Z  δA
δA δB  δB A, B ≡ d4 x , + µ δAa δχa (x) δκa (x) µ (x) δζa (x)

(6.35)

6. R ENORMALIZATION OF GAUGE THEORIES

219

we therefore have
S, S = 0 ,
Γ, Γ = 0 .

(6.36)

The first equation may be viewed as a constraint on the terms that can appear in the classical action, while the second equation constrains which divergences may appear in higher orders. Let us now write the effective action as a loop expansion, Γ ≡S+

∞ X

Γl ,

(6.37)

l=1

where S is given in eq. (6.34), and the subsequent terms Γ l are of order l in h. ¯ The Zinn-Justin equation at order L thus reads X
Γ p, Γ q = 0 . (6.38) p+q=L

The renormalization procedure amounts to correcting order by order with counterterms the classical action S, S

→

S(L) ,

(6.39)

such that S(L) contains counterterms up to order L, and gives finite Γ l ’s for l ≤ L (but in general not beyond the order L). The first step
is to prove that it is possible to find counterterms such that the equation S, S = 0 is preserved at every order. Let us assume that we have achieved this up to the order L − 1. All Γ l for l ≤ L − 1 are now finite, while Γ L still contains a divergent part, that we denote Γ L,div . We can rewrite the Zinn-Justin equation at order L as follows, L−1 X


S, Γ L + Γ L , S = − Γ l , Γ L−l .

(6.40)

l=1

Only the left hand side contains divergences, and we therefore have

S, Γ L,div + Γ L,div , S = 0 ,

(6.41)

which constrains the structure of the divergences at order L. A natural candidate for the counterterm at order L is to simply add −Γ L,div to the classical action, S

→

S − Γ L,div ,

(6.42)

220

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

since this automatically cancels the superficial divergence of Γ L without affecting anything in the lower orders. However, this modified classical action does not obey exactly the Zinn-Justin equation, since h



i
S−Γ L,div , S−Γ L,div = S, S − S, Γ L,div + Γ L,div , S + Γ L,div , Γ L,div . | {z } | {z } | {z } 0 from lower orders

6=0

0 from eq. (6.41)

(6.43)

Note that the non-zero term in the right hand side is of order strictly greater than L. It is possible to make it vanish by adding to the shift of eq. (6.42) some terms of higher order than L, that do not change anything for any order ≤ L. The conclusion of this inductive argument is that one can shift the classical action at each order
in such a way that the divergences in Γ are canceled, while always preserving S, S = 0.

6.3.2

Allowed terms in the classical action

The second step in the discussion of the renormalization of Yang-Mills theory is to determine the terms
that are allowed in the classical action. This action must satisfy the constraint S, S = 0, as well as Lorentz invariance, global gauge symmetry and ghost number conservation. In addition, from the power counting of the section 6.1 and Weinberg’s theorem, we know that all the ultraviolet divergences in Yang-Mills theory will occur in local operators of dimension 4 at most. In order to discuss the form of the allowed terms, let us first list the mass dimension and ghost number of the various fields that enter in S: field mass dimension ghost number

Aµ a 1 0

χa 1 +1

χa 1 -1

ζµ a 2 -1

κa 2 -2

All the allowed terms in S must obey the following conditions: • mass dimension 4 or less, • ghost number 0, • Lorentz invariance, • global gauge invariance. c sileG siocnarF

In addition, eq. (6.31) implies that the χ and ζ dependences come in the form of a dependence on the combination ζµ − χ

∂G = ζµ + ∂ µ χ , ∂Aµ

(6.44)

6. R ENORMALIZATION OF GAUGE THEORIES

221

where in the right hand side we have assumed the covariant gauge condition G(A) = ∂µ Aµ
and anticipated an integration by parts. Finally, the Zinn-Justin equation S, S = 0 must be satisfied. Since the sources ζµ a and κa have mass dimension 2, at most two of them may appear. However, terms with two such sources cannot contain any other field since the mass dimension 4 is already reached, and they cannot have ghost number zero. Therefore, S can only contain terms that have degree 0 or 1 in ζµ a and κa . The source ζµ must be combined with another combination of fields that have a one Lorentz index, one colour index, mass dimension at most 2, and ghost number +1. The only operators that fulfill these conditions are b fabc ζµ a Aµ χc

and

ζµ a ∂µ χa .

(6.45)

ζµ a

Once the dependence on is fixed, the dependence on the antighosts will be completely known from eq. (6.44). Likewise, κa must be combined with an object that has one colour index, mass dimension at most 2 and ghost number +2. The only possibility is fabc κa χb χc .

(6.46)

From the information gathered so far, the classical action must have the following general form: Z h
b µ S[A, χ, χ; ζ, κ] = Σ[A] + d4 x gα fabc ζµ a + ∂ χa Aµ χc i
γ abc µ + ∂ ∂ χ + +β ζµ χ f κ χ χ , (6.47) µ a a b c a a 2

where α, β, γ are three arbitrary constants. The term Σ cannot depend on the sources ζµ a and κa because we have already constructed explicitly all the allowed terms that contain these sources, and cannot depend on χ because the antighost dependence is µ already encapsulated in the combination ζµ a + ∂ χa . A dependence on χ in Σ is also forbidden because χ would be the only field in Σ with a non-zero ghost number. Our next step is to constrain the coefficients gα, β, γ and the functional Σ[A] in order to satisfy the Zinn-Justin equation (6.33). The functional derivatives that enter in (6.33) are given by: δS δAµ a δS δζµa δS δχa δS δκa

=

δΣ − gα fabc (ζµb + ∂µ χb ) χc , δAµ a

=

µ gα fade Aµ d χe + β ∂ χa ,

=

µ abc κb χc , gα fabc (ζµb + ∂µ χb ) Aµ c + β (ζµa + ∂µ χa ) ∂ + γ f

=

γ ade f χd χe . 2

(6.48)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Thus, the Zinn-Justin equation reads 0 =

Z

d4 x

h δΣ   ade µ Ad χe + β ∂µ χa µ gα f δAa 
µ +(ζµb + ∂µ χb ) − gα fabc χc gα fade Aµ d χe + β ∂ χa
 µ γ ade + gα fabc Aµ χd χe c + β δab ∂ 2 f i γ2 abc ade f f κb χc χd χe . (6.49) + 2

Using the Jacobi identity satisfied by the structure constants, one may first check that the last term, in κχχχ, is identically zero, and therefore does not provide any constraint. Consider now the terms in ζAχχ:  gα ζµb − gα fabc fade Aµ d χc χe + abc ade

= gα (γ − gα) f

f

γ 2

µ abc ade |f {zf } Ac χd χe

−fabd faec −fabe facd ζµb Aµ d χc χe .

 (6.50)

Since this is the only term containing this combination of fields, it cannot be canceled by other terms, and therefore we must have gα = γ .

(6.51)

Let us now study the terms in ζχ(∂χ),   γ β ζµb −gα fabc χc ∂µ χa + fbde ∂µ (χd χe ) = β (γ−gα) fbac ζµb (∂µ χa ) χc . 2 (6.52) Thus, the cancellation of this term does not bring any additional constraint beyond eq. (6.51). At this point, all the terms containing ζµ a have been canceled (and by extension also the terms with ∂µ χa ), and the Zinn-Justin equation reduces to Z  δΣ  µ gα facb Aµ (6.53) 0 = d4 x c χb + β ∂ χa . δAµ a Let us first rewrite the second factor as follows
β ∂µ δab − igαβ−1 (−ifcab )Aµ χ , c | {z } b

(6.54)

(Aµ adj )ab

and note that it has the structure of an adjoint covariant derivative acting on χb , µ
Dadj ab ≡ ∂µ δab − igαβ−1 (Aµ (6.55) adj )ab .

223

6. R ENORMALIZATION OF GAUGE THEORIES Thus, eq. (6.53) is equivalent to Z δΣ µ
0 = d4 x µ Dadj ab χb . δAa

(6.56)

The second factor may be viewed as the variation of the gauge field under an infinitesimal gauge transformation, µ
µ Aµ (6.57) a → Aa + ϑ Dadj ab χb ,

where we have introduced a constant Grassmann variable ϑ to make the second term a commuting object. Therefore, for the integral to be zero for an arbitrary χb (x), the functional Σ[A] must be invariant under this transformation. Recalling our discussion of the local gauge invariant operators of mass dimension four or less, we conclude that the only possible form for Σ is Z δ µν a Σ[A] = − d4 x Fa Fµν , (6.58) 4 µν

µ

where F is the field strength constructed with the covariant derivative D and δ another constant. Given all the above constraints, we must have S[A, χ, χ; ζ, κ] =

Z

h d4 x −

δ 4


adj
µν a µ Fa Fµν + β ζµ a + ∂ χa Dµ ab χb i abc f κ χ χ + gα . (6.59) a b c 2

Up to rescalings of the various fields and of the coupling constant g, this is structurally identical to the bare classical action of eq. (6.34). Note that this equation implies that the field renormalization factors for the gauge field Aµ a and for the source κa are equal, ZA = Zκ . c sileG siocnarF

6.4 Background field method 6.4.1

Rescaled fields

In this section, we describe the calculation of the one-loop quantum corrections to the coupling constant by a method based on the quantum effective action combined with the so-called background field method. The first step of this method is to rescale the gauge field by the inverse of the coupling constant: g Aµ

→

Aµ .

(6.60)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

By doing this, the various objects that appear in the Yang-Mills action are transformed as follows: Fµν Dµ

→

→


1 µ ν ∂ A − ∂ν Aµ − i [Aµ , Aν ] g ∂µ − i Aµ .

(6.61)

In other words, up to a rescaling in the case of the field strength Fµν , these objects are transformed into their counterparts for a coupling equal to unity. In the rest of this section, the notation Aµ , Dµ , Fµν will refer to the rescaled quantities. In terms of the rescaled fields, the Yang-Mills action simply reads Z 1 a (6.62) SYM = − 2 d4 x Fµν a Fµν , 4g | {z } no g

where all the dependence on the coupling constant appears now in the prefactor g−2 . This action has a local non-Abelian gauge invariance analogous to the original one, but with g = 1: Aµ

6.4.2

→

† † AΩ µ ≡ Ω Aµ Ω + i Ω ∂µ Ω .

(6.63)

Background field gauge

The background field method consists in choosing a background field Aa µ (x), and in writing the gauge field Aa (x) as a deviation around this background µ a a Aa µ ≡ Aµ + aµ .

(6.64)

In this decomposition, the background field Aµ is not a dynamical field: it will just act as a parameter that we shall not quantize, and the path integration is thus only on the deviation aa µ (one may thus view this as a shift of the integration variable). In a terms of Aa µ and aµ , the field strength that enters in the Yang-Mills action can be written as

Fµν = Fµν + ∂µ aν − i [Aµ , aν ] − ∂ν aµ − i [Aν , aµ ] − i [aµ , aν ] , (6.65)

where Fµν is the field strength constructed with the background field. With explicit colour indices, this reads
µ
µ ν ν µν abc µ ν Fµν ab ac , (6.66) a = Fa + Dadj ab ab − Dadj ab ab + f  µ µ where Dµ adj = ∂ − i A , ·] is the adjoint covariant derivative associated to the background field Aµ . If we view the background field as a constant, the original gauge transformation on Aµ corresponds to the following transformation on aµ , aµ

→

Ω† aµ Ω + Ω† Aµ Ω − Aµ + i Ω† ∂µ Ω .

(6.67)

225

6. R ENORMALIZATION OF GAUGE THEORIES

If we parameterize Ω = exp(iθa ta ) and expand to first order in θa , an infinitesimal gauge transformation of aµ a reads µ
abc aµ → aµ θ b aµ (6.68) c . a a − Dadj ab θb + f This invariance leads to the same pathologies as in the original theory, and we must fix the gauge in order to have a well defined path integral. The background field gauge corresponds to the following condition on aa µ,
b (6.69) Ga (A) ≡ Dµ adj ab aµ = ωa . Let us recall that a gauge fixing function Ga (A) leads to the following terms in the effective Lagrangian: ξ Ga (A)Ga (A) 2 g2
∂Ga = − χa Dadj µ bc χc b ∂Aµ

LGF = −

(gauge fixing term)

LFPG

(Fadeev-Popov ghosts) .

With the choice of eq. (6.69), the Fadeev-Popov term becomes



µ adj adj LFPG = −χa Dµ adj ab Dµ bc χc = Dadj χ a Dµ χ a ,

(6.70)

(6.71)

where in the second equality we have anticipated an integration by parts and used the adj µ notation (Dadj µ χ)a ≡ (Dµ )ab χb (and a similar notation for (Dadj χ)a ).

6.4.3

Residual symmetry of the gauge fixed Lagrangian

The effective Lagrangian LYM + LGF + LFPG possesses a residual gauge symmetry that corresponds to gauge transforming in the same way the background field Aµ and the total field Aµ , Aµ Aµ

→ Ω† Aµ Ω + i Ω† ∂µ Ω , → Ω† Aµ Ω + i Ω† ∂µ Ω .

(6.72)

Indeed, under this joint transformation we have aµ Dµ Dµ

→ Ω† aµ Ω , → Ω† Dµ Ω ,

→ Ω† Dµ Ω ,

χ → Ω† χ , χ → χΩ , G(A) → Ω† G(A) Ω .

(6.73)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

From this, we conclude that the gauge fixing Lagrangian LGF and the Fadeev-Popov Lagrangian LFPG are both invariant in this transformation, as well as the Yang-Mills Lagrangian. Since the path integration measure over aµ , χ, χ is also invariant under this transformation, the result of the path integral must be invariant under local gauge transformations of the background field Aµ .

6.4.4

One-loop running coupling

Let us now turn to the calculation of the quantum effective action at one-loop. For this, we use the results of the section 2.6.5, where we have shown that these oneloop corrections are obtained by expanding the classical action to quadratic order in deviations with respect to a background field, and by performing the resulting Gaussian path integration with respect to the deviations (which gives a functional determinant). The first step is to expand the three terms of the gauge fixed Lagrangian to second order in the deviation aµ . In this calculation, we choose the gauge fixing parameter ξ = 1. The quadratic terms in the combined Yang-Mills and gauge fixing terms read LYM + LGF

=

= =

−


2 1 1 ν ν µ (Dµ adj a )a −(Dadj a )a 2 2 2g


 µ c a 2 +fabc Faµν ab µ aν + (Dadj aµ ) i  1 h 2 µν − 2fabc Fbµν acν − 2 aa µ − Dadj )ac g 2g i  (1) 1 h ρσ 2 µν µν − D ) g + (F ) (M ) acν , − 2 aa adj ac ac ρσ µ adj 2g (6.74) (1)

where we have introduced (Mρσ )µν ≡ i(δρ µ δσ ν − δρ ν δσ µ ) the generators of the Lorentz transformations for 4-vectors (the Lorentz transformation corresponding to (1) the transformation parameters ωρσ reads Λµν = exp( 2i ωρσ (Mρσ )µν )). For the ghost term, the quadratic part is h
2 i LFPG = χa − Dadj ab χb . (6.75)

Note that the operator that appears between the two ghost fields is the spin-0 analogue of the one that appears in eq. (6.74), since the generators of Lorentz transformations (0) for spin-0 objects are identically zero (Mρσ ≡ 0). Although we have not considered fermions so far in this chapter, the Dirac Lagrangian would give a contribution equal / or equivalently the square root of the determinant of (iD) / 2. to the determinant of iD,

227

6. R ENORMALIZATION OF GAUGE THEORIES Noting that / iD


2


[γµ , γν ] Dµ Dν

=

−D2 + i

=

−D2 + (Fρσ ) Mρσ

i 2

(1/2)

,

(6.76)

(1/2)

where the Mρσ ≡ 4i [γρ , γσ ] are the generators of Lorentz transformations for spin-1/2 fields. Note that the covariant derivatives and the field strength are in the fundamental representation (assuming fermions that transform according to the fundamental representation, like quarks). Therefore, for each of the fields that appear in the quantum effective action (gauge fields, ghosts, fermions), we get a determinant ∆r,s of an operator containing −D2 (in the representation r corresponding to the field under consideration) plus a “spin connection”2 made of the contraction of the field strength with the Lorentz generators corresponding to the spin s of the field:   (1) ρσ gauge fields : ∆adj,s=1 ≡ det − D2adj + Fadj Mρσ   (0) ρσ ghosts : ∆adj,s=0 ≡ det − D2adj + Fadj Mρσ | {z } =0   (1/2) ρσ 2 . (6.77) fermions : ∆f,s=1/2 ≡ det − Df + Ff Mρσ In terms of these determinants, the 1-loop quantum effective action is given by Γ [A, χ, ψ] = Sr + ∆S +

i i nf ln ∆adj,s=1 − ln ∆f,s=1/2 − i ln ∆adj,s=0 , (6.78) 2 2

where ∆S denotes the 1-loop counterterms, and nf is the number of fermion flavours. Using the invariance with respect to local gauge transformations of the background field, we must have Z i a + ··· , (6.79) ln ∆r,s = Cr,s d4 x Faµν Fµν 4 where the dots represent higher dimensional gauge invariant operators. Being of dimension higher than four, these operators do not contribute to the renormalization of the coupling. The constant Cr,s depends on the group representation r and spin s of the field. These coefficients are ultraviolet divergent, Cr,s = cr,s ln

Λ2 , κ2

(6.80)

where Λ is an ultraviolet scale and κ the typical scale of inhomogeneities of the background field. After combining them with the counterterms from ∆S, the ultraviolet 2 This terms describes the coupling between the magnetic moment of the particle and the background field. Its detailed form depends on the spin of the particle.

228

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

scale is replaced by a renormalization scale µ, Cr,s

→

Cr,s = cr,s ln

µ2 . κ2

(6.81)

From eq. (6.78), we see that the 1-loop renormalized coupling at the scale µ and the bare coupling must be related by 1 g2b

= =

nf 1 1 Cf,1/2 − Cadj,0 + Cadj,1 − g2r (µ) 2 2 1  µ2 1 nf c − c − c + ln 2 . adj,1 adj,0 f,1/2 g2r (µ) 2 2 κ

(6.82)

The explicit calculation of the constants cr,s requires to expand the logarithm of the functional determinants to second order in the background field strength Fµν . Thanks to the organization of eqs. (6.77), this calculation needs to be performed only once, for generic gauge group and Lorentz representations. This leads to cr,s =

i 1 h1 d(s) − 4C(s) N(r) , 3 2 (4π)

(6.83)

where d(s) is the number of spin components (respectively 1, 4, 4 for scalars, fermions, and vector particles), C(s) is the normalization of the trace of two Lorentz generators3 ,
(s) (s)
tr Mρσ Mαβ = C(s) (gρα gσβ − gρβ gσα ,

(6.84)

and N(r) is the normalization of the trace of two generators of the Lie algebra in representation r,
b tr ta = N(r) δab . (6.85) r tr For the fundamental and adjoint representations of su(N), we have N(f) = 12 and N(adj) = N. Therefore, the constants involved in the 1-loop running coupling are cadj,0 =

N , 3(4π)2

cadj,1 = −

20 N 3(4π)2

cf,1/2 = −

4 , 3(4π)2

(6.86)

and the coupling evolves according to 1 1  11 1 = + N− g2r (µ) (4π)2 |3 {z g2b

2 3

>0 for nf ≤

3 For

spin-0,

1 2

nf }

11 N 2



ln

µ2 . κ2

and 1, this constant is respectively 0, 1 and 2.

(6.87)

229

6. R ENORMALIZATION OF GAUGE THEORIES

Given two scales µ and µ0 , the renormalized couplings at these scales are related by 1 1  11 1 − = N− g2r (µ) g2r (µ0 ) (4π)2 3

2 3

 µ2 nf ln 2 , µ0

(6.88)

which may be rewritten as g2 (µ0 )

g2 (µ) = 1+

g2 r (µ0 ) (4π)2

11 3

N−

2 3

.
2 nf ln µ 2 µ

(6.89)

0

In quantum chromodynamics, where the gauge group is SU(3) (i.e. N = 3) and where there are 6 flavours of quarks in the fundamental representation, the coefficient in front of the logarithm is positive, which indicates that the coupling constant decreases as the scale µ increases. The coupling constant in fact goes to zero when µ → ∞, a property known as asymptotic freedom. Thanks to the formula (6.83), it would have been easy to determine the one-loop running of the coupling in the presence of matter fields in arbitrary representations. c sileG siocnarF

230

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 7

Renormalization group In quantum field theory, the renormalization group refers to a set of tools for investigating the changes of a system when observed at varying distance scales, akin to varying the magnifying power of a microscope in order to uncover new features that were not visible at lesser resolution scales. For renormalizable theories, such a change of scale merely amounts to a change in a few parameters of the theory (masses, coupling, field normalization), but the use of the renormalization group is not limited to this class of theories, as we shall discuss in the last section.

7.1 Callan-Symanzik equations Let us consider a renormalizable quantum field theory, for instance a scalar theory with a φ4 interaction (renormalizable in d ≤ 4 space-time dimensions). For simplicity, assume firstly that this field is massless, and denote by M the scale at which the renormalization conditions are imposed. For instance, these conditions can be chosen as follows: Γ (4) (p1 , p2 , p3 , p4 ) = −iλ , for (p1 +p2 )2 = (p1 +p3 )2 = (p2 +p4 )2 = −M2 , Π(p)|p2 =−M2 = 0 , dΠ(p) =0, dp2 2 2

(7.1)

p =−M

where Π(p) is the self-energy and Γ (4) the 1-particle irreducible 4-point function. There is a large amount of freedom in the choice of the renormalization conditions. Two sets of renormalization conditions may correspond to the same physical theory

231

232

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

provided that the bare Green’s functions, expressed in terms of the bare parameters of the Lagrangian, are identical. Indeed, the renormalization scale M appears only √ when we replace the bare field φb by the renormalized field φr ≡ φb / Z and the bare coupling constant λb by the renormalized coupling constant λr . The bare and renormalized Green’s functions are related by (n)

G(n) (x1 , · · · , xn ) = Z−n/2 Gb (x1 , · · · , xn ) . r

(7.2)

In order to have the same physical theory, we must change Z and λ when varying the renormalization scale M. With such a variation of the scale M, we can write: (n)

(n)

(n)

∂Gr ∂Gr dGr ∂λ = + . dM ∂M ∂λ ∂M

(7.3)

On the other hand, we may obtain this derivative from the right hand side of eq. (7.2) and the fact that bare Green’s functions must remain unchanged: (n)

dGr n ∂Z (n) =− G . dM 2 Z ∂M r Combining the previous two results, we obtain   ∂ ∂ M +β + nγ G(n) =0, r ∂M ∂λ

(7.4)

(7.5)

where we have defined β≡M

∂λ ∂M

,

γ≡

M ∂Z . 2 Z ∂M

(7.6)

Eq. (7.5) is known as the Callan-Symanzik equation, or renormalization group (RG) equation. The quantity β, called the beta function of the theory, controls how the coupling constant varies with the scale M. γ, known as the anomalous dimension of the field φ, controls the rescaling of the field when the scale M is changed. The “group” terminology comes from the following considerations. Formally, the solutions of eq. (7.5) can be written as follows: G(n) (· · · ; M) = U(M, M0 ) G(n) (· · · ; M0 ) , r r

(7.7)

where the evolution operator U(M, M0 ) is a Green’s function of the operator between the square brackets in the right hand side of eq. (7.5). A 1-dimensional group structure can be attached to this evolution by noting that U(M2 , M0 ) = U(M2 , M1 ) U(M1 , M0 ) .

(7.8)

In other words, a finite rescaling can be broken down into several smaller rescalings without affecting the final result.

233

7. R ENORMALIZATION GROUP

7.1.1

One-loop calculation of β and γ

In practice, one can determine the anomalous dimension γ and the β function at oneloop from the wavefunction and vertex counterterms δZ and δλ . Since Z = 1 + δZ , we can directly write M ∂δZ . (7.9) 2 ∂M In order to determine the β function for a 4-legs vertex, one should start from the (4) renormalized 4-point function Gr (p1 , · · · , p4 ). Diagrammatically, this function reads   X  , (7.10)  = + + + + γ=

i

where the first term in the right hand side is the tree-level vertex, the second and third terms are respectively the 1PI vertex correction and the associated counterterm. The fifth and sixth terms are the self-energy corrections on the external lines and the corresponding counterterms. Up to one-loop, this equation can be written as follows: ! Y i h (4) Gr (p1 , · · · , pn ) = − iλb p2i i (4)

+Γb

− iδλ i X 1 (2) (Γb (pi ) − p2i δiZ ) . (7.11) −iλb 2 pi i

In this equation, the first line is the tree-level 4-point function, the second line contains the one-loop 1PI vertex correction and the vertex counterterm (necessary in order to fulfill the renormalization condition for the vertex at the scale M), and the last line is the sum of the 1-loop corrections on the external lines (the counterterms δiZ are determined by the normalization condition of the propagator at the scale M). The dependence of this renormalized Green’s function on the renormalization scale M arises from the counterterms δλ and δiZ . By applying the Callan-Symanzik equation to this Green’s function, we obtain at leading order ! X λ X ∂δiZ ∂ i δλ − λ =0, (7.12) δZ + β + M M ∂M 2 ∂M i

i

where we have replaced the anomalous dimensions γi attached to the external lines by their expression given by eq. (7.9) in terms of the corresponding counterterms δiZ . Therefore, we obtain the following formula for the β function: ! λX i ∂ −δλ + β=M δZ . (7.13) ∂M 2 i

234

7.1.2

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS (2)

Solution for the 2-point function Gr

In a massless theory, we may always parameterize the 2-point function as follows: G(2) r (p) =

i g(−p2 /M2 ) , p2

(7.14)

where g(−p2 /M2 ) is a function so far arbitrary. Since the M dependence arises solely from the ratio −p2 /M2 , we can rewrite the derivative with respect to M in the Callan-Symanzik equation in the form of a derivative with respect to p:   ∂ ∂ p −β + 2 − 2γ G(2) (7.15) r (p) = 0 . ∂p ∂λ In order to solve this equation, let us introduce a function λ(p, λ) defined by: dλ(p, λ) = β(λ) , d ln(p/M)

λ(M, λ) = λ .

(7.16)

In other words, λ is the running coupling constant that takes the value λ at the momentum scale M. We can then write the solution of the Callan-Symanzik equation in the following form:  p  Z ′ dp i  γ(λ(p′ , λ)) , (7.17) G(2) r (p) = 2 G(λ(p, λ)) exp 2 p p′ M

where G(λ(p, λ)) is an arbitrary function that cannot be determined from the renormalization group equations1 . This function must be determined order by order from perturbative calculations. In the case of the 2-point function, we have G(λ(p, λ)) = 1+O(λ). The exponential in eq. (7.17) is the cumulative field renormalization between the scales M and p. In particular, for a constant anomalous dimension, this factor is (p/M)2γ , and we see that it alters the power law dependence of the propagator with respect to momentum, changing a power −2 into −2 + 2γ. c sileG siocnarF

7.2 Correlators containing composite operators 7.2.1

Callan-Symanzik equations

A very useful extension of the previous formalism concerns the case of correlators that contain one of more composite operators, i.e. made of several fields evaluated at 1 An

arbitrary function of the running coupling is allowed as a prefactor, since we have:   ∂ ∂ G(λ(p, λ)) = 0 . −β p ∂p ∂λ

235

7. R ENORMALIZATION GROUP

the same space-time point. Similarly to the case of elementary operators, we must introduce a renormalization factor ZO , determined order by order in perturbation theory in order to fulfill a certain renormalization condition at the scale M. The renormalized operator Or is related to the bare operator Ob by the relationship Or = Ob /ZO . Let us consider now a renormalized correlation function involving a composite operator O and n elementary fields: G(n;1) (x1 , · · · , xn ; y) ≡ hφ(x1 ) · · · φ(xn )O(y)i . r

(7.18)

The corresponding bare correlation function is given by (n;1)

G(n;1) (x1 , · · · , xn ; y) = Z−n/2 Z−1 r O Gb

(x1 , · · · , xn ; y) .

(7.19)

By requesting that the bare correlation function remains unchanged upon changes of the renormalization scale M, we obtain the following equation satisfied by the renormalized correlation function   ∂ ∂ M =0, (7.20) +β + nγ + γO G(n;1) r ∂M ∂λ where we have defined the anomalous dimension of the composite operator O as follows γO ≡

7.2.2

M ∂ZO . ZO ∂M

(7.21)

Anomalous dimension of the operator O

The practical determination of the anomalous dimension γO of a composite operator O made of m elementary fields φ can be done by studying the correlation function (m;1) Gr and by applying to it the Callan-Symanzik equation. This method is identical to the one used in the determination of the expression (7.13) for the beta function, and leads to ! 1X i ∂ −δO + δZ , (7.22) γO = M ∂M 2 i

where δO is the counterterm that one must adjust in order to satisfy the renormalization condition of the operator O at the scale M.

7.2.3

Anomalous dimension of a conserved current

A very useful example in practice is that of a current such as Jµ ≡ ψγµ ψ .

(7.23)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The anomalous dimension of such an operator is given by γJ = M


∂ −δJ + δψ = 0 . ∂M

(7.24)

The equality of the counterterms δJ and δψ is a consequence of the Ward identities, i.e. of the gauge symmetry associated to charge conservation and to the conservation of the current Jµ .

7.2.4

Renormalization of operators of arbitrary dimensions

Let us denote by LM the renormalized Lagrangian at the scale M. Consider now adding to this Lagrangian the following sum of interaction terms: X ci Oi (x) , (7.25) LM → LM + i

where the Oi ’s are arbitrary local operators, not necessarily renormalizable in four dimensions. The Callan-Symanzik equation for a correlator containing n elementary fields φ and an arbitrary number of these new interaction terms reads: " # X ∂ ∂ ∂ M G(n) =0. (7.26) +β + nγ + γi ci r ∂M ∂λ ∂ci i

In this equation, γi is the anomalous dimension of the operator Oi and the operator (n) ci ∂/∂ci counts the number of occurrences of Oi inside the function Gr . If di is the dimension of the operator Oi (in mass units), it is convenient to define a dimensionless coupling constant ρi by the following relation, ci ≡ ρi M4−di .

(7.27)

Thanks to this definition, the previous Callan-Symanzik equation becomes: " # X ∂ ∂ ∂ M G(n) =0, +β + nγ + βi r ∂M ∂λ ∂ρi

(7.28)

i

where we denote βi ≡ ρi (γi + di − 4). With these notations, we see that the additional couplings ρi play exactly the same role as the original coupling λ. We can therefore mimic the explicit solution found in the case of the two-point function in the section 7.1.2. Let us first introduce running couplings λ, ρi , as solutions of the following differential equations dλ(p, λ) = β(λ, ρi ) , d ln(p/M) dρi (p, ρi ) = βi (λ, ρi ) , d ln(p/M)

λ(M, λ) = λ , ρi (M, ρi ) = ρi .

(7.29)

237

7. R ENORMALIZATION GROUP In the weak coupling limit, the functions βi are given at lowest order by βi ≈ (di − 4)ρi ,

(7.30)

and the solution of the previous equations for ρi reads ρi (p) = ρi (M)

 p di −4 . M

(7.31)

This result sheds some light on the fact that all fundamental interactions (except gravity, for which the proper quantum theory is not known) appear to be described by renormalizable quantum field theories at the energy scales relevant for the Standard Model (i.e. p . 1 TeV). Indeed, let us assume that there exists at a much higher scale (typically M ∼ 1016 GeV, the conjectured scale for the unification of all couplings) a more fundamental quantum field theory, comprising all sorts of interactions and whose couplings are of order one (at this unification scale, couplings that are allowed by symmetries have no reason to be much smaller than unity). After evolving the scale down to the sub-TeV scale of the Standard Model, all the couplings for which di − 4 > 0, i.e. all the operators that are not renormalizable in four spacetime dimensions, have become much smaller than the others and have effectively disappeared from the Lagrangian. c sileG siocnarF

7.3 Operator product expansion 7.3.1

Introduction

The operator product expansion (OPE) is a tool that allows to study the renormalization flow at the level of the operator themselves, instead of encapsulating them inside a correlator (although the derivation still requires that we consider a correlator). The intuitive idea is that a non-local product of operators may be approximated by a local composite operator when the separations between the original operators go to zero, possibly with a numerical prefactor that depends on the separation between the operators in the original product. However, since limits of operators are difficult to handle, it is convenient to consider a weaker form of limit, in which the product of operators under consideration is encapsulated into a correlation function of the form (n)

G12 (x; y1 , · · · , yn ) ≡ hA1 (x)A2 (0)φ(y1 ) · · · φ(yn )i ,

(7.32)

where A1 and A2 are local operators, and φ an elementary field. Let us consider a limit where the coordinates yi are fixed, while x → 0. We can already note that, since the product of operators at the same point is ill-defined in general, we may expect divergences in this limit. c sileG siocnarF

238

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS (n)

It turns out that the behaviour of G12 when x → 0 is entirely determined by the operators A1 and A2 themselves, in a way that does not depend on the other fields φ(yi ) (provided they are kept at a finite distance from the the points 0 and x). In order to determine this behaviour, Wilson proposed to expand the product A1 (x)A2 (0) as a sum of composite local operators, with x dependent coefficients: X Ci12 (x) Oi (0) , (7.33) A1 (x)A2 (0) = i

where the Oi are a basis of composite local operators that have the same quantum numbers as the product A1 A2 . All the x dependence is carried by the Wilson coefficients Ci12 (x). This decomposition can then be used in any correlation function (n) where the product A1 (x)A2 (0) appears. For instance, the correlation G12 introduced at the beginning of this section would read X (n) (n) G12 (x; y1 , · · · , yn ) = Ci12 (x) Gi (y1 , · · · , yn ) , (7.34) i

where we denote (n)

Gi (y1 , · · · , yn ) ≡ hOi (0)φ(y1 ) · · · φ(yn )i .

7.3.2

(7.35)

Callan-Symanzik equation for Ci12 (x)

Let us assume that we have defined the normalization of the operators A1 , A2 , Oi at the scale M. The coefficients Ci12 (x) in eq. (7.33) should a priori also depend on M. In order to determine this dependence, let us firstly write the Callan-Symanzik (n) equation for the renormalized correlator2 G12 :   ∂ ∂ (n) (7.36) +β + nγ + γA1 + γA2 G12 = 0 , M ∂M ∂λ where γ, γA1 and γA2 are the anomalous dimensions of the operators φ, A1 and A2 , (n) respectively. Concerning the correlation functions Gi that enter in the right hand side of eq. (7.34), we have the following equations:   ∂ ∂ (n) M +β + nγ + γi Gi = 0 , (7.37) ∂M ∂λ where γi is the anomalous dimension of Oi . The left hand side and right hand sides of eq. (7.34) are consistent provided that the coefficients Ci12 obey the following 2 In the rest of this chapter, we do not write explicitly the subscript r to indicate the renormalized quantities, in order to simplify the notations. From the context, it is always clear when a quantity is renormalized.

239

7. R ENORMALIZATION GROUP equation:   ∂ ∂ M +β + γA1 + γA2 − γi Ci12 = 0 . ∂M ∂λ

(7.38)

This equation confirms a posteriori the fact that the coefficients Ci12 must depend on the renormalization scale M. Moreover, we see that this dependence only depends on the anomalous dimensions of the operators A1 , A2 and Oi , but not on the specific (n) correlation function G12 that was used in the derivation (in particular, eq. (7.38) does not depend on the number n of fields φ, nor on their anomalous dimension). It is this property that renders the operator product expansion universal.

7.3.3

Separation dependence of Ci12 (x)

If the dimensions of A1 , A2 and Oi are respectively D1 , D2 and di , then the dimension of Ci12 is D1 + D2 − di .Therefore, we may write Ci12 (x; M) ≡

1 ei (M|x|) , C |x|D1 +D2 −di 12

(7.39)

ei (Mx) is a dimensionless function of the sole variable M|x|. One can where C 12 determine this function similarly to the case of the 2-point function considered in the section 7.1.2, by introducing the running coupling λ(1/|x|). We obtain the following structure for the coefficient Ci12 : Ci (λ(1/|x|)) Ci12 (x; M) = 12D +D −d |x| 1 2 i   M Z ′ dp   × exp  (γi (λ(p′ )) − γA1 (λ(p′ )) − γA2 (λ(p′ ))) , p′ 1/|x|

(7.40)

where Ci12 is a function of the running coupling that can be obtained by a matching to perturbative calculations. We see that the leading short distance behaviour is controlled by the prefactor |x|di −D1 −D2 , that becomes singular if di < D1 + D2 . Moreover, the contribution of the operators Oi whose dimension obeys di > D1 +D2 goes to zero when x → 0. One does not need to consider such operators in the OPE when studying the short distance limit. In asymptotically free theories where the coupling goes to zero at short distance, such as QCD, we may carry a bit further the determination of the Wilson coefficients.

240

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Indeed, at the first order of perturbation theory, the anomalous dimensions are proportional to g2 , and we may write the anomalous dimension of any operator O as follows: γO ≡ −aO

g2 , (4π)2

(7.41)

where aO is a numerical constant (the minus sign is conventional). Therefore, we have αs , (7.42) γi − γA1 − γA2 = (aA1 + aA2 − ai ) 4π with αs ≡ g2 /4π. At one loop, the running coupling αs is given by: αs (Q2 ) = 4π

β0 ln

1 

Q2

Λ2 QCD

,

(7.43)

where β0 is the first Taylor coefficient of the QCD β function. From this, we get Ci12 (x; M) =

Ci12 (g(1/|x|)) |x|D1 +D2 −di

"

2

2

ln(1/|x| ΛQCD ) ln(M2 /Λ2QCD )

A1 −aA2 # ai −a2β 0

.

(7.44)

We see that, besides the trivial power law prefactor in |x|di −D1 −D2 , there are corrections in the form of powers of logarithms that may be large when x → 0. When di = D1 + D2 , these logarithms are in fact the main source of |x| dependence.

7.3.4

Operator mixing

It may happen that several of the operators Oi that enter in the OPE basis for the product A1 (x)A2 (0) mix under the evolution of the scale M. This means that the anomalous dimensions γi are in fact a matrix γij (when there is no mixing, this matrix is diagonal and the γi ’s that we have used so far are its diagonal elements) and (n) the Callan-Symanzik equations for the correlators Gi are coupled:   X  ∂ ∂ (n) δij M (7.45) +β + n γ + γij Gj = 0 . ∂M ∂λ i

(n)

The equation for G12 is unchanged, and we obtain the following equation for the Wilson coefficients   X ∂ ∂ M +β + γA1 + γA2 Cj12 − γij Ci12 = 0 . (7.46) ∂M ∂λ i

241

7. R ENORMALIZATION GROUP

Note that when the operators A1 and A2 are conserved currents, their anomalous dimensions are zero, and this equation simplifies into 

 X ∂ ∂ M Cj12 − +β γij Ci12 = 0 . ∂M ∂λ

(7.47)

i

This situation turns out to be quite frequent in applications of the OPE. c sileG siocnarF

7.4 Example: QCD corrections to weak decays 7.4.1

Fermi theory

In order to illustrate the use of the operator product expansion on a concrete case, let us consider the weak interactions between quarks and leptons. In the standard model, the interactions between charged currents take the following form: LI =

g2 µ J (0) Dµν (0, x) Jν† (x) + h.c. , L 2 L

(7.48)

where Jµ is the left handed charged current (containing a leptonic term and a term L due to quarks) and Dµν (0, x) is the propagator of the W ± boson between the points 0 and x. At low energy, we may neglect the momentum carried by the W ± boson propagator in front of the W ± mass. In this approximation, the propagator becomes momentum independent, and its Fourier transform is proportional to δ(x). We may then replace the non-local interaction term of eq. (7.48) by a 4-fermion (local) contact interaction, which is nothing but the interaction term of Fermi’s √ theory. The prefactor of this interaction term, g2 /2M2W , is usually denoted 4GF / 2 where GF is Fermi’s constant: 4G Lint ≈ √ F Jµ (0) Jν† (0) + h.c. . L 2 L

(7.49)

Thanks to the operator product expansion, one may study in greater detail the limit from the electroweak theory to Fermi’s theory, i.e. the process by which one replaces the non-local product of two currents by one or more local interaction terms. This example will also illustrate how this decomposition in local operators depends on the energy scale of the processes under consideration, by including the strong interaction corrections at one loop. Let us discuss first two trivial cases regarding the effect of QCD corrections at one loop. Firstly, purely leptonic weak interactions are not affected by strong

242

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

interactions at this order since leptons do not couple directly to gluons (but QCD corrections do exist at two loops and beyond). The other simple case is that of semi-leptonic weak interactions, involving a leptonic current and a current made of quarks. Indeed, the leptonic current is not renormalized by strong interactions. The quark current, conserved at leading order, is also not affected by strong interactions since its anomalous dimension is zero. Finally, a gluon cannot connect the lepton and the quark currents. Thus, semi-leptonic weak interactions are not affected by QCD corrections at one loop. The only non-trivial case, to which we will devote the rest of this section, is that of weak interactions between quark currents, i.e. the non-leptonic weak interactions. As an example, let us consider the QCD corrections to the weak decay of the strange quark, which in Fermi’s theory comes from the following coupling: (dL γµ uL ) (uL γµ sL ).

7.4.2

Operator product expansion

Let us consider the OPE of the following product of currents Aµ 1 (x)A2µ (0), with: µ Aµ 1 ≡ dL γ uL ,

µ Aµ 2 ≡ uL γ sL .

(7.50)

When going from the standard model to Fermi’s theory, the non-local dependence of the W ± propagator is captured by the Wilson coefficients Ci12 (x). Therefore, (since the mass MW is the only dimensionful the typical separation x is x ∼ M−1 W parameter in the propagator). On the other hand, the scale M characteristic of Kaon decays is of the order of the mass of a Kaon, around 500 MeV. The simplest operators on which we may expand the product A1 (x)A2 (0) are the following: O1 ≡ (dL γµ uL )(uL γµ sL ) , O2 ≡ (dL γµ sL )(uL γµ uL ) ,

(7.51)

where in the second one two quark operators of different flavours have been interchanged. Note that the mass dimension of the operators A1 and A2 is 3, while that of O1 and O2 is 6. Therefore, we have dA1 + dA2 − di = 0, which means that the x dependence of the Wilson coefficients comes entirely from the logarithms in the expression (7.44). The more complicated operators that may enter in this expansion all have a larger mass dimension, so that dA1 + dA2 − di < 0. Thanks to the prefactor in eq. (7.44), the corresponding Wilson coefficients are very small since M|x| ∼ M/MW ≪ 1. Thus, one can restrict the OPE of A1 (x)A2 (0) to the sole operators O1 and O2 when applied to the physics of Kaon decays.

7.4.3

Wilson coefficients

In order to determine the Wilson coefficients Ci12 for the operators Oi with equation (7.44), we first need to calculate the anomalous dimensions γA1 , γA2 , as well as γ1 ,

243

7. R ENORMALIZATION GROUP

γ2 , for the operators A1 , A2 , O1 and O2 . Since A1 and A2 are conserved at the first order, their anomalous dimension is zero: γA1 = γA2 = 0 .

(7.52)

In order to obtain the anomalous dimensions of the operators O1 and O2 , let us introduce the following graphical representation for these operators:

u

d

O1 =

u ,

u

d

O2 =

.

u

s

(7.53)

s

This representation renders explicit the fact that these operators are products of two currents. Thanks to eq. (7.22), the anomalous dimension of these operators is obtained by calculating the vertex counterterm and the counterterms associated to the external lines. All the order-g2 strong interaction corrections are listed in the figure 7.1 in the case of O1 . The contributions to γ1 of the first three diagrams on the first line u

d

u

s

Figure 7.1: Order-g2 QCD corrections to the operator O1 .

cancel, because their sum gives the anomalous dimension of a conserved (at first order) current. The same conclusion holds for the remaining three graphs of the first line. Thus, we need only to consider the diagrams of the second line. In Feynman gauge, the expression of the first diagram of the second line is given by: u

d

= u

s

(−ig)2

Z

  / dD k −i ik µ a λ d γ t γ u L L (2π)D k2 (k + p)2 f  × uL γλ ta f

/ ik γµ sL (k − q)2



,

(7.54)

244

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where p and q are the (incoming) momenta carried by the quark lines to which the gluon is attached. The ta f are the generators of the fundamental representation of the / and q / have su(3) algebra, that holds the quarks. In the numerator, some terms in p been dropped because they do not contribute to the ultraviolet divergence of the graph. The integral over k can be rewritten as follows3 : Z

′

dD k kν kν (2π)D k2 (k + p)2 (k − q)2

=

=

=

gνν d

Z

′

dD k 1 D 2 (2π) (k + p) (k − q)2 1 Z D ′ Z gνν d k 1 dx d (2π)D (k2 + ∆)2 0

gνν i d

′

Z1

dx

Γ (2 − D 1 2) , (7.55) D/2 2−D/2 (4π) ∆

0

where we denote k ≡ k + xp − (1 − x)q and ∆ ≡ x(1 − x)(p + q)2 . Since the renormalization scale is M, we may impose that the Lorentz invariant quantity (p + q)2 is equal to −M2 , so that ∆ is proportional to M2 . Since the power 2 − D/2 to which the denominator ∆ is raised goes to zero in four dimensions, we may neglect the prefactor x(1 − x) inside ∆ and the integral over the Feynman parameter x simply gives a factor equal to unity. If we take the limit D → 4 in all the factors that do not diverge and do not depend on M, we obtain: u

d

= u

s

 g2 Γ (2 − D 1  λ a 2) dL γµ γν ta f γ uL [uL γλ tf γν γµ sL ] . (7.56) 2 4−D 4 (4π) M

The contribution of this graph to the counterterm for the normalization of O1 is given by the opposite of this result. c sileG siocnarF

In order to simplify the combination of spinors, Dirac and colour matrices that appear in the result of eq. (7.56), it is useful to use the chiral representation (also known as Weyl’s representation) since only the left handed component of the spinors enter in this expression. In this representation, the Dirac matrices are given by ! ! µ 0 σ −1 0 γµ = , γ5 = , (7.57) σµ 0 0 1 with σµ ≡ (1, σ) and σµ ≡ (1, −σ) where σ is a vector made of the three Pauli matrices. In this representation, the left handed projector PL ≡ (1 − γ5 )/2 and the 3 The

first equality disregards some terms that are ultraviolet finite.

245

7. R ENORMALIZATION GROUP right handed one PR ≡ (1 + γ5 )/2 simplify into: ! ! 1 0 0 0 PL = , PR = , 0 0 0 1

(7.58)

so that any 4-component spinor can be viewed as two 2-components spinors, one of which is right handed and the other one left handed: ! ψL ψ= . (7.59) ψR Using this representation, we can for instance easily obtain µ ν λ a dL γµ γν γλ ta f uL = dL σ σ σ tf uL .

(7.60)

This equation contains a small abuse of notations, since it contains the 4-component spinors (ψL , 0) in the left hand side, while the right hand side contains only the 2-component left handed spinors ψL . In order to reduce the combination of spinors that appear in eq. (7.56), we need a to simplify the products (σµ )αβ (σµ )γδ and (σµ )αβ (σµ )γδ as well as (ta f )ij (tf )kl . In both cases, this can be done by using the Fierz identity for the generators of the fundamental representation of the su(n) algebra, introduced in the section 4.1.6. Let us recall this identity here:   1 1 a a (tf )ij (tf )kl = δil δjk − δij δkl . (7.61) 2 n For the contraction of colour matrices ta f , we can apply it directly with n = 3:   1 1 a δ δ − δ δ (ta ) (t ) = (7.62) il jk ij kl . f ij f kl 2 3 For the contraction of the σµ or the σµ , let us recall that the Pauli matrices σi are related to the su(2) fundamental generators τi by σi = 2 τi .

(7.63)

Using this relation and the Fierz identity for the fundamental representation of su(2), we obtain: (σµ )αβ (σµ )γδ = (σµ )αβ (σµ )γδ

= = =

δαβ δγδ − 4(τi )αβ (τi )γδ   1 δαβ δγδ − 2 δαδ δβγ − δαβ δγδ 2 (7.64) 2 [δαβ δγδ − δαδ δβγ ] .

246

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Thanks to eqs. (7.62) and (7.64), we obtain4 

 λ a dL γµ γν ta f γ uL [uL γλ tf γν γµ sL ]

2 = 2(uL γµ uL )(dL γµ sL ) − (dL γµ uL )(uL γµ sL ) . 3

(7.65)

We recognize the operators O1 and O2 in this expression. We are therefore in a situation where renormalization introduces a mixing between operators. The second diagram is identical to the one we have just calculated. The third diagram of the second line reads: u

d

= u

s

(−ig)2

Z

  / dD k −i ik µ a λ d γ t γ u L L (2π)D k2 (k + p)2 f 

/ a −ik t γλ sL × uL γµ (k − r)2 f



,

(7.66)

where r is the momentum that flows into the diagram by the line carrying the s quark. The integration over k is similar to the previous case, and leads to u

d

=− u

s

 g2 Γ (2 − D 1  λ a 2) dL γµ γν ta f γ uL [uL γµ γν tf γλ sL ] . 2 4−D 4 (4π) M (7.67)

Likewise, we can simplify the Dirac and colour matrices by using Fierz identities: 

 λ a dL γµ γν ta f γ uL [uL γµ γν tf γλ sL ] 8 = 8(uL γµ uL )(dL γµ sL ) − (dL γµ uL )(uL γµ sL ) , 3

(7.68)

which is again a linear combination of O1 and O2 . The last diagram gives the same result. c sileG siocnarF

By combining the four contributions, we obtain the following form for the operator O1 , renormalized at the scale M, in terms of the bare operators: O1r = O1b − δ11 O1b − δ12 O2b , 4 The

derivation can be made easier by using the graphical form (4.84) of the Fierz identity.

(7.69)

247

7. R ENORMALIZATION GROUP where the counterterms δij are given by δ11 ≡

g2 Γ (2 − D 2) 2 4−D (4π) M

,

δ12 ≡ −3

g2 Γ (2 − D 2) . 2 4−D (4π) M

(7.70)

By calculating in the same way the one-loop corrections to the operator O2 , we obtain the counterterms δ22 and δ21 , that are equal to δ21 = δ12

,

δ22 = δ11 .

(7.71)

Because of the mixing, the anomalous dimensions for the operators O1,2 form a non-diagonal matrix ! −2 6 g2 ∂δij . (7.72) = γij = M ∂M (4π)2 6 −2 In order to solve the coupled Callan-Symanzik equations (7.47), we must find a basis of operators in which the matrix of anomalous dimensions becomes diagonal. This is achieved by choosing5 : 1 [O1 − O2 ] , 2 1 ≡ [O1 + O2 ] . 2

O1/2 ≡ O3/2

(7.73)

The corresponding eigenvalues of the matrix γij are γ1/2 = −8

g2 (4π)2

γ3/2 = 4

,

g2 . (4π)2

(7.74)

Using the equation (7.44) (the functions Ci12 are equal to 1 at the first order of perturbation theory) at a distance scale x ≈ M−1 , we obtain the following values for W the Wilson coefficients:

1/2 C12 (M−1 ; M) W

3/2 C12 (M−1 ; M) W

=

=

"

ln(M2W /Λ2QCD )

"

ln(M2W /Λ2QCD )

ln(M2 /Λ2

QCD

)

ln(M2 /Λ2QCD )

# β4

0

,

#− β2

0

.

(7.75)

5 The subscripts 1/2 and 3/2 are related to the isospin variation in the s quark decay mediated by these operators.

248

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Since MW ≫ M and β0 = 11 − 2Nf /3 is positive6 , the operator A1 (x)A2 (0) responsible for the weak decay of the quark s receives a larger contribution from the operator O1/2 than from O3/2 (roughly by a factor 3.6 if we use M ≈ 500 MeV, ΛQCD ≈ 150 MeV, and 5 quark flavours). This calculation qualitatively7 corroborates the empirical observation that weak decays of Kaons correspond predominantly to an isospin variation of 1/2.

7.5 Non-perturbative renormalization group Until now, our discussion of renormalization has been strictly rooted in perturbation theory and limited to the context of renormalizable theories, at the exception of the section 7.2.4 where we discussed the running of the couplings in front of operators of any dimension. In this framework, the renormalization flow is formalized by the Callan-Symanzik equations, that describe the scale dependence of correlation functions. However, the ideas behind renormalization have a much wider range of application: they are also relevant non-perturbatively, and they may be applied directly at the level of actions rather than correlation functions. In this section, we first develop heuristically some general concepts related to the renormalization flow in an abstract space of theories. These ideas are then made more tangible in the form of a functional flow equation for the quantum effective action, whose solution interpolates between the classical action and the full quantum action.

7.5.1

Kadanoff’s blocking for lattice spin systems

The general concepts of renormalization that we aim at introducing in this section can be first exposed by considering the simple example of a system of spins on a lattice, the simplest of which is the Ising model in two dimensions, which is exactly solvable for interactions among nearest neighbors. This model is known to have a disordered phase at high temperature, a ferromagnetic order at low temperature (where spins align with an external magnetic field, no matter how small), and a second order phase transition at a critical temperature T∗ . At the second order transition, the correlation length of the system becomes infinite, despite the fact that the interactions are short ranged. Roughly speaking, a measure of the complexity of the study of a discrete physical system (at least if one attempts to do it from the theory that describes the interactions among the microscopic degrees of freedom) is the number of elementary degrees of freedom per correlation length. By this account, second order phase transitions are among the hardest problems to analyze. 6 In this problem, N = 5 flavours of quarks should be taken into account in the running of the strong f coupling constant, in order to include all the quarks up to mass of the W ± bosons. 7 The measured imbalance between the isospin variations 1/2 and 3/2 is even larger, but a quantitative explanation would involve non-perturbative aspects of QCD.

7. R ENORMALIZATION GROUP

Figure 7.2: Kadanoff’s blockspin renormalization. Top: the spins are grouped into 3 × 3 blocks. Middle: each block of 9 spins is replaced by a single spin determined by the rule of majority. Bottom: the lattice is scaled down (new spins come into the picture, that where previously outside of the represented area).

249

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Kadanoff devised a method, called block-spin renormalization to facilitate the study of such a situation. The basic ideas of this method are illustrated in the figure 7.2. Firstly, one groups the spins into connected sets, for instance in 3 × 3 blocks as shown in the figure. Then, the spins inside each of these blocks are replaced by some sort of average spin. One possibility is to use the “rule of majority”: the new spin is chosen to be up if five or more of the original spins were up, and down otherwise. The physical motivation for this replacement is that the calculation of macroscopic observables (e.g. the total magnetization in a large sample of the material under consideration) does not require to know in detail the value of each of the elementary spins, and should be doable from these coarse grained variables. Of course, one should adjust carefully the interactions among the newly introduced averaged spins, so that the macroscopic properties of the system are unchanged. One may for instance require that the partition function of the system is unmodified. In general, even if the original Hamiltonian had only short range nearest neighbors interactions, the Hamiltonian that describes the coarse-grained spins may have long range interactions. The block-spin renormalization comprises a third step, that consists in a rescaling of distances so that each of the coarse-grained spin occupy the same area as one of the original elementary spins (this step is necessary for the transformation to have fixed points). The combination of these three steps, called a (discrete) renormalization group step R, may be viewed as transforming a bare action S0 into a renormalized action: Sr ≡ R S0 .

(7.76)

However, the real power of this idea comes by iterating the renormalization group steps R until there are only a few of the coarse-grained spins in a macroscopic area of the system. Under such a sequence of renormalization steps, the actions are sequentially transformed as follows: S0 7−→ S1 7−→ S2 7−→ S3 7−→ · · ·

(7.77)

S∗ = R S∗ .

(7.78)

R

R

R

R

The behaviour of the mapping Rn for large n contains all the information we may need about the macroscopic properties of the system. In particular, a critical point, where the system has an infinite correlation length and is self-similar, corresponds to a fixed point of this transformation, i.e. to an action S∗ that satisfies

The concept of renormalization group introduced so far in the case of a discrete system, consisting in a coarse graining followed by a rescaling, can be generalized to a continuous system such as a quantum field theory. In this case, one introduces a length scale ℓ, and the renormalization group transformation consists in integrating out the smaller length scales. One may denote τ ≡ ln(ℓ/ℓ0 ), where ℓ0 is the initial

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short distance scale. Thus, τ = 0 corresponds to the bare action at short distance, and τ = +∞ corresponds to macroscopic distances, and the discrete steps of eq. (7.77) are replaced by an equation of the form ∂ τ Sτ = H S τ ,

(7.79)

where the RG flow for an infinitesimal step ∆τ is R = 1 + ∆τ H. c sileG siocnarF

7.5.2

Wilsonian RG flow in theory space

One may view a given action S as a point in an abstract space, where each axis corresponds to the coupling constant in front of a given operator. For instance, in the case of a lattice spin system, there would an axis for the strength of the interactions among nearest neighbors, an axis for the strength of the interactions among sites √ whose distance is 2 lattice units, and so on... In a scalar quantum field theory, these could be the couplings for the operators φφ, φ2 , φ4 , φ6 , ... A renormalization group transformation such as (7.76) defines a mapping of the points in this theory space, either discrete or continuous depending of the system. We have illustrated this in the continuous case in the figure 7.3, where the thick gray line shows how a bare action S0 at short distance flows as the distance scale ℓ increases, leading to a theory that may have very different couplings at macroscopic scales. Note that only three out of many (possibly infinitely many for a continuous system) dimensions are shown in the figure. c sileG siocnarF

As we have already mentioned in the previous section, a critical point must be a fixed point of this mapping, e.g. the point S∗ in the figure 7.3. Important properties of the renormalization group flow may be learned by linearizing the flow in the vicinity of such a fixed point, by writing S ≡ S∗ + ∆S

,

H S = L ∆S + · · · ,

H S∗ = 0 ,

(7.80)

where L is a linear mapping. Then, one may define the eigenoperators of L, L On = λn On ,

(7.81)

where λn is the corresponding eigenvalue. In the vicinity of the fixed point, we thus have X S ≈ S∗ + cn eλn τ On , (7.82) n

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S0

S*

Figure 7.3: Renormalization group flow in theory space (the arrows go from UV to IR scales). The black dot is a critical fixed point S∗ . The gray surface is the critical surface, i.e. the universality class made of all the theories that flow into the critical point. The light colored line, flowing away from the critical point, corresponds to the direction of a relevant operator. The thick gray line illustrates the flow from a generic initial action S0 .

7. R ENORMALIZATION GROUP

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where the cn are coefficients determined by initial conditions. This expression leads to the following classification of operators8 : • λn < 0 : such an operator corresponds to an attractive direction in the vicinity of the fixed point. Even if the action contains this operator at some short distance scale, its coupling vanishes as one gets close to the critical point. This operator is said to be irrelevant, because it plays no role in the long distance critical phenomena. • λn > 0 : this operator corresponds to a repulsive direction in the vicinity of S∗ . Any admixture of this operator will grow as one goes to larger distance scales. An operator with a positive eigenvalue is called relevant. • λn = 0 : such an operator is called marginal. Usually, it means that the operator may either grow or shrink, but slower than exponentially (and a more refined calculation that goes beyond this linear analysis is necessary in order to decide between the two behaviours). The previous discussion, based on a linear analysis near the critical point, may be extended globally as follows. One defines the critical surface as the domain of theory space which is attracted into the critical point as the length scale goes to infinity. All the bare actions that lie in this domain (the shaded surface in the figure 7.3) describe systems that have the same long distance behaviour. Despite the fact that these systems may correspond to completely different microscopic degrees of freedom and interactions, they are described by the same action S∗ at large distances. For this reason, this domain is also called the universality class of the critical point. The relevant operators correspond to the directions of theory space that are “orthogonal” to the critical surface. The term relevant follows from the fact that the coupling of these operators must be fine-tuned in order to be on the critical surface: in other words, the relevant couplings matter for making the system critical. A remarkable aspect of phase transitions is that the number of these relevant operators is small9 , despite the fact that the microscopic interactions may require a very large number of distinct couplings. Heuristically, this follows from a dimensional argument: since the action is dimensionless, the coupling constants of higher dimensional operators must have a negative mass dimension, and therefore they scale as inverse powers of 8 This discussion does not exhaust all the possibilities. Firstly, in a theory space with two or more dimensions, eigenvalues can be complex valued, corresponding to RG trajectories that spiral around the fixed point (spiraling inwards if the real part is negative and outwards if it is positive). Another possibility is limit cycles (i.e. closed RG trajectories), that play a role for instance in the Efimov effect (a scaling law in the binding energies of 3-boson bound states when the 2-body interaction is too weak to have a two-body bound state). 9 In the case of 2-dimensional Ising model, the only parameters that need to be adjusted in order to reach the critical point are the temperature (T∗−1 ≈ 0.44) and the external field (equal to zero).

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the ultraviolet cutoff. Thus, these operators are irrelevant. Only operators of low dimensionality can be relevant, and there is usually a (small) finite number of them10 . Let us now consider the domain that originates from the fixed point (the light colored line in the figure 7.3), sometimes called the ultraviolet critical surface. This is the domain spanned by the renormalization group flow if one starts from an infinitesimal region around the fixed point. Any theory that lies on the UV critical surface is renormalizable, since it evolves into the fixed point at short distance: this indeed means that one may safely send the ultraviolet cutoff to infinity in such a theory (this corresponds to moving in the direction opposite to the arrows in the figure 7.3). Note also that theories on the UV critical surface transform into one another under the renormalization flow, but the couplings of the various relevant operators depend on the scale. The following situations may occur: • For such a theory to be renormalizable in the perturbative sense, the couplings should remain small all the way to the ultraviolet scales. This happens when the fixed point is a Gaussian fixed point, whose action S∗ contains only a kinetic term (i.e. is Gaussian in the fields). This is the case for quantum chromodynamics, thanks to asymptotic freedom. • It may also happen that around a Gaussian fixed point, the only relevant operators are quadratic in the fields, like mass and kinetic terms. In this case, there is no interacting renormalizable action, and the theory is said to suffer from triviality. There is nowadays strong evidence that, in a pure real scalar field theory, the operator φ4 is not relevant in four space-time dimensions (it is relevant in three dimensions or less) and therefore such a field theory is trivial because only the non-interacting theory makes sense. • When the fixed point is a non-trivial interacting fixed point instead of a Gaussian one, the theories on the UV critical surface are also renormalizable, but their high energy behaviour cannot be studied by perturbative means. This situation is called asymptotic safety11 . To conclude this discussion, let us say a word about generic RG trajectories, i.e. neither located on the critical surface nor on the UV critical surface, such as the line originating from the short distance action S0 in the figure 7.3. Generically, when evolving towards larger length scales, the irrelevant couplings decrease and the relevant ones increase, and the action approaches that of a renormalizable theory. This sets in a more general framework our observation of the section 7.2.4 (there, it was largely based on dimensional analysis). Moreover, if the microscopic action S0 starts 10 An exception to this assertion is the renormalization group on the light-cone used in the study of deep inelastic scattering. There, peculiarities of the kinematics lead to an infinite number of relevant operators. 11 The concept of asymptotic safety was introduced by Weinberg, as a logical possibility for a renormalizable quantum field theory of gravity.

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close to but not exactly on the critical surface, the theory firstly approaches the critical point upon increasing the length scale, but instead of reaching it, it departs from it on even larger scales to follow one of the repulsive directions. In such a system, the correlation length may be large but not infinite as it would be at the critical point (the turning point between the approach of the critical point and the subsequent departure from it happens roughly when the RG scale equals the correlation length).

7.5.3

Functional RG equation for scalar theories

The block-spin renormalization procedure that we have discussed in the section 7.5.1 can be extended to the case of a continuous system such as a quantum field theory. Moreover, while our discussion has been so far qualitative, we shall now derive an explicit RG flow equation for the quantum effective action, the solution of which would provide the full quantum content from tree level contributions only. Reminders about the quantum effective action : Let us first recall some basic results about the quantum effective action Γ [φ], taken from the section 2.6. It is related to the generating functional W[j] of connected Feynman graphs by Z i Γ [φ] = W[jφ ] − i d4 x jφ (x)φ(x) . (7.83) where the current jφ is defined implicitly by δΓ [φ] + jφ (x) = 0 . δφ(x) or equivalently in terms of W by δW[j] φ(x) = . i δj(x) j=j

(7.84)

(7.85)

φ

In other words, jφ (x) is the external source such that the expectation value of the field is φ(x). By combining the path integral representation of W, Z Z h i   (7.86) eW[j] = Dφ(x) exp iS[φ(x)] + i d4 x j(x)φ(x) ,

with eqs. (7.83) and (7.84) we obtain the following functional equation satisfied by the effective action Γ : Z Z i h   δΓ [ϕ] φ(x) . (7.87) ei Γ [ϕ] = Dφ(x) exp iS[φ + ϕ] − i d4 x δϕ(x)

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(We have performed a shift φ → φ + ϕ in the dummy functional integration variable.) Although this equation formally defines the quantum effective action, its use is not convenient because it still contains a path integration. Physically, this difficulty is related to the fact that the equation integrates out all the length scales at once. The functional RG equation that we derive now circumvents this problem by integrating out quantum fluctuations only in a small range of scales at a time. c sileG siocnarF

Regularized generating functional : Let us introduce a momentum scale κ and define eWκ [j]

≡ =

h δ i exp i ∆Sκ Z[j] iδj Z  Z

    Dφ(x) exp i S[φ] + ∆Sκ [φ] + i jφ ,

(7.88)

where Z[j] is the usual generating functional for time ordered correlation functions and ∆Sκ is defined in terms of the Fourier transform of the fields as follows: ∆Sκ [φ] ≡

Z

d4 p e e φ(−p) Rκ (p) φ(p) . (2π)4

(7.89)

Rκ is an ordinary function that plays the role of a cutoff in momentum. At low momentum p/κ ≪ 1, it should be positive in order to give a mass for the soft modes, and thus provide an infrared regulator: lim Rκ (p) = µ2 > 0 .

p/κ→0

Moreover, this function is assumed go to zero when the scale κ → 0, lim Rκ (p) = 0 ,

κ→0

(7.90)

(7.91)

which means that the cutoff plays no role in this limit and we recover the full quantum theory. This is the limit we aim at reaching at the end of the RG flow. In contrast, it should become large when κ → ∞: lim Rκ (p) = ∞ .

κ→∞

(7.92)

This property ensures that when κ is large, the right hand side of eq. (7.88) is dominated by the saddle point, so that the corresponding effective action equals the classical action.

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Scale dependence of Wκ : By denoting τ ≡ ln(κ/Λ) (where Λ is the ultraviolet scale at which the classical action is defined), we have

 i ∂τ Wκ [j] = i ∂τ ∆Sκ [ φ κ ] + 2

Z

d4 p e κ (p) , ∂τ Rκ (p) G (2π)4

(7.93)

where Gκ (p) is the connected 2-point function obtained from Wκ [j], Gκ (x, y) ≡

δ2 Wκ [j] , iδj(x)iδj(y)

(7.94)

 and φ κ is the corresponding 1-point function:

 δWκ [j] . φ(x) κ ≡ iδj(x)

(7.95)

Scale dependent effective action : Let us now alter the definition (7.83) in order to make it depend on the scale κ, by writing Z Γκ [φ] + ∆Sκ [φ] = −i Wκ [jφ ] − d4 x jφ (x) φ(x) . (7.96) The left hand side is written as Γκ + ∆Sκ in order not to include in the definition of the effective action the unphysical regulator ∆Sκ . Like in the original definition, the field φ and the current jφ are related by δWκ [j] φ(x) = . (7.97) iδj(x) j=jφ

In terms of Γκ this relationship reads jφ (x) +

δΓκ [φ] h e i + Rκ φ (x) = 0 . δφ(x)

(7.98)

Differentiating eq. (7.97) with respect to j(y) and eq. (7.98) with respect to φ(y), and multiplying the results, we obtain the following identity: " # Z δ2 Wκ [j] δ2 Γκ [φj ] 4 +Rκ (x, y) , (7.99) i δ(x − y) = d z iδj(y)iδj(z) δφj (z)δφj (x) {z } | | {z } Gκ (y,z)

that generalizes eq. (2.94).

Γκ,2 (z,x)

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Flow equation for Γκ : Now, we can differentiate eq. (7.96) with respect to the scale: Z ∂τ Γκ [φ] = −∂τ ∆Sκ [φ] − i ∂τ Wκ [jφ ] − d4 x φ(x) ∂τ jφ (x) h i = −∂τ ∆Sκ [φ] − i ∂τ Wκ [j] j=jφ Z Z δW [j ] κ φ 4 ∂τ jφ (x)− d4 x φ(x)∂τ jφ (x) −i d x δjφ (x) Z 1 d4 p e κ (p) . = ∂τ Rκ (p) G (7.100) 2 (2π)4 In the second line, we have made explicit the fact that Wκ [jφ ] contains both an intrinsic scale dependence and an implicit one from the κ dependence of its argument jφ . Using eq. (7.99), this can be put into the following form: i ∂τ Γκ = Tr 2



(∂τ Rκ )

"

δ2 Γκ [φ] + Rκ δφδφ

#−1 

,

(7.101)

that depends only on Γk (the integral over the momentum p has been written compactly in the form of a trace). Let us make a few remarks concerning this equation: • It describes the renormalization group trajectory of the effective action in theory space, starting from the bare classical action at κ = ∞ and going to the full quantum effective action when κ → 0. c sileG siocnarF

• This equation is a functional differential equation, that does not involve any functional integral, unlike eq. (7.87). Nevertheless, it cannot be solved exactly in general, and various truncation schemes have been devised in order to obtain physical results. • The term Rκ in the denominator provides an infrared regularization (by adding a kind of mass term to the inverse propagator). • The factor ∂τ Rκ is peaked around momentum modes of order κ. Thus, the right hand side is rather localized in momentum space, in contrast with the equation (7.87) that includes all the momentum scales at once. • The choice of the regularizing function Rκ is not unique, provided that it fulfills the conditions (7.90-7.92). Consequently, the renormalization group trajectories depend somehow on this choice (this may be viewed as a dependence on the renormalization scheme). However, the fixed points of the renormalization group flow do not depend on this choice.

Chapter 8

Effective field theories Until now, we have discussed various quantum field theories (the electroweak theory and quantum chromodynamics) that are believed to provide a unified description of all particle physics up to the scale of electroweak symmetry breaking, i.e. roughly ΛEW ∼ 200 GeV. However, it is hard to imagine that there isn’t some kind of new physical phenomena (new particles, new interactions) at higher energy scales (so far out of reach of experimental searches). An interesting question is therefore to understand why the Standard Model is such a good description of physics below the electroweak scale, despite the fact that it does not contain any of the physics at higher scale. In other words, despite the fact that there is distinct physics on scales that span many orders of magnitude, why can “low energy” phenomena be described by ignoring most of the higher scales? The same question could be asked in other areas: for instance, why can chemistry (i.e. phenomena of atomic bonding in molecules) get away without any of the complications of quantum electrodynamics? The general question is that of the separation between various physical scales. c sileG siocnarF

In the context of quantum field theory, such a low energy description is called an effective theory. The basic idea is that most of the details of an underlying more fundamental (i.e. valid at higher energy) description are not important at lower energies, except for a small number of parameters. As we shall see in this chapter, effective field theories may occur in several situations: • Top-down : the quantum field theory which is valid at higher energy is known, but it is unnecessarily complicated to describe phenomena at lower energy scales. A typical example is that of a theory that contains particles that are much heavier than the energy scale of interest (e.g., the top quark in quantum chromodynamics, while one is interested in interactions at the GeV scale). In this case, the effective theory “integrates out” the higher mass particles in order to obtain a simpler theory.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS • Bottom-up : we have a theory believed to be valid at a given energy scale, but have no clear idea of what may exist at higher scales. In this case, one may view the existing theory as an effective description of some (so far unknown) more fundamental theory at higher energy, and try to complete it by adding new (higher dimensional, and therefore usually non-renormalizable in four dimensions) local interactions to it. • Symmetry driven : even when the underlying theory is known, its direct application may be rendered very impractical because the physics of interest involves some non-perturbative phenomena, such as the formation of bound states (for instance, in QCD at low energy, the quarks and gluons cease to be the relevant degrees of freedom and the physical excitations are the light hadrons). An effective theory for these bound states may be constructed from the requirement that it should be consistent with the symmetries of the underlying theory. This case differs from the top-down approach in the sense that the low energy description is not constructed by integrating out the high scales, but solely from symmetry considerations.

In the top-down approach, where the fundamental underlying theory is known, the goal of obtaining an effective description for low energy phenomena could in principle be achieved by the renormalization group. In particular, the functional renormalization group introduced in the section 7.5.3 allows to evolve from an ultraviolet classical action towards a low energy quantum effective action, by progressively integrating out layers of lower and lower momentum. There is nothing wrong with this approach, but one has to keep in mind that the effective action obtained in this way is usually extremely complicated and cumbersome to use in practical applications (in particular, it could have infinitely many effective interactions, all of which are in general nonlocal). In a sense, the quantum effective action that results from the RG evolution is much more complex that the original ultraviolet action, and the gain in terms of simplicity is rather dubious. In contrast, the concept of effective theory that we are aiming at in this chapter is a field theory in which the ultraviolet physics is encapsulated into a finite number of local operators, with coupling constants that may depend on the energy scale and on the properties of the degrees of freedom that have been integrated out.

8.1 General principles of effective theories 8.1.1

Low energy effective action

For the purpose of this general discussion, let us consider a quantum field theory in which the fields are collectively denoted φ (this may be a single field, or a collection of several fields) and a classical action S[φ]. We view this theory as the high energy

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theory, and we wish to construct an alternative description applicable to low energy phenomena, below some energy scale Λ. To this effect, let us assume that we can split the field into a low frequency part (soft) and a high frequency part (hard), φ ≡ φS + φH .

(8.1)

This separation may be achieved by a cutoff in Fourier space, but the details of how this is done are not important at this level of discussion. The classical action of the original theory is thus a function of φS and φH and the path integration is over the soft and the hard components of the field. Now, assume that we are interested in calculating the expectation value of an observable that depends only on the soft component of the field, O(φS ). Then, we may write Z Z     (8.2) hOi = DφS DφH eiS[φS ,φH ] O(φS ) = DφS eiSΛ [φS ] O(φS ) ,

where in the second equality we have defined Z   iSΛ [φS ] e ≡ DφH eiS[φS ,φH ] .

(8.3)

SΛ [φS ] is the action of the low energy effective theory. Using the operator product expansion, it may be written as a sum of local operators, possibly infinitely many of them: Z X SΛ [φS ] ≡ dd x λn On . (8.4) n

8.1.2

Power counting

The behaviour of the couplings λn can be inferred from dimensional analysis. For the sake of this discussion, let us consider the case where φ is a scalar field, whose mass dimension is φ ∼ (mass)(d−2)/2 in d spacetime dimensions. If the operator On contains Nn powers of the field φ and Dn derivatives, its dimension is On ∼ (mass)dn

with dn = Dn + Nn

d−2 2

,

(8.5)

and it must be accompanied with a coupling λn whose dimension is (mass)d−dn . Assuming that the cutoff Λ is the only dimensionful parameter that enters in the construction of the effective theory (except for the field operator and derivatives, that enter in the operators On ), we must have λn = Λd−dn gn , where gn is a dimensionless constant, whose numerical value is typically of order one. Consider now the application of this effective theory to the study of a phenomenon characterized by a single energy scale E. On dimensional grounds, we have Z dd x On ∼ Edn −d . (8.6)

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Combined with the corresponding coupling constant, the contribution of this operator would be of order  d−dn Z Λ d . (8.7) λn d x On ∼ gn E This estimate is the basis of the following classification of the operators that may enter in the action of the effective theory: • dn > d : the contribution of these operators is suppressed at low energy, i.e. when E ≪ Λ. For this reason, these operators are called irrelevant. This does not mean that their contribution is not important and interesting, since there may be observables for which they are the sole contribution. Note also that these operators are non-renormalizable by the standard power counting rules. c sileG siocnarF

• dn = d : the contribution of these operators does not depend on the ratio of scales E/Λ, except perhaps via logarithms. These operators are called marginal, and correspond to renormalizable operators. • dn < d : the contribution of these operators becomes more and more important as the energy scale decreases. These operators, called relevant, are superrenormalizable. Recall also that a higher dimension dn corresponds to operators of greater complexity (since in d > 2 the dimension increases with more powers of the field or more derivatives). Therefore, there is in general only a finite number of operators whose dimension is below a given value. For a given a cutoff Λ and an energy scale E, one must therefore only consider a finite number of operators in order to reach a given accuracy. In a conventional quantum field theory, one usually insists on including only renormalizable operators, in order to avoid the proliferation of new couplings at each order or perturbation theory, and the usual statement of renormalizability amounts to saying that all infinities may be absorbed into the redefinition of a finite number of parameters of the theory, at every order of perturbation theory. In contrast, since a low energy effective theory may contain operators of dimension dn > d, it is usually not renormalizable in this usual sense, but the cutoff Λ provides a natural way of keeping all the contributions finite. In this case, the power counting is organized by the fact that the cutoff Λ is also the dimensionful scale that enters in the couplings of negative mass dimension that come with operators of mass dimension greater than four. For instance, an operator of dimension 6 has a coupling constant that scales as Λ−2 , and physical observables may be expanded in powers of E/Λ, where E is some low energy scale. In the presence of such higher dimensional operators, the usual statement of renormalizability must now be replaced by a weaker assertion: namely, that all the ultraviolet divergences that occur at a given order in E/Λ can be

8. E FFECTIVE FIELD THEORIES

263

absorbed into the redefinition of a finite number of parameters. More precisely, in order to calculate consistently effects of order Λ−r , we must include all operators up to a mass dimension of 4 + r. Thus, the number of constants that must be adjusted in the renormalization process grows as we go to higher order. In the case of top-down effective theories, the renormalizability of the underlying field theory implies that the low energy physics depends on the ultraviolet only through the values of the relevant and marginal couplings. In addition, a small number of irrelevant couplings may matter in certain specific observables (e.g., if an irrelevant operator is the only one that contributes). In fact, if the cutoff of the effective theory is high enough compared to the physical energy scale of interest, the effective theory can have a very strong predictive power, despite the fact that it a priori contains an infinity of operators. But conversely, in a bottom-up approach where we try to extend a renormalizable theory by adding to it higher dimensional operators, the fact that the low energy theory is renormalizable implies that it is not sensitive to the scale of new physics (in other words, a renormalizable low energy theory cannot predict at which high energy scale it breaks down and is superseded by another theory).

8.1.3

Relevant operators

In fact, in an effective theory, the relevant operators (super-renormalizable) are often more troublesome than the irrelevant ones (non-renormalizable). Consider for instance the operator φ2 , that corresponds to the mass term in the effective Lagrangian and has dimension φ2 ∼ (mass)d−2 , and whose corresponding coupling has dimension (mass)2 , i.e. λ = g Λ2 . Thus, small masses are not natural in a low energy effective theory: the natural scale of a mass is that of the cutoff Λ (the dimensionless coupling g is generically of order one). In order to obtain small masses in a low energy effective field theory, there must be some symmetry that prevents the corresponding mass term, such as: • A gauge symmetry for spin 1 particles. • A chiral symmetry for fermions. • A spontaneous breaking of symmetry, so that some scalars are the corresponding massless Nambu-Goldstone bosons. • Supersymmetry may also forbid certain types of mass terms (if unbroken, the mass must be strictly zero, and if broken, the mass will settle to a value close to the scale of supersymmetry breaking). By that account, the Standard Model (without any supersymmetric extension) is not natural, since it does not contain any mechanism to prevent the mass of the Higgs

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scalar boson to be at a cutoff scale (possibly much higher than the electroweak scale) where the Standard Model is superseded by a more fundamental theory. c sileG siocnarF

Likewise, relevant interaction terms have a large contribution to low energy observables, that scales like  d−dn Λ ≫ 1 with d > dn . (8.8) E Therefore, the existence of relevant interaction terms implies that the dynamics is strongly coupled at low energy. This may lead to the formation of bound states or condensates, which calls for a low energy effective theory that contains different degrees of freedom. An example is that of the identity operator, which is not forbidden by any symmetry and has mass dimension 0 (therefore, it is a relevant operator). Although this operator has no effect if added to the Lagrangian of a field theory (since it amounts to adding a constant to the potential energy), its coefficient becomes a cosmological constant if this field theory is minimally coupled to gravity1 . From the power counting of the previous section, the natural value of the coupling constant in front of this operator is Λd . Thus, if we view the Standard Model as an effective theory, the cosmological constant should be at least as large as the fourth (d = 4) power of the cutoff at which the Standard Model is replaced by some other theory. This in sharp contrast with observations. Indeed, if the dark energy inferred from the measured acceleration of the expansion of the Universe is attributed to a cosmological constant, its value is many orders of magnitude below its natural value in quantum field theory (its corresponds to an energy density of the vacuum of the order of 10−47 GeV4 ).

8.2 Example: Fermi theory of weak decays As a first illustration of the concept of effective field theory, let us consider the case of Fermi’s theory of weak interactions. Historically, this model was constructed before the advent of the electroweak gauge theory, and therefore it may be viewed as a bottom-up construction. Nowadays, since the electroweak theory provides us a more fundamental description of weak interactions, we may derive Fermi’s theory in a top-down fashion, as a low energy approximation of a known high energy theory.

8.2.1

Fermi theory as a phenomenological description

If we consider the Standard Model at a scale of the order of the nucleon mass, i.e. around a GeV, it contains only the leptons, the light quarks, and the massless gauge 1 This example illustrates an ambiguity one faces when coupling a field theory to gravity: only energy differences matter for the dynamics of the field theory, but the absolute value of the energy enters in the energy-momentum tensor that acts as a source in Einstein’s equations.

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bosons (photon and gluons). Thus, this low energy truncation has no mechanism for weak decays. Nevertheless, one may write an effective coupling involving a proton, a neutron (here, we prefer to use hadrons, that are the states encountered in actual experimental situations), an electron and the corresponding neutrino. The most general local operator combining these four fields may be written as

g12 ψp Γ1 ψn ψe Γ2 ψν , 2 Λ

(8.9)

where g12 is a dimensionless constant, Λ is a dimensionful scale, and Γ1,2 are matrices chosen in the following set  Γ1,2 ∈ 1, γ5 , γµ , γµ γ5 , 4i [γµ,γν ] . (8.10) | {z } σµν

Note that σµν γ5 is not linearly independent from these matrices, since σµν γ5 ∝ ǫµνρσ σρσ , and therefore need not be included in this list. Thus, the most general Lorentz invariant Lagrangian involving these four fields reads

Leff = ψp γµ ψn ψe γµ (CV + CV′ γ5 )ψν

+ ψp γµ γ5 ψn ψe γµ γ5 (CA + CA′ γ5 )ψν {z } | vector, axial

+ ψp ψn ψe (CS + CS′ γ5 )ψν

+ ψp γ5 ψn ψe γ5 (CP + CP′ γ5 )ψν | {z } scalar, pseudo-scalar

+ ψp σµν ψn ψe σµν (CT + CT′ γ5 )ψν . | {z } tensor

(8.11)

Note that the presence of certain terms violate some discrete symmetries. For instance, ′ the primed terms CV,S,P,T all violate parity, and T -invariance requires that the ratio ′ Ci /Ci be real for all i ∈ {V, A, S, P, T }. On the other hand, by confronting this effective Lagrangian with the existing data on weak decays, we learn that CV = Λ−2

with Λ ∼ 350 GeV ,

CA ≈ 1.25 × CV ,

CV ∼ CV′ ,

CA ∼ CA′ ,

′

CS,P,T CS,P,T , . 1% . CV CV

(8.12)

The first of these results is an indication of the energy scale at which the Fermi theory breaks down and should be replaced by a more accurate microscopic description of

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weak decays, and the second one implies that this underlying theory is chiral . The ′ fact that CV,A ∼ CV,A is a sign of parity violation in weak interactions. Finally, the last property tells us that this microscopic interaction is not mediated by a scalar or a tensor with a mass less than ∼ 2 TeV. All these informations may be used in constraining the possible form of the theory that describes weak interactions at higher energies.

8.2.2

Fermi theory from the electroweak model

Let us now consider the opposite exercise: namely, start from the Lagrangian of the Standard Model and obtain the low energy effective theory of weak interactions by a matching procedure. We know that the W ± bosons responsible for weak decays couple to left-handed fermions arranged in SU(2) doublets:   ν  d e , (8.13) , u L e L where we have written only the relevant doublets for the decay n → peνe . In addition, we have to keep in mind that the mass eigenstates are misaligned with the weak interaction eigenstates in the quark sector. Thus, the vertex Wud contains a factor Vud from the CKM matrix. With these ingredients, the tree level decay amplitude d → ueνe reads: A=

   i g2 µ µ uγ (1 − γ )d eγ (1 − γ )ν Vud 2 5 5 e , 8 k − M2W

(8.14)

where kµ is the 4-momentum carried by the intermediate W boson. In the low momentum limit, k2 ≪ M2W , this amplitude becomes independent of the momentum transfer and could have been generated by the following contact interaction    G Leff = √F Vud ψu γµ (1−γ5 )ψd ψe γµ (1−γ5 )ψν 2

g2 G . with √F ≡ 8 M2W 2 (8.15)

In order to obtain from this the physical decay amplitude n → peνe , we need the matrix element 

p ψu γµ (1 − γ5 )ψd n (8.16)

with initial and final nucleons instead of quarks. In the low momentum limit, it may be related to a similar matrix element with the spinors of the proton and neutron by  

p ψu γµ (1 − γ5 )ψd n = p ψp γµ (gV − gA γ5 )ψn n + O(kµ ) , (8.17)

8. E FFECTIVE FIELD THEORIES

267

where gV,A are two constants that may be viewed as the zero momentum limit of some form factors. Then, by comparing the decay amplitudes obtained from the low energy effective theory guessed on the basis of phenomenological considerations, and the one obtained by starting from the electroweak theory, we obtain CV = −CV′ = gV

1 g2 Vud = 2 , 2 8 MW Λ

CA = −CA′ = −gA

g2 Vud , 8 M2W

′ CS,P,T = CS,P,T =0.

(8.18)

In this top-down approach, we see that the parity violation inferred from experimental evidence is in fact maximal in the electroweak theory, and that the scalar and tensor contributions are exactly zero. Note also that the scale Λ that we introduced by hand in the low energy effective theory does not coincide exactly with the mass of the heavy particle which is integrated out (in the present case, the W boson), but has the same order of magnitude. Finally, even though we performed here the matching at tree level, it is in principle possible to correct the coefficients of the low energy effective theories by electroweak and QCD loop corrections.

8.3 Standard model as an effective field theory 8.3.1

Standard Model

The Standard Model unifies the strong and electroweak interactions into a unique renormalizable field theory. Although it agrees with most observed phenomena2 , it is unreasonable to expect that the Standard Model remains an accurate description of particle physics to arbitrarily high energy scales. A more modest point of view is to consider the Standard Model as a low energy approximation of some more fundamental theory that we do not yet know. In this perspective, it would just be the zeroth order of some expansion, L = LSM + L(1) + L(2) + · · · , |{z} |{z} |{z} Λ0

Λ−1

(8.19)

Λ−2

and a natural endeavor is to construct the terms L(1,2,··· ) , made of operators with mass dimension greater than four. By power counting, these operators must be suppressed by coupling constants that are inversely proportional to powers of some high energy scale Λ at which corrections to the Standard Model become important. In the construction of these corrections, one usually abides by the following constraints: 2 One exception is the fact that neutrinos have masses, that does not have a very compelling explanation in the Standard Model – we shall return to this issue in the next subsection.

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Figure 8.1: Left: a higher dimensional operator provides a correction (light) to an observable which is non-zero in the Standard Model (dark). Right: the higher dimensional operator allows a process that was impossible in the Standard Model. In the latter case, experiments usually provide an upper value for the yield of these very rare processes, that decreases as the sensitivity improves, thereby pushing higher up the energy scale of this new physics.

• Lorentz invariance is preserved to all orders in Λ−1 , • The SU(3) × SU(2) × U(1) gauge symmetry of the Standard Model remains a symmetry of the higher order corrections (the idea being that whatever is the more fundamental theory that underlies the Standard Model, it is more symmetric, not less), • The corrections are built with the degrees of freedom of the Standard Model, • The vacuum expectation value of the Higgs is not modified by the corrections. As we have mentioned earlier, since the Standard Model is renormalizable, there is no way to determine the scale Λ within the Standard Model itself. Instead, one should enumerate the higher dimensional operators up to a certain mass dimension (which corresponds to a certain order in Λ−1 ) and investigate their possible observable consequences. Experiments can then search for these effects, and either provide the values of some of the parameters introduced in L(1,2,··· ) , or give lower bounds on the scale of new physics in case of a null observation. Note that there are two main classes of higher dimensional operators, illustrated in the figure 8.1: c sileG siocnarF

• Operators that lead to corrections to processes already allowed in the Standard Model. These corrections may become potentially visible in more precise experiments. • Operators that allow processes that were forbidden in the Standard Model. In this case, what is needed are more sensitive experiments, able to detect extremely rare events.

8. E FFECTIVE FIELD THEORIES

8.3.2

269

Dimension 5 operators and neutrino masses

The right handed neutrinos are singlet under SU(3) and SU(2) and have a null electrical charge, which means that they do not feel any of the interactions of the Standard Model. As a consequence, all the neutrinos detected in experiments (via their weak interactions with the matter of the detector) are left handed neutrinos, implying that there is no direct evidence for the existence of right handed neutrinos. For this reasons, right handed neutrinos are usually not considered as a part of Standard Model. The observation of neutrino oscillations, i.e. the fact that the flavour of a neutrino can change as it propagates, implies that there are non-zero mass differences between neutrinos3 . Therefore, at most one of the neutrinos can be massless, and at least two of them must be massive. Neutrino masses from the Higgs mechanism : Since the electroweak theory is chiral (right handed leptons are SU(2) singlet, while the left handed ones belong to SU(2) doublets), a naive Dirac mass term of the form mD ψL ψR is not invariant under SU(2). However, we may construct such a Dirac mass in the same way as for the other leptons, by starting from a Yukawa coupling involving the Higgs boson:
λ ψL ,iα ǫij Φ∗j ψR ,α , (8.20)

where i, j are indices in the fundamental representation of SU(2) and α is a Dirac index. The matrix ǫ ≡ it2 is the second generator of the fundamental representation of SU(2). Thanks to the contraction of the left handed spinor doublet with the Higgs field, we now have an SU(2) invariant combination. Then, spontaneous symmetry breaking gives a non-zero expectation value v to the Higgs field, and this interaction term becomes a Dirac mass term for the neutrino, with a mass mD = λ v. Generating the neutrino mass by this mechanism would place the neutrinos almost on the same footing as the other leptons, provided we add right handed neutrinos to the degrees of freedom of the Standard Model4 . The only distinctive feature of the right handed 3 Consider for instance a β decay: it produces an electron anti-neutrino (i.e. a weak interaction eigenstate) of definite momentum. If mass eigenstates are misaligned with the weak interaction eigenstates, then this neutrino may project on several mass eigenstates. Since the time evolution of the phase of a wavefunction depends on the mass of the particle, these mass eigenstates evolve slightly differently in time (unless all the neutrino masses are identical). At the detection time, this leads to a flavour decomposition which is different from the one at the time of production. Thus, the original electron anti-neutrino will be a mixture of electron, muon and tau anti-neutrinos. Conversely, the observation of this change of flavour implies mass differences in the neutrino sector. 4 Whether this type of term is “beyond the Standard Model” is to a large extent a matter of definition. Before the observation of neutrino oscillations, the Standard Model was most often defined without right handed neutrinos, and therefore massless neutrinos. But it would have been equally acceptable to include right handed neutrinos from the start, with Yukawa couplings so small that their masses were too small to detect.

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neutrinos would be that they do not feel any of the gauge interactions of the Standard Model. For this reason, they are sometimes called sterile neutrinos. The main drawback of this solution is that it requires an even larger range of values of the Yukawa couplings, with no natural explanation. Majorana neutrino masses : An alternative would be to have a Majorana mass for the left handed neutrinos of the Standard Model. Instead of introducing this mass term by hand, it can be generated via spontaneous symmetry breaking from a Weinberg operator:

c ψtL ,iα ǫij Φj Cαβ Φtk ǫkl ψL ,lβ , Λ

(8.21)

where C ≡ γ0 γ2 is the charge conjugation operator. Firstly, note that this operator has mass dimension 5, hence the coupling constant proportional to Λ−1 . In fact, this operator is the only lepton number violating 5-dimensional operator that obeys the constraints listed in the previous section5 . After spontaneous symmetry breaking, the Higgs field acquires a vacuum expectation value, leading to a Majorana mass term for the left handed neutrinos, c v2 t ν C νL , Λ L

(8.22)

which corresponds to a Majorana mass mM = cv2 Λ−1 . The appeal of this mechanism, is that a small mass of the neutrinos is naturally explained by a high scale Λ for the new physics. For instance, a neutrino mass of the order of 1 eV or below corresponds to Λ & 1013 GeV. As we have already mentioned, the operator in eq. (8.21) does not conserve lepton number, since it is not invariant under the following global transformation ψ → eiα ψ ,

ψ† → e−iα ψ† ,

ψt → eiα ψt .

(8.23)

For this reason, this alternative mechanism is clearly beyond the Standard Model. However, as long as gauge symmetries are preserved, the violation of lepton number is not considered particularly dramatic. In a sense, one may view the lepton conservation that exists in the Standard Model as accidental, being a consequence of the fact that only dimension-four operators are included. Weinberg operator from the low energy limit of another QFT : In the spirit of the bottom-up construction of an effective theory, the operator of eq. (8.21) can 5 ψt

ǫij Φj and Φtk ǫkl ψL ,lβ are both SU(2) invariant (but not Lorentz invariant), and the combination Cαβ ψL ,lβ is Lorentz invariant. This combination is SU(3) invariant only for the leptons L ,iα (not for the quarks). L ,iα

ψt

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8. E FFECTIVE FIELD THEORIES

be obtained by exploring all the possibilities for dimension 5 operators built with the degrees of freedom of the Standard Model and some symmetry requirements. However, this operator can also be obtained in the low energy limit of a renormalizable quantum field theory. Consider an extension of the field content of the Standard Model, where we add a right handed neutrino νR with a very large Majorana mass MR (much heavier than the electroweak scale), that also couples to the SU(2) doublet containing the left handed neutrino and to the Higgs field via a Yukawa coupling, L = LSM + LνR ,



/ νR − y ψL ǫΦ∗ νR − y∗ νR Φt ǫ† ψL L νR ≡ i ν R ∂
1 + MR νtR C νR + M∗R νt∗ C ν∗R . R 2

(8.24)

With two instances of the Yukawa coupling and a propagator of the heavy Majorana Φ

Φ

Φ

ψL

ψL p

p n :

ta ij φcj 6= 0 ,

ta ij φcj = 0 .

Thus, the matrix R(g) can be written as ! n X a a R(θ) = exp i . θ t

(8.68)

(8.69)

a=1

The value of the potential does not change under the action of G on φc , and we are free to choose the value of its minimum to be V(φc ) = 0. Thus, the action becomes Z
µ
1 S= dd x φci ∂µ R−1 ∂ Rkj (θ) φcj ik (θ) 2 Z
1 dd x φci Aµ (θ)Aµ (θ) ij φcj , (8.70) =− 2

where in the second expression we have introduced Aµ ≡ R−1 ∂µ R (an element of the algebra). Eq. (8.70) gives the action in terms of the “coordinates” θa on the coset G/H, corresponding to a certain choice of the generators ta . However, it is interesting to express the action in terms of a completely arbitrary system of coordinates on G/H, that we may denote ϑm . Since eq. (8.70) has only two derivatives ∂µ · · · ∂µ , the same must be true of its expression in any system of coordinates. On the other hand, it may contain terms of arbitrarily high degree in ϑ. Thus, the most general action is of the form Z

1 dd x gmn (ϑ) ∂µ ϑm ∂µ ϑn , (8.71) S= 2

where the coefficients gmn (ϑ) can be related to R(θ) as follows:       b −1 ∂R a b a −1 ∂R tr t R . gmn (ϑ) ≡ −4 φci tik tkj φcj tr t R ∂ϑm ∂ϑn

(8.72)

They form a metric tensor on G/H, if the coset is viewed as a Riemannian manifold. Indeed, if we use a different system of coordinates ̟p on G/H, gmn (ϑ) would be replaced by       b −1 ∂R b a −1 ∂R tr t R gpq (̟) ≡ −4 φci ta t φ tr t R ik kj cj ∂̟p ∂̟q  m  n  ∂ϑ ∂ϑ gmn (ϑ) , (8.73) = ∂̟p ∂̟q

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8. E FFECTIVE FIELD THEORIES

which is indeed the expected transformation law of a metric tensor under a change of coordinates. The field theory described by the action (8.71) is called a non-linear sigma model. Note that the derivative ∂µ ϑm of the coordinate ϑm is a vector that lives on the tangent space to the manifold G/H at the point ϑ. Therefore, the action (8.71), in which the tensor gmn is contracted with two vectors, is a scalar – invariant under changes of coordinates on the manifold. c sileG siocnarF

The Taylor expansion of the metric in powers of the field ϑ determines which couplings exist in the classical action. Interestingly, even though the kinetic term of the original action was quadratic in the fields, we now have a term with two derivatives and possibly arbitrarily high orders in the field. Loosely speaking, this is due to the fact that spontaneous symmetry breaking has restricted the fields from a space n in which the symmetry G was linearly realized, down to a curved manifold in which it is realized non-linearly. In addition, it is worth stressing that the final action is uniquely determined from eq. (8.69), but may take various explicit forms depending on the choice of coordinates ϑm on G/H. In other words, the non-linear sigma model has an intrinsic geometrical meaning, that does not depend on the system of coordinates one uses.

❘

Path integral quantization : The quantization of the non-linear sigma model can be achieved via path integration. The action is quadratic in derivatives of the field, but with the unusual feature that these derivatives are multiplied by a function of the field. In order to ascertain the consequence of this property, it is necessary to start from the Hamilton formulation of the path integral, and to perform explicitly the integral over the conjugate momenta. For a Lagrangian density L=



1 gmn (ϑ) ∂µ ϑm ∂µ ϑn , 2

(8.74)

the conjugate momenta read πm ≡

∂L = gmn (ϑ) ∂0 ϑn , ∂∂0 ϑm

(8.75)

and the Hamiltonian is given by


1 1 H = πm ∂0 ϑm −L = gmn (ϑ) πm πn + gmn (ϑ) ∇ϑm · ∇ϑn , (8.76) 2 2

where gmn is the inverse of the metric tensor, gmn gnp = δm p . The Hamiltonian is quadratic in the momenta, but since the coefficient in front of πm πn depends on the field, the determinant produced in the Gaussian integration over the momenta cannot be disregarded. After this integral has been performed, the generating functional is given by the following formula  Z  Z p   Y   i g(x) Dϑm (x) exp , (8.77) Z[jm ] = dd x L(ϑ) + jm ϑm h ¯ m

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φc

G/H

Figure 8.7: Perturbative expansion in the non-linear sigma model: only field configurations near φc are explored.

where we denote g(x) ≡ det (gmn (ϑ(x))). Interestingly, the field dependence  of Q m Dϑ gmn (ϑ) alters the path integral in a rather natural way: the measure is m  √ Q replaced by g m Dϑm , which is invariant under changes of coordinates on the manifold G/H. Note that in eq. (8.77), we have introduced an explicit h, ¯ that will be useful later to keep track of the number of loops. The perturbative expansion in the non-linear sigma model corresponds to an expansion in powers of h. ¯ From the path integral, we can infer that the typical field amplitudes scale as √ ϑ∼ h ¯ , (8.78) which means that the perturbative expansion is also an expansion around ϑ = 0 (i.e. around φ = φc ). For such small fields, the effects of the curvature of the manifold are perturbative, and we can expand the metric tensor in powers of the field (an explicit choice of coordinates must be made for this). The bare propagator of the ϑ fields is given by Gmn (p) =

i δmn . + i0+

p2

(8.79)

Renormalization : Dimensional analysis tells that the field ϑ has the dimension ϑ ∼ (mass)(d−2)/2

(8.80)

(in a system of units where h ¯ = 1). From this, we see that there are three cases regarding the ultraviolet power counting in the non-linear sigma model:

8. E FFECTIVE FIELD THEORIES

289

• d < 2 : the Taylor coefficients of the metric tensor all have a positive mass dimension, and are therefore super renormalizable. • d = 2 : the Taylor coefficients are dimensionless, and the theory is renormalizable. • d > 2 : the Taylor coefficients have a negative mass dimension and are all non-renormalizable by power counting. The most interesting situation is therefore the two-dimensional case. It differs somewhat from the renormalization of the quantum field theories we have encountered until now, since the action contains an infinite series of terms (of increasing degree in ϑ), and an important question is whether the action (8.71) conserves its structure under renormalization. Recall that the fields ϑm transform under a non-linear representation of the group G. Thus, their variation under an infinitesimal transformation of parameters ǫa may be written as δϑm ≡ ǫa Tam (ϑ) ,

(8.81)

Tam (ϑ)

where the are smooth functions of the fields. Under the same transformation, the variation of the action reads Z δS δS = ǫa d2 x Tam (ϑ) m , (8.82) δϑ (x) and the invariance of the action under G thus requires that

∂Tap ∂Tap ∂gmn + g + g =0. (8.83) pn pm ∂ϑp ∂ϑm ∂ϑn In other words, the possible forms of the metric tensor are constrained by the symmetry G. Indeed, the coset G/H is an homogeneous space12 , i.e. a manifold that possesses additional symmetries that reduce the dimension of the space of allowed metrics. More precisely, an homogeneous space is such that given any pair of points ϑ and ϑ ′ on the manifold, there is an isometry (i.e. a distance preserving transformation) that maps ϑ to ϑ ′ . If in addition the space is isotropic, then it is said to be maximally symmetric13 . In an N-dimensional maximally symmetric space, there is a particularly simple relationship between the metric and curvature tensors: Tap

Rmn

=

Rmnpq

=

R gmn (R ≡ Rm m ) , N
R gmp gnq − gmq gnp . N(N − 1)

(8.84)

12 Thanks to their connections to Lie algebras, a systematic classification of homogeneous spaces is possible. 13 A maximally symmetric manifold of dimension N has N(N + 1)/2 distinct isometries. In Euclidean space, this corresponds to N translations and N(N − 1)/2 rotations, but this maximal number of isometries is the same in N-dimensional manifolds with curvature.

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(These two identities imply that the scalar curvature R is constant over the entire manifold for a dimension N > 2.) A possible strategy for studying the renormalization of the sigma model is to introduce an analogue of the BRST transformation of non-Abelian gauge theories, and the associated Slavnov-Taylor identities obeyed by the quantum effective action. These identities, combined with dimensional and symmetry arguments that restrict the terms that may arise in the renormalized action, are sufficient to show that the renormalized action is structurally identical to eq. (8.71), with a group-invariant metric tensor that obeys a renormalized version of eqs. (8.83). c sileG siocnarF

Example of G = O(n) : A scalar field φi with n components has an O(n) symmetry if the action depends only on the combination φi φi . Potentials with non-trivial minima (i.e. at φ 6= 0) in fact have infinitely degenerate minima that form a (n − 1)-dimensional sphere Sn−1 (see the figure 8.6 for an illustration in the case n = 3). Each minimum has a stabilizer subgroup H = O(n − 1) (the smaller group of rotations around the direction fixed by this minimum), and we indeed have Sn−1 = O(n)/O(n − 1). A possible explicit parameterization of the field φ consists in writing  φ ≡ σ, ξ , (8.85)

where σ has one component and ξ has n − 1 components. Assuming the parameters of the potential are adjusted so that the sphere Sn−1 of minima has radius φ = 1, we must impose the constraint σ2 + ξ2 = 1, which means that σ may be viewed as a dependent field that depends non-linearly on ξ. Usually, these coordinates are
chosen in such a way that the symmetry-breaking vacuum is φc = σ = 1, ξ = 0 . In the vicinity of φc , σ is the “radial” massive field, while the ξi are the “angular” variables corresponding to the massless Nambu-Goldstone bosons. Then, we may split the generators of the o(n) algebra into those of the stabilizer o(n − 1) and the complementary set of generators: • The generators of o(n−1) act linearly on ξ. More precisely, they leave σ2 +ξ2 invariant by leaving both σ and ξ2 unchanged (thus simply rotating the n − 1 components of ξ). • In contrast, the generators of the complementary set preserve σ2 + ξ2 , but mix σ and ξ as follows: σ ξi

→ σ − ǫ i ξi , q i i 1 − ξ2 , → ξ +ǫ

and therefore they act non-linearly on ξ.

(8.86)

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8. E FFECTIVE FIELD THEORIES

σ Figure 8.8: Illustration of the (σ, ξ) coordinates for an O(3) model. The dark circle corresponds to the transformations that preserve σ and act linearly on ξ (as an O(2) rotation). The light colored circles are the transformations that mix σ and ξ (and transform the latter non-linearly).

ξ1

ξ2

The most general O(n)-invariant action with σ2 + ξ2 = 1 reads S

= =

Z

 1 dd x (∂µ σ)(∂µ σ) + (∂µ ξ)(∂µ ξ) 2 Z 1 dd x gij (ξ) (∂µ ξi )(∂µ ξj ) , 2

(8.87)

where in the second line we have eliminated σ and we have defined gij (ξ) ≡ δij +

ξi ξ j 1 − ξ2

.

(8.88)

The tensor gij is the metric on the Sn−1 sphere, in the system of coordinates provided by the ξi . The couplings of this theory are determined by the Taylor expansion of the metric tensor, which in this case is completely specified by the choice of the coordinates and by the symmetries of the problem. In d = 2 dimensions, this theory is renormalizable by power counting. Although it contains an infinite number of couplings, it is not necessary to renormalize each of them individually. Instead, the renormalization preserves the structure of the action (8.87) with a metric tensor that remains dictated by the O(n) symmetry.

8.5.3

Nonlinear sigma model on a generic Riemannian manifold

We have derived the non-linear sigma model as the effective action that describes the dynamics of the massless Nambu-Goldstone bosons after a spontaneous breaking of symmetry. In this case, the fields of the non-linear sigma model live on a manifold which is also a homogeneous space thanks to the symmetries of the original problem.

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These symmetries severely constrain the possible forms of the metric, and play an important role in constraining the form of the loop corrections. However, it is possible to consider an action of the form (8.71) for fields ϑm living on a generic smooth Riemannian manifold that does not possess any special symmetry. The power counting argument made earlier is unchanged, and we expect that this more general kind of sigma model is also renormalizable in 2 dimensions. For these generalized models, it has been shown that the dependence of the metric tensor (i.e. the function that defines all the couplings of the model) on the renormalization scale µ is governed by the following Callan-Symanzik equation: µ

1 mn 1 ∂ mn g =− R − 2 Rmpqr Rn pqr + higher orders . ∂µ 2π 8π

(8.89)

Note that if we apply this equation in the case of a maximally symmetric space, for which the curvature tensors have simple expressions in terms of the metric tensor, it reduces to µ

h i R R ∂ mn g =− gmn 1 + + ··· . ∂µ 2π N 2π N(N − 1)

(8.90)

Thus, in this special case, the metric is rescaled but retains its form under changes of scale (because it is constrained by the isometries of the manifold). On a generic manifold, the scale evolution governed by eq. (8.89) explores a much broader space of metrics. Generally speaking, the renormalization flow tends to expand the regions of negative curvature and to shrink those of positive curvature.

Figure 8.9: Left to right : successive stages of the Ricci flow on a 2-dimensional manifold.

There is an interesting analogy between the renormalization group eq. (8.89) and the Ricci flow, ∂τ gmn = −2 Rmn ,

(8.91)

8. E FFECTIVE FIELD THEORIES

293

introduced independently in mathematics by Hamilton in 1981 as a tool for studying the geometrical classification of 3-dimensional manifolds14 . In a sketchy way, the idea is to start with a generic metric tensor on the manifold, and to smoothen this metric by evolution with the Ricci flow (the Ricci flow is somewhat analogous to a heat equation, that tends to uniformize the temperature distribution). For instance, if the metric evolves into one that has a constant positive curvature, one would have proved that the original manifold is homeomorphic to a sphere. For 2-dimensional manifolds, this is indeed what happens: the Ricci flow evolves the metric tensor into one that has a constant scalar curvature, corresponding to one of the three possible geometries. Applications of Ricci flow to 3-dimensional manifolds turned out to be complicated by singularities that develop as the metric evolves, and required additional steps known as “surgery” to excise the singularities. There is nowadays some speculation about whether the additional terms in eq. (8.89) compared to eq. (8.91) have a regularizing effect that may prevent the appearance of these singularities and thus make the surgical steps unnecessary.

14 In 2 dimensions, connected manifolds are known to fall into three geometrical classes: flat, spherical or hyperbolic, depending on their curvature. More precisely, any such 2-dimensional manifold can be endowed with a metric that has a constant scalar curvature, either null, positive or negative. Thurston geometrization conjecture proposed a similar –but much more complicated– classification of 3-dimensional manifolds. In particular, this conjecture contains as a special case Poincar´e’s conjecture, stating that every closed simply connected 3-dimensional manifold is homeomorphic to a 3-sphere. The geometrization conjecture was proved in 2003 by Perelman, with techniques in which the Ricci flow plays a central role.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 9

Quantum anomalies Noether’s theorem states that for each continuous symmetry of a classical Lagrangian, there exists a corresponding conserved current. By construction, this conservation law holds at tree level, and a very important question is whether it is preserved by quantum corrections in higher orders of the theory. Quantum anomalies are situations where a classical symmetry is violated by quantum effects. We have already encountered anomalies in the section 3.5, where we saw that the fermionic functional measure is not invariant under chiral transformations of massless fermions, which had interesting connections with the index of the Dirac operator (its zero modes in the presence of an external field). When such an anomaly arises in a global symmetry like chiral symmetry, its effect is just to introduce a corrective term into the conservation equation of the corresponding current (which may have some physical consequences, however). But when it affects a local gauge symmetry, its effects are devastating, since the renormalizability and unitarity of gauge theories relies on the validity to all orders of the gauge symmetry. In general, gauge theories with an anomalous gauge symmetry do not make sense, and it is therefore of utmost importance to check that no such gauge anomaly is present in theories of phenomenological relevance. c sileG siocnarF

9.1 Axial anomalies in a gauge background 9.1.1

Two dimensional example: Schwinger model

The simplest example of theory that exhibits a quantum anomaly is quantum electrodynamics in two dimensions with massless fermions, also known as the Schwinger model. The Lagrangian of this theory reads: 1 / Ψ − Fµν Fµν , L ≡ i Ψ¯ D 4

(9.1)

295

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where Dµ ≡ ∂µ − ie Aµ and Fµν ≡ ∂µ Aν − ∂ν Aµ . This Lagrangian is invariant under (local) U(1) transformations, Ψ(x) Aµ (x)

eieχ(x) Ψ(x) , Aµ (x) + ∂µ χ(x) ,

→ →

(9.2)

which, by Noether’s theorem, implies the existence of a conserved electromagnetic current: Jµ ≡ −ie Ψ γµ Ψ

∂µ J µ = 0 .

,

(9.3)

(In the following, this current will be called a vector current.) Being a gauge symmetry, this invariance is crucial for the unitarity of the theory, since it ensures that longitudinal photons do not contribute as initial or final states of physical amplitudes. Because the fermions are massless, this theory has another symmetry. In order to see it, let us introduce1 a matrix γ5 , γ5 =

1 ǫµν γµ γν = γ0 γ1 , 2

(9.4)

where ǫµν is the 2-dimensional completely antisymmetric tensor, normalized by ǫ01 = +1. Using γ5 , one may decompose Ψ in its left and right handed components:

Ψ = Ψ R + ΨL ,

ΨR ≡

1 + γ5 Ψ, 2

ΨL ≡

1 − γ5 Ψ, 2

(9.5)

and the fermionic part of the Lagrangian can be rewritten as / Ψ = i Ψ†R γ0 D / ΨR + i Ψ†L γ0 D / ΨL . i ΨD

(9.6)

In other words, the kinetic term does not mix the left and right components (this would not be true with a mass term). As a consequence, the Lagrangian is invariant if we multiply the left and right components by independent phases, ΨR

→

eiα ΨR

,

ΨL

→

eiβ ΨL .

(9.7)

Note that this is a global invariance, unlike the gauge symmetry discussed previously. Equivalently, the massless Dirac Lagrangian is invariant under the following global transformation, Ψ 1 It

5

→

eiθγ Ψ ,

is possible to define γ5 ≡

ir−1

(2r)!

γ5

in any even space-time dimension D = 2r, as follows

ǫµ1 µ2 ···µ2r γµ1 γµ2 · · · γµ2r .

(9.8)

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9. Q UANTUM ANOMALIES

that amounts to multiplying by conjugate phases the left and right components (because of the γ5 in the exponential). Since this is a continuous symmetry, Noether’s theorem also applies here and tells us that the axial current is conserved: 5 µ Jµ 5 ≡ −ie Ψ γ γ Ψ

,

∂µ J µ 5 =0.

(9.9)

Figure 9.1: Left: 1-loop contribution to the vector current in a background gauge potential (the wavy line terminated by a cross represents the background field). Right: 1-loop contribution to the axial current.

The conservation laws (9.3) and (9.9) have been obtained with Noether’s theorem, from the fact that the classical Lagrangian possesses certain continuous symmetries. Let us now study how the vector and axial currents are modified at 1-loop. Here, we consider a fixed configuration of the gauge potential Aµ (x), that acts as a background external field (this also means that the photon kinetic term plays no role in this discussion). The lowest order 1-loop graphs that contribute to these currents are shown in the figure 9.1. The expectation values of the currents resulting from these graphs can be written as

µ 

µ  eJ (q) = Πµν (q) A e ν (q) , eJ (q) = Πµν (q) A e ν (q) , (9.10) 5 5

(the tilde denotes the Fourier transform of the external field) where the self-energies Πµν and Πµν 5 are given by µν

iΠ

(q) ≡

i Πµν 5 (q) ≡


/γν (k /+q /) tr γµk dD k e , (2π)D (k2 + i0+ )((k + q)2 + i0+ )
Z D /γν (k /+q /) tr γ5 γµk d k . e2 D 2 + 2 (2π) (k + i0 )((k + q) + i0+ ) 2

Z

(9.11)

(The only difference between them is the γ5 inside the trace, that comes from the definition of the axial current). In order to secure the subsequent manipulations, let us assume that some regularization has been performed on the momentum integrals, without specifying it for now. The denominators can be arranged into a single factor by using Feynman’s parameterization, Z1 1 1 = , (9.12) dx 2 (k2 + i0+ )((k + q)2 + i0+ ) (l + ∆(x))2 0

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where we have introduced l ≡ k + x q and ∆(x) ≡ x(1 − x) q2 . After calculating the trace, the vector-vector self-energy can be written as follows: Πµν (q) = A(q2 ) gµν − B(q2 )

qµ qν , q2

(9.13)

where the coefficients A(q2 ) and B(q2 ) are given by the following integrals: 2

A(q ) ≡ B(q2 ) ≡

2

−iDe

−iDe2

Z

Z

dD l (2π)D dD l (2π)D

Z1 0

Z1 0


2 ∆(x) + D − 1 l2 dx , (l2 + ∆(x))2 dx

(l2

2 ∆(x) . + ∆(x))2

(9.14)

In D = 2 spacetime dimensions, the second integral is finite and gives: B(q2 ) =

D=2

e2 , π

(9.15)

while the first integral is ambiguous. Indeed, the term in l2 in the numerator leads 2 − 1 that vanishes to an ultraviolet divergence, but it is multiplied by the factor D precisely when D = 2. If we use a cutoff as ultraviolet regulator, this term would vanish and we would have A = B/2, which would violate the conservation of the vector current at one-loop. In dimensional regularization, in contrast, the factor 2 D − 1 compensates a pole in 1/(D − 2) that comes from evaluating the integral in D dimensions, leaving a finite but non-zero result. In fact, in dimensional regularization we obtain A = B, and the conservation of the vector current holds at one-loop. No matter which regularization procedure we adopt, it must give A = B for vector current conservation, i.e. for preserving gauge symmetry at 1-loop. c sileG siocnarF

Let us now turn to the axial-vector self-energy we obtain
tr γ5 γµ γν = −D ǫµν ,

Πµν 5 .

Using the definition of γ5 ,

(9.16)

and

/ νB / tr γ5 γµAγ




= =




/ + Bν tr γ5 γµA / − A · B tr γ5 γµ γν Aν tr γ5 γµ B h i −D ǫµσ Bσ Aν + Aσ Bν − (A · B) gσ ν . (9.17)

This identity leads to h qσ qν i µσ Πσ ν (q) = −ǫµσ A(q2 ) gσ ν − B(q2 ) 2 , Πµν 5 (q) = −ǫ q

(9.18)

299

9. Q UANTUM ANOMALIES

where A and B are the same coefficients as in eq. (9.13). Therefore, the divergence of the axial current is given by

 2 µν e ν (q) . qµ eJµ qµ A (9.19) 5 (q) = −A(q ) ǫ

If we have adopted a regularization that preserves gauge symmetry, i.e. such that A = B, this divergence is non-zero and reads

 e2 µν e ν (q) , ǫ qµ A qµ eJµ (q) = − 5 π

(9.20)

or, going back to coordinate space:

 e2 µν e2 µν ∂ µ Jµ ǫ ∂µ Aν (x) = − ǫ Fµν (x) . 5 (x) = − π 2π

(9.21)

The non-conservation of the axial current at one loop is the unavoidable conclusion in any regularization scheme that preserves the conservation of the vector current. Moreover, since when this is the case A becomes equal to the ultraviolet finite coefficient B, it does not suffer from any scheme dependence, and the above result may thus be viewed as a scheme-free result. The result (9.21) is known as an axial anomaly. A somewhat milder conclusion of this 2-dimensional exercise is that it not possible to preserve both vector and axial current conservation at one-loop. We could in principle adopt a regularization scheme that conserves the axial current, which requires A = 0. But the price to pay would be the loss of gauge invariance at 1-loop. Since gauge invariance is deemed more fundamental (in particular, it ensures the unitarity of the theory), this route is generally not considered further. Note that ultraviolet divergences are necessary2 for the existence of this anomaly. Indeed, at the classical level, the Lagrangian density is invariant under the global transformation: 5

Ψ → eiθγ Ψ

,

5

Ψ† → Ψ† e−iθγ .

(9.22)

The Feynman graphs that contribute to the expectation value of the axial current in a background electromagnetic field have an equal number of Ψ’s and Ψ† ’s (this statement is true to all orders of perturbation theory). Since the axial symmetry is global, when we apply the above axial transformation to a graph, all the factors exp(±iθγ5 ) should naively cancel, leaving a result that does not depend on θ. This conclusion would indeed be correct if all the integrals were finite, but may be invalidated by the subtraction procedure necessary to obtain finite results in the presence of divergences. In the explicit example that we have studied, the ultraviolet regularizations that are consistent with gauge symmetry all spoil axial symmetry. 2 In a certain sense, the axial anomaly is also an infrared effect since it exists only for massless fermions (for massive fermions, there is no axial symmetry to begin with). Moreover, as we have already seen when discussing the Atiyah-Singer index theorem, the axial anomaly is related to the zero modes of the Dirac operator in a background field.

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Beyond one loop, a graph contributing to the expectation value of the axial current may contain subgraphs that are ultraviolet divergent. However, since QED is renormalizable, these sub-divergences will all have been made finite thanks to counterterms calculated in the previous orders of the perturbative expansion. Thus, we need only to study the intrinsic ultraviolet divergence of the graph under consideration, an indicator of which is given by its superficial degree of divergence. For the sake of definiteness, let us assume that the graph G has nψ fermion propagators, nγ photon propagators, nV photon-fermion-fermion vertices, nA insertions of the external electromagnetic field and nL loops (plus one extra vertex where the axial current is attached). These quantities are not independent, but obey the following identities: 2nγ = nV , 2nψ = 2 + 2(nV + nA ) , n L = nψ + nγ − nA − nV .

(9.23)

Using these relations, the superficial degree of divergence of the graph reads: ω(G) ≡ 2nL − nψ − 2nγ = 2 − nψ .

(9.24)

The simplest graph that contributes to the axial current, shown in the figure 9.1, has nψ = 2 and therefore has a logarithmic ultraviolet divergence. More complicated graphs, either with more insertions of the external field or with more than one loop, all have nψ > 2 and are therefore convergent after all their sub-divergences have been subtracted. This argument, although it lacks some rigor, indicates that the axial anomaly does not receive any correction beyond the one-loop result, and that eq. (9.21) is therefore an exact result. An alternate justification of this property is based on the derivation of the axial anomaly from the fermionic path integral, which gives the determinant of the Dirac operator in the background field. Indeed, as we have seen in the section 2.5, functional determinants correspond to 1-loop diagrams.

9.1.2

Axial anomaly in four dimensions

γ5 in four dimensions : Let us now turn to a more realistic 4-dimensional example, that also has some relevance in understanding the decay of pseudo-scalar mesons like the π0 . The setup is exactly the same as in the previous section, except that we consider now four space-time dimensions. The main modification is the definition of the γ5 matrix, γ5 =

D=4

i ǫµνρσ γµ γν γρ γσ = i γ0 γ1 γ2 γ3 . 4!

(9.25)

The traces of a γ5 with any odd number of ordinary Dirac matrices are all zero,
tr γ5 γµ1 · · · γµ2n+1 = 0 . (9.26)

301

9. Q UANTUM ANOMALIES

In order to evaluate the traces of γ5 with an even number of Dirac matrices, let us firstly recall the general formula for a trace of an even number of Dirac matrices: X Y
sign (P) gµs1 µs2 , (9.27) tr γµ1 · · · γµ2n = D pairings P

s∈P



where a pairing P is a set of pairs P = (s1 s2 ), (s1′ s2′ ), · · · made of the integers in [1, 2n]. The signature of P, denoted sign (P), is the signature of the permutation that reorders the sequence s1 s2 s1′ s2′ · · · into 1234 · · · . Since the Minkowski metric tensor gµν is diagonal, each Lorentz index carried by one of the Dirac matrices must coincide with the Lorentz index of another matrix in order to obtain a non vanishing result. Hence, we have

tr γ5 = i tr (γ0 γ1 γ2 γ3 = 0 . (9.28) The same is true if the γ5 is accompanied by only two ordinary Dirac matrices,

tr γ5 γµ γν = i tr (γ0 γ1 γ2 γ3 γµ γν = 0 , (9.29)

and the simplest non-zero trace is tr (γ5 γµ γν γρ γσ ). By the previous argument, each of the indices µνρσ must match one of the indices 0123 hidden in γ5 = i γ0 γ1 γ2 γ3 . Therefore, µνρσ must be a permutation of 0123. Since the four Dirac matrices are all distinct, they all anticommute, and the result is completely antisymmetric in µνρσ, so that we have
tr γ5 γµ γν γρ γσ = A ǫµνρσ . (9.30) c sileG siocnarF

In order to calculate the prefactor, we just need to evaluate the trace for a particular assignment of the indices, for instance µνρσ = 3210,

3210 5 3 2 1 0 0 1 2 3 3 2 1 0 Aǫ | {z } = tr γ γ γ γ γ = i tr γ γ γ |γ {zγ } γ γ γ = −4 i . (9.31) +1

This gives A = −4 i, i.e.
tr γ5 γµ γν γρ γσ = −4 i ǫµνρσ .

|

|

|

−1

{z

+1

{z

−1

{z

−1

}

}

} (9.32)

Order 1 in the external field : Let us now turn to the calculation of the expectation value of the axial current in four dimensions. The simplest graph to consider is again the graph on the right of the figure 9.1. Its contribution to axial current is

µ  eJ (q) = Πµν (q) A e ν (q) , (9.33) 5 5

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 9.2: Graph contributing to the chiral anomaly in a gauge background in four space-time dimensions.

with i Πµν 5 (q)

≡ =


/γν (k /+q /) tr γ5 γµk dD k e (2π)D (k2 + i0+ )((k + q)2 + i0+ )
Z D Z1 / )γν (l/ + (1 − x)q /) tr γ5 γµ (l/ − xq d l 2 e dx , (2π)D 0 (l2 + ∆(x))2 (9.34) 2

Z

where we have introduced the Feynman parameterization in the second line, and the new integration variable l ≡ k + xq. The trace that appears in the numerator is proportional to ǫµανβ (l − xq)α (l + (1 − x)q)β ∝ ǫµανβ lα qβ ,

(9.35)

and is therefore odd in the momentum l. Therefore, the momentum integral vanishes, and this graph does not contribute to the axial current. Order 2 in the external field : At second order in the external field, we encounter the graph of the figure 9.2. Its contribution to the expectation value of the axial current reads Z 4 4

µ  eJ (q) = 1 d k1 d k2 (2π)4 δ(q + k1 + k2 ) 5 2! (2π)8 e ν (k1 )A e ρ (k2 ) , ×Γ µνρ (q, k1 , k2 ) A (9.36) 5

where we have introduced the following three-point function:

i Γ5µνρ (q, k1 , k2 ) ≡
Z D /+a / +k /1 )γν (k /+a / )γρ (k /+a / −k /2 ) tr γ5 γµ (k d k 3 ≡e (2π)D ((k+a+k1 )2 +i0+ )((k+a)2 +i0+ )((k+a−k2 )2 +i0+ )
Z D /+b / +k /2 )γρ (k /+b / )γν (k /+b / −k /1 ) tr γ5 γµ (k d k 3 +e . (2π)D ((k+b+k2 )2 +i0+ )((k+b)2 +i0+ )((k+b−k1 )2 +i0+ ) (9.37)

303

9. Q UANTUM ANOMALIES

The two terms correspond to the two ways of attaching the fields with momenta k1 and k2 to the external photon lines. For a reason that will become clear later, we have taken the freedom to introduce independent shifts a and b of the integration variables in the two terms. Such shifts would of course have no effect on convergent integrals, since they just correspond to a linear change of variable. However, we are here in the presence of linearly divergent integrals, and these shifts have a nontrivial interplay with the ultraviolet regularization. Note that since {γ5 , γα } = 0, we may move the γ5 just before the matrices γν or γρ without changing the integrand, as if the axial current was attached at the other summits of the triangle (where the momenta k1 or k2 enter, respectively). Next, in order to test the conservation of the axial current, we contract this amplitude with qµ , that we may rewrite as follows: qµ

=

−(k1 + k2 )µ

=

(k + a − k2 )µ − (k + a + k1 )µ

=

(k + b − k1 )µ − (k + b + k2 )µ .

(9.38)

This leads to Z D d k ανβρ qµ Γ5µνρ (q, k1 , k2 ) = 4e3 ǫ (2π)D  (k1 )α (k + a)β × 2 ((k+a) + i0+ )((k+a+k1 )2 +i0+ ) (k2 )α (k + a)β ((k+a)2 + i0+ )((k+a−k2 )2 +i0+ ) (k1 )α (k + b)β − ((k+b)2 + i0+ )((k+b−k1 )2 +i0+ ) +

(k2 )α (k + b)β − ((k+b)2 + i0+ )((k+b+k2 )2 +i0+ )



.

(9.39)

By taking a = b = 0, and assuming a regularization that preserves Lorentz invariance, each term leads to a vanishing integral. Consider for instance the first term. Since k1 is the only 4-vector that enters in the integrand besides the integration variable k, the result of its integral is proportional to ǫανβρ (k1 )α (k1 )β = 0. Since the same reasoning applies to the four terms, we would therefore naively conclude that the axial current is conserved. However, we should make sure that the vector currents are also conserved. For this, we also need to calculate (k1 )ν Γ5µνρ and (k2 )ρ Γ5µνρ . The

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

same method as above gives (k1 )ν Γ5µνρ (q, k1 , k2 ) ×



3

= −4e

Z

dD k αµβρ ǫ (2π)D

(k + a)α (k + a − k2 )β ((k+a)2 + i0+ )((k+a−k2 )2 +i0+ )

(k + a + k1 )α (k + a − k2 )β ((k+a+k1 )2 + i0+ )((k+a−k2 )2 +i0+ ) (k + b + k2 )α (k + b − k1 )β + ((k+b+k2 )2 + i0+ )((k+b−k1 )2 +i0+ )  (k + b + k2 )α (k + b)β . − ((k+b)2 + i0+ )((k+b+k2 )2 +i0+ ) −

(9.40)

and Z D d k αµβν (k2 )ρ Γ5µνρ (q, k1 , k2 ) = −4e3 ǫ (2π)D  (k + a + k1 )α (k + a − k2 )β × ((k+a+k1 )2 + i0+ )((k+a−k2 )2 +i0+ ) (k + a + k1 )α (k + a)β ((k+a+k1 )2 + i0+ )((k+a)2 +i0+ ) (k + b)α (k + b − k1 )β + ((k+b)2 + i0+ )((k+b−k1 )2 +i0+ ) −

(k + b + k2 )α (k + b − k1 )β − ((k+b+k2 )2 + i0+ )((k+b−k1 )2 +i0+ )



.

(9.41)

It turns out that the choice a = b = 0 leads to non vanishing results for the conservation of the vector currents. Consider for instance (k1 )ν Γ5µνρ . With a = b = 0 and a regularization that preserves Lorentz invariance as well as reflection symmetry k → −k, we have: Z D d k αµβρ µνρ 3 (k1 )ν Γ5 (q, k1 , k2 ) = −8e ǫ (2π)D (k + k2 )α (k − k1 )β × ((k+k2 )2 + i0+ )((k−k1 )2 +i0+ ) ∝

ǫαµβρ (k2 )α (k1 )β 6= 0 .

(9.42)

A systematic search indicates that the only choice of a and b that gives a null result for both eqs. (9.40) and (9.41) is a = −b = k2 − k1 .

(9.43)

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9. Q UANTUM ANOMALIES

Since the conservation of the vector current is necessary in order to preserve gauge symmetry, and that the latter is a requirement for unitarity, we must adopt this choice. Returning to eq. (9.39) for the axial current with these values of a and b, we obtain: qµ Γ5µνρ (q, k1 , k2 ) = 16e3

Z

dD k ανβρ (k1 )α (k+k2 −k1 )β ǫ . (2π)D (k+k2 )2 +i0+ (k+k2 −k1 )2 +i0+ (9.44)

Let us define Fνρ (k) ≡ ǫανβρ

(k1 )α (k−k1 )β , k2 +i0+ (k−k1 )2 +i0+

(9.45)

and note that Z D d k νρ F (k) = 0 . (2π)D

(9.46)

(because with a Lorentz invariant regularization the result can only depend on the vector k1 , which would unavoidably give zero when contracted with the two free slots of the ǫανβρ .) Therefore, we can write qµ Γ5µνρ (q, k1 , k2 ) = =

Z

dD k (2π)D Z D d k 16e3 (2π)D

16e3

h h

i Fνρ (k + k2 ) − Fνρ (k) kσ 2

i ∂Fνρ (k) kσ kτ ∂2 Fνρ (k) + 2 2 + ··· . σ σ τ ∂k 2 ∂k ∂k (9.47)

Since the integrand now contains only derivatives, we can use Stokes’s theorem in order to rewrite the divergence of the axial current as a surface integral on the boundary at infinity of momentum space. If we view this boundary as the limit k∗ → ∞ of a sphere of radius k∗ , the “area” of this boundary grows like k3∗ in D = 4. On the other hand, the function Fνρ (k) behaves as k−3 , and each subsequent derivative decreases faster by one additional power of k−1 . Therefore, the result is given in full by the first term of the expansion: qµ Γ5µνρ (q, k1 , k2 ) = =

=

16e3

Z

dD k σ ∂Fνρ (k) k (2π)D 2 ∂kσ

16ie3 ανβρ ǫ (k1 )α (k2 )σ lim k∗ →∞ (2π)4

−i

e3 νραβ ǫ (k1 )α (k2 )β , 2π2

Z

S3 (k∗ )

|

d3 S

kσ kβ k k4

{z

}

π2 gσβ 2

(9.48)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

In the second line, S3 (k∗ ) is the 3-sphere of radius k∗ (i.e. the boundary of a 4-ball of radius k∗ ), kσ /k is the unit vector normal to the sphere, and the factor i arises when going to Euclidean momentum space. Note that we have anticipated the limit k → ∞ in order to simplify the function Fνρ (k). Therefore, the contribution of the triangle graph to the divergence of the axial current reads

 qµ eJµ 5 (q)

=

or in coordinate space

−i

Z

d4 k1 d4 k2 δ(q + k1 + k2 ) (2π)4 e ν (k1 )A e ρ (k2 ) , ×(k1 )α (k2 )β A

e3 νραβ ǫ 4π2

 e3 ανβρ ∂ µ Jµ ǫ Fαν (x) Fβρ (x) . 5 (x) = − 16π2

(9.49)

(9.50)

This is the main result of this section, namely the existence of an anomalous divergence of the axial current in the presence of a background electromagnetic field. In the course of the calculation, we have seen that depending on the labeling of the integration momentum, we can make the anomaly appear in any of the three external currents. In the situation considered here, with one axial current corresponding to a global symmetry, and two vector currents stemming from a local gauge symmetry, we must enforce the conservation of the vector currents and therefore assign in full the anomaly to the axial one. But the same calculation would arise in the context of a chiral gauge theory (where the left and right handed fermions belong to different representations of the gauge group). In this case, the natural choice would be to regularize the triangle so that the symmetry among the three currents is preserved, and the anomaly would then be equally shared by the three currents. c sileG siocnarF

Corrections : Let us now discuss potential corrections to the result (9.50). Firstly, we should examine one-loop graphs with more than two photons in addition to the insertion of the axial current. A simple dimensional argument can exclude that such graphs contribute to the divergence of the axial current. Indeed, ∂µ Jµ 5 has mass dimension 4. In an abelian gauge theory, each external photon must appear in the right hand side in the form of the field strength Fµν , that has mass dimension 2. A term with n photons would thus have mass dimension 2n, and require a prefactor of mass dimension 4 − 2n to be a valid contribution to the divergence of the axial current. But since the fermions we are considering are massless and the coupling constant is dimensionless in four dimensions, there is no dimensionful parameter in the theory for making up such a prefactor. Let us now consider higher loop corrections. From the calculation that led to eq. (9.50), the anomaly results from the integration over the momentum that runs in the fermion loop, provided that the integrand has mass dimension 4 or higher. Note

9. Q UANTUM ANOMALIES

307

that some of the higher order corrections just renormalize the objects that appear in the right hand side of eq. (9.50), such as the photon field strength and the coupling constant, without changing the structure of the anomaly (including the numerical prefactor). Quite generally however, adding an internal photon line requires to add more fermion propagators in the main loop, which reduces its degree of ultraviolet divergence. Of course, the integration over the momentum of this internal photon may itself be ultraviolet divergent, but it can be regularized in a way that does not interfere with axial symmetry and thus does not contribute to the anomaly.

9.2 Generalizations 9.2.1

Axial anomaly in a non-abelian background

In the previous section, we have discussed axial anomalies in an abelian gauge theory. However, a similar anomaly arises in the presence of a non-abelian background gauge field. Let us assume that the fermions are in a representation of the gauge algebra where the generators are ta . The calculation of the triangle graph proceeds almost in the same way as in the abelian case, except for the Lie algebra generators, and eq. (9.50) becomes




e3 ∂ µ Jµ tr ta tb ǫανβρ ∂α Aa ∂β Ab ν (x) ρ (x) . 5 (x) = − 2 4π

(9.51)

This is not gauge invariant, but it is easy to guess what should be the right hand side to restore gauge invariance:


e3 b ∂ µ Jµ tr ta tb ǫανβρ Fa αν (x) Fβρ (x) . 5 (x) = − 2 16π

(9.52)

The same dimensional argument that we have used in the abelian case also applies here: there cannot be contributions to the anomaly of degree higher than two in the field strength. Note that when expanded in terms of the gauge potential Aa µ, eq. (9.52) contains terms of degree 3 and 4, that exist only in a non-abelian background. Diagrammatically, they correspond to contributions coming from the following two diagrams:

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

(But the direct extraction of the anomaly contained in these graphs would be very cumbersome, due to the numerous terms arising from permutations of the external gauge fields.)

9.2.2

Axial anomaly in a gravitational background

Another situation where an axial anomaly is present is the case of a gravitational background. Of course, this is to a large extent an academic exercise since the resulting anomaly is extremely small, due to the weakness of the gravitational coupling at the usual scales of particle physics. Nevertheless, since every field is in principle coupled to gravity, the anomalies caused by a gravitational background are unavoidable unless the matter fields of the theory are arranged in a specific way. Interestingly, the calculation of this gravitational anomaly can be performed even if we do not have a consistent quantum theory of gravity, since it does not involve quantum fluctuations of the gravitational field (the only loop is a fermion loop). At tree level, the couplings between gravity and ordinary fields are determined from the principle of general covariance. Let us sketch here how such a calculation is done, without entering into too many technical detail. The first step is to obtain a generally covariant generalization of the Dirac operator, for an arbitrary metric tensor gµν , from which we can read off the coupling of the fermion to the background gravitational field. In a curved spacetime, we wish to generalize the Dirac matrices so that they satisfy  µ γ (x), γν (x) = 2 gµν (x) . (9.53)

(In this section, we use the Greek letters µ, ν, ρ, σ for indices related to curved coordinates, and Greek letters from the beginning of the alphabet α, β, γ, δ for indices related to flat Minkowski coordinates.) In a curved spacetime, the covariant derivative of the metric tensor vanishes, and it is therefore natural to request the same for the Dirac matrices. However, this requires that we introduce a spin connection, which is a matrix Γµ defined so that λ ∇µ γν ≡ ∂µ γν − Γµν γλ − Γµ γν + γν Γµ = 0 ,

(9.54)

λ where Γµν is the usual Christoffel’s symbol. The covariant derivative acting on a spinor is (∂µ − Γµ ) Ψ and the generally covariant Dirac equation for a massless fermion reads c sileG siocnarF

i γµ (∂µ − Γµ ) Ψ = 0 .

(9.55)

In order to construct a Lagrangian that transforms as a scalar, we need a matrix Γ such that ψ† Γψ is a real scalar. This is the case if the following conditions are satisfied Γ = Γ† , Γ γµ = 㵆 Γ , ∇µ Γ = ∂µ Γ + Γµ† Γ + Γ Γµ .

(9.56)

309

9. Q UANTUM ANOMALIES We then define Ψ ≡ Ψ† Γ , and the Lagrangian density is √ L ≡ i −g Ψ γµ ∇µ Ψ .

(9.57)

(g is the determinant of the metric tensor.) The vector current and its conservation law generalize into J µ ≡ Ψ γµ Ψ

∇ µ Jµ = 0 .

,

(9.58)

In the massless case, we can in addition define a conserved axial current: 5 µ Jµ 5 ≡ Ψγ γ Ψ

,

∇ µ Jµ 5 =0.

(9.59)

However, as we shall see, this conservation law suffers from an anomaly in a curved spacetime. Firstly, let us introduce a representation of the Dirac matrices for a generic curved spacetime, that makes an explicit connection with the metric tensor. This is achieved by introducing four vector fields eα µ (x) (called a vierbein, or tetrad) such that3 gµν (x) = ηαβ eα µ (x) eβ ν (x) ,

(9.60)

where in this section we use the notation ηαβ for the Minkowski metric tensor. This is equivalent to introducing at each point x a local Minkowski frame with coordinates yα . Note that eα µ transforms as a vector under diffeomorphisms (a coordinate vector) with respect to the index µ, and as an ordinary 4-vector under Lorentz transformations (called a tetrad vector in this context) with respect to the index α. The indices α, β, · · · are raised and lowered with the Minkowski metric tensor, while the indices µ, ν, · · · are raised and lowered with the curved space metric gµν (x). Since in the right hand side of eq. (9.60) the indices α and β are contracted with the Lorentz tensor ηαβ , the result is a scalar under Lorentz transformations, but a rank-2 tensor under diffeomorphisms. The Dirac matrices in curved spacetime (γµ (x)) can then be related to those in flat spacetime (γα ) by γµ (x) = eα µ (x) γα ,

(9.61)

and a spin connection Γµ that satisfies eq. (9.54) (and reduces to zero in flat spacetime) is given by 1 Γµ (x) = − γα γβ eαρ (x) ∇µ eβ ρ (x) , 4

(9.62)

ν β with ∇µ eβ ρ = ∂µ eβ ρ − Γµρ e ν (since eβ ρ is a coordinate vector with respect to the index ρ). A matrix Γ that fulfills eqs. (9.56) is the flat spacetime γ0 , and the matrix γ5 is still given in terms of the flat spacetime Dirac matrices by γ5 = i γ0 γ1 γ2 γ3 . 3 In this section, we denote η αβ ≡ diag (1, −1, −1, −1) the flat spacetime Minkowski metric, in order to distinguish it from gµν .

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 9.3: Graph contributing to the chiral anomaly in a gravitational background in four space-time dimensions.

We have now a representation of the Dirac operator in an arbitrary curved spacetime, expressed in terms of the vierbein eα µ that encodes the curved metric, from which we may read off the coupling between a spin 1/2 field and the external gravitational field. We will not go into the technology required for calculating a fermion loop like the graph of the figure 9.3, and just quote the final result for the divergence of the axial current:

 ∇ µ Jµ 5 (x) =

1 ǫαβγδ Rαβ µν (x) Rγδµν (x) , 384π2

(9.63)

where Rµνρσ is the curvature tensor (it plays in gravity the same role as the field strength Fµν in a non-abelian gauge theory). This formula indicates that a curved spacetime, i.e. an external gravitational field, leads to an anomalous contribution to the divergence of the axial current. This effect is of course tiny in ordinary situations where gravity is weak. But it should in principle be kept in mind when attempting to construct an anomaly free chiral gauge theory, if one wishes this theory to remain consistent all the way up to the Planck scale.

9.2.3

Anomalies in chiral gauge theories

In all the examples that we have considered until now in this chapter, the anomaly appeared in the conservation of a current associated to a global symmetry such as chiral symmetry. Although it indicates a violation of this symmetry by quantum corrections, the anomaly does not make the theory inconsistent in this case. However, in graphs mixing the axial current and insertions of external gauge fields, we made sure that the ultraviolet regularization does not spoil the Ward identity associated to the gauge symmetry. But we may also consider chiral gauge theories, in which the left and right handed components of the fermions belong to different representations of the gauge algebra. This is for instance the case in the Standard Model, where the electroweak interaction is chiral (the left handed fermions form SU(2) doublets, while the right handed fermions are singlet under SU(2)). In such a theory, the gauge coupling between

311

9. Q UANTUM ANOMALIES

fermions and gauge fields involve the left or right projectors PR,L ≡ (1 ± γ5 )/2, and the generators of the Lie algebra that appear in these vertices are ta , respectively R,L (the left and right generators would be equal in a theory where the two fermion chiralities belong to the same representation). The triangle diagram that gave the axial anomaly in four dimensions is replaced by a graph with three external gauge bosons, with chiral couplings to the fermion loop. When the fermion in the loop is massless, the left and right chiralities do not mix, and the multiple occurrences of the projectors simplify into a single one, thanks to PR PL = 0 , PL2 = PL ,   PR,L , γµ γν = 0 ,

PR2 = PR ,



/ 1 PR γ ν p / 1 γν p / 2 γρ p /3 , / 2 PR γ ρ p / 3 = tr PR γµp tr PR γµp

/ 1 γν p / 2 γρ p /3 . / 1 PL γ ν p / 2 PL γ ρ p / 3 = tr PL γµp tr PL γµp

(9.64)

The γ5 contained in the projectors PR,L may lead to an anomaly, with a relative sign between the right and left chiralities. The calculation is almost identical to the case of a global axial symmetry, except that now we should choose the shifts a and b so that the resulting 3-point function is symmetric in the external fields, since they play identical roles. But this choice does not eliminate the anomaly; it just distributes it evenly among
the three external currents, leading to an anomaly proportional to tr ta {tb , tc } . When there are both right and left fermions in the loop, the anomaly is proportional to

dabc ≡ tr ta {tb , tcR } − tr ta {tb , tcL } . (9.65) R R L L Obviously, this is zero in a vector theory, where the right and left fermions couple in the same way to the gauge bosons. c sileG siocnarF

Anomaly cancellation in the Standard Model : Unlike anomalies of global symmetries, an anomaly of a gauge symmetry makes it immediately inconsistent because it would for instance spoil its unitarity and renormalizability. For this reason, most chiral gauge theories do not make sense. The only ones that actually do are those for which the fermion fields are arranged in representations of the gauge group such that dabc = 0. This turns out to be the case for the Standard Model with its known matter fields: all the gauge anomalies cancel (within each generation of fermions) thanks in particular to the peculiar values of the weak hypercharges of the quarks and leptons. In order to proceed with this verification, we need the the quantum numbers listed in the table 9.1 for the fermions of the Standard Model. The weak isospin and hypercharge are the quantum numbers of the fermion under SU(2) × U(1). Both of these gauge interactions are chiral, since the charges T3 and

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Weak isospin T3

Weak hypercharge Y

Elec. charge

Left handed fermions νe , νµ , ντ

+ 12

−1

0

e, µ, τ

− 12

−1

−1

u, c, t

+ 12

+ 31

+ 32

d, s, b

− 12

+ 31

− 31

Right handed fermions eR , µR , τR

0

−2

−1

uR , cR , tR

0

+ 34

+ 32

dR , sR , bR

0

− 32

− 31

Table 9.1: Weak isospin, hypercharge and electrical charge of the fermions of the Standard Model.

Y of the left and right handed fermions are different. After spontaneous symmetry breaking via the Higgs mechanism, the fields Bµ 3 (third component of SU(2)) and Aµ (U(1)) mix to give the Z boson and the photon fields. The electrical charges of the fermions are then given by Q = T3 + Y2 (since the electrical charges are the same for left and right fermions, the resulting U(1)em of electromagnetism is a non-chiral gauge interaction). The simplest case of anomaly cancellation is the 3-gluon triangle, which is not anomalous because the strong interaction vertex is a vector coupling: su(3) c su(3) a b

R − L cancellation (see eq. (9.65)).

su(3)

For the triangle involving three SU(2) bosons, the anomaly cancels thanks to a peculiar identity obeyed by the su(2) generators: su(2) k su(2) i j

trsu(2) (ti {tj , tk }) = 0 .

su(2)

In triangles that have a single SU(3) or a single SU(2) boson, the anomaly cancels because the corresponding generators are traceless:

313

9. Q UANTUM ANOMALIES su(2)

u(1)

j su(3)

su(3) a

trsu(3) (ta ) = 0 ,

a i

su(2)

u(1)

su(3)

u(1)

b su(2)

su(2) i

trsu(2) (ti ) = 0 .

i a

su(3)

u(1)

In triangles with a single U(1) boson and a pair of SU(2) or SU(3) bosons, the anomaly cancels thanks to the specific linear combination of weak hypercharges one gets by summing over all the allowed fermions in the loop: su(3) b

P

u(1) a

quarks su(3)

  y = 2 − 31 + 34 − 23 = 0 ,

su(2) j

P

u(1) i

left handed fermions

su(2)

  y = 3 − 13 +1 = 0 .

(In the first of these cancellations, there is a factor of 2 in the first term to account for the fact that the left handed quarks form SU(2) doublets, and in the second equality the first term has a factor 3 because the quarks can have three colours.) Note also that loops with left handed fermions should be counted with a minus sign, according to eq. (9.65). Finally, the triangle with three U(1) bosons has no anomaly, thanks to the fact that the sum of the cubes of the weak hypercharges over all fermions is zero: u(1)

u(1)

u(1)

P

3  3  3  3  y3 = 6 − 13 +3 43 +3 − 32 +2+ −2 = 0 .

(Again, the numerical prefactors count the number of SU(3) and SU(2) states for each fermion.) Interestingly, gravitational anomalies also cancel in the standard model. Indeed, an anomaly may potentially exist in the triangle with a U(1) boson and two gravitons. But this anomaly would be proportional to the sum of the weak hypercharges of all fermions, which turns out to be zero: G

u(1)

G

P

        y = 6 − 13 +3 43 +3 − 32 +2+ −2 = 0 .

One can see the crucial role played by the weak hypercharges assigned to the various fermions of the Standard Model in these cancellations. Conversely, one may try to

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

determine these hypercharges so that all anomalies cancel. Up to a permutation of the right handed quarks (uR and dR in the first family), there are only two solutions, say U(1)A and U(1)B (one of them corresponds to the Standard Model). Moreover, these two solutions cannot be mixed in the same theory, because one would have a non-canceling anomaly in a triangle that mixes the U(1)A and U(1)B gauge bosons.

9.3 Wess-Zumino consistency conditions 9.3.1

Consistency conditions

In the subsection 9.2.1, where we have derived the axial anomaly in a non-abelian background field, we first obtained a partial answer with only the terms quadratic in the external field, and then we used gauge symmetry in order to reconstruct the missing terms (of order 3 and 4 in the external field). However, how to promote such a partial result into the full expression of the anomaly is not always so obvious, for instance in the case of chiral gauge theories where the gauge symmetry itself is anomalous (in this case, we cannot invoke gauge invariance to restore the full answer). The Wess-Zumino consistency conditions are a set of equations satisfied by the anomaly function, that are powerful enough to allow reconstructing the anomaly from the knowledge of its lowest order in the gauge fields. Even in the case where the anomalous symmetry is global, it is convenient to couple a (fictitious in that case) gauge field Aµ to the corresponding current Jµ whose conservation is violated by the anomaly. By doing this, we promote the symmetry to a local gauge invariance (violated by the anomaly), and we may return to a global symmetry by letting the gauge coupling go to zero. Let us denote Γ [A] the effective action for the gauge field (i.e. the effective action in which the fermions are included only in the form of loop corrections). In the absence of anomaly, Γ [A] would be invariant under gauge transformations of the field Aµ , 0

=

no anomaly

δθ Γ [A] = =

Z

 δΓ [A] 
d4 x Dadj θ (x) b µ ab δAa µ (x) Z
δ − d4 x θb (x) Dadj Γ [A] . µ ba δAa µ (x) | {z }

(9.66)

i Tb (x)

When this symmetry is spoiled by an anomaly, the effective action is no longer invariant, and we may write Ta (x) Γ [A] ≡ Ga [x; A] ,

(9.67)

315

9. Q UANTUM ANOMALIES

where the function Ga [x; A] encodes the anomaly. This function is closely related to the non-zero right hand side of the anomalous conservation law for the current associated to the symmetry, since the effective action and the current are related by Jµa (x) +

δΓ [A] =0, δAa µ (x)

(9.68)

which implies Dadj µ




ba

Jµa (x) = −Gb [x; A] .

(9.69)

Since the anomaly is local, Gb [x; A] should be a local (at the point x) polynomial in the gauge field and its derivatives. One may then check that the operators Ta (x) obey the following commutation relation, 

 Ta (x), Tb (y) = i g fabc δ(x − y) Tc (x) ,

(9.70)

where the fabc are the structure constants of the gauge group. From this, we deduce the following identity Ta (x) Gb [y; A] − Tb (y) G[x; A] = i g fabc δ(x − y) Gc [x; A] ,

(9.71)

called the Wess-Zumino consistency conditions. Since this identity is linear in the anomaly function Ga , it cannot constrain its overall normalization (for this, it is usually necessary to compute the triangle diagram). However, this equation is strong enough to fully constrain its dependence on the gauge field from the term of lowest order in A. c sileG siocnarF

9.3.2

BRST form of the Wess-Zumino condition

The consistency condition can be recasted into a more convenient form that involves BRST symmetry. Let us introduce a ghost field χa , and recall that the BRST transformation reads: adj QBRST Aa µ (x) = Dµ




ab

χb (x) ,

g QBRST χa (x) = − fabc χb (x) χc (x) . 2 (9.72)

Then, let us encapsulate the anomaly function into the following local functional of ghost number +1: Z G[A, χ] ≡ d4 x χa (x) Ga [x; A] . (9.73)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

We obtain: QBRST G[A, χ] =

=

Z i d4 xd4 y χa (x)χb (y) Tb (y)Ga [x; A] Z g d4 x fabc χa (x)χb (x) Gc [x; A] − 2 Z

i d4 xd4 y χa (x)χb (y) Tb (y)Ga [x; A] − Ta (x)Gb [y; A] 2  +i g δ(x − y) fabc Gc [x; A] . {z } | =0

(9.74)

Therefore, the Wess-Zumino consistency conditions are equivalent to the statement that the functional G[A, χ] is BRST-invariant: QBRST G[A, χ] = 0 .

(9.75)

Since QBRST is nilpotent, a trivial solution of this equation is of course G[A, χ] = QBRST h[A] ,

(9.76)

where h[A] does not depend on the ghost field (indeed, QBRST increases the ghost number by one unit, and G[A, χ] must have ghost number unity). But since h[A] is a local functional of the gauge field, it may be subtracted from the action to cancel the anomaly. Thus, genuine anomalies are given by local functionals G[A, χ] of ghost number +1 that satisfy the consistency condition (9.75), modulo a term obtained by acting with QBRST on a functional of A only. Note that if we write G[A, χ] as the integral of a local density, Z G[A, χ] ≡ d4 x G(x) , (9.77) then the BRST action on the density should be a total derivative QBRST G(x) = ∂µ ζµ .

9.3.3

(9.78)

Solution of the consistency condition

In order to determine how the Wess-Zumino equation constrains G(x), the language of differential forms introduced in the section 4.5.3 is very handy, as a way to encapsulate both Lorentz and group indices in compact objects. The 1-forms dxµ anticommute among themselves under the exterior product ∧. In addition, they also anticommute

317

9. Q UANTUM ANOMALIES

with the ghost field and the BRST generator QBRST . The volume element weighted by the fully antisymmetric tensor ǫµνρσ can therefore be written as d4 x ǫµνρσ = dxµ ∧ dxν ∧ dxρ ∧ dxσ .

(9.79)

Then, given a vector Vµ and the corresponding 1-form V ≡ Vµ dxµ ,

(9.80)

we may write in a compact manner Z Z 4 µνρσ d xǫ Vµ Vν Vρ Vσ = V ∧ V ∧ V ∧ V .

(9.81)

The exterior derivative d ≡ ∂µ dxµ ∧ satisfies d2 = 0 ,

QBRST d + dQBRST = 0 .

(9.82)

If we also denote a µ A ≡ ig Aa µ t dx

,

χ ≡ ig χa ta ,

(9.83)

(for later convenience, we absorb a factor i in the definitions of A and χ) the BRST transformations take the following form QBRST A = −dχ + A ∧ χ + χ ∧ A , QBRST χ = χ ∧ χ .

(9.84)

On dimensional grounds, the anomaly function G[A, χ] may contain the following terms: Z


G[A, χ] = −iC d4 x ǫµνρσ χa tr ta ∂µ Aν (∂ρ Aσ )

+ia1 ∂µ Aν Aρ Aσ + ia2 Aµ ∂ν Aρ Aσ + ia3 Aµ Aν (∂ρ Aσ )  −b Aµ Aν Aρ Aσ . (9.85)

The term on the first line comes from the triangle diagram, whose explicit calculation gives the overall coefficient C. The terms of the second and third lines come from the square and pentagon diagrams, respectively. Alternatively, they can be obtained from the consistency conditions. Firstly, the previous equation may be rewritten as a sum of forms: Z

 G[A, χ] = γ tr χ ∧ (dA) ∧ (dA)) +α1 (dA) ∧ A ∧ A + α2 A ∧ (dA) ∧ A

 +α3 A ∧ A ∧ (dA) + β A ∧ A ∧ A ∧ A , (9.86)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where γ, α1,2,3 , β are constants related to C, a1,2,3 , b. Consider first the BRST transform of the last term,



 QBRST tr χ ∧ A ∧ A ∧ A ∧ A = tr χ ∧ χ ∧ A ∧ A ∧ A ∧ A + terms in χ ∧ (dχ) ∧ A ∧ A ∧ A . (9.87)

Since QBRST cannot increase the degree in A, the term in χ ∧ χ ∧ A ∧ A ∧ A ∧ A cannot be canceled by the terms in α1,2,3 , and therefore we must have β = 0. We need then to evaluate the BRST transformation of the other terms. For instance,



QBRST tr χ ∧ (dA) ∧ (dA) = tr − χ ∧ χ ∧ (dA) ∧ (dA) +χ ∧ (dχ) ∧ A ∧ (dA) −(dχ) ∧ χ ∧ (dA) ∧ A −A ∧ χ ∧ (dA) ∧ (dχ)  −χ ∧ A ∧ (dχ) ∧ (dA) . (9.88) By evaluating similarly the BRST transforms of the other terms, one can check that when α1 = −α2 = α3 = −1/2 the BRST transform of the anomaly functional is the integral of an exact form and therefore vanishes: Z Z QBRST G[A, χ] = γ dF = γ F=0. (9.89)

❘

4

∂

❘

4

This is in fact the only possibility. Introducing the field strength 2-form, F ≡ dA − A ∧ A =

ig a a t Fµν dxµ dxν , 2

(9.90)

the anomaly functional for these values of the coefficients can then be rewritten as Z

h i 1 G[A, χ] = γ tr χ ∧ d A ∧ F + A ∧ A ∧ A . (9.91) 2

Therefore, except for the prefactor γ whose determination requires to calculate the triangle diagram, the consistency relations completely determine the dependence of the anomaly function on the gauge field. c sileG siocnarF

9.4 ’t Hooft anomaly matching Some models of physics beyond the Standard Model conjecture that the quarks and leptons are bound states of more fundamental degrees of freedom, confined by some

319

9. Q UANTUM ANOMALIES

strong gauge interaction at a scale Λ ≫ Λelectroweak . A difficulty with this picture is to explain the fact that quarks and leptons are light (in fact, massless, if it were not for electroweak symmetry breaking), while being bound states of some strong interaction at a much higher scale. Indeed, the naive mass of these confined states is naturally of order Λ (the Goldstone mechanism cannot give light fermions, only scalar particles). As shown by ’t Hooft, one way this may happen is to have in the underlying fundamental theory a global chiral symmetry with generators T a , such that the anomaly function tr (T a {T b , T c }) is non-zero. In the low energy sector of the spectrum of this theory, there must be spin 1/2 massless bound states, on which this chiral symmetry acts with generators a , and whose anomaly coefficients are identical to the high energy ones:

❚

tr

❚a ❚b , ❚c
= tr


T a T b, T c .

(9.92)

The proof of this assertion goes as follows. Let us first couple a fictitious weakly coupled gauge boson to the generators T a . We also introduce additional fictitious massless fermions coupled only to the fictitious gauge boson, but not to the strongly interacting gauge bosons responsible for the confinement, tuned so that their contribution exactly cancels the anomaly: h


i tr T a T b , T c

physical high energy

h 
i + tr T a T b , T c =0. fictitious

(9.93)

fermions

Let us now examine the low energy part of the spectrum of this theory, i.e. at energies much lower than the strong scale Λ. Since they are not coupled in any way to the strong sector, this low energy spectrum contains the fictitious gauge bosons and massless fermions, unmodified compared to what we have introduced at high energy. In addition, this spectrum contains the bound states made of the trapped fermions and strongly interacting gauge bosons. For consistency, this low energy description must also be anomaly-free, which means that the bound states must transform under the chiral symmetry with generators a , such that

❚

h

tr

❚a ❚b , ❚c


i

physical bound states

h 
i + tr T a T b , T c =0. fictitious

(9.94)

fermions

The crucial point in this argument is that the contribution of the fictitious fermions is the same in the equations (9.93) and (9.94), because these fermions are not coupled to the strongly interacting sector. Eqs. (9.93) and (9.94) immediately give (9.92). In other words, the anomalies of the trapped elementary fermions must be mimicked by those of the massless spin 1/2 bound states they are confined into.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

9.5 Scale anomalies 9.5.1

Classical scale invariance

Until now, the quantum anomalies we have encountered in this chapter are related to chiral couplings of fermions. But there exist another anomaly, ubiquitous in most quantum field theories, related to quantum violations of scale invariance. Consider a quantum field theory whose Lagrangian does not contain any dimensionful parameter. In four spacetime dimension, this means that it contains only operators of mass dimension exactly equal to 4, which excludes all mass terms. This is the case for instance for a massless scalar field theory with a quartic coupling, whose action reads:

S[φ] ≡

Z

d4 x

g

µν

2

ν (∂µ x φ(x))(∂x φ(x)) −

λ 4  φ (x) . 4!

(9.95)

A scaling transformation amounts to multiplying all length scales by some factor xµ

→

yµ ≡ eϑ xµ .

(9.96)


d In this transformation, a field φ(x) of dimension mass φ and its derivative transform as: φ(x) ∂µ x φ(x)

→

→

φ ′ (y) ≡ e−ϑ dφ φ(e−ϑ y) ,

′ −ϑ (dφ +1) µ φ(x) ∂µ φ (y) = e ∂ x y

x=e−ϑ y

,

(9.97)

while the integration measure over spacetime is rescaled by d4 x

→

d4 y = e4 ϑ d4 x .

(9.98)

Consider now the transformed action, Z

g λ ′4  µν ′ ν ′ (∂µ φ (y))(∂ φ (y)) − φ (y) S[φ ′ ] = d4 y y y 2 4! Z 

gµν µ −4ϑdφ λ 4 (∂x φ(x))(∂ν φ (x) = e4ϑ d4 x e−2ϑ(1+dφ ) x φ(x)) − e 2 4! = S[φ] , (9.99) where we have used the fact that the mass dimension of φ is dφ = 1 in four spacetime dimensions. The action defined in eq. (9.95) is thus invariant under scale transformations. The same conclusion holds for any classical action that does not contain any dimensionful parameter, provided the appropriate dimension dφ is used for each field. This is for instance the case of pure Yang-Mills theory in four dimensions, or quantum chromodynamics in which we neglect the quark masses.

321

9. Q UANTUM ANOMALIES

9.5.2

Dilatation current

Since the transformation (9.97) is continuous, Noether’s theorem implies that there is a corresponding conserved current. On the one hand, the infinitesimal variation of the field is δφ(x) ≡ φ ′ (x) − φ(x) = −ϑ dφ + xµ ∂µ ) φ(x) + O(ϑ2 ) .

(9.100)

On the other hand, the scale transformation (9.96) directly applied to the integrand of the action gives a variation h i δ d4 x L(x)

= =


−ϑ d4 x 4 + xµ ∂µ L(x) + O(ϑ2 )   −ϑ d4 x ∂µ xµ L(x) + O(ϑ2 ) .

(9.101)

It is important to include the measure in this calculation, since it is not invariant under scale transformations. The variation of the measure gives the 4 in the first line, which is crucial for obtaining a total derivative in the second line. Then, from the derivation of Noether’s theorem, we conclude that ∂µ



 ∂L dφ + xν ∂ν )φ − xµ L = 0 . ∂(∂µ φ) | {z }

(9.102)

Dµ

The vector Dµ is called the dilatation current. In the case of the scalar field theory used earlier as an example, the explicit form of Dµ is   Dµ = xν (∂µ φ)(∂ν φ) − gµν L + φ(∂µ φ) . (9.103) | {z } | {z } Θµν

1 µ 2 2∂ φ

In this formula, we recognize that the factor multiplying xν is the energy-momentum tensor Θµν , whose divergence is zero thanks to translation invariance, i.e. ∂µ Θµν = 0. This observation facilitates the calculation of ∂µ Dµ , since we have c sileG siocnarF

∂µ Dµ

=

Θµ µ + (∂µ φ)(∂µ φ) + φ(φ)

=

2(∂µ φ)(∂µ φ) + φ(φ) − 4 L   λ φ φ + φ3 . | {z6 }

=

=0

The final zero follows from the classical equation of motion of the field.

(9.104)

322

9.5.3

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Link with the energy-momentum tensor

In the previous section, we have seen that the energy-momentum tensor appears in the expression of the dilatation current. More precisely, this energy-momentum tensor is the canonical one (i.e. the one obtained as the Noether’s current associated to translation invariance). Then, the divergence of the dilatation tensor is the trace of this energy-momentum tensor (Θµ µ = − 21  φ2 ), plus an additional term that turns out to cancel it exactly. In fact, in such a scale invariant theory, it is possible to introduce a traceless definition of the energy-momentum tensor, that we shall denote T µν , such that T µµ = 0 ,

∂µ T µν = 0 ,

(9.105)

and a valid definition of the dilatation current is Dµ ≡ xν T µν .

(9.106)

(As we shall see shortly, this new dilatation current gives the same conserved charge as the current Dµ introduced earlier.) The tracelessness of T µν is then equivalent to the conservation of Dµ . In the case of a massless φ4 scalar field theory in four dimensions, this improved energy-momentum tensor reads
1 µν T µν ≡ Θµν + g  − ∂µ ∂ν φ2 . (9.107) 6

This tensor is traceless, because the trace of the additional term is + 21  φ2 . Moreover, this additional term has a null divergence, and therefore we have ∂µ T µν = 0. Note also that the component µ = 0 of the added term is a total spatial derivative, 
2 − ∂i ∂i φ2 (ν = 0) 0ν 0 ν g −∂ ∂ φ = , (9.108) i 0 2 −∂ ∂ φ (ν = i)

which implies that the conserved charges (i.e. the momenta Pν ) obtained from Θµν and T µν are the same. Since T µν is traceless, the current Dµ = xν T µν is conserved. But we should also check that we have not modified the corresponding conserved charge. We have D0 = xν T 0ν

= = = =


1 xν g0ν  − ∂0 ∂ν φ2 6
1 i 0 i 0ν x ∂ ∂ − x0 ∂i ∂i φ2 xν Θ + 6 1 0ν i i 0 xν Θ + − (∂ x )∂ + ∂i (xi ∂0 − x0 ∂i )) φ2 |{z} 6 = −3 h1 i 1 0 2 0ν xν Θ + ∂ φ +∂i (xi ∂0 − x0 ∂i )φ2 . (9.109) 6 {z 2 } | xν Θ0ν +

D0

9. Q UANTUM ANOMALIES

323

The first two terms are identical to the original D0 of eq. (9.103), and the third term is a total spatial derivative. Therefore, when we integrate this charge density over all space, the new definition of the dilatation current gives the same conserved charge as eq. (9.103).

9.5.4

Energy-momentum tensor via coupling to gravity

The discussion of the previous section highlights the fact that, even in classical field theory, the energy-momentum tensor is not uniquely defined. It is possible to add a term that does not alter its conservation and does not change the conserved charges, but that modifies its trace. There are also cases (e.g., Yang-Mills theory), where the canonical energy-momentum tensor Θµν is not even symmetric, but can be improved into a symmetric one. An alternate method of deriving the energy-momentum tensor, that leads directly to a symmetric tensor, is to minimally couple the theory to gravity, and to vary the metric. To that effect, consider an infinitesimal spacetime dependent translation xµ

→

xµ + ξµ (x) .

(9.110)

Under such a transformation, the metric tensor varies by4 δgµν (x) = ∇µ ξν (x) + ∇ν ξµ (x) ,

(9.111)

where ∇µ is the covariant derivative. Let us recall for later use an important identity Z Z

√ √ (9.112) d4 x −g A ∇µ B = − d4 x −g ∇µ A B ,

where g ≡ det(gµν ). Since eq. (9.111) is merely a change of a dummy integration variable, the action is not modified. Therefore, we may write ! Z
δS δS 4 δφ(x) 0 = δS = d x ∇µ ξν (x) + ∇ν ξµ (x) + {z } δgµν (x) | δφ(x) {z } | δgµν (x) =0   Z √ 2 δS = − d4 x −g ∇µ √ ξν (x) . (9.113) −g δgµν (x) c sileG siocnarF

In the first line, the second term vanishes when the field φ is a solution of the classical equation of motion. For this to be true for an arbitrary variation ξν (x), we must have ∇µ T µν = 0 ,

2 δS with T µν ≡ √ . −g δgµν

(9.114)

4 Note that, although xµ is not a vector, the infinitesimal variation ξµ (x) is a vector, tangent to the coordinate manifold at the point x. Therefore, it makes sense to act on it with a covariant derivative.

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By construction, this tensor is symmetric and (covariantly) conserved, and the nature of the coordinate transformation (9.111) makes it clear that it is related to translation invariance5 . In order to obtain the flat space energy-momentum tensor, one should set gµν to the Minkowski metric tensor after evaluating the derivative. Moreover, if we apply a scale transformation to the coordinates, xµ

→

eϑ xµ ,

(9.115)

the metric tensor is simply rescaled: gµν

→

e−2ϑ gµν .

(9.116)

Moreover, if the classical action does not contain any dimensionful parameter, it is invariant under this rescaling, and we can write ! Z δS δS e−2ϑ gµν (x) + 0 = δS = d4 x δφ(x) . (9.117) δgµν (x) δφ(x) | {z } =0

This equation implies that the derivative of the action with respect to the metric, and therefore the energy-momentum tensor T µν , is traceless. In order to illustrate this method, let us consider Yang-Mills theory, whose action coupled to gravity reads Z √ 1 ρσ S=− (9.118) d4 x −g gµρ gνσ Fµν a Fa . 4 In order to calculate √ the derivative of this action with respect to the metric, we need the variation of −g, that can be obtained as follows: det (gµν + δgµν )

=

etr ln(gµν +δgµν )

= ≈

etr ln(gµρ (δ ν +g δgσν ))
det (gµν ) 1 + gµν δgµν .

ρ

Hence,

ρσ

√ gµν ∂ −g √ = −g , ∂gµν 2

(9.119)

(9.120)

and we obtain the following expression for the energy-momentum tensor: T µν = Fµα a Fα ν a −

gµν a αβ a F F , 4 αβ

(9.121)

whose trace is obviously zero. 5 It is important to note that the derivation implicitly assumes that the parameters in the action, such as the coupling constants, do not depend explicitly on the position.

9. Q UANTUM ANOMALIES

325

Scale anomaly and β function

9.5.5

Until now, our analysis of the dilatation current has been purely classical, since we have shown its conservation from the classical action. At this level, it follows from the absence of any dimensionful parameters in the theory. However, this main not remain true when loop corrections are taken into account, because of ultraviolet divergences. This is quite clear if we regularize these divergences by introducing an ultraviolet cutoff, but it is also true in dimensional regularization. In the latter case, the fact that d 6= 4 implies that the coupling constants become dimensionful, which also breaks scale invariance. Taking the trace of eq. (9.121) in d dimensions, we obtain 4 − d a αβ a Fαβ F . (9.122) 4 The expectation value in the right hand side has ultraviolet divergences that, in dimensional regularization, become poles in (d − 4)−1 . These terms cancel the prefactor, leaving a non-zero result for the trace of the energy-momentum tensor even in the limit d → 4. Another point of view is to introduce counterterms to subtract the ultraviolet divergences, and then remove the regulator that controlled the ultraviolet behaviour. But after this procedure, the bare coupling constant in the action becomes scale dependent, which also breaks scaling symmetry. From this hand-waving discussion, we expect that the divergence of the dilatation current, i.e. the trace of the energymomentum tensor, is related to the β function that controls the running of the coupling: Tµµ =

c sileG siocnarF

µ

∂g = β(g) . ∂µ

(9.123)

Moreover, even if the classical scale invariance is broken by the renormalization group flow, it should be recovered at the fixed points of the RG flow. For instance, a quantum field theory is scale invariant at critical points. In Yang-Mills theory, we can derive the form of this trace in the following (nonrigorous) manner. Let us start from the Yang-Mills action, written in terms of rescaled fields, so that the coupling appears in the form of a prefactor g−2 : Z h √ 1 a µν a i F F , (9.124) S = d4 x −g − 4 g2b µν

where gb is the bare coupling constant. When this theory is regularized by an gauge invariant cutoff µ (e.g., a lattice regularization), the bare coupling becomes cutoff dependent in order for the renormalized quantities to have a proper ultraviolet limit. Then, consider again the scaling transformation defined in eqs. (9.115) and (9.116). With a scale dependent coupling, the physics is invariant provided we also change the scale at which the coupling is evaluated gb (µ)

→

gb (e−ϑ µ) .

(9.125)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The infinitesimal form of this transformation is δgµν = −2 ϑ gµν ,

∂gb . δgb = −ϑ µ ∂µ | {z }

(9.126)

β function

Then, by writing explicitly the two sources of ϑ dependence in the variation of the action, we get Z h 1 β(gb ) a µν a i . (9.127) F F 0 = δS = −ϑ d4 x 2 T µ µ − gµν =ηµν gb 2 g3b µν Therefore, we obtain the following form of the anomalous divergence of the dilatation current: ∂µ Dµ = T µ µ =

β(g) a µν a F F . 2 g µν

(9.128)

This derivation is only heuristic, but a more rigorous treatment using properly renormalized operators would lead to the same result. This anomaly can also be derived in perturbation theory, from the loop corrections to the dilatation current,

.

(The dotted line terminated by the dark blob denotes the vertex between two gluons and the dilatation current.) Note that, thanks to asymptotic freedom, Yang-Mills theory becomes better and better scale invariant as the energy scale increases. Finally, when one adds quarks in order to obtain QCD, the right hand side of the previous equation contains also terms in m ψψ, due to the explicit breaking of scale invariance (already in the classical theory, therefore this is not a quantum anomaly) by the masses of the quarks.

Chapter 10

Localized field configurations All the applications of quantum field theory we have encountered so far amount to study situations that may be viewed as small perturbations above the vacuum state; i.e. interactions involving states that contain only a few particles. Besides the fact that these situations are actually encountered in scattering experiments, their importance stems from the stability of the vacuum, that makes it a natural state to expand around. In this chapter, we will study other field configurations, classically stable, that may also be sensible substrates for expansions that differ from the standard perturbative expansion that we have studied until now. However, under normal circumstances, a localized “blob” of fields is not stable: it will usually decay into a field which is zero everywhere. As we shall see, the stability of the field configurations considered in this chapter is due to topological obstructions that prevent a smooth transformation between the field configuration of interest and the null field that corresponds to the vacuum. These field configurations can be classified according to their space-time structure: • Event-like : localized both in time and space (e.g., instantons). These may be viewed as local extrema of the 4-dimensional action, and therefore may give a (non-perturbative) contribution to path integrals. • Worldline-like : localized in space, independent of time (e.g., skyrmions, monopoles). These field configurations behave very much like stable particles (at least classically), and their non-trivial topology confers them conserved charges. • Strings, Domain walls : extended in one or two spatial dimensions, independent of time. c sileG siocnarF

327

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

50 45 40 35

Figure 10.1: Quartic potential (10.1) exhibiting spontaneous symmetry breaking.

V(φ)

30 25 20 15 10 5 0 -10

-5

0

5

10

φ

10.1 Domain walls A domain wall is a 2-dimensional1 interface between two regions of space where a discrete symmetry is broken in different ways. Their simplest realization arises in a real scalar field theory, symmetric under φ → −φ, but with a potential that leads to spontaneous symmetry breaking, such as V(φ) ≡ V0 −

µ2 2 λ 4 φ + φ , 2 4!

(10.1)

where the constant shift V0 is chosen so that the minima of this potential are 0. There are two such minima, at field values r 6 µ2 φ = ±φ∗ , φ∗ ≡ . (10.2) λ In order to simplify the discussion, let us consider field configurations that depend only on x, and are independent of time, as well as of the transverse coordinates y, z. We seek field configurations that obey the classical field equation of motion, −∂2x φ + V ′ (φ) = 0 , and have a finite energy (per unit of transverse area), Z +∞ 


2 dE = dx 21 ∂x φ(x) + V(φ(x)) < ∞ . dydz −∞

(10.3)

(10.4)

This energy density is the sum of two positive definite terms (since we have adjusted the potential so that its minima are V(±φ∗ ) = 0. For the integral over x to converge 1 This is for 4-dimensional spacetime. In D-dimensional spacetime, domain walls have dimension D − 2.

329

10. L OCALIZED FIELD CONFIGURATIONS

when x → ±∞, it is necessary that φ(x) becomes constant when |x| → ∞, and that this constant be +φ∗ or −φ∗ . There are therefore four possibilities for the values of the field at x = ±∞: (i)

:

(ii)

:

(iii) (iv)

: :

φ(−∞) = +φ∗ ,

φ(−∞) = −φ∗ ,

φ(−∞) = −φ∗ , φ(−∞) = +φ∗ ,

φ(+∞) = +φ∗ ,

φ(+∞) = −φ∗ ,

φ(+∞) = +φ∗ , φ(+∞) = −φ∗ .

(10.5)

The first two of these possibilities do not lead to stable field configurations of positive energy, because they can be continuously deformed (while holding the asymptotic values unchanged) into the constant fields φ(x) = +φ∗ , or φ(x) = −φ∗ , respectively, that have zero energy. Physically, this means that if one creates a field configuration with these boundary values, it will decay into a constant field (i.e., the regions where the field was excited to values different from ±φ∗ will dilute away to |x| = ∞).

The interesting cases are encountered when the field takes values corresponding to opposite minima at x = −∞ and x = +∞. If one holds the asymptotic values of the field fixed, then it is not possible to deform continuously such a field configuration into one that would have zero energy. Thus, there must be stable field configurations of positive energy with these boundary values. A very handy trick, due to Bogomol’nyi, is to rewrite the energy density as follows: Z  2 Z φ(+∞) p p dE 1 +∞ dx ∂x φ(x)± 2 V(φ(x)) ∓ dφ 2 V(φ) . (10.6) = dydz 2 −∞ φ(−∞) In the cases i, ii, the second term vanishes, and the energy density is allowed to be zero, by having a constant field equal to ±φ∗ . Let us consider now the case iii. In this case, it is convenient to choose the minus sign in the first term, so that Z  2 Z +φ∗ p p 1 +∞ dE = dx ∂x φ(x) − 2 V(φ(x)) + dφ 2 V(φ) . (10.7) dydz 2 −∞ −φ } | ∗ {z >0

The second term is now strictly positive, and does not depend on the details of φ(x) (except its boundary values). Since the first term is the integral of a square, this implies that there is no field configuration of zero energy with this boundary condition. The minimal energy density possible with this boundary condition is Z +φ∗ p dE (10.8) = dφ 2 V(φ) , dydz min −φ∗ reached for a field configuration that obeys p ∂x φ(x) = 2 V(φ(x)) .

(10.9)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

φ

x

Figure 10.2: Domain wall profile corresponding to the potential of the figure 10.1.

Taking one more derivative implies that ∂2x φ =

(∂x φ) V ′ (φ) p = V ′ (φ) , 2 V(φ)

(10.10)

which is nothing but the classical equation of motion (10.3). Solutions of this equation with prescribed boundary values ±φ∗ at x = ±∞ interpolate between the two ground states of the potential of the figure 10.1. The ground state φ = +φ∗ is realized at x → +∞, while the other ground state is realized at x → −∞. Since these two vacua correspond to two different ways to spontaneously break the φ → −φ symmetry, there must exist an interface between the two phases, called a domain wall. From eq. (10.9), we may write x(φ) = x0 +

Zφ 0

p

dξ 2 V(ξ)

,

(10.11)

where x0 is an integration constant that can be interpreted as the coordinate where the field φ is zero. In other words, x0 is the location of the center of the domain wall that separates the regions of different vacua. The domain wall is a local minimum of the energy density (and the absolute minimum for the mixed boundary conditions iii). Moreover, it is separated from the (lower energy) configurations i, ii that have a constant field by an infinite energy barrier2 . Indeed, going from iii to i implies shifting the value of the field from −φ∗ to +φ∗ in the (infinite) vicinity of x = −∞. 2 From

this fact, we may infer that domain walls are also stable quantum mechanically.

331

10. L OCALIZED FIELD CONFIGURATIONS

In the middle of this process, the field in this region will be φ = 0, at which V(φ) = V0 > 0, a configuration that has an infinite energy density. Thus, the domain wall solution is stable, except for shifts of x0 (since the energy density is independent of x0 ): the domain wall may move along the x axis, but cannot disappear. Let us finish by a note on the y, z dependence that has been neglected so
far. Reintroducing the transverse dependence adds the term 12 (∂y φ)2 + (∂z φ)2 to the integrand of the energy density in eq. (10.4). This term is positive, or zero for fields that do not depend on y and z. Therefore, the minimum of energy density is reached for domain walls that are invariant by translation in the transverse directions. Domain walls that are not translation invariant are not stable, but will relax to this
y, z-invariant configuration. Physically, one may view the term 12 (∂y φ)2 + (∂z φ)2 as a surface tension energy, and the energetically favored configurations are those for which the interface has the lowest curvature. c sileG siocnarF

10.2 Skyrmions Skyrmions are field configurations that arise in models resulting from a spontaneous symmetry breaking, such as a non-linear sigma model. Consider for instance the following action, Z

1 X 

S[ξ] = dD x gab (ξ) ∂i ξa ∂i ξb + · · · , (10.12) 2 a,b

where the fields ξa are the Nambu-Goldstone bosons of a broken symmetry from the symmetry group G down to H. The matrix gab (ξ) is positive definite, and in general field dependent. The dots represent terms with higher derivatives, that we have not written explicitly. In such a model, the Nambu-Goldstone fields ξa may be viewed as elements of the coset G/H. In order to have a finite action, the derivatives of the fields should decrease faster than |x|−D/2 at large distance, a ∂i ξ (x) . |x|−D/2 , (10.13) |x|→∞

which means that the field ξa (x) should go to a constant, with a remainder that decreases faster than |x|1−D/2 .

The constant value of ξa at infinity can be chosen to be some fixed predefined element of G/H. Thus, we may view the field ξa (x) as a mapping ξa :

SD 7→ G/H ,

(10.14)

where SD is the D-dimensional sphere, which is topologically equivalent to the euclidean space D with all the points |x| = ∞ identified as a single point. This

❘

332

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 10.3: Stereographic projection that maps the plane 2 to the sphere S2 . All the points at infinity in the plane are identified, and mapped to the north pole of the sphere.

❘

equivalence may be made manifest by a stereographic projection, illustrated for D = 2 in the figure 10.3. These mappings, taking a fixed value at |x| = ∞, can be organized into topological classes containing functions that can be continuously deformed into one another. The set of these classes is a group, known as the D-th homotopy group of G/H, denoted πD (G/H). The original version of this model was intended to describe nucleons as a topologically stable configuration of the pion field. In this case, there are D = 3 spatial dimensions, and the chiral symmetry SU(2) × SU(2) is spontaneously broken to SU(2). The coset in which ξa lives is SU(2), and the relevant homotopy group is π3 (SU(2)) = ❩. The integer that enumerates the topological classes is then identified with the baryon number. Note that the model defined by eq. (10.12), with only second order derivatives, cannot have stable solutions, a result known as Derricks’ theorem. In order to see this, consider a skyrmion solution ξa (x), and construct another field by a rescaling: ξa (x) ≡ ξa (x/R) . R

(10.15)

The action becomes S[ξR ] = RD−2 S[ξ]. In D > 2 dimensions, we may make it decrease continuously to zero, despite the fact that ξa and ξa have the same topology. R Such a solution may be stabilized by adding a term with higher derivatives, such as Z    (10.16) V[ξ] ≡ dD x habcd (ξ) ∂i ξa ∂i ξb ∂j ξc ∂j ξd . Under the same rescaling, we now have V[ξR ] = RD−4 V[ξ]. In D = 3 spatial dimensions, the term with second derivatives decreases to zero when R → 0, while the above quartic term increases to +∞. Their sum therefore exhibits an extremum at

10. L OCALIZED FIELD CONFIGURATIONS

333

some finite scale R∗ . Although we obtain in this way non-trivial stable solutions, there is a priori no reason to limit ourselves to terms with four derivatives, and therefore the predictive power of such a model is limited by the many possible choices for these higher order terms.

10.3 Monopoles 10.3.1

Dirac monopole

Magnetic monopoles are not forbidden in quantum electrodynamics, but their existence would automatically lead to the quantization of electrical charge, as first noted by Dirac. Let us reproduce here this argument. Consider the radial magnetic field of a would-be monopole: B=g

b x . |x|2

(10.17)

Maxwell’s equation ∇ · B = 0 implies that we cannot find a vector potential A for this magnetic field in all space. But it is possible to find one that works almost everywhere, for instance A(x) = g

1 − cos θ eφ , |x| sin θ

(10.18)

where θ is the polar angle, φ the azimuthal angle, and eφ is the unit vector tangent to the circle of constant |x| and θ. This vector potential is not defined on the semi-axis θ = π (i.e. the semi-axis of negative z). One may argue that on this semi-axis, we have in addition to the monopole field a singular Bz whose magnetic flux precisely cancels the magnetic flux of the monopole, so that the total flux on any closed surface containing the origin is zero, as illustrated in the figure 10.4. Thus, in this solution, the magnetic flux Φm ≡ 4π g of the monopole is “brought from infinity” by an infinitely thin “solenoid”. Even if it is infinitely thin, such a solenoid may in principle be detected by looking for interferences between the wavefunctions of charged particles that have propagated left and right of the solenoid (this corresponds to the AharonovBohm effect). For a particle of electrical charge e, the corresponding phase shift is eΦm = 4πeg. Dirac pointed out that this interference is absent when the phase shift is a multiple of 2π, i.e. when the electric and magnetic charges are related by ge =

n ,n ∈ ❩ . 2

(10.19)

Thus, electrodynamics can perfectly accommodate genuine magnetic monopoles, provided this condition is satisfied, since the annoying solenoid that comes with

334

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

θ

φ

er

eφ eθ

Figure 10.4: Left: notations for the polar coordinates local frame used in eq. (10.18). Right: magnetic field lines of the Dirac monopole, corresponding to the vector potential of eq. (10.18).

the above vector potential is totally undetectable. In particular, this implies that all electrical charges should be multiples of some elementary quantum of electrical charge if monopoles exist. Note that in quantum electrodynamics, while the electric and magnetic charges must be related by eq. (10.19), there is no constraint a priori on the mass of monopoles and it should be regarded as a free parameter. c sileG siocnarF

Let us mention briefly an alternative argument, that does not involve discussing the detectability of Dirac’s solenoid. Instead of the vector potential of eq. (10.18), one could instead have chosen A ′ (x) = −g

1 + cos θ eφ , |x| sin θ

(10.20)

that has a singularity on the semi-axis θ = 0. When Dirac’s quantization condition is satisfied, one may patch eqs. (10.18) and (10.20) in order to obtain a vector potential which is regular in all space (except at the origin, where the monopole is located). To see this, consider a region Ω1 corresponding to 0 ≤ θ ≤ 3π/4 and a region Ω2 corresponding to π/4 ≤ θ ≤ π. Then, we choose A in Ω1 and A ′ in Ω2 . In the overlap of the two regions, π/4 ≤ θ ≤ 3π/4, we have (A − A′ ) · dx = 2g dφ ,

(10.21)

and we can write A − A′ = ∇χ with χ(φ) ≡ 2g φ .

(10.22)

335

10. L OCALIZED FIELD CONFIGURATIONS

For this to be an acceptable gauge transformation, the phase by which it multiplies the wavefunction of a charged particle should be single-valued, i.e. eie χ(φ+2π) = eie χ(φ) ,

(10.23)

which is precisely the case when the condition (10.19) is satisfied. This argument can even be made without any reference to the explicit solutions (10.18) and (10.20). Let us consider a large sphere surrounding the origin, divide it in an upper and lower hemispheres (see the figure 10.7), and denote A and A ′ the vector potentials that represent the monopole in these two hemispheres. On the equator, their difference should be a pure gauge, A − A′ =

i † Ω (x) ∇ Ω(x) . e

Along the equator, we have  Ω(φ) = Ω(0) exp

− ie

Z

(10.24)

A−A γ[0,φ]

′




· dx



,

(10.25)

where the integration path γ[0, φ] is the portion of the equator that extends between the azimuthal angles 0 and φ. After a complete revolution, we have Ω(2π) = =

Ω(0) exp Ω(0) exp



− ie

I

A−A

Equator

− ie ΦU + ΦL | {z } flux =4πg




′




· dx



= Ω(0) e−4πi eg .

(10.26)

To obtain the first equality on the second line, we use Stokes’s theorem to rewrite the contour integrals of A and A ′ as surface integrals of the corresponding magnetic field. Therefore, we obtain the magnetic fluxes through the upper and lower hemispheres, respectively, whose sum is the total flux 4πg of the monopole. Requesting the single-valuedness of Ω leads to Dirac’s condition on eg.

10.3.2

Monopoles in non-Abelian gauge theories

There are also non-abelian field theories that exhibit U(1) magnetic monopoles, as classical solutions whose stability is ensured by topology. The simplest example is an SU(2) gauge theory coupled to a scalar field in the adjoint representation3 , whose 3 This model is known as the Georgi-Glashow model. It was considered at some point as a possible candidate for a field theory of electroweak interactions, until the neutral vector boson Z0 was discovered. Here, we use it as a didactical example of a theory with classical solutions that are magnetic monopoles.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Lagrangian density reads

with


1 1 Dµ Φa Dµ Φa ) − V(Φ) , Fa,µν + L ≡ − Fa 4 µν 2

(10.27)


2 λ Φa Φa − v2 , 8 c Dµ Φa = ∂µ Φa − e ǫabc Ab µΦ ,

V(Φ) ≡

a a b c Fa µν = ∂µ Aν − ∂ν Aµ − e ǫabc Aµ Aν ,

(10.28)

where we have written explicitly the structure constants of the su(2) algebra. In order to study static classical solutions, it is simpler to consider the minima of the energy: Z

1  

 a a a a Ea E ≡ d3 x Di Φa + V(Φ) , (10.29) i Ei + Bi Bi + Di Φ 2 a a where Ea i ≡ F0i is the (non-abelian) electrical field and Bi ≡ magnetic field.

1 2

ǫijk Fa jk is the

It is possible to choose a gauge (called the unitary gauge) in which the scalar field triplet takes the form
Φa = 0, 0, v + ϕ . (10.30)

In this equation, we have anticipated spontaneous symmetry breaking, that will give to the scalar field a vacuum expectation value v, and we have made a specific choice about the orientation of the vacuum in SU(2). The field ϕ is thus the quantum fluctuation of the scalar about its expectation value. In this process, the fields A1,2 µ will become √ massive (with a mass3MW = e v), as well as the scalar field (with mass MH = λ v), while the field Aµ remains massless (it corresponds to a residual unbroken U(1) symmetry). The classical vacuum of this theory corresponds to ϕ=0,

Aa µ =0.

(10.31)

Now, we seek stable classical field configurations that are local minima of the energy, but are not equivalent to the vacuum in the entire space. To prove the existence of such fields, it is sufficient to exhibit a field configuration of non-zero energy that cannot be continuously deformed into the null fields of eq. (10.31) (up to a gauge transformation). In order to have a finite energy, the scalar field Φa should reach a minimum of the potential V(Φ) at large distance |x| → ∞ (we have shifted the potential so that its minimum is zero), but it may approach different minima depending on the direction b x in space. The allowed asymptotic behaviours of Φa x, for three spatial dimensions) define a mapping from the sphere S2 (the orientations b c sileG siocnarF

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10. L OCALIZED FIELD CONFIGURATIONS

Figure 10.5: Cartoon of the hedgehog configuration of eq. (10.32). Each needle indicates the internal orientation of Φa at the corresponding point on the sphere.

sileG siocnarF

to the sphere Φa Φa = v2 of the minima of V(Φ). Since su(2) is 3-dimensional, the set of zeroes of the scalar potential is also a sphere S2 , and it is natural to consider the following configuration4 : Φa (b x) ≡ v b xa ,

(10.32)

sometimes called a “hedgehog field” because the direction of internal space pointed to by the scalar field is locked to the spatial direction, as shown in the figure 10.5. Any smooth classical field Φa that obeys this boundary condition at infinite spatial distance must vanish at some point in the interior of the sphere. Therefore, it cannot simply be a gauge transform of the constant field Φa = v δ3a (the expectation value of the scalar field in the vacuum). Once again, the classes of fields that can be continuously deformed into one another are given by a homotopy group, in this case the group π2 (M0 ) where M0 is the manifold of the minima of the scalar potential. For the SU(2) group, M0 is topologically equivalent to the 2-sphere S2 , and the equivalence classes of the mappings S2 7→ S2 are indexed by the integers, since π2 (S2 ) = ❩. The hedgehog field of eq. (10.32) has topological number +1, while the vacuum has topological number 0. At spatial infinity, the hedgehog configuration (10.32) is gauge equivalent to the standard scalar vacuum aligned with the third colour direction, Φa = v δ3a . In order to see this, let us introduce the following SU(2) transformation, that depends on the

4 Here, we see that it is crucial that the scalar potential has non-trivial minima. If Φa ≡ 0 was the only minimum, it would not be possible to construct solutions of finite energy that are not topologically equivalent to the vacuum.

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polar angle θ and azimuthal angle φ as follows5 :   θ θ θ Ω(θ, φ) ≡ − cos sin φ + 2i sin t1f + cos cos φ t3f , 2 2 2

(10.33)

where the ta f are the generators of the fundamental representation of su(2). Then, one may check explicitly that   δ3a Ω† ta Ω = sin θ cos φ t1f + sin φ t2f + cos θ t3f = b xa ta (10.34) f . Thus, Ω transforms the usual scalar vacuum into the hedgehog configuration at infinity. Note that (10.33) is not a valid gauge transformation over the entire space because it is not well defined at the origin.

The choice of eq. (10.32) for the asymptotic behaviour of the scalar field was motivated by the requirement that the potential V(Φ) gives a finite contribution to the
2 energy. The term in Di Φa should also give a finite contribution. However, note that
v ia δ −b xi b xa (10.35) ∂i Φa (b x) = |x| is not square integrable. We must therefore adjust the asymptotic behaviour of the gauge potential in order to cancel this term in the covariant derivative, by requesting that xc ǫabc Ab i b

=

|x|→∞

which is satisfied if Ab i

=

|x|→∞

δia − b xi b xa , e |x|

xd ǫibd b + term in b xb . e |x|

(10.36)

(10.37)

The corresponding field strength and magnetic field are given by Fa ij Ba i

=

|x|→∞

=

|x|→∞


d 1  xi b xa b xj − ǫjad b x − ǫijd b 2 ǫija + 2 ǫiad b xd , 2 e |x| b xa xi b . (10.38) e |x|2

Therefore, at large distance (these considerations do not give the precise form of the fields at finite distance) there is a purely radial magnetic field that vanishes like |x|−2 , 5 When

an SU(2) transformation in the fundamental representation is written as

Ω ≡ u0 + 2i ua ta f ,

its unitarity (Ω† Ω = 1) is equivalent to u20 + u21 + u22 + u23 = 1.

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10. L OCALIZED FIELD CONFIGURATIONS

i.e. according to Coulomb’s law, thus suggesting that a magnetic monopole is present at the origin. For a more robust interpretation, we should apply a gauge transformation that maps the asymptotic Hedgehog scalar field into the usual scalar vacuum, aligned with the third colour direction. Thanks to eq. (10.34), we see that in this process the magnetic field of eq. (10.38), proportional to b xa , will become proportional to δ3a . But the third colour direction precisely corresponds to the gauge potential that remains massless in the spontaneous symmetry breaking SU(2) → U(1). Therefore, eq. (10.38) is indeed the magnetic field of a U(1) magnetic monopole. Its flux through a sphere surrounding the origin is Φm =

4π , e

(10.39)

equivalent to that of a magnetic charge g ≡ e−1 at the origin. Until now, we have only discussed the implications of requiring a finite energy on the asymptotic form of the scalar field and of the gauge potentials. In order to obtain their values at finite distance, one may make the following ansatz: Φa (x) = v b xa f(|x|) ,

Aa i (x) = ǫiab

b xb g(|x|) , e |x|

(10.40)

where f, g are two functions that can be determined from the classical equations of motion. From this solution over the entire space, one sees that the monopole is an extended object made of two parts: • A compact core, of radius Rm ∼ M−1 , in which the SU(2) symmetry is W unbroken and the vector bosons are all massless. One may view the core as a cloud of highly virtual gauge bosons and scalars. • Beyond this radius, a halo in which the SU(2) symmetry is spontaneously broken. In this halo, up to a gauge transformation, the scalar field is that of the ordinary broken vacuum, the vector bosons A1,2 are massive, and the A3 field is massless, with a tail that corresponds to a radial U(1) magnetic field. Given these fields, the total energy of the field configuration can be identified with the mass (in contrast with Dirac’s point-like monopole in quantum electrodynamics, whose mass is not constrained) of the monopole (since it is static). It takes the form Mm =

4π M C(λ/e2 ) , e2 W

(10.41)

where C(λ/e2 ) is a slowly varying function of the ratio of coupling constants, of order unity. Note that the core and the halo contribute comparable amounts to this mass. Interestingly, the size M−1 of this monopole is much larger (by a factor W

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

α−1 = 4π/e2 ) than its Compton wavelength M−1 m . Therefore, when α ≪ 1, the monopole receives very small quantum corrections and is essentially a classical object. We have argued earlier that the topologically non-trivial configurations of the scalar field that lead to a finite energy can be classified according to the homotopy group π2 (S2 ). Since this group is the group ❩ of the integers, there are monopole solutions with any magnetic charge multiple of e−1 (the solution we have constructed explicitly above has topological number 1), i.e. ge = n ,

n∈❩.

(10.42)

Therefore, in this field theoretical monopole solution, the electrical charge would also be naturally quantized. At first sight, eqs. (10.42) and (10.19) appear to differ by a factor 1/2. Note however, that in the SU(2) model we are considering in this section, it is possible to introduce matter fields in the fundamental representation6 that carry a U(1) electrical charge ±e/2 (this is the smallest possible electrical charge in this model). Thus, if rewritten in terms of this minimal electrical charge, the monopole quantization condition (10.42) is in fact identical to Dirac’s condition. Although the Georgi-Glashow model studied in this section is no longer considered as phenomenologically relevant, theories that unify the strong and electroweak interactions into a unique compact Lie group (such as SU(5) for instance) do have magnetic monopoles. c sileG siocnarF

10.3.3

Topological considerations

In the previous two subsections, we have encountered two seemingly different topological classifications of magnetic monopoles. The Dirac monopole appeared closely related to the mappings from a circle (the equator between the two hemispheres in the figure 10.7) to the group U(1), whose classes are the elements of the homotopy group π1 (U(1)) = ❩. In contrast, the monopole discussed in the Georgi-Glashow model was related to the behaviour of the scalar field at large distance, i.e. to mappings from the 2-sphere S2 to the manifold M0 of the minima of the scalar potential V(Φ), whose equivalence classes are the elements of the homotopy group π2 (M0 ) = ❩. Let us now argue that these two ways of viewing monopoles are in fact equivalent. In order to make this discussion more general, consider a gauge theory with internal group G, coupled to a scalar, spontaneously broken to a residual gauge symmetry of group H. Let us denote M0 the manifold of the minima of the scalar potential. This manifold is invariant under transformations of G. Given a minimum Φ0 , the other minima can be obtained by multiplying Φ0 by the elements of G:

 M0 = Φ Φ = ΩΦ0 ; Ω ∈ G . (10.43) 6 If

Ψ is a doublet that lives in this representation, the covariant derivative acting on it reads:   e 3 1 0 a Ψ. Dµ Ψ = ∂µ Ψ − i e Aa t Ψ = ∂ Ψ − i A µ µ f 2 µ 0 −1

341

10. L OCALIZED FIELD CONFIGURATIONS

Figure 10.6: Illustration of the symmetry breaking pattern. H is the residual invariance after choosing a minimum Φ0 . The coset G/H is the manifold that holds the minima of V(Φ) (a 2-sphere in the case of the Georgi-Glashow model).

Φ0 G/H

H

(Here, we are assuming that there are no accidental degeneracies among the minima, i.e. no minima Φ0 and Φ0′ that are not related by a gauge transformation.) The manifold defined in eq. (10.43) is in fact the coset G/H, M0 = G/H .

(10.44)

This pattern of spontaneous symmetry breaking is illustrated in the figure 10.6. The first way of classifying monopoles is to consider the gauge field on a sphere, as was done in the subsection 10.3.1. At large distance compared to the inverse mass of the bosons that became massive due to spontaneous symmetry breaking, only the massless gauge bosons contribute, and the corresponding gauge fields live in the algebra h of the residual group H. We can reproduce the argument made at the end of the subsection 10.3.1. The gauge potentials in the upper and lower hemispheres are related on the equator by a gauge transformation Ω(φ) ∈ H, that must be single-valued as the azimuthal angle φ wraps around the equator. Ω(φ) is therefore a mapping from the circle S1 to the residual gauge group H. These mappings can be grouped into classes that differ by their winding number. In this general setting, we may adopt the winding number as the definition of the product eg of the electric charge by the magnetic charge comprised within the sphere. Note that π1 (H) is discrete, and therefore the winding number can vary only by finite jumps7 . Moreover, the mapping Ω(φ) on the equator is a smooth function of the azimuthal angle φ and of the radius R of the sphere. Consequently, the winding number must be independent of the radius R. From this fact, two different situations may arise: • The relevant gauge fields belong to h all the way down to zero radius. In this case, the magnetic charge is independent of the radius of the sphere at all R, 7 Therefore, it must be conserved by time evolution. Indeed, time evolution is continuous, and the only way for a discrete quantity to evolve continuously is to be constant.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

A∈h Ω(φ) ∈ H

Figure 10.7: Decomposition of the sphere into two hemispheres with gauge potentials A and A ′ .

A' ∈ h

which means that the monopole is a point-like singularity at the origin, like the original Dirac monopole. • There exists a short-distance core in which the gauge fields live in an algebra which is larger than h (possibly the algebra g before symmetry breaking). Inside this core, the above argument is no longer valid, and the magnetic charge inside the sphere may vary continuously with the radius. In this case, the monopole is an extended object whose size is the radius of the core (its magnetic charge is spread out in the core). Alternatively, we may construct a monopole as a non-trivial classical field configuration that minimizes the energy, by starting from the behaviour at infinity of the scalar field. In order to have a finite energy, the scalar field should go to a minimum of V(Φ) when |x| → ∞. The asymptotic scalar field is therefore a mapping from the 2-sphere S2 to M0 = G/H, and it leads to a classification of the classical field configurations based on the homotopy group π2 (G/H). The correspondence between the two points of view is based on the following relationship, π2 (G/H) = π1 (H)/π1 (G) .

(10.45)

For a simply connected Lie group G (e.g., all the SU(N)), the first homotopy group is trivial, π1 (G) = {0}, and we have π2 (G/H) = π1 (H) , hence the equivalence between the two ways of classifying monopoles.

(10.46)

10. L OCALIZED FIELD CONFIGURATIONS

343

10.4 Instantons Until now, all the extended field configurations we have encountered were time independent. After integration over time, their action is infinite, and therefore they do not contribute to path integrals. In this section, we will discuss field configurations of finite action, called instantons, that are localized both in space and in time. Consider a Yang-Mills theory in D-dimensional Euclidean space, whose action reads Z 1 ij S[A] ≡ dD x Fij (10.47) a (x)Fa (x) . 4 (We use latin indices i, j, k, · · · for Lorentz indices in Euclidean space.) Instantons are non-trivial (i.e. not pure gauges in the entire spacetime) gauge field configurations that realize local minima of this action.

10.4.1

Asymptotic behaviour

In order to have a finite action, these fields must go to a pure gauge when |x| → ∞, Aia ta

→

|x|→∞

i † Ω (b x) ∂i Ω(b x) , g

(10.48)

where Ω(b x) is an element of the gauge group that depends only on the orientation b x. Since multiplying Ω(b x) by a constant group element Ω0 does not change the asymptotic gauge potential, we can always arrange that Ω(b x0 ) = 1 for some fixed x0 . Note that a gauge potential such as (10.48), that becomes a pure gauge orientation b at large distance, must decrease at least as fast as |x|−1 . More precisely, we may write Aia (x)ta =

i † Ω (b x) ∂i Ω(b x) + ai (x) . g {z } | {z } | |x|−1

(10.49)

≪|x|−1

The field strength associated to such a field decreases faster than |x|−2 , and therefore the corresponding action is finite in D = 4 dimensions. There is in fact a scaling argument showing that instanton solutions can only exist in four dimensions. Given an instanton field configuration Ai (x) and a scaling factor R, let us define AiR (x) ≡

1 i A (x/R) . R

(10.50)

Since classical Yang-Mills theory is scale invariant, the field AiR is also an extremum of the action (i.e. a solution of the classical Yang-Mills equations) if Ai is. The action of this rescaled field is given by S[AR ] = RD−4 S[A] .

(10.51)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Instanton

Figure 10.8: Cartoon of an instanton (the illustration is for D = 3, although instantons actually exist in D = 4). The sphere S3 is in fact infinitely far away from the center of the instanton. Pure gauge

F ij = 0

S3

Therefore, given an instanton Ai (x), we may continuously deform it into another field configuration AiR (x) whose action is multiplied by RD−4 . Unless D = 4, this action has a higher or lower value, in contradiction with the fact that Ai was a local extremum8 . Thus, non-trivial local extrema of the classical Euclidean Yang-Mills action can only exist in D = 4. In four dimensions, if Ai is an instanton, then AiR is also an instanton (with the same value of the action). Thus, classical instantons can exist with any size. But this degeneracy is lifted by quantum corrections, that introduce a scale into Yang-Mills theory via the running coupling.

10.4.2

Bogomol’nyi inequality and self-duality condition

In the study of instantons, a useful variant of Bogomol’nyi trick is to start from the following obvious inequality, Z 1 a 2 0 ≤ d4 x (Fa (10.52) ij ∓ ǫijkl Fkl ) , 2 which leads to   Z 1 a a 4 a a a a 0 ≤ d x Fij Fij ∓ ǫijkl Fij Fkl + ǫijkl ǫijmn Fkl Fmn 4   Z 1 a a a a 4 a a = d x Fij Fij ∓ ǫijkl Fij Fkl + (δkm δln − δkn δlm )Fkl Fmn 2 Z
a a a = d4 x 2Fa (10.53) ij Fij ∓ ǫijkl Fij Fkl . 8 The only exception to this reasoning occurs if S[A] = 0. But this happens only in the trivial situation where Ai is a pure gauge in the entire spacetime.

10. L OCALIZED FIELD CONFIGURATIONS

345

By choosing appropriately the sign, this can be rearranged into a lower bound for the action: Z 1 a S[A] ≥ ǫijkl d4 x Fa (10.54) F ij kl , 8

known as Bogomol’nyi’s inequality. Interestingly, we recognize in the right hand side an integral identical to the one that enters in the θ-term of Yang-Mills theories (see the section 4.5) or in the anomaly function (see the section 3.5 and the chapter 9). This equality becomes an equality when: c sileG siocnarF

1 a Fa ij = ± ǫijkl Fkl . 2

(10.55)

A solution that obeys this condition is by construction a minimum of the Euclidean action S[A], and therefore a solution of the classical Yang-Mills equations. But like in the case of domain walls, finding field configurations that fulfill this self-duality condition is somewhat simpler than solving directly the Yang-Mills equations. Thus, from now on, we will look for gauge fields that fulfill eq. (10.55) and go to a pure gauge as |x| → ∞.

10.4.3

Topological classification

In D = 4, the functions Ω(b x) that define the asymptotic behaviour of instantons map the 3-sphere S3 into the gauge group G, Ω :

S3 7→ G ,

(10.56)

with a fixed value Ω(b x0 ) = 1. These functions can be grouped into topological classes, such that mappings belonging to the same class can be continuously deformed into one another. The set of these classes can be endowed with a group structure, called the third homotopy group of G and denoted π3 (G) (for any SU(N) group with N ≥ 2, we have π3 (G) = ❩). Note that the asymptotic forms of the fields Ai and AiR are identical, implying that these two instantons belong to the same topological class. Since their actions are identical in four dimensions, this scaling provides a continuous family of instantons that belong to the same topological class and have the same action. This is in fact more general: we will show later that the action of an instanton depends only on the topological class of the instanton, and therefore can only vary by discrete amounts.

10.4.4

Minimal action

Let us assume that we have found a self-dual gauge field configuration, that realizes the equality in eq. (10.54). In order to calculate its action, we can use the fact that

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

a ǫijkl Fa ij Fkl is a total derivative,

h  g abc a b c  i 1 ijkl a a a − F . ǫ Fij Fkl = ∂i ǫijkl Aa f Aj Ak Al j kl 2 3 {z } |

(10.57)

Ki

(This property was derived in the section 4.5.) The vector Ki can also written as a trace of objects belonging to the fundamental representation:   2ig Ki = 2 ǫijkl tr Aj Fkl + Aj Ak Al . 3

(10.58)

Since the integrand in the right hand side of eq. (10.54) is a total derivative, one may use Stokes’s theorem in order to rewrite the integral as a 3-dimensional integral extended to a spherical hypersurface SR of radius R → ∞: Z Z 1 1 4 a a Smin [A] ≡ ǫijkl d x Fij Fkl = lim d3 Si Ki . (10.59) R→∞ 4 S 8 R

Thus, the minimum of the action depends only on the behaviour of the gauge field at large distance (this does not mean that the action does not depend on details of the gauge field in the interior, but more simply that the gauge fields that realize the minima are fully determined in the bulk by their asymptotic behaviour). From the earlier discussion of the asymptotic behaviour of instanton solutions, we know that Ai (x)

∼

|x|→∞

|x|−1 ,

Fij (x)

≪

|x|→∞

|x|−2 .

(10.60)

Therefore, in the current Ki , the term Aj Fkl is negligible in front of the term Aj Ak Al at large distance, and we can also write Smin [A] = =

ig R→∞ 3 lim

Z

d3 Si ǫijkl tr Aj Ak Al SR

1 R→∞ 3 g2 lim

Z

SR





d3 Si ǫijkl tr Ω† (∂j Ω)Ω† (∂k Ω)Ω† (∂l Ω) ,

(10.61)

where Ω(b x) is the group element that defines the asymptotic pure gauge behaviour of the gauge potential in the direction b x. In this expression, each derivative brings a factor R−1 , while the domain of integration scales as R3 . The result is therefore independent of the radius of the sphere and we can ignore the limit R → ∞. On this sphere, let us choose a system of coordinates made of three variables (θ1 , θ2 , θ3 ), such that the volume element in SR is dθ1 dθ2 dθ3 . To rewrite the previous integral more explicitly in terms of these variables, it is convenient to

347

10. L OCALIZED FIELD CONFIGURATIONS

introduce a fourth –radial– coordinate θ0 ≡ |x|. The coordinates (θ0 , θ1 , θ2 , θ3 ) are thus coordinates in 4 , and d4 x = dθ0 dθ1 dθ2 dθ3 . The volume element on the xi = ∂θ0 /∂xi , we can write sphere SR is dθ1 dθ2 dθ3 = d4 x δ(θ0 − R). Noting that b

❘

d3 Si = b xi dθ1 dθ2 dθ3 =

∂θ0 ∂θ0 dθ1 dθ2 dθ3 = d4 x δ(θ0 − R) ∂xi ∂xi

(10.62)

and the minimal action becomes Z 1 ∂θ0 ∂θa ∂θb ∂θc Smin [A] = d4 x δ(θ0 − R) ǫijkl 3 g2 ∂xi ∂xj ∂xk ∂xl

∂Ω(θ) † ∂Ω(θ) ∂Ω(θ) † † , Ω (θ) Ω (θ) ×tr Ω (θ) ∂θa ∂θb ∂θc (10.63) where we have rewritten the derivatives with respect to xi in terms of derivatives with respect to θa (the implicit sums on a, b, c run over the indices 1, 2, 3 only, because the group element Ω depends only on the orientation b x). Finally, we may use:   ∂θ0 ∂θa ∂θb ∂θc ∂(θ0 θ1 θ2 θ3 ) ǫlijk ǫ0abc . = det (10.64) ∂xi ∂xj ∂xk ∂xl ∂(x1 x2 x3 x4 ) | {z } =ǫabc

The determinant is nothing but the Jacobian of the coordinate transformation {xi } → {θa }. Therefore, we obtain

Z 1 ∂Ω(θ) † ∂Ω(θ) ∂Ω(θ) † † Smin [A] = 2 dθ1 dθ2 dθ3 ǫabc tr Ω (θ) Ω (θ) Ω (θ) . 3g ∂θa ∂θb ∂θc (10.65)

10.4.5

Cartan-Maurer invariant

Definition : In order to calculate the integral that appears in eq. (10.65), let us make a mathematical diggression. Consider a d-dimensional manifold S, of coordinates (θ1 , θ2 , · · · , θd ), a manifold M that may be viewed as a matrix representation of a Lie group, and a mapping Ω from S to M: (θ1 , θ2 , · · · , θd ) ∈ S −→ Ω(θ1 , θ2 , · · · , θd ) ∈ M .

(10.66)

The Cartan-Maurer form F[Ω] is an integral that generalizes the one encountered earlier:

Z ∂Ω(θ) ∂Ω(θ) , (10.67) F[Ω] ≡ dθ1 · · · dθd ǫi1 ···id tr Ω† (θ) · · · Ω† (θ) ∂θi1 ∂θid

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where ǫi1 ···id is the d-dimensional completely antisymmetric tensor, normalized according to ǫ12···d = +1. In d dimensions, this tensor tranforms as follows under circular permutations: ǫi1 ···id = (−1)d−1 ǫi2 ···id i1 .

(10.68)

Using the cyclicity of the trace, we conclude that F[Ω] = 0 if the dimension d is even. In the following, we thus restrict the discussion to the case where d is odd9 . Coordinate independence : Consider now another system of coordinates on S, that we denote θ′i . We have:   ∂θ′jd ∂θ′j1 ∂(θ′i ) i1 ···id ǫj1 ···jd , (10.69) ··· = det ǫ ∂θi1 ∂θid ∂(θj ) and the determinant in the right hand side is the Jacobian of the coordinate transformation. We thus obtain   Z ∂Ω(θ) ∂Ω(θ) · · · Ω† (θ) , (10.70) F[Ω] = dθ′1 · · · dθ′d ǫj1 ···jd tr Ω† (θ) ∂θ′j1 ∂θ′jd which is identical to eq. (10.67), except for the fact that it is expressed in terms of the new coordinates θ′i . This proves that F[Ω] is independent of the choice of the coordinate system on S, and is a property of the manifold S itself. Change under a small variation of Ω : Let us now study the change of F[Ω] when we vary the mapping Ω by δΩ. Thanks to the cyclicity of the trace, the variation of each factor Ω† ∂Ω/∂θi gives the same contribution to the variation of F[Ω]. Therefore, it is sufficient to consider one of these variations, and to multiply its contribution by the number of factors, d:   Z ∂Ω(θ) ∂Ω(θ) · · · δ Ω† (θ) . δF[Ω] = d dθ1 · · · dθd ǫi1 ···id tr Ω† (θ) ∂θi1 ∂θid (10.71) The variation of the last factor inside the trace can be written as   ∂Ω(θ) ∂δΩ(θ) ∂Ω(θ) δ Ω† (θ) +Ω† (θ) = −Ω† (θ)δΩ(θ) Ω† (θ) ∂θid ∂θid ∂θid | {z } −

= 9 This

Ω† (θ)

∂Ω† (θ) Ω(θ) ∂θi d

∂δΩ(θ)Ω† (θ) Ω(θ) . ∂θid

(10.72)

is the case in the study of instantons, since in this case the manifold S is the 3-sphere S3 .

349

10. L OCALIZED FIELD CONFIGURATIONS Then, integrating by parts with respect to θid , we obtain:

Z δF[Ω] = −d dθ1 · · · dθd ǫi1 ···id  

∂ ∂Ω(θ) † ∂Ω(θ) † × tr Ω (θ) · · · Ω (θ) δΩ(θ)Ω† (θ) . ∂θid ∂θi1 ∂θid−1 (10.73) All the terms containing a factor ∂2 Ω/∂θid ∂θia vanish because the second derivative is symmetric under the exchange of of the indices id and ia , while the prefactor ǫi1 ···id is antisymmetric. The remaining terms are those where the derivative with respect to θd act on one of the factors Ω† . There are d − 1 such terms, which after some reorganization can be written as

Z X ∂Ω† ∂Ω ∂Ω† ǫi1 iσ(2) ···iσ(d) tr δF[Ω] = −d dθ1 · · · dθd ··· δΩ . ∂θi1 ∂θi2 ∂θid σ

|

cyclic perm. of 2···d

{z 0

}

(10.74)

ǫi1 ···id changes sign under a one-step cyclic permutation of its last d − 1 indices. Therefore, the d − 1 terms in the sum exactly cancel since d − 1 is even, and we have δF[Ω] = 0 .

(10.75)

Therefore, F[Ω] is invariant under small changes of Ω, which implies that F[Ω] can only vary by discrete jumps. In particular, when S is the d-sphere Sd , F[Ω] depends only on the homotopy class of Ω. These classes form a group πd (M). Moreover, F[Ω] provides a representation of πd (M): if Ω denotes the homotopy class to which Ω belongs, we have F[Ω1 × Ω2 ] = F[Ω1 ] + F[Ω2 ] .

(10.76)

(We denote by × the group composition in πd (M).) As a consequence, if there exists an Ω for which the Cartan-Maurer invariant is nonzero, then all its integer multiples can also be obtained, thereby proving that the homotopy group πd (M) contains Z. c sileG siocnarF

Case of a Lie group target manifold : Let us now specialize to the case where the target manifold M is a d-dimensional Lie group H, and exploit its group structure in order to obtain simpler expressions. In this case, the θa ’s can also be used as coordinates on H. Consider two elements Ω1 and Ω2 of H, represented respectively by the coordinates θa and φa . Their product Ω2 Ω1 is an element of H of coordinates ψ(θ, φ) (the group multiplication determines how ψ depends on θ and φ). Since we

350

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

have shown that the choice of cordinates on S is irrelevant, we may choose them in such a way that the function Ω(θ) is a representation of the group H, i.e. Ω(φ)Ω(θ) = Ω(ψ(θ, φ)) .

(10.77)

By differentiating this equality with respect to ψj at fixed φ, we obtain ∂Ω(θ) ∂θi ∂Ω(ψ) = , ∂θi ∂ψj ∂ψj

Ω(φ)

(10.78)

and after left multiplication by Ω† (ψ), this leads to Ω† (θ)

∂ψj † ∂Ω(ψ) ∂Ω(θ) = Ω (ψ) . ∂θi ∂θi ∂ψj

(10.79)

Using (10.69), the integrand of F[Ω] at the point θ can be expressed as ǫ



∂Ω(θ) ∂Ω(θ) † tr Ω (θ) · · · Ω (θ) ∂θi ∂θid

 1  ∂Ω(ψ) ∂Ω(ψ) ∂(ψ) , ǫj1 ···jd tr Ω† (ψ) · · · Ω† (ψ) = det ∂(θ) ∂ψj1 ∂ψjd (10.80)

i1 ···id

†

where ψ can be any fixed reference point in the group. In the right side, the integration variable θ now appears only inside the determinant. The Lie group H being a smooth manifold, it can be endowed with a metric tensor γij (θ), that transforms as follows in a change of coordinates γij (ψ) =

∂θk ∂θl γkl (θ) . ∂ψi ∂ψj

(10.81)

Given a mapping Ω(θ) between coordinates and group elements, a possible choice for the metric is given by10

1 ∂Ω(θ) ∂Ω(θ) † γij (θ) = − tr Ω† (θ) Ω (θ) . (10.82) 2 ∂θi ∂θj Moreover, for any such metric γij (θ), we have:  s  det γ(θ) ∂(ψ) = . det ∂(θ) det γ(ψ)

(10.83)


the algebra of a compact Lie group, the Killing form K(X, Y) ≡ tr adX adY is a negative definite inner product, from which one can define a distance on the group manifold in the vicinity of the origin (see the section 4.2.4). Eq. (10.82) extends this definition globally to the entire group, in a way which is invariant under left and right group action. 10 In

10. L OCALIZED FIELD CONFIGURATIONS

351

Therefore, the Cartan-Maurer invariant F[Ω] takes the following form F[Ω] =

∂Ω(ψ) ∂Ω(ψ) · · · Ω† (ψ) ǫj1 ···jd tr Ω† (ψ) ∂ψj1 ∂ψjd Z p 1 ×p dd θ det γ(θ) , det γ(ψ)

(10.84)

in which all the terms p that do not depend on θ have been factored out in front of the integral. In fact, dd θ det γ(θ) is an invariant measure on the Lie group, and the integral is therefore the volume of the group. In other words, the previous formula exploits the group invariance in order to rewrite the Cartan-Maurer invariant as the product of the integrand evaluated at a fixed point by the volume of the group. Since ψ is arbitray in this expression, we may choose the value ψ0 that corresponds to the group identity. Furthermore, groups elements in the vicinity of the identity may be written as Ω(ψ)

≈

ψ→ψ0

1 + 2i (ψ − ψ0 )a ta ,

(10.85)

where the ta ’s are the generators of the Lie algebra h. Then, the derivatives read simply ∂Ω(ψ) = 2i ta . (10.86) ∂ψa ψ0

From this, we obtain the following compact expression for F[Ω]: Z p  1 F[Ω] = (2i)d ǫi1 ···id tr ti1 · · · tid p dd θ det γ(θ) . (10.87) det γ(ψ0 ) Cartan-Maurer invariant for H = SU(2) : Consider the following mapping from the 3-sphere S3 to the fundamental representation of SU(2):   θ4 + iθ3 θ2 + iθ1 = θ4 + 2i θa ta , (10.88) Ω(θ) = −θ2 + iθ1 θ4 − iθ3

with t1,2,3 the generators of the su(2) algebra (for the fundamental representation, the Pauli matrices divided by 2) and θ21 + θ22 + θ23 + θ24 = 1. The following identities hold: det Ω(θ) = 1 , ∀i ∈ {1, 2, 3},

Ω† (θ) = θ4 − 2i θa ta , θi ∂Ω(θ) = 2i ti − p . ∂θi 1 − θ2

(10.89)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

(We denote θ2 ≡ θ21 + θ22 + θ23 .) In the evaluation of eq. (10.82), we need traces of products of up to four ta matrices. In the fundamental representation, they can all be obtained from tr (ti ) = 0 , i 1 ti tj = ǫijk tk + δij , 2 4

(10.90)

which leads to 1 ij δ , 2 i tr (ti tj tk ) = ǫijk , 4 1 tr (ti tj tk tl ) = (δij δkl + δil δjk − δik δjl ) . 8

tr (ti tj ) =

(10.91)

Then, the metric tensor of eq. (10.82) reads γij (θ) = δij +

θi θj 1 − θ2

,

(10.92)

and its determinant is det γ(θ) =

1 1 − θ2

.

(10.93)

Combining the above results, we obtain the following expression for the CartanMaurer invariant of the homotopy class of Ω in π3 (SU(2)) Z 2 d3 θ . (10.94) F[Ω] = (2i)3 ǫabc tr (ta tb tc ) p 1 − θ2

The factor 2 comes from the fact that there are two allowed values of θ4 for each θ1,2,3 . Finally, we have F[Ω] = 96π

Z1 0

dθ θ2 √ = 24π2 . 1 − θ2

(10.95)

In fact, the mapping of eq. (10.88) wraps only once in SU(2), and the above result therefore corresponds to the topological index +1. Since 24π2 is non-zero, there are other classes of Ω’s whose Cartan-Maurer invariants are the integer multiples of this result, and the second homotopy group is π3 (SU(2)) = ❩. Note also that this result extends to any Lie group that contains an SU(2) subgroup. c sileG siocnarF

10. L OCALIZED FIELD CONFIGURATIONS

10.4.6

353

Explicit instanton solution

In a gauge theorie whose gauge group contains an SU(2) subgroup, the mapping of eq. (10.88)) can be used to construct the asymptotic form of an instanton of topological index +1, Ai (x)

=

|x|→∞

i † Ω (b x)∂i Ω(b x) , g

(10.96)

with Ω(b x) ≡ b x4 + 2i b xi ti . One may then prove that the self-dual field configuration in the bulk that has this large distance behaviour is given by Ai (x) =

i r2 Ω† (b x)∂i Ω(b x) , g r 2 + R2

(10.97)

with an arbitrary radius R. From the result (10.95) of the previous subsection, we find that the minimum of the action that corresponds to this solution is: Smin [A] =

8π2 . g2

(10.98)

Up to translations, dilatations or gauge transformations, this is the only field configuration that gives this action. The field strength corresponding to eq. (10.97) is localized in Euclidean spacetime, with a size of order R. One may also superimpose several such solutions. Provided that their centers are separated by distances much larger than R, this sum is also a solution of the classical equations of motion, and its action is a multiple of 8π2 /g2 .

10.4.7

Instantons and the θ-term in Yang-Mills theory

Since we have uncovered classical field configurations of non-zero topological index with finite action, a legitimate question is their role in an Euclidean path integral, since functional integration a priori sums over all classical field configurations. For more generality, we may assume that in the path integral the fields of topological index n are weighted with a factor P(n) that may vary with n (this generalization would allow for instance to exclude fields of topological index different from zero). Thus, the expectation value of an observable O may be written as Z X −1 P(n) [DA]n O[A] e−S[A] , (10.99) hOi = Z n∈❩

where [DA]n is the functional measure restricted to gauge fields of topological index n. The normalization factor Z is given by the same path integral without the observable.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The dependence of P(n) on the topological index cannot be arbitrary. In order to see this, let us consider two spacetime subvolumes Ω1 and Ω2 , non overlapping and such that Ω1 ∪ Ω2 = 4 . Assume further that the support of the observable O is entirely inside Ω1 . The topological number, that may be obtained as the integral a 11 over spacetime of ǫijkl Fa ij Fkl , is additive and we may define topological numbers n1 and n2 for Ω1 and Ω2 , respectively. The total topological number n is given by n = n1 + n2 . In the expectation value of eq. (10.99), we can therefore split the integration into the domains Ω1 and Ω2 as follows Z Z X P(n1 + n2 ) [DA]n1 O[A] e−SΩ1 [A] [DA]n2 e−SΩ2 [A] , hOi = Z−1

❘

n1 ,n2 ∈❩

(10.100)

where [DA]ni is the functional measure for gauge fields with topological number ni in the domain Ωi . Since the observable is localized inside the domain Ω1 , we should be able to remove any dependence on the domain Ω2 from its expectation value. This dependence cancels between the numerator and the factor Z−1 in the previous expression provided that the weight P(n1 + n2 ) factorizes as follows: P(n1 + n2 ) = P(n1 )P(n2 ) ,

(10.101)

which implies that P(n) = e−nθ ,

(10.102)

where θ is an arbitrary constant. From the previous results, the topological number of a field configuration is given by the integral Z g2 a d4 x ǫijkl Fa (10.103) n= ij Fkl . 64π2

Therefore, we may capture the effect of the topological weight P(n) by adding to the Lagrangian density the following term

θ g2 ijkl a a ǫ Fij Fkl . (10.104) 64π2 After this term has been added, it is no longer necessary to split the path integral into separate topological sectors. The previous Lagrangian is nothing but the θ-term that we have already encountered in the discussion of non-Abelian gauge theories. There, it appeared as a term that cannot be excluded on the grounds of gauge symmetry. In the present discussion, we see that the θ-term results from a non-uniform weighting of the field configurations of different topological index (θ = 0 corresponds to a path integration where all the fields are weighted equally, regardless of their topological index). Lθ ≡

11 But note that this integral does not have to be an integer when the integration domain is not the entire spacetime. However, it is approximately an integer when the size of the domain is much larger than the instanton size.

10. L OCALIZED FIELD CONFIGURATIONS

10.4.8

355

Quantum fluctuations around an instanton

Consider an instanton solution Aµ n,α (x), that provides a local minimum of the Euclidean action, where the subscript n is the topological index of the instanton, and α collectivey denotes all the other parameters that characterize the instanton (its center, its size, its orientation in colour space). The expectation value of an observable reads Z Z

 O = Z−1 [DA] e−S[A] O(A) = Z−1 [Da] e−S[An,α +a] O(An,α + a) ,

(10.105)

where we denote aµ the difference Aµ − Aµ n,α . Since the instanton is an extremum of the action, the dependence of the action on aµ begins with quadratic terms: S[An,α + a] =

8π2 |n| 1 + g2 2

Z

d4 xd4 y G−1 nα,mβ (x, y) a(x)a(y) + · · · (10.106)

It is important to note that the action has flat directions in the space of field configurations, that correspond to changing the parameters of the instanton inside its topological class. For instance, changing the center coordinates of the instanton does not modify the value of its action. Along these directions, the second derivative of the action vanishes. This means that the matrix of second-order coefficient G−1 nα,mβ (x, y) has a number of vanishing eigenvalues, corresponding to these flat directions. If we expand the action only to quadratic order in aµ , which amounts to a oneloop approximation in the background of the instanton, a typical contribution to the expectation value of eq. (10.105) is a product of dressed propagators Gnα,mβ (x, y) connecting pairwise the gauge fields contained in the observable O and a determinant: 

 1/2 Y

 2 2 G + ··· . O = Z−1 e−8π |n|/g det G

(10.107)



(10.108)

Our goal here is simply to extract the dependence of such an expectation value on the topological index n. Besides the obvious exponential prefactor, a dependence on n hides in the determinant. Le us rewrite it as a product on the spectrum of G−1 det G

1/2

=

Y

λ−1/2 , s

s

where the λs are the eigenvalues of G−1 . If we rescale the gauge fields by a power of the coupling g, g A → A, the only dependence on g in the Yang-Mills action is a prefactor g−2 , and all the eigenvalues λs are also proportional to g−2 . Moreover, as explained above, we should remove the zero modes from this product, since they do

356

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

not give a quadratic term in eq. (10.106). If we are interested only in the powers of g, we may write 

det G

1/2

∼

Y

all modes

g

Y

g−1 .

(10.109)

zero modes

The first factor, that involves a (continuous) infinity of modes, is not well defined but it does not depend on the details of the instanton background. In contrast, the second factor brings one factor of g−1 for each collective coordinate of the instanton. For an instanton of topological number n = 1, these collective coordinates are: • the 4 coordinates of the center of the instanton, • the size R of the instanton, • 3 angles that determine the orientation of the instanton, • for SU(2), 3 parameters defining a global gauge rotation. c sileG siocnarF

Of the last 6 parameters, 3 correspond to simultaneous spatial and colour rotations that produce the same instanton solution, and they should not be counted. There are therefore 8 collective coordinates for the n = 1 SU(2) instanton12 , and its contribution to expectation values scales as

 2 2 O n=1 ∼ e−8π /g g−8 .

(10.110)

Because of the exponential factor that contains the inverse coupling, all the Taylor coefficients of this function are vanishing at g = 0. Thus, such a contribution never shows up in perturbation theory.

12 This counting is more involved for an SU(3) instanton. In this case, there are 7 collective coordinates corresponding to rotations and gauge transformations, hence a total of 12 collective coordinates.

Chapter 11

Modern tools for tree level amplitudes 11.1 Shortcomings of the usual approach Transition amplitudes play a central role in quantum field theory, since they are the building blocks of most observables. Their square gives transition probabilities, that enter in measurable cross-sections. Until now, we have exposed the traditional way of calculating these amplitudes. Starting from a classical action that encapsulates the bare couplings of a given quantum field theory, one can derive Feynman rules for propagators and vertices (listed in the figure 5.2 for Yang-Mills theory in covariant gauges), whose application provides a straightforward algorithm for the evaluation of amplitudes. However, the use of these Feynman rules is very cumbersome for the following reasons: • Even at tree level, the number of distinct graphs contributing to a given amplitude increases very rapidly with the number of external lines, as shown in the table 11.1 for amplitudes with external gluons only. • The internal gluon propagators of these diagrams carry unphysical degrees of freedom, which contributes to the great complexity of each individual diagram. • The Feynman rules are sufficiently general to compute amplitudes with arbitrary external momenta (not necessarily on-shell) and polarizations (not necessarily physical), although this is not useful for amplitudes that will be used in crosssections. One would hope for a leaner formalism, that only calculates what is strictly necessary for physical quantities.

357

358

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

# of gluons 4 5 6 7 8 9 10

# of diagrams 4 25 220 2,485 34,300 559,405 10,525,900

n! 24 120 720 5,040 40,320 362,880 3,628,800

Table 11.1: Number of Feynman diagrams contributing to tree level amplitudes with external gluons only. (The third column indicates the values of n!, for comparison. We see that the number of graphs grows faster than the factorial of the number of external gluons.)

The situation becomes even worse with loop diagrams. Another situation with an even higher degree of complexity, even at tree-level, is that of gravity. It would be desirable to be able to calculate tree-level amplitudes with gravitons, since they enter for instance in the study of the scattering of gravitational waves by a distribution of masses. But because the graviton has spin 2, the corresponding Feynman rules are considerably more complicated (especially the self-couplings of the graviton) than those of Yang-Mills theory. It turns out that physical on-shell amplitudes in gauge theories are considerably simpler than one may expect from the Feynman rules and the intermediate steps of their calculation by the usual perturbation theory, and a legitimate query is whether there is a more direct route to reach these compact answers. The goal of this chapter is to give a glimpse (in particular, our discussion will be restricted to tree-level amplitudes, but a significant part of the many recent developments deal with loop corrections) of some of the recent developments that led to powerful new methods for calculating amplitudes. A recurring theme of these methods is to avoid as much as possible references to the Lagrangian, which may be viewed as the main source of the complications in standard perturbation theory (for instance, the gauge invariance of the Lagrangian is the reason why non-physical gluon polarizations appear in the Feynman rules). Instead, these methods try to gather as much information as possible on amplitudes based on symmetries and kinematics.

11.2 Colour ordering of gluonic amplitudes Let us firstly focus on the colour structure of tree Feynman diagrams, in order to organize and simplify it. Although the techniques we expose here can be extended to quarks, we consider tree amplitudes that contain only gluons for simplicity, in the case of the SU(N) gauge group. The structure constants fabc of the group appear in the three-gluon and four-gluon vertices. The first step is to rewrite the structure constants in terms of the generators ta f of the fundamental representation of su(N).

359

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES Using the following relations among the generators, 

 b abc c ta tf f , tf = i f

b tr (ta f tf ) =

,

δab , 2

(11.1)

we can write

b c b a c i fabc = 2 tr (ta f tf tf ) − 2 tr (tf tf tf ) ,

(11.2)

which has also the following diagrammatic representation   c c     abc . if =2 −    b b a

(11.3)

a

The black dots indicate the fundamental representation generators ta f . Note that the “loops” in this representation are not actual fermion loops, they are just a graphical cue indicating how the indices carried by the ta f ’s are contracted in the traces. We may also apply this trick to the 4-gluon vertex, which from the point of view of its colour structure (but not for what concerns its momentum dependence) is equivalent to a sum of three terms with two 3-gluon vertices, a a

a

b

= d

c

a

b

b

b

+

+

.

(11.4)

c

d

d

c

d

c

Since the gluon propagators are diagonal in colour (i.e proportional to a δab ), the ta f that are attached to the endpoints of the internal gluon propagators have their colour indices contracted and summed over. The result of this contraction is given by the following su(N) Fierz identity: j

i

a (ta f )ij (tf )kl =

= k

l

1 2

−

1 2N

. (11.5)

Thus, it seems that these contractions produce 2n terms for n internal gluon propagators, but this can in fact be simplified tremendously by noticing that the second term of the Fierz identity corresponds to the exchange of a colourless object1 , that does not couple to gluons. All these terms in 1/N must therefore cancel in purely gluonic amplitudes (this is not true anymore if quarks are involved, either as external lines or via loop corrections). c sileG siocnarF

1A

more rigorous justification is to note that SU(N) × U(1) = U(N), where U(N) is the group of the

360

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

We illustrate in the following equation a few of the colour structures generated by this procedure in the case of a tree-level five-gluon diagram:

=

+

+ . . . (11.6)

Each of the terms contains a single trace of five ta f , one for each external gluon (the colour matrices attached to the internal gluon lines have all disappeared when using the Fierz identity). The terms in the right hand side correspond to the various ways of choosing the clockwise or counterclockwise loop for each fabc (see eq. (11.3)). “Twists” such as the one appearing in the second term of the previous equation arise when two such adjacent loops have opposite orientations. Quite generally, any n-gluon tree amplitude Mn (1 · · · n) can be decomposed as a sum of terms corresponding to the allowed colour structures. These colour structures are single traces of fundamental representation colour matrices carrying the colour indices of the external gluons. A priori, these matrices could be reshuffled by an arbitrary permutation in Sn , but thanks to the cyclic invariance of the trace we can reduce the sum to the quotient set Sn /❩n of permutations modulo a cyclic permutation2 : X a a Mn (1 · · · n) ≡ 2 tr (tf σ(1) · · · tf σ(n) ) An (σ(1) · · · σ(n)) , (11.7) σ∈Sn /❩n

where the prefactor 2 combines the factors 2 from eq. (11.2) and the factors 12 from the first term of the Fierz identity (11.5). The object An (σ(1) · · · σ(n)) is called a colour-ordered partial amplitude. By construction, it depends only on the momenta and polarizations of the external gluons, but not on their colours since they have already been factored out in the trace. Therefore, the partial amplitudes are gauge N × N unitary matrices. For the fundamental generators of the u(N) algebra, the Fierz identity is j

i

=

u(N)

k

1 2

.

l

The U(N) gauge theory differs from the SU(N) one by the extra U(1), and the comparison of their Fierz identities indicates that the term in 1/2N in eq. (11.5) is due to this U(1) factor. Being Abelian, this extra factor corresponds to a photon-like mode that does not couple to gluons. 2 This is equivalent to considering permutations that have the fixed point σ(1) = 1, i.e. permutations that only reshuffle the set {2 · · · n}. For n external gluons, there are (n − 1)! independent colour structures. The basis provided by these traces is over-complete, and there exist linear relationships among the tree-level partial amplitudes, known as the Kleiss-Kuijf relations. These relations reduce the number of partial amplitudes from (n − 1)! to (n − 2)!. Additional relationships known as the Bern-Carrasco-Johansson relations further reduce this number to (n − 3)!,

361

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

Table 11.2: Comparison between the number of Feynman graphs and the number of cyclic-ordered graphs for tree level amplitudes with external gluons only.

# (gluons) 4 5 6 7 8 9 10

# (graphs) 4 25 220 2,485 34,300 559,405 10,525,900

# (cyclic-ord.) 3 10 38 154 654 2,871 12,925

invariant. From eq. (11.7), the squared amplitude summed over all colours can be written as 2 X Mn (1 · · · n) =

colours

4

X

X

σ,ρ∈Sn /❩n colours

a

a

a

a

tr (tf σ(1) · · · tf σ(n) ) tr∗ (tf ρ(1) · · · tf ρ(n) )

× An (σ(1) · · · σ(n)) A∗n (ρ(1) · · · ρ(n)) . (11.8)

The sum over colours of the product of two traces that appears in the first line can be performed using the su(N) Fierz identity (11.5). For instance

tr (ta tb tc td te ) tr∗ (tb ta tc td te ) =

,

(11.9)

which can be then expressed as a function of N by repeated use of the Fierz identity. At this point, we have isolated the colour dependence of the amplitude, from its momentum and polarization dependences that are factorized into the partial amplitudes. Of course, calculating the latter is still not easy, but the task is significantly reduced for two reasons: • The colour-ordered partial amplitudes only receive contributions from planar graphs where the gluons are cyclic-ordered, whose number grows much slower than the total number of graphs, as shown in the table 11.2. The graphs contributing to the 4, 5, 6-point colour ordered amplitudes are listed in the figure 11.1. • The Feynman rules for calculating the cyclic colour-ordered amplitudes, listed in the figure 11.2, are much simpler than the original Yang-Mills Feynman rules because the vertices are stripped of all their colour factors.

362

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

3

2

4

4

1

5

5

3 2 1

4 3 2

6

1

Figure 11.1: Diagrams contributing to the 4-point, 5-point and 6-point colour ordered amplitudes in Yang-Mills theory. The external points are labeled 1 to n = 4, 5, 6 in the counterclockwise direction. The solid lines represent gluons.

=

−i gµν i + p2 + i0+ p2 + i0+

=

 g gµν (k − p)ρ

p



1−

1 ξ



pµ pν p2

µ

k

q p

ν

µ

ρ

ν

=

ρ

+ gνρ (p − q)µ + gρµ (q − k)ν

− i g2 (2 gµρ gνσ − gµσ gνρ − gµν gρσ )

σ

Figure 11.2: Rules for colour-ordered graphs in Yang-Mills theory.



363

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

In the case of the 4-gluon vertex, we have included only the terms that correspond to the cyclic ordering µνρσ (note that it is invariant under cyclic permutations, i.e. the Feynman rule is the same for the vertices νρσµ, ρσµν and σµνρ). We can already see a considerable simplification of the Feynman rules, since all the colour factors have disappeared, and the Lorentz structure of the 4-gluon vertex is also much simpler than in the original Feynman rules. But even after having isolated the colour structure, the remaining colour-ordered amplitudes are still complicated. As an illustration of the colour-ordered Feynman rules, let us consider the partial amplitude A4 (1, 2, 3, 4) that contributes to one of the colour structures in the gg → gg amplitude. Because of colour ordering, only three graphs contribute to this partial amplitude: 2

2

3

2

3

+

A4 (1, 2, 3, 4) = 1

+

4

. 1

1

3

(11.10)

4

4

For definiteness, let us assume that the external momenta p1 · · · p4 are defined as incoming, and denote ǫ1 · · · ǫ4 the four polarization vectors. Using the rules listed in the figure (11.2), we obtain: A4 (1, 2, 3, 4) = −i g2 h (2p2 + p1 ) · ǫ1 ǫλ2 − (2p1 + p2 ) · ǫ2 ǫλ1 = (p1 + p2 )2 ih +ǫ1 · ǫ2 (p1 − p2 )λ (p3 + 2p4 ) · ǫ3 ǫ4λ

−(2p3 + p4 ) · ǫ4 ǫ3λ + ǫ3 · ǫ4 (p3 − p4 )λ −i g2 h (p2 + 2p3 ) · ǫ2 ǫλ3 − (2p2 + p3 ) · ǫ3 ǫλ2 + (p2 + p2 )2 ih +ǫ2 · ǫ3 (p2 − p3 )λ (2p1 + p4 ) · ǫ4 ǫ1λ

−(p1 + 2p4 ) · ǫ1 ǫ4λ + ǫ1 · ǫ4 (p4 − p1 )λ

i

i

h i −i g2 2(ǫ1 · ǫ3 )(ǫ2 · ǫ4 ) − (ǫ1 · ǫ4 )(ǫ2 · ǫ3 ) − (ǫ1 · ǫ2 )(ǫ3 · ǫ4 ) . (11.11)

Although this is considerably simpler than the full 4-gluon amplitude, it remains quite difficult to extract physical results from such an expression.

364

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

11.3 Spinor-helicity formalism 11.3.1

Motivation

Part of the complexity of eq. (11.11) lies in the fact that this formula still contains a large amount of redundant and unnecessary information, since each polarization may be shifted by a 4-vector proportional to the momentum of the corresponding external gluon, thanks to gauge invariance. For instance, the transformation µ µ ǫµ 1 → ǫ1 + κ p1 ,

(11.12)

leaves the amplitude unchanged. However, it is not clear how to optimally choose the polarization vectors in order to simplify an expression such as eq. (11.11). In other words, the question is how to represent the spin degrees of freedom of the external particles in order to make the amplitude as simple as possible. In the traditional approach to the calculation of amplitudes, one usually refrains from introducing any explicit form for the polarization vectors. Instead, one first squares the amplitude written in terms of generic polarization vectors, such as eq. (11.11), and then the sum over the polarizations of the external gluons is performed by using X

physical pol.

ǫµ∗ (p)ǫν (p) = −gµν +

pµ nν nµ pν + , p·n p·n

(11.13)

where nµ is some arbitrary light-like vector. Note that this is the formula for summing over all physical polarizations, which is necessary when calculating unpolarized crosssections. For cross-sections involving polarized particles, one would perform only a partial sum, which leads to a different projector in the right hand side. If the amplitude is a sum of Nt terms, then this process generates 3N2t terms in the squared amplitude summed over polarizations. In contrast, the spinor-helicity method that we shall expose below aims at obtaining the amplitude with explicit polarization vectors, for a given assignment of the helicities {h1 = ±, · · · , hn = ±} of the external gluons, in the form of an expression made of Nt terms that can be easily evaluated (numerically at least). The sum of these Nt terms is done first, and then squared, which is an O(1) computational task (simply squaring a complex number). Thus, the total cost scales as 2n Nt in this approach. Since Nt grows very quickly with n, this is usually better.

11.3.2

Representation of 4-vectors as bi-spinors

In the previous section, we have seen how the adjoint colour degrees of freedom may be represented in terms of the smaller fundamental representation. Likewise, we will now represent the Lorentz structure associated to spin-1 particles in terms of spin-1/2 variables. From a mathematical standpoint, this representation exploits the fact that

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365

elements of the Lorentz group SO(3, 1) can be mapped to 2 × 2 complex matrices of unit determinant, i.e. elements of the group SL(2, ). Likewise, 4-momenta can be mapped to 2 × 2 complex matrices. In order to make this mapping explicit, let us introduce a set of four matrices σµ defined by

❈

σµ ≡ (1, σi ) ,

(11.14)

where σ1,2,3 are the usual Pauli matrices. In terms of these matrices, a 4-vector pµ can be mapped into   p0 + p3 p1 − ip2 µ µ p → P ≡ pµ σ = . (11.15) p1 + ip2 p0 − p3

(In the second equality, we have used the explicit representation of the Pauli matrices.) For amplitudes involving only external gluons, the momentum pµ has a vanishing invariant norm, pµ pµ = 0, which translates into 0 = =

pµ pµ = p20 − p21 − p22 − p23 (p0 + p3 )(p0 − p3 ) − (p1 + ip2 )(p1 − ip2 ) = det (P) .

(11.16)

Thus, the massless on-shell condition is equivalent to the determinant of the matrix P being zero. For a 2 × 2 matrix, a null determinant means that the matrix can be factorized as the direct product of two vectors: Pab = λa ξb ,

(11.17)

where λ, ξ are complex vectors known as Weyl spinors. An explicit representation of these vectors is ! ! √ √ p0 + p3 p0 + p3 , ξb ≡ . (11.18) λa ≡ p1 +ip2 p1 −ip2 √ √ p0 +p3

p0 +p3

For a real valued 4-vector, λa and ξa are mutual complex conjugates. However, when we later analytically continue the external momenta in the complex plane, this will no longer be the case. To make the notations more compact, it is customary to introduce the following notations: 

p = λa , p = ξa , (11.19) so that the matrix P may be written as:  P = p p .

(11.20)

It is also convenient to define spinors with raised indices, related to the previous ones as follows,   λa ≡ ǫab λb = p , ξa ≡ ǫab ξb = p , (11.21)

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where ǫab is the completely antisymmetric tensor in two dimensions, normalized with ǫ12 = +1. From these spinors with raised indices, we may define a 2 × 2 matrix representation of the 4-vector pµ with raised indices:  P ≡ p p . (11.22) Note that this alternative representation corresponds to the definition3 P ≡ pµ σµ ,

(11.23)

with σµ ≡ (1, −σi ). In the Weyl representation, where the Dirac matrices read   0 σµ µ , (11.24) γ = σµ 0 we thus have / ≡ pµ γµ = p



0 P

 P . 0

(11.25)

The fact that we are dealing with on-shell momenta is already built in the factorized representation of eq. (11.17). Amplitudes depend on kinematical invariants such as (p + q)2 , for which it is straightforward to check that4

  (p + q)2 = 2 p · q = pq pq , (11.26)

where the brackets are defined by contracting upper and lower spinor indices, as in

 a pq ≡ p a q . (11.27)



 These brackets are antisymmetric ( pq = − qp ), since they may also be written as:

 pq = ǫab ξa (p)ξb (q) . (11.28)

 Note that the mixed brackets are zero, pq

=  0,as well  as the angle and square brackets with twice the same momentum, pp = pp = 0. c sileG siocnarF

It is useful to work out the form of momentum conservation in the spinor formalism. For an amplitude with external momenta {pi }, chosen to be all incoming, let us  denote i , i , · · · the corresponding spinors. For any arbitrary on-shell momenta p and q, we may then write

X  X   0 = p p i i q . (11.29) Pi q = i

3 We

4 For

ǫac ǫbd δdc

i

may use = δab and ǫac ǫbd σidc = −σiab . real momenta, angle and square brackets are complex conjugates, and (p + q)2 is a real quantity.

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Another interesting identity follows from the fact spinors  that  three 2-component  cannot be linearly independent. Thus, given p , q and r , we must have a relationship of the form:    r = α p + β q .

(11.30)



Contracting this equation with p and q gives the explicit expression of the coefficients α and β:



 pr qr



. , β= (11.31) α= qp pq

This leads to

      p qr + q rp + r pq = 0 ,

(11.32)

known as the Schouten identity. A similar identity holds with square brackets:       p qr + q rp + r pq = 0 .

11.3.3

(11.33)

Polarization vectors

At this point, we have a representation in terms of spinors for the on-shell momenta that appear on the external legs of amplitudes. We also need a similar representation for the polarization vectors. The polarization vectors for a gluon of momentum p with positive and negative helicities may be represented as follows:

µ 

µ  q σ p p σ q µ √ √   , ǫµ (p; q) ≡ − , ǫ (p; q) ≡ − (11.34)

 + − 2 qp 2 pq

where q is an arbitrary reference momentum, whose presence is due to the gauge invariance (eq. (11.12)). It does not have to correspond to any of the physical momenta upon which the amplitude depends, and can be chosen in such a way that it simplifies the amplitude. This auxiliary vector can be different for each external line, but it must be the same in each contribution to a given process (this is because a single graph usually does not give a gauge invariant contribution when considered alone). Let us mention a useful Fierz identity for contracting two of the numerators that appear in the above polarization vectors5 :

5 We

 

µ   1 σ 2 3 σµ 4 = 2 13 24 , may use (σµ




ab

(σµ




cd

= 2(δab δcd − δad δbc ) = 2 ǫac ǫbd .

(11.35)

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from which we obtain the contractions between polarization vectors ′

′

ǫ+ (p; q) · ǫ+ (p ; q ) = ǫ− (p; q) · ǫ− (p ′ ; q ′ ) = ǫ+ (p; q) · ǫ− (p ′ ; q ′ ) =

 ′  ′  pp qq



, qp q ′ p ′

′  ′  pp qq    , qp q ′ p ′

′  ′  qp pq

 . qp p ′ q ′

(11.36)

Using eq. (11.25), we also obtain the following identities: p · ǫ± (p; q) = q · ǫ± (p; q) = 0 ,

  qk kp k · ǫ+ (p; q) = − √  , 2 qp

  pk kq k · ǫ− (p; q) = − √   . 2 pq

11.3.4

(11.37)

Three-point amplitudes in Yang-Mills theory

Let us now discuss the very important case of 3-particle amplitudes in the massless case, since they will appear later as the building blocks of more complicated amplitudes. Such an amplitude depends on three on-shell momenta p1,2,3 such that p1 + p2 + p3 = 0. This implies that

  12 12 = 2 p1 · p2 = (p1 + p2 )2 = p23 = 0 . (11.38)

  

 Therefore, either 12 = 0 or 12 = 0. Let us assume that 12 6= 0. We also have: 



   

   12 23 = 1 P2 3 = − 1 P1 +P3 3 = − 11 13 − 13 33 = 0 , (11.39) |{z} |{z} 0

0

    which implies

that 23 = 0. Likewise, 13 = 0. Therefore, all the square brackets are zero if 12 6= 0. Conversely, all the angle brackets would be zero if instead we had assumed that 12 6= 0. From this discussion, we conclude that massless on-shell 3-point amplitudes may depend either on square brackets or on angle brackets, but not on a mixture of both. Recall now that, for real momenta, angle and square brackets are related by complex conjugation. Thus, 3-point amplitudes can only exist for complex momenta. This is of course a trivial consequence of kinematics: momentum conservation p1 + p2 + p3 = 0 is impossible for three real-valued light-like momenta, except on a measure-zero subset of exceptional configurations.

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

369

Let us now be more explicit and calculate the 3-point amplitudes in Yang-Mills theory. For generic polarization vectors ǫ1,2,3 , the second Feynman rule of the figure (11.2) leads to h i A3 (123) = 2g (ǫ1 ·ǫ2 )(p1 ·ǫ3 )+(ǫ2 ·ǫ3 )(p2 ·ǫ1 )+(ǫ3 ·ǫ1 )(p3 ·ǫ2 ) , (11.40)

where we have used pi · ǫi = 0 to cancel several terms. Consider first the helicities − − +. Using eqs. (11.35) and (11.37), we obtain √ 2g − − +  

 A3 (1 2 3 ) = −  q1 1 q2 2 q3 3

     

 

× 12 q1 q2 q3 1 13 + 2q3 q2 3 12 2q1   

 (11.41) + q3 1 3q1 23 3q2 .

Each of the three terms contains

in  the

numerator

 an angle bracket between the external momenta (respectively 12 , 12 and 23 ). Therefore, for this amplitude to be non-zero, we must adopt the choice of spinor representation where it is the square   brackets that are zero. With this choice, the first term vanishes since it contains 13 :    

   

√ 2q3 q2 3 12 2q1 + q3 1 3q1 23 3q2   

 A3 (1 2 3 ) = − 2 g . q1 1 q2 2 q3 3 (11.42) − − +

Using momentum conservation (11.29) in the form of     

 11 1q1 + 12 2q1 + 13 3q1 = 0 , |{z}

(11.43)

0

and the Schouten identity (11.32), we arrive at    √  q1 3 q2 3 − − +  . A3 (1 2 3 ) = 2 g 12  q1 1 q2 2

Momentum conservation also implies

     12 q2 3 q1 3  =  ,  = q1 1 23 q2 2

 12

, 31

(11.44)

(11.45)

which leads to a form of the amplitude that does not contain the auxiliary vectors q1,2 anymore:

3 √ 12 − − + (11.46) A3 (1 2 3 ) = 2 g   . 23 31

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

We have thus obtained a remarkably compact expression of the 3-point amplitude in terms of spinor variables, which is explicitly independent of all the auxiliary vectors qi . Likewise, a similar calculation would give the following answer for the + + − amplitude: + + −

A3 (1 2 3 ) =

√

 3 12 2 g    . 23 31

(11.47)

(The + + + and − − − amplitudes are zero in Yang-Mills theory, as argued in the next subsection.) Eqs. (11.46) and (11.47) are both much simpler than the Feynman rule for the 3-gluon vertex. This is the simplest illustration of an assertion we made at the beginning of this section, namely that on-shell amplitudes with physical polarizations are much simpler than one may expect from the traditional perturbative expansion. In the case of the 3-gluon amplitude, we may think that the simplicity comes from the fact that it receives contributions from a single diagram. However, this is not true. As a teaser for the next section, let us give the answers for some 4-gluon and 5-gluon amplitudes in the spinor-helicity formalism: − − + +

A4 (1 2 3 4 ) = A4 (1− 2− 3+ 4+ 5+ ) =

3 √ 2  12   , i( 2 g) 23 34 41

3 √ 2 3  12    , i ( 2 g) 23 34 45 51

(11.48)

that appear to generalize trivially eq. (11.46) although they result from the sum of 3 and 10 Feynman graphs (see the figure 11.1), respectively. In this section, we have followed a pedestrian approach that consists in starting from the usual Feynman rules, and translating all their building blocks in the spinor-helicity language. However, the simplicity of the results provides an important hint: there must be a better way to obtain them, that bypasses the traditional Feynman rules and provides the answer much more directly. c sileG siocnarF

11.3.5

Little group scaling

It turns out that massless on-shell 3-point amplitudes are almost completely constrained by a scaling argument, except for an overall prefactor. Thus, the Lagrangian is in a sense not necessary for specifying their form (it only plays a marginal role in setting their normalization). From eqs. (11.20) and (11.22), it is clear that the representation of massless on-shell 4-momenta as bi-spinors is invariant under the following rescaling:     p → λ p , p → λ−1 p , (11.49)

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

371

known as little group scaling. The terminology follows from the fact that there is a one-parameter SO(2) subgroup (the rotations in the plane transverse to p) of the Lorentz group that leaves invariant the vector pµ . Such a residual symmetry that leaves a vector invariant is called little group. In the spinor formulation, this residual symmetry precisely corresponds to the transformation of eq. (11.49).   Under little group scaling of p and p , the polarization vectors of eq. (11.34) scale as follows: −2 µ ǫµ ǫ+ (p; q) , + (p; q) → λ

2 µ ǫµ − (p; q) → λ ǫ− (p; q) ,

(11.50)

i.e. a scaling by a factor λ−2h for a helicity h. Note that the polarization vectors are invariant under little group scaling of the auxiliary vector q. In an amplitude, the internal ingredients (propagators and vertices) are not affected by little group scaling. Therefore, if we apply the little group scaling λi to an external momentum i of an amplitude, its expression in terms of square and angle spinors must transform as An (1 · · · ihi · · · n)

→

i λ−2h An (1 · · · ihi · · · n) , i

(11.51)

where hi is the helicity of the external line i (we do not need to specify the helicities of the other external lines). It turns out that the structure of all 3-point amplitudes6 is completely fixed by this property. Let us start from the following generic expression

 α β γ A3 (1h1 2h2 3h3 ) = C 12 23 31 ,

(11.52)

with α, β, γ undetermined exponents and C a numerical prefactor. Little group scaling implies that −2 h1 = α + γ ,

−2 h2 = α + β ,

−2 h3 = β + γ ,

(11.53)

whose solution is α = h3 − h1 − h2 ,

β = h1 − h2 − h3 ,

γ = h2 − h3 − h1 .

(11.54)

Therefore the 3-point amplitude must have the following structure

h −h −h h −h −h h −h −h A3 (1h1 2h2 3h3 ) = C 12 3 1 2 23 1 2 3 31 2 3 1 , (11.55)

in which only the numerical prefactor remains to be determined. The − − + 3-gluon amplitude derived in the previous subsection indeed has this structure. 6 This reasoning cannot be extended to higher n-point amplitudes because they can depend on both square and angle brackets, and because the number of constraints provided by the helicities of the external lines is not sufficient to fix the unknown exponents.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Note that instead of eq. (11.52), we could have chosen an ansatz that involves the square brackets,  α ′  β ′  γ ′ 31 . 23 A3 (1h1 2h2 3h3 ) = C 12

(11.56)

(This is the only alternative, since we are not allowed to mix square and angle brackets in a 3-point amplitude for massless particles.) Little group scaling would now lead to α ′ = −h3 + h1 + h2 ,

β ′ = −h1 + h2 + h3 ,

γ ′ = −h2 + h3 + h1 , (11.57)

and consequently  −h3 +h1 +h2  −h1 +h2 +h3  −h2 +h3 +h1 A3 (1h1 2h2 3h3 ) = C 12 23 31 . (11.58)

The expected dimension of the amplitude is sufficient to choose between eqs. (11.55) and (11.58). Indeed, both angle and square brackets have mass dimension 1, while the 3-gluon amplitude should have dimension 1 in 4-dimensional Yang-Mills theory (for which the coupling constant is dimensionless). Since all the kinematical dependence is carried by the brackets, the prefactor C can only be made of coupling constants and numerical factors, and must therefore be dimensionless in Yang-Mills theory. Consider first the − − + amplitude: eq. (11.55) gives a mass dimension +1, while eq. (11.58) gives a mass dimension −1. Therefore, the − − + amplitude must be expressed by eq. (11.55) in terms of angle brackets. The same argument tells us that the + + − amplitude must be given by eq. (11.58), in terms of square brackets. Let us consider now the − − − amplitude, for which the little group scaling tells us that

   A3 (1− 2− 3− ) = C 12 23 31 .

(11.59)

Therefore, the prefactor C should have mass dimension −2, which cannot be constructed from the dimensionless coupling constant of Yang-Mills theory, unless C = 0 (the same conclusion holds if we try to construct this amplitude with square brackets). Likewise, we conclude that the + + + amplitude is zero as well.

11.3.6

Maximally Helicity Violating amplitudes

Let us consider a tree Feynman diagram contributing to a n-point amplitude, with n3 3-gluon vertices, n4 4-gluon vertices and nI internal propagators. These quantities are related by: n + 2 nI nI

= =

3 n3 + 4 n4 , n3 + n4 − 1 .

(11.60)

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11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

The second equation is the statement that this graph has no loops. From these equation we get the following identities: n = n3 + 2 n4 + 2 ,

n3 − 2 n I = 4 − n .

(11.61)

The contribution of this Feynman graph to the amplitude is made of n polarization vectors, nI denominators coming from the internal propagators, and n3 powers of momentum in the numerator, that come from the 3-gluon vertices7 : i ih Q hQ νj
n n3 n µi L ǫ
4−n mass 3 j=1 j i=1 i ∼ An (1 · · · n) ∼ ∼ mass . (11.62) QnI
2n 2 mass I k=1 Kk

Firstly, we see that the mass dimension of the n-point amplitude is 4 − n. Moreover, the amplitude An does not carry any Lorentz index. Therefore, in the numerator all the Lorentz indices µi and νj must be contracted pairwise. These contractions lead to three type of factors: ǫi · ǫi ′ ,

ǫi · Lj ,

Lj · Lj ′ .

(11.63)

Only-+ amplitude : Now, consider an amplitude with only + helicities. From eqs. (11.36), we see that all contractions between polarization vectors are proportional to

 (11.64) ǫ+ (i; qi ) · ǫ+ (i ′ ; qi ′ ) ∝ qi qi ′ .

By choosing the auxiliary momenta qi to be all equal to q, we make all these contractions vanish. Therefore, to obtain a non-zero contribution, it is necessary to contract all the polarization vectors with momenta from the 3-gluon vertices, ǫi · kj . But from the first of eqs. (11.61), we see that n > n3 , which means that it is impossible to contract all the n polarization vectors with the n3 momenta from the vertices. Thus, the all-plus amplitude is zero: An (1+ 2+ · · · n+ ) = 0 .

(11.65)

By the same reasoning, we conclude that the all-minus amplitude is also zero. We can see here the power that stems from the freedom of choosing the auxiliary vectors qi ; for generic qi ’s, this amplitude would still be zero (since it does not depend on the qi ’s), but this zero would result from intricate cancellations among the many graphs that contribute to An . Instead, with a smart choice of the auxiliary vectors, we can make this cancellation happen graph by graph. c sileG siocnarF

7 We assume for simplicity Feynman gauge, in which the numerator of the gluon propagator does not depend on momentum.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

− + · · · + amplitude : Consider now an amplitude with one − helicity carried by the first external leg, and n − 1 + helicities carried by the external legs 2 to n. Now, it is convenient to choose the auxiliary vectors as follows q2 = q3 = · · · = qn = p1 .

(11.66)

Again, all the contractions between pairs of polarization vectors cancel, since we have ǫ+ (i; qi ) · ǫ+ (i ′ ; qi ′ ) = 0 for i, i ′ ≥ 2, and ǫ− (1; q1 ) · ǫ+ (i; qi ) = 0 for i ≥ 2. Since n > n3 , it is not possible to contract all the polarization vectors with momenta from the 3-gluon vertices, and these amplitudes also vanish at tree level: An (1− 2+ · · · n+ ) = 0 .

(11.67)

(We also have An (1+ 2− · · · n− ) = 0 at tree level.) Maximally Helicity Violating amplitudes : Let us flip one more helicity, e.g. with the assignment 1− 2− 3+ · · · n+ . This time, a useful choice of auxiliary vectors is q1 = q2 = pn

,

q3 = q4 = · · · = qn = p1 .

(11.68)

With this choice, all the contractions of polarization vectors are zero, except: ǫ− (2; q2 ) · ǫ+ (i; qi ) 6= 0

for i = 3, · · · , n − 1 .

(11.69)

Thus, this time, we need to contract the remaining n − 2 polarization vectors with the n3 momenta from the 3-gluon vertices, which is possible (provided that n4 = 0, which means that diagrams containing 4-gluon vertices do not contribute to the − − + · · · + amplitude for our choice of auxiliary vectors). Therefore, this assignment of helicities gives a non-zero amplitude: An (1− 2− 3+ · · · n+ ) 6= 0 .

(11.70)

These amplitudes, called the Maximally Helicity Violating (MHV) amplitudes, are the simplest non-zero amplitudes. As we shall see later, they are given at tree level by very compact formulas in terms of square and angular brackets (note that up to n = 5 external lines, all the non-zero amplitudes are MHV amplitudes). Generically, the complexity of amplitudes increases with the number of − helicities, culminating with amplitudes that have comparable numbers of − and + helicities (increasing further the number of − helicities then reduces the complexity).

11.4 Britto-Cachazo-Feng-Witten on-shell recursion 11.4.1

Main idea

As we have seen, the main obstacle to the calculation of amplitudes by the usual Feynman rules is the proliferation of graphs as one increases the number of external

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

375

legs. This problem remains true even after one has factorized the colour factors, even if it is somewhat mitigated by the fact that the number of cyclic-ordered graphs grows at a slower pace. This issue could be avoided if there was a way to break down a tree amplitude into smaller pieces (themselves tree amplitudes) that have a smaller number of external legs. It turns out that an amplitude naturally factorizes into two sub-amplitudes when one of its internal propagators goes on-shell. The physical reason of such a factorization is that on-shell momenta correspond to infinitely long-lived particles. Thus, the two sub-amplitudes on each side of this on-shell propagator do not talk to one another. The other advantage of this situation is that the two sub-amplitudes would themselves be on-shell, and therefore we may use for them spinor-helicity formulas that could have been previously obtained for amplitudes with fewer external legs. If this were possible, we would thus obtain a recursive relationship (in the number of external legs) for on-shell amplitudes.

11.4.2

Analytical properties of amplitudes with shifted momenta

Unfortunately, with fixed generic external momenta, tree amplitudes do not have internal on-shell propagators. The trick is to consider a one-parameter complex deformation of the external momenta, adjusted in order to make an internal denominator vanish: An (12 · · · n)

→

An (12 · · · n; z) ,

(11.71)

where z is a complex variable that controls the deformation. The singularities of tree Feynman graphs come from the zeroes of the denominators of its internal propagators, which give poles in z. Our goal will be to choose this deformation in such a way that the total momentum remains conserved, and the deformed external momenta are still on-shell. With such a choice, we will be able to reuse the on-shell formulas obtained for smaller amplitudes. Let us consider the ratio An (· · · ; z)/z. Besides the poles coming from the internal propagators, the ratio also has a simple pole at z = 0. Let us assume that An (· · · ; z) vanishes when |z| → ∞, so that the integral of An (· · · ; z)/z on a contour at infinity in the complex plane vanishes. Then, we may write I dz An (· · · ; z) 0 = = An (· · · ; 0) z γ 2πi X An (· · · ; z) (11.72) + Res . z zi zi ∈{poles of An }

The first term, An (· · · ; z = 0), is nothing but the amplitude we aim at calculating. This formula therefore expresses it in terms of the residues of An (· · · ; z)/z at the

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

simple poles corresponding to the internal propagators of the amplitude. Moreover, these residues will be factorizable into smaller on-shell amplitudes, precisely because the poles zi correspond to the on-shellness of some internal propagator. c sileG siocnarF

11.4.3

Minimal momentum shifts

There are many ways to implement a complex shift of the external momenta, but all of them must fulfill the following conditions: • The sum of the shifted incoming momenta should remain zero. Therefore, we must shift at least two momenta (and the simplest is to shift only two). • The shifted momenta should stay on-shell at all z. • The amplitude evaluated at the shifted momenta should go to zero as |z| → ∞.

The condition of momentum conservation is trivially satisfied by choosing two momenta i, j to be shifted, and by giving them opposite shifts: pi pj

→

→

bi = pi (z) ≡ pi + z k , p bj = pj (z) ≡ pj − z k , p

(11.73)

where we denote with a hat the shifted momenta. All the momenta pk for k 6= i, j are bi,j are satisfied provided that left unmodified. The on-shell conditions for p k2 = 0 ,

pi · k = 0 ,

pj · k = 0 .

(11.74)

It turns out that these equations have two solutions (up to an arbitrary prefactor), provided we allow complex momenta. In the spinor notation, the first condition is automatically satisfied if K can be factorized as in eq. (11.17), while the second and third conditions become

 

  ik ik = 0 , jk jk = 0 . (11.75)  µ µ This  explains why we need a complex momentum k . Indeed, for a real k , k and reduce conditions  to ik = 

k are related by complex conjugation, and the above jk = 0. With two-component spinors, this implies k ∝ i and k ∝ j , which  is in general impossible. By allowing a complex momentum kµ , we let k and k be independent, which allows to solve the above conditions by having for instance:     k = i , k = j . (11.76)

(The other independent solution consists in exchanging the roles of i and j.) The bi-spinors corresponding to the shifted momenta are    
b i = i i +z j i = i +z j P i ,

  
b j = j j −z j i = j P j −z i ,

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11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

Figure 11.3: Propagators affected by the momentum shift (shown in dark) in a tree amplitude. The lighter colored lines do not depend on z. The propagators on the external lines are not actually part of the expression of the amplitude.

i

j

(11.77) from which we read the shifted spinors:

11.4.4

     ^ı = i , ^ = j − z i ,      ^ı = i + z j , ^ = j .

(11.78)

Behaviour at |z| → ∞

Until now, our description of this method has been completely generic and applicable to all sorts of quantum field theories, since no reference has been made to the details of its Lagrangian. These details become important when discussing the condition that An (· · · ; z) vanishes at infinity. Let us discuss the behaviour at large z in the case of Yang-Mills theory. Firstly, a z dependence enters in the polarization vectors of the external lines i and j. For generic auxiliary vectors, we have:

µ  q σ ^ı =−√  ∼z , 2 q^ı

µ  ^ı σ q √ ǫµ (^ ı ; q) = −   ∼ z−1 , − 2 ^ıq ǫµ ı; q) + (^

µ  q σ ^ = − √  ∼ z−1 , 2 q^

µ  ^ σ q µ ǫ− (^; q) = − √   ∼ z . 2 ^q (11.79)

ǫµ ; q) + (^

Inside a graph contributing to this n-point amplitude, we can follow a string of propagators that all carry shifted momenta, from the external line i to the external line j, as illustrated in the figure 11.3. For all these propagators, since k2 = 0, the denominators are linear in z. In addition, the 3-gluon vertices along this string of propagators are linear in the momenta, and therefore scale as z. Along this string, there are s vertices (3-gluon or 4-gluon vertices), and s − 1 propagators, hence a

378

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

global behaviour at most ∼ z at large z (obtained when all these vertices are 3-gluon vertices, that scale as z). For the assignment {hi = −, hj = +} of polarizations, we thus find an overall behaviour8 in z−1 , valid graph by graph. For other combinations of polarizations on the lines i, j, this diagrammatic argument suggests that they do not go to zero. However, the actual behaviour for {hi = −, hj = −} and {hi = +, hj = +} is better than the one suggested by this graph by graph estimate. Firstly, note that this problem is reminiscent of the eikonal approximation, in which a hard on-shell particle punches through a background of much softer particles that very mildly disturb its motion. This can be studied by splitting the gauge field Aµ into a hard component aµ that describes the gluons along the string with shifted momenta and a soft background Aµ (describing the unshifted gluons attached to the hard ones), Aµ ≡ Aµ + aµ .

(11.80)

When rewriting the Yang-Mills Lagrangian in terms of these fields, it is sufficient to keep terms that are quadratic in the hard field, since in our problem exactly two external lines are shifted: 

ig   1  LYM = · · ·− tr Dµ aν −Dν aµ Dµ aν −Dν aµ + tr aµ , aν Fµν +· · · , 4 2 (11.81) where the covariant derivative Dµ is constructed with the background field. When splitting the gauge potential as in eq. (11.80), one may fix independently the gauge for the background and for the fluctuation aµ . For the latter, a convenient choice is the background field gauge, c sileG siocnarF

Dµ aµ (x) = ω(x) .

(11.82)

After adding the gauge fixing term, the quadratic part of the Lagrangian becomes 

ig   1  LYM+GF = · · · − tr Dµ aα Dµ aα + tr aµ , aν Fµν + · · · (11.83) 4 2

In this equation, the first term possesses an extended Lorentz symmetry, since it is invariant under independent Lorentz transformations of the fluctuations and of the background, while the second term is only invariant under simultaneous transformations of Aµ and aµ . Let us denote Mαβ [A] the propagator of the fluctuation aµ , amputated of its final lines. This propagator contains 3-gluon couplings to the background field, that 8 When the shifted amplitude decreases faster than z−1 , one may obtain a more compact expression by integrating An (· · · ; z)(1 − z/z∗ )/z, where z∗ is one of the poles of An . There is no boundary term thanks to the faster decrease of An , and the additional subtraction removes the contribution from the pole z∗ , leading to an expression with one less term.

379

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

come from the first term of eq. (11.83), and 4-gluon couplings to the background field coming from the second term. With only 3-gluon couplings, we have Mαβ ∼ z (because there is one more vertex than propagators), and each 4-gluon vertex removes one power of z. Given the Lorentz structure of eq. (11.83), we may write
Mαβ = c1 z + c0 + c−1 z−1 + · · · gαβ + Aαβ + z−1 Bαβ + · · · (11.84)

In this formula, the first term comes entirely from the first term of eq. (11.83), whose extended Lorentz symmetry leads to the factor gαβ . All the coefficients in this expansion are functionals of the soft field. The term Aαβ , that comes from a single insertion of the 4-gluon vertex, is antisymmetric. The subsequent terms correspond to 2 or more insertions of the 4-gluon vertex. These terms have no definite symmetry, but they are not needed in the discussion. The amputated 2-point function Mαβ also obeys the following on-shell Ward identities: β bα p ) = 0 , i Mαβ ǫhj (b

bβ ǫα ı) Mαβ p hi (b j =0,

(11.85)

with shifted on-shell momenta and polarization vectors. Note that, unlike in an Abelian gauge theory, it is necessary to contract one side of the function with a physical polarization vector for the identity to hold. The shifted amplitude An is obtained by keeping n − 2 powers of the background field in Mαβ , and by contracting with the appropriate polarization vectors: An ∼ ǫα ı; q) Mαβ ǫβ ; q ′ ) . hi (b hj (b

(11.86)

Choosing the auxiliary vectors to be q ≡ pi and q ′ ≡ pj , the explicit form of the polarization vectors is9 √ √ 2 ∗µ 2 µ
µ µ ǫ− (bı; q) = −   kµ , ǫ+ (bı; q) = −  k + z pj , ji ij √ √
2 µ 2 µ µ ′ ′ ǫ+ (b; q ) = −  k , ǫ− (b; q ) = −   k∗µ − z pµ i . ij ji (11.87) Note that with this choice of auxiliary vectors, we have lost a power of z in the denominators of ǫµ ı; q) and ǫµ ; q ′ ). This will not change the final results, since − (b + (b on-shell amplitudes do not depend on the auxiliary vectors. As a check of the insight gained from Feynman diagrams, let us first consider the case {hi = −, hj = +}. The shifted amplitude behaves as An;−+ ∼ (k · k)(c1 z + · · · ) + kα kβ Aαβ +O(z−1 ) ∼ O(z−1 ) |{z} | {z } 0

(11.88)

0

9 In order to obtain these expressions, one may contract the polarization vectors with σ

corresponding 2 × 2 matrix, which can be done by using σµ  and i j = k∗µ σµ .


ab

σµ




cd

µ to first obtain the  = 2 δa d δb c , j i = kµ σµ ,

380

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The first term vanishes because k is on-shell, and the second one thanks to the antisymmetry of Aαβ . Next, consider the case {hi = −, hj = −}, for which we obtain An;−−

∼

kα Mαβ ǫβ ; q ′ ) − (b

=

β −z−1 pα ; q ′ ) i Mαβ ǫ− (b

∼

∗β −1 z−1 pi · (k∗ −z pi )(c1 z + · · · ) + z−1 pα −z pβ ) i (k i )Aαβ + O(z

∼

O(z−1 ) .

(11.89)

The second line is obtained by using the Ward identity, and in the third line all terms that could be larger than z−1 vanish due to p2i = pi · k∗ = 0 and thanks to the antisymmetry of Aαβ . The case {hi = +, hj = +} is very similar and leads to An;++

∼

ǫα ı; q) Mαβ kβ + (b

=

z−1 ǫα ı; q) Mαβ pβ + (b j

∼

β −1 z−1 (k∗ +z pj ) · pj (c1 z + · · · ) + z−1 (k∗α +z pα ) j )pj Aαβ + O(z

∼

O(z−1 ) .

(11.90)

Finally, in the last case, {hi = +, hj = −}, we obtain An;+− ∼ O(z3 ), and therefore we cannot use such a shift in eq. (11.72).

11.4.5

Recursion formula

Since all-+ amplitudes are zero, the assignment of helicities that we shall consider is generically of the form 1− · · · r− (r + 1)+ · · · n+ , and the shift applied to the lines i = 1, j = n leads to a vanishing amplitude when |z| → ∞. We can therefore apply eq. (11.72) and write the amplitude in the following way: X An (· · · ; z) (11.91) Res An (· · · ) = − . z zi zi ∈{poles of An }

As explained earlier, the poles zi come from the vanishing denominators of the internal propagators, i.e. one of the dark colored propagators in the figure 11.4. Let us denote KI the momentum (before the shift) carried by the propagator producing the pole, with the convention that it is oriented in the same direction as p1 . The shift changes this momentum into KI

→

b ≡ K + zk , K I I

(11.92)

and the condition that the denominator of the propagator vanishes after the shift is b 2 = K2 + 2 z K · k , 0=K I I I I

i.e. zI = −

K2I . 2 KI · k

(11.93)

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11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

Figure 11.4: Setup for the BCFW recursion formula with shifts applied to the external lines 1 and n. The pole comes from the propagator carrying the momentum KI , highlighted in dark. This singular propagator divides the graph into left and right sub-amplitudes, AL and AR .

KI AL

3

AR

2

n−2

n−1 1−

n+

The singular propagator divides the amplitude into left and right sub-amplitudes, so that we may write: An (^ 1 2 · · · (n−1)^ n; z) ≡

X

h=±

b +h ; z) AL (^ 1 2 · · ·− K I

i b −h · · · (n−1)^ AR (K n; z) , I 2 b KI (11.94)

with a sum over the helicity h of the intermediate gluon10 . From this expression, the residue at the pole zI of An (· · · ; z)/z takes the form Res

X An (· · · ; z) b −h · · · (n−1)^ b +h ; z ) i A (K n; zI ) . AL (^ 1 2 · · ·− K =− I I z K2I R I zI h=±

(11.95)

Both AL and AR have strictly less than n external lines, which means that the formula is recursive: it expresses an amplitude in terms of smaller amplitudes, eventually breaking it down to 3-point amplitudes. Moreover, the crucial point here is that, b 2 = 0, the left and right sub-amplitudes when evaluated at the value zI that gives K I have only on-shell (but complex) external momenta. Therefore, this recursion never requires off-shell amplitudes, which is of utmost importance for keeping out of the calculation unnecessarily complicated kinematics and unphysical degrees of freedom. Since each internal z-dependent propagator can be singular for some z, eq. (11.91) contains one term for each such propagator. There are at most n − 3 terms in this sum, corresponding to the partitions of [2, n−1] = [2, l]∪[l+1, n−1] with 2 ≤ l ≤ n−2. c sileG siocnarF

b and the AL and AR are defined with all gluons incoming. This is why one has argument −K I b other one +KI . For the same reason, the helicity is +h on one side and −h on the other side. 10 Both

382

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

+

+

+

+

KI 3

AL

+

+

−

AR

2−

n−1 1

11.4.6

n−2

−

n

+

+

+

Figure 11.5: Setup for applying the BCFW recursion formula to the calculation of the − − + · · · + MHV amplitude. We have indicated explicitly all the helicities. Note that only h = + is allowed in the sum over the helicity of the singular propagator (otherwise the right-side sub-amplitude would be a vanishing all-+ amplitude).

Parke-Taylor formula for MHV amplitudes

MHV recursion formula : As an illustration of the BCFW recursion formula, let us determine the explicit expression of the MHV amplitudes11 An (1− 2− 3+ · · · n+ ). We show all the helicity assignments, including those of the singular propagator, in the figure 11.5. In order to avoid having an all-+ sub-amplitude on the right, we must choose h = +. This choice makes AR an − + · · · + amplitude, which is also zero unless it is a 3-point amplitude. Thus, the BCFW formula reduces to a single term: An (1− 2− 3+ · · · n+ ) =

b+; z ) i An−1 (^ 1− 2− 3+ · · · (n − 2)+ − K I I K2I b − (n − 1)+ n ×A3 (K ^ +; z ) , (11.96) I

I

where the momentum carried by the singular propagator is (before the shift) KI = −(pn−1 + pn ) .

(11.97)

In the right hand side of eq. (11.96), the factor on the right is an already known 3-point amplitude, and the factor on the left is an MHV amplitude with n − 1 external legs. Four-point MHV amplitude : Let us now calculate the first few iterations of this recursion, in order to guess a formula for the MHV amplitude that will be 11 This assignment of helicities, with the negative helicities carried by adjacent lines, is the simplest. MHV amplitudes with non-adjacent negative helicities are also given by the Parke-Taylor formula, but the proof is a bit more complicated in this case.

383

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

our hypothesis for an inductive proof. Firstly, consider the − − ++ 4-point MHV amplitude, In this case, the BCFW recursion formula gives b+; z ) A4 (1− 2− 3+ 4+ ) = A3 (^ 1− 2− − K I I

i b − 3+ ^4+ ; z ) , A3 (K I I K2I

and both amplitudes in the right hand side are known. This gives:  3

3 ^ 3^4 12 1 − − + + 2 A4 (1 2 3 4 ) = 2 i g



     . b K b ^ b K b 3 2K 1 12 12 ^4K I I I I Using the fact that      ^ 1 = 1 , ^ 1 = 1 + z 4 ,

we obtain

   ^ 4 = 4 − z 1 ,

    b = 1 1 + 2 2 + z 1 4 , b KI K I I    

b 4 = 21 14 , b K 2K I I    

b 3 = 12 23 , b K 1K I I

(11.98)

(11.99)

 ^4] = 4 , (11.100)

(11.101)

which leads to

 3 34 A4 (1 2 3 4 ) = 2 i g     . 41 12 23 − − + +

2

(11.102)

This formula, that depends only on square brackets, can also be expressed in terms

3 of angle brackets. Let us multiply the numerator and denominator by 12 . Then, momentum conservation leads to       41 12 = − 43 32 ,

 

  12 23 = − 14 43 ,       12 12 = (p1 + p2 )2 = (p3 + p4 )2 = 34 34 , (11.103) and we finally obtain

3 12 A4 (1 2 3 4 ) = 2 i g    . 23 34 41 − − + +

2

(11.104)

This formula could in principle have been obtained from eq. (11.11), by putting the external lines on-shell and by using − − ++ polarization vectors, at the cost of considerable effort. We see here the power of on-shell recursion: since one only manipulates on-shell sub-amplitudes with physical polarizations, the complexity of all the intermediate expressions is comparable to that of the final result, unlike with the standard method.

384

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Five-point MHV amplitude : Consider now the amplitude A5 (1− 2− 3+ 4+ 5+ ). The BCFW recursion formula (11.96) now reads: A5 (1− 2− 3+ 4+ 5+ ) = =

b − 4+ ^5+ ; z ) b + ; z ) i A3 (K A4 (^ 1− 2− 3+ − K I I I I K2I

3   3 ^12 4^5 √ ( 2 g)3 i2 

 , 

     b 4 b 1 45 45 ^5K b K b K 23 3K I I I I (11.105)

   where we have chosen to express K2I as (p4 + p5 )2 = 45 45 . This time, we use         ^ 1 = 1 + z 5 , ^5 = 5 − z 1 , 1 = 1 , ^     b = − 4 4 − 5 5 + z 1 5 , b KI K I I 

 

 b K b ^ 3K 5 = − 34 45 , I I  

 

b 4 = − 51 45 , b K ^ 1K I

I

 ^5] = 5 ,

(11.106)

which gives

3 √ 3 2  12    . A5 (1 2 3 4 5 ) = ( 2 g) i 23 34 45 51 − − + + +

(11.107)

This remarkably simple formula, that encapsulates the sum of 10 cyclic-ordered Feynman diagrams (in QCD, this corresponds to 25 diagrams before colour ordering), in fact exhausts all the possibilities for 5-point functions (the + + − − − amplitude is given by the same formula with square brackets instead of angle brackets).

Parke-Taylor formula : The previous results for 3, 4 and 5-point MHV amplitudes lead us to conjecture the following general formula:

3 √ 12 n−2 n−3  

  , An (1 2 3 · · · n ) = ( 2 g) i 23 34 · · · (n − 1)n n1 (11.108) − − +

+

known as the Parke-Taylor formula. Let us assume the formula to be true for all p < n, and consider now the case of the n-point MHV amplitude. The BCFW

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

385

recursion formula reads: An (1− 2− 3+ · · · n+ ) =

=

b+; z ) i An−1 (^ 1− 2− 3+ · · · (n−2)+ − K I I 1 − + + b (n−1) n ^ ; zI ) × 2 A3 (K I KI

3 ^12 √ ( 2 g)n−2 in−3  



b 1 b K 23 · · · (n−2)K I I  3 (n−1)^ n 1 

 ×  , b K b (n−1) (n−1)n (n−1)n n ^K I I (11.109)

where we have used our induction hypothesis for the (n − 1)-point MHV amplitude that appears in the left sub-amplitude. The spinor manipulations that are necessary to simplify this expression are the same as in the case of the 5-point amplitude, and lead to:  

  b n b K ^ = − (n−2)(n−1) (n−1)n , (n−2)K I I  

 

b (n−1) = − n1 (n−1)n , b K ^ 1K



I

I

(11.110)

thanks to which we obtain eq. (11.108) for n points. Up to 5-points, all amplitudes are MHV (or anti-MHV, i.e. + + − − −). Beyond 5-points, there exist non-MHV amplitudes, that are not given by the Parke-Taylor formula. However, multiple MHV amplitudes can be sewed together in order to construct the non-MHV ones, with a set of rules known as the Cachazo-Svrcek-Witten (CSW) rules, derived in the section 11.6. Such an expansion is much more efficient that the textbook perturbation theory, because it is in terms of on-shell gauge-invariant building blocks (the MHV amplitudes) that already encapsulate a lot of the underlying complexity.

11.5 Tree-level gravitational amplitudes 11.5.1

Textbook approach for amplitudes with gravitons

In the previous section, we have derived the BCFW recursion formula and applied it to the calculation of the tree-level MHV amplitudes in Yang-Mills theory. However, the validity of this recursion is by no means limited to a gauge theory with spin-1 bosons such as gluons. It may in fact be applied to any quantum field theory provided that: • we have expressions for the on-shell 3-point amplitudes,

386

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS • the shifted amplitudes vanish when |z| → ∞. c sileG siocnarF

In particular, it could be interesting to apply it to the calculation of scattering amplitudes that involve gravitons12 . The Feynman rules for Einstein gravity can be obtained from the Hilbert-Einstein action, SHE ≡

Z

d4 x

√

−g

2 gµν gρσ gµν m2 2  R− Fµρ Fνσ + (∂µ φ)(∂ν φ) − φ , 2 κ 4 2 2 (11.111)

where gµν is the metric tensor, R is the Ricci curvature and κ is a coupling constant related to Newton’s constant by κ2 = 32π GN . In this action, we have also added the minimal coupling to a gauge field and to a scalar field, in order to investigate gravitational interactions with light and matter. The rules for the propagators and vertices involving gravitons are obtained by expanding the metric around flat space: gµν = ηµν + κ hµν .

(11.112)

(ηµν is the flat space Minkowski metric.) Let us make a remark on dimensions: Newton’s constant has mass dimension −2, κ has mass dimension −1, the Ricci curvature has mass dimension 2, and hµν has mass dimension +1 (like the scalar φ and the photon Aµ ). The expansion in powers of hµν leads to an infinite series of terms (because the Ricci tensor contains the inverse gµν of the metric tensor, and also √ because of the expansion of the square root −g). Schematically, the expansion of the Hilbert-Einstein action starts with the following terms: SHE

∼

Z

d4 x h∂2 h + κ h2 ∂2 h + κ2 h3 ∂2 h + · · ·

+κ hφ∂2 φ + κ h F2 + · · ·



.

(11.113)

This sketch only indicates the number of powers of h and the number of derivatives contained in each term, but of course the actual structure of these terms is much more complicated. For instance, the vertex describing the coupling φφh between two scalars and a graviton reads: Γ µν (p1 , p2 ) = −

i iκ h µ ν µ µν 2 p1 p2 + pν p − η (p · p − m ) , 1 2 1 2 2

(11.114)

where p1,2 are the momenta carried by the two scalar lines (since the graviton has spin 2, the graviton attached to this vertex carries two Lorentz indices). But the 12 At tree-level, these amplitudes are completely prescribed by the equivalence principle and general relativity, and their calculation does not require to have a consistent theory of gravitational quantum fluctuations.

387

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

γγh coupling is far more complicated, and the hhh tri-graviton vertex is even more complex, leading to extremely cumbersome perturbative calculations if performed within the traditional approach. It turns out that tree amplitudes in Einstein gravity have a simple form in the spinor-helicity formalism, very much like their Yang-Mills analogue. The goal of this section is to illustrate on two examples the use of the spinor-helicity formalism, combined to the BCFW recursion, in order to calculate some amplitudes that have a relevance in gravitational physics: (1) gravitational bending of light by a mass, and (2) scattering of a gravitational wave by a mass. In both examples, the mass acting as a source of gravitational field is taken to be a scalar particle. In the approach based on conventional Feynman perturbation theory, these processes are given by the diagrams shown in the figure 11.6. In particular, the second example (bending of a

Aφγ→φγ ∼

Aφh→φh ∼

Figure 11.6: Feynman diagrams contributing to the gravitational photon-scalar and graviton-scalar scattering amplitudes. The wavy double lines represent gravitons.

gravitational wave by a mass) would be an extremely difficult calculation, because of the complexity of the 3-graviton vertex.

11.5.2

Three-point amplitudes with gravitons

In order to obtain these amplitudes with the formalism previously exposed in the case of Yang-Mills theory, the first step is to obtain the 3-point amplitudes involving scalars, photons and gravitons. External scalar particles must have helicity h = 0, photons can have helicities h = ±1 and gravitons can have helicities h = ±2 with polarization vectors that are “squares” of the gluon polarization vectors: µ ν ǫµν 2h (p; q) = ǫh (p; q) ǫh (p; q) .

(11.115)

388

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

For 3-point amplitudes that involve only massless particles (photons and gravitons), little group scaling is sufficient to constrain completely their form. We obtain: Ahγγ (1±2 2+ 3+ ) = Ahγγ (1±2 2− 3− ) = 0 ,  4  −2 , Ahγγ (1+2 2+ 3− ) = − κ2 12 23     −2 4 Ahγγ (1+2 2− 3+ ) = − κ2 23 31 ,



4 −2 Ahγγ (1−2 2+ 3− ) = − κ2 23 31 ,

4 −2 . Ahγγ (1−2 2− 3+ ) = − κ2 12 23

(11.116)

In order to obtain the zeroes of the first line, and to choose between square and angle brackets for the non-zero results, we use the fact that the 3-point amplitude must have mass dimension +1, with a prefactor made up only of numerical constants and one power of κ (that has mass dimension −1). The value √ of the prefactor is obtained by inspecting the term of order κ in the expansion of −g F2 . For the 3-graviton amplitudes, little group scaling leads to Ahhh (1+2 2+2 3+2 ) = Ahhh (1−2 2−2 3−2 ) = 0 ,

6 −2 −2 Ahhh (1−2 2−2 3+2 ) ∝ κ 12 23 31 ,       6 −2 −2 Ahhh (1+2 2+2 3−2 ) ∝ κ 12 23 31 .

(11.117)

Interestingly, the kinematical part of the non-zero 3-graviton amplitudes is simply the square13 of that of the 3-gluon amplitudes with like-sign helicities (see eqs. (11.46) and (11.47)), despite a considerably more complicated Feynman rule for the 3-graviton vertex. This is yet another illustration of the fact that traditional Feynman rules carry a lot of unnecessary information that disappears in on-shell amplitudes with physical polarizations. For the φφh amplitude, we cannot rely on little group scaling because the scalar field is massive. Instead, we simply contract eq. (11.114) with the polarization vector (11.115) of the graviton, and take the external momenta on mass-shell. For a graviton of helicity +2, we have Aφφh (10 20 3+2 ) = =



−i κ p1 · ǫ+ (p3 ; q) p2 · ǫ+ (p3 ; q)

  i κ q P1 p3 q P2 p3 , − 2

2 qp

(11.118)

3

13 This property of 3-point purely gravitational amplitudes has a generalization for n-point amplitudes, known as the Kawai-Lewellen-Tye (KLT) relations. These relations have also been interpreted as a form of colour-kinematics duality by Bern, Carrasco and Johansson.

389

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES where p1 + p2 + p3 = 0. With a graviton of helicity −2, we have

  i κ p3 P1 q p3 P2 q 0 0 −2 . Aφφh (1 2 3 ) = − 2  2 qp

(11.119)

3

Note that,







  p3 P2 q = − p3 P1 q − p3 P3 q = − p3 P1 q , | {z } 0



 

 q P2 p3 = − q P1 p3 − q P3 p3 = − q P1 p3 , | {z }

(11.120)

0

which allows the following simplification of the above 3-point amplitudes: 0 0 +2

Aφφh (1 2 3

2 i κ q P1 p3 )= 2 ,

2 qp3

0 0 −2

Aφφh (1 2 3

2 i κ p3 P1 q )= 2 .  2 qp3 (11.121)

Note that since p21 = m2 6= 0, the bi-spinor P1 does not admit a factorized form, and this cannot be simplified further. c sileG siocnarF

11.5.3

Gravitational bending of light

Shifted momenta : Consider now the amplitude Aγγφφ (1+ 2− 30 40 ), and apply the shift to the lines 2 and 3, as illustrated in the figure14 11.7: b2 ≡ p2 + z k , p 2

k =0,

b3 ≡ p3 − z k , p

k · p2 = k · p3 = 0 .

(11.122)

Since p2 is massless, the condition p2 · k = 0 can be satisfied by choosing for instance:   k = 2 (11.123) However, since p23 = m2 , the bi-spinor P3 that represents the momentum p3 cannot be factorized. Instead, we may write

 0 = 2 k · p3 = − k P3 k , (11.124)

14 Note that the factorization with one scalar and one photon on each side of the singular propagator is not allowed: indeed, the intermediate propagator would need to carry a scalar, and we would have two φφγ sub-amplitudes, that are zero per our assumption that the scalar field is not electrically charged.

390

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

0

4

1

AL

KI

Figure 11.7: BCFW shift for the calculation of the γγφφ amplitude.

AR

30

2

which can be solved by15   k = P3 2 .

(11.125)

The shifted bi-spinors read    
b 2 = 2 2 + z k k = 2 + z P3 2 2 , P {z } |  b2   b 3 = P3 − z k k = P3 − z P3 2 2 . P

(11.126)

Note that the second one is not factorizable, because the line 3 carries a massive particle. Scattering amplitude : With this choice of shifts, the BCFW recursion formula can be written as follows Aγγφφ (1+ 2− 30 40 ) =

i X b +h ; z ) Ahφφ (K b −h ^30 40 ; z ) , Aγγh (1+ ^2− − K I I I I K2I h=±2

(11.127)

where the shifted momenta in the 3-point amplitudes are evaluated at the zI for which b of the intermediate graviton is on-shell. The condition for the shifted momentum K I the intermediate momentum to be on-shell reads

    b 2 = (p1 + p b2 )2 = 2 p1 · p b2 = 12 , (11.128) 0=K 12 + zI 1 P3 2 I {z } |   ^ =0 12





 15 m2 22 = 0. When p3 is massless, P3 factorizes as P3 =  We use k P3 k = 2 P3 P 3 2 = 

 3 3 , and this solution becomes k = 3 32 . Up to a rescaling, this is the solution we have previously used in the massless case.

391

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

    whose solution is zI = − 12 / 1 P3 2 . Plugging in the results for the 3-point amplitudes and summing explicitly over the two helicities of the intermediate graviton, the γγφφ amplitude can be written as

+ − 0 0

Aγγφφ (1 2 3 4 ) =

 2 b P4 q b 1 4 K K I I   2  b 2 1^2 qK I

 2  b ^2 4 q P4 b KI K + I 2 . (11.129)

 b 2 1^2 qK

κ2 1

  4 12 12



I

For the first term, we may write

 


4

4   2  b 1 4 K b 1 4 b 1 4 b2 K 2 p1 · p 1^2 1^2 K I I I =0.  2 =  2

4 =   2

4 =

 b 1 b 1 b 1 4 1^ 2 1^ 2 K 1^ 2 K K I I I



(11.130)

The final zero occurs when we evaluate the expression at zI , as a consequence of eq. (11.128). Therefore, the amplitude reduces to a single term. Furthermore, we are still free to choose the auxiliary vector q. A convenient choice turns out to be q = p2 , which leads to: Aγγφφ (1+ 2− 30 40 ) = Then, notice that

 2 b 2 2 2 P4 b KI κ2 K I .

3   4 12 12

  



b 2 = 2 P4 1 12 , KI K 2 P4 b I

(11.131)

(11.132)

which gives the following extremely compact form for the amplitude: Aγγφφ (1+ 2− 30 40 ) =

2 κ2 2 P4 1

  . 4 12 12

(11.133)

†  Cross section and deflection angle : Using 2 P4 1 = 1 P4 2 , the modulus square of the amplitude is 2 κ4 2 P 12 1 P 22 4 4 + − 0 0 . Aγγφφ (1 2 3 4 ) =

2   2 16 12 12

(11.134)

392

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Note that

  2 P4 1 1 P4 2

= =

ab  cd  ab  cd  2 a P4 1 b 1 c P4 2 d = P4 1 b 1 c P4 2 d 2 a

tr P4 P1 P4 P2 = p4µ p1ν p4ρ p2σ tr σµ σν σρ σσ

=

2 p4µ p1ν p4ρ p2σ ηµν ηρσ − ηµρ ηνσ

=

+ηµσ ηνρ − i ǫµνρσ
2 2 (p1 · p4 )(p2 · p4 ) − p24 (p1 · p2 )

=

s13 s14 − m4 ,




(11.135)

where we have introduced the Lorentz invariants sij ≡ (pi +pj )2 and used s24 = s13 and s12 + s13 + s14 = 2 m2 (both follow from momentum conservation). Therefore, the squared amplitude reads 2 κ4 (s s − m4 )2 13 14 . Aγγφφ (1+ 2− 30 40 ) = 16 s212

(11.136)

The differential cross-section with respect to the solid angle of the outgoing photon is given by 2 dσ 1 (11.137) = Aγγφφ (1+ 2− 30 40 ) . 2 dΩ 64π s14

Let us now consider the limit of long wavelength photons, namely ω = |p1,2 | ≪ m. In this limit, the Lorentz invariants that appear in the cross-section simplify into16 s12 s13 s14

≈

≈

≈

4 ω2 sin2

θ 2

,

m2 − 2 m ω − 4 ω2 sin2 2

m + 2mω ,

θ 2

, (11.138)

where ω is the photon energy and θ its deflection angle in the center of mass frame (which is also the frame of the massive scalar particle in this limit). For large enough impact parameters, the deflection angle is small, θ ≪ 1. Thus we obtain in this limit 16 G2N m2 dσ . ≈ dΩ θ4

(11.139)

In order to determine the deflection angle as a function of the impact parameter b, consider a flux F of photons along the z direction, with the massive scalar at rest at 16 If we are only interested in the limit of small energy ω and small deflection angle θ, then the somewhat complicated calculation of the numerator done in eq. (11.135) can be avoided. Indeed, in this limit, the ab massive scalar is at rest and P4 ≈ m δab . Moreover, the 3-momenta of the incoming and outgoing √  



2ω
photons are nearly parallel to the z axis. This implies that 1 a ≈ 1 a ≈ − 2 a ≈ − 2 a ≈ , 0

  and 2 P4 1 ≈ 1 P4 2 ≈ −2mω.

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

393

the origin. Out of this flux, consider specifically the incoming photons in a ring of radius b and width db. The number of photons flowing per unit time through this ring is 2π b F db .

(11.140)

All these photons are scattered in the range of polar angles [θ(b) + dθ, θ(b)] (note that dθ is negative for db > 0, because the deflection angle decreases at larger b), which corresponds to a solid angle:
dΩ = −2π sin θ(b) dθ .

(11.141)

By definition, the number of scattering events is the flux times the cross-section, i.e. c sileG siocnarF

2π b F db = F

dσ dΩ , dΩ

(11.142)

that can be integrated for small angles into θ(b) =

4 GN m . b

(11.143)

(The integration constant is chosen so that the deflection vanishes when b → ∞.) This is indeed the standard formula from general relativity, that can be derived by considering geodesics in the Schwarzchild metric.

11.5.4

Scattering of gravitational waves by a mass

Let us now study the scattering amplitude between a scalar and a graviton, whose low energy limit will provide us information about the scattering of a long wavelength gravitational wave by a mass. A priori, each of the two gravitons may have a helicity ±2, but the cases {+2, +2} and {−2, −2} correspond to a helicity flip of the graviton, which is suppressed at low frequency. Therefore, let us consider the amplitude Ahhφφ (1−2 2+2 30 40 ). When writing the BCFW recursion for this amplitude, the simplest shift is one that affects the lines 1 and 2, more specifically:      ^ 2 = 2 − z 1 , 2 = 2 , ^      ^ 1 = 1 + z 2 , ^ 1 = 1 ,

(11.144)

Because the polarization vectors of the gravitons are squares of the spin-1 ones, this shift can be proven to lead to a vanishing amplitude when |z| → ∞ simply by power counting. With the shift (11.144), the intermediate propagator carries a scalar, and

394

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

40

30

K23

AL

30

AR

1−2

40

AL

2+2

K24

AR

1−2

2+2

Figure 11.8: BCFW shift for the calculation of the hhφφ amplitude.

therefore it has only the h = 0 helicity. The BCFW recursion formula contains two terms, Ahhφφ (1−2 2+2 30 40 ) =

i b0 ) Ahφφ (^2+2 30 − K 23 − m2 i b0 ) b0 ) , Ahφφ (^2+2 40 − K + Ahφφ (^ 1−2 30 K 24 24 2 K24 − m2 (11.145) b0 ) Ahφφ (^ 1−2 40 K 23

K223

that differ by a permutation of the external scalars. In the above equation, we have b 23 ≡ p b2 + p3 in the first term and made explicit the intermediate momentum, K b 24 ≡ p b2 + p4 in the second one. The explicit forms of the first and second terms are K

2 2 b 0 )Ahφφ (^ b0 ) −iκ2 ^1 P4 q q ′ P3 ^2 Ahφφ (^ 1−2 40 K 2+2 30 − K 23 23 i =  2

2 , K223 − m2 4(K223 −m2 ) ^1q q ′ ^2

2 2 b0 ) b 0 )Ahφφ (^ −iκ2 ^1 P3 q q ′ P4 ^2 Ahφφ (^ 2+2 40 − K 1−2 30 K 24 24 = i  2

2 . K224 − m2 4(K2 −m2 ) ^1q q ′ ^2 24

(11.146)

A convenient choice of auxiliary vectors is q = p2 and q ′ = p1 , which leads to −2 +2 0 0

Ahhφφ (1

2

3 4 ) =

=



4  κ2 1 P3 2 1 1 −i 2  2 + 4 12 12 K223 − m2 K224 − m2

4 κ2 1 P3 2 1

  . (11.147) i 16 12 12 (p2 · p3 )(p2 · p4 )

395

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

The square of this amplitude can be related to that of photon-scalar gravitational scattering by  2 2 −2 +2 0 0 − + 0 0 Ahhφφ (1 2 3 4 ) = Aγγφφ (1 2 3 4 ) 1−

m2 s12 (s13 −m2 )(s14 −m2 )



.

(11.148)

In the limit of a graviton of small energy (i.e. a gravitational wave of long wavelength) and small deflection angle (i.e. at large impact parameter), the second factor in the right hand side becomes equal to 1, and we have 2 Ahhφφ (1−2 2+2 30 40 )

2 − + 0 0 ≈ (1 2 3 4 ) A . γγφφ ω≪m

(11.149)

θ≪1

This implies that in this limit the bending of a gravitational wave by a mass is the same as the bending of a light ray (but there are some differences beyond this limit).

11.6 Cachazo-Svrcek-Witten rules 11.6.1

Off-shell continuation of MHV amplitudes

Our proof of the Parke-Taylor formula, based on BCFW recursion, is not faithful to the actual chronology, since the formula was conjectured in 1986 and a proof was found in 1988 using an off-shell recursion derived by Berends and Giele, well before on-shell recursion. The Cachazo-Svrcek-Witten rules, also anterior to BCFW recursion, provide a way to construct the tree-level non-MHV amplitudes by an expansion in which the MHV ones play the role of vertices, as in the following diagram:

-

+ -

+ +

+

-

,

where the two “vertices” are + + −− 4-point MHV amplitudes, sewed together in order to make a contribution to a non-MHV 6-point amplitude. In this section, we will use ideas inspired of the derivation of on-shell recursion in order to establish these rules. Firstly, for energy-momentum conservation to hold in the vertices of such a diagram, the intermediate propagator linking the two vertices must generically carry an off-shell momentum. This means that in such a construction, we need first to

396

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

generalize the MHV amplitudes to external off-shell momenta. As we shall see shortly, the CSW rules only use as vertices the MHV amplitudes that have exactly two negative helicities, i.e. those that are expressible in terms of angle brackets only. Thus, we would like to have an off-shell extension of these brackets. The main issue here is that for an off-shell momentum pµ , the 2 × 2 matrix P by which it is represented in the spinor-helicity method  has a non-vanishing determinant, and is therefore not factorizable as P = p p . Let us assume that ηµ is a light-like auxiliary vector, then we may always write pµ ≡ Pµ + zηµ

(11.150)

with P2 = 0. The on-shellness of Pµ determines uniquely the value of z, z=

p2 . 2p · η

(11.151)

Since Pµ is on-shell, the corresponding matrix 

P is factorizable into the direct product of a square and angle spinors, P = P P . Then, we may write        (11.152) η P = η|P + z η|η = η|P P + z η|η η , | {z } 0

which gives



η P . P =  η|P

(11.153)

Note that this identity contains the angle spinor P in the numerator and the square spinor P in the denominator, consistent with the fact that rescaling one with λ and the other with λ−1 gives an equally valid representation. For this reason we may simply ignore the denominator and define

 P = η P . (11.154)

In the following, we adopt this formula as the definition of the angle spinor associated with the off-shell momentum pµ . Note that when pµ is on-shell, then p = P, and the angle spinor P defined in eq. (11.154) is indeed proportional to the usual p . c sileG siocnarF

11.6.2

CSW rules for next-to-MHV amplitudes

Consider now the case of amplitudes with exactly three negative helicities, carried by the external lines i, j, k, and consider the following deformed square spinors  

  bı ≡ i + z jk η ,  

  b ≡ j + z ki η ,  

  b k ≡ k + z ij η , (11.155)

397

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

k− Figure 11.9: Propagators affected by the shift of eq. (11.155) in a generic amplitude.

i− j−

while the corresponding angle spinors are left unchanged17 . The shifted external momenta are defined as direct products of these shifted square spinors and the original angle spinors, and are therefore still on-shell. Note also that    bı i + b j + b k k

=

   i i + j j + k k | {z } 0      +z η jk i + ki j + ij k . {z } | 0

(11.156)

The first zero is due to momentum conservation in the unshifted amplitude, and the second one follows from Schouten identity. Thus, the above shift preserves momentum conservation. The propagators affected by the shift of eq. (11.155) form three lines starting at the three external points of negative helicity, that meet somewhere inside the graph. Since the shift modifies only the square spinors, the polarization vectors of negative helicity scale as z−1 . Moreover, from the figure 11.9, we see that there are p + 1 vertices and p propagators along the affected lines. Even in the worst case where all these vertices are 3-gluon vertices that scale like z, the overall scaling of the graph is bounded by z−3+(p+1)−p ∼ z−2 , and therefore it goes to zero as |z| → ∞. Then, we proceed like in the derivation of BCFW’s recursion formula, by integrating An (· · · ; z)/z over z on a circle of infinite radius. The behaviour of the deformed amplitude at large z 17 Recall that the usual BCFW shift acts only on a pair i, j of external lines, by shifting the angle spinor of one and the angle spinor of the other.

398

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

ensures that there is no boundary term, and we obtain: An (· · · ) = =

X

−

Res

z∗ ∈{poles of An }

X

An (· · · ; z) z z∗

b −h ; z ) i A (· · · K bh; z ) , AL (· · · − K I R I I I 2 K I I,h=±

(11.157)

i∈A L

where KI is the momentum of the intermediate propagator producing the pole. By construction, the singular propagator separating the left and right sub-amplitudes must be one of the dark propagators in the figure 11.9. Thus, when writing this factorized formula, we may decide that the external line i belongs to the left sub-amplitude, and at least one of j, k belongs to the right sub-amplitude. The propagator carrying the momentum KI can be on any of the three branches highlighted in the figure 11.9. In all three cases, there is only one choice of the intermediate helicity h that gives a non-zero result, and this h is such that both AL and AR are MHV amplitudes with two negative helicities. The next-to-MHV amplitude under consideration takes the following diagrammatic form: i

k

An = i

- -

+

-

- + -

+ j

k

j

-

- + -

k +

-

j,

(11.158)

i

where we have only indicated the relevant helicity assignments (all the thin lines carry positive helicities). Thus, as far as the helicities are concerned, the vertices in these graphs are MHV amplitudes with exactly two negative indices. Note that in eq. (11.158) the shift has no influence on the external lines because the MHV amplitudes with two negative helicities depend only on the spinors. The

angle b that corresponds MHV vertices in this equation also depend on the angle spinor K I to the on-shell (because we evaluate the amplitude at the value of z for which the intermediate propagator is singular) shifted momentum. Consider for instance the first diagram, obtained when the singular propagator is on the line that stems from the external line i. In this case, we have 

  b = K + z∗ jk η i , b =K b KI K (11.159) I I I  where z∗ is the value of z at the pole. By contracting this relation with η , we obtain 

  b K b = η K . ηK I I I

(11.160)

b in the MHV vertices resulting from the theorem of Thus, the angle spinor K I  residues is proportional to the off-shell extension η KI that we have proposed in the

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

399

  b all cancel, because the line previous subsection. Finally, note that the factors ηK I of momentum KI has opposite helicities on either side of the propagator that links the two MHV amplitudes18 . Therefore, in the MHV diagrams of eq. (11.158), we do not need to find the poles z∗ and we may directly evaluate  the MHV amplitudes that play the role of vertices with the off-shell angle spinor η KI .

Let us summarize here the CSW rules for calculating amplitudes with exactly three negative helicities: • Start from the three skeleton diagrams of eq. (11.158), and interpret the vertices as MHV amplitudes with one off-shell external leg. Note that the actual number of MHV graphs depends on the number of positive helicity external lines, since they may be attached to either of the two MHV vertices (provided we do not change the cyclic ordering of the external lines, and that all the vertices obtained in this way have at least three lines). • The intermediate propagator is simply a scalar propagator i/K2I , with the value of KI determined by momentum conservation at the vertices. • Replace the MHV vertices by their Parke-Taylor  with the angle

expression, ≡ η K (from now on, K spinor of the intermediate off-shell line given by I I

we omit the hat on KI , since shifting the momenta was just an intermediate device for establishing the CSW rules). Note that in this derivation of the CSW rules, we are guaranteed that the final result  final result does not depend on the choice of the spinor η introduced in the shift. Indeed, the pole at z = 0 if An (· · · ; z)/z is by construction the amplitude we are looking for. The main advantage of the CSW rules is that they express amplitudes in terms of high-level building blocks that already contain a large number of colour ordered graphs, as illustrated in the following example of a 2-vertices MHV graph contributing to the 1− 2− 3− 4+ 5+ 6+ six-point amplitude: 6 I

3

6

1

5 4

I 2

5 4

3

18 Recall that under a rescaling by a factor λ of an angle spinor, the MHV amplitudes with exactly two negative helicities scale by λ2 if the scaling affects an external line of negative helicity, and by λ−2 if it affects an external line of positive helicity.

400

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

11.6.3

Examples

Four-point − − −+ amplitude : let us first use the CSW rules to evaluate the 1− 2− 3− 4+ amplitude. Of course, we know beforehand that the result should be zero, so this is no more than a trivial illustration of the rules. In this case, the CSW rules give the following graphs: 3

4 + -

A4 (1− 2− 3− 4+ ) =

-

+

1

1 - + -

+ 2

3

4 + -

2

+ 3

4

- + + 1

-

2,

(11.161) but the last one is trivially zero because one of the vertices (−−) does not exist. In the first graph, the intermediate momentum (oriented  from left to right) is KI = p1 + p4 = −(p2 + p3 ), and its angle spinor KI ≡ η KI obeys     KI 1 = − η4 14 ,

    KI 3 = − η2 23 ,



    KI 2 = η3 23 ,

    KI 4 = η1 14 ,



while the CSW rules give − − − +

A4,1 (1 2 3 4 ) = =

3 3 1KI 23 1 

   2

2ig

KI 4 41 KI 3KI KI 2  3

 η4 14 2       . −2ig η1 η2 η3 23 2

(11.162)



(11.163)

In the second graph, the intermediate momentum is LI = p1 + p2 = −(p3 + p4 ), and the corresponding angle spinor satisfies

    LI 1 = − η2 12 ,

    LI 3 = η4 34 ,

This time, the CSW rules give − − − +

A4,2 (1 2 3 4 ) = =



    LI 2 = η1 12 ,

    LI 4 = − η3 34 .

3

3 LI 3 12 1  2ig   2 

2LI LI 1 LI 34 4LI  3

 η4 34 2       . 2ig η1 η2 η3 12

(11.164)

2

(11.165)

401

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES Therefore, the sum of these two contributions is     3 34 14 η4 − − − + 2       −   = 0 , A4 (1 2 3 4 ) = 2ig η1 η2 η3 12 23 | {z }

(11.166)

0

where the final cancellation is due to momentum conservation. c sileG siocnarF

Six-point − − − + ++ amplitude : Let us consider now a genuine example of next-to-MHV amplitude, the six-point [123]− [456]+ amplitude. The MHV graphs that contribute to this function are: 6

A6 ([123]− [456]+ ) =

5 + + - -

1

4 3 1 6 5 4 + + + + - + - - , + - + 2 2 3

(11.167)

where the shaded ellipses indicate that the lines (4, 5) (in the left graph) or (5, 6) (in the right graph) can either be attached to the left or right MHV vertex, provided the cyclic order is not modified. Each term in eq. (11.167) therefore corresponds to three MHV graphs, i.e. a total of six19 . For this amplitude, the CSW rules give: A6 ([123]− [456]+ )  5 X

= −4ig4 i=3

+

6 X j=4

3 1K  i  

Ki i+1



2Lj

i+1i+2 ··· 61

3 12 

  Lj j+1 ··· 61

1 L2 j

1 K2 i



3 23  

Ki 2

34 ··· iKi

3 L 3

 j 

34 ··· j−1j

jLj







,

(11.168)

where the intermediate momenta are respectively Ki ≡ −(p2 + p3 + · · · + pi ) and ≡ η Ki K Lj ≡ −(p + p + · · · + p ), and the corresponding angle spinors are i 3 4 j  and Lj ≡ η L j . For a numerical evaluation of this amplitude, one may take any auxiliary spinor η , since the amplitude does not depend on this choice.  A  somewhat simpler analytic expression may also be obtained by choosing η ≡ 2 . Indeed, since Ki = Li − p2 , this choice leads to the following simplification,

  Ki j = Li j , for all i, j , (11.169)

19 This number should be contrasted with the 38 six-point colour ordered graphs (see the figure 11.1). Moreover, each of these colour-ordered graphs is considerably more complicated than the MHV diagrams.

402

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

thanks to which many factors become identical in the two terms of eq. (11.168). With this choice, the terms i = 4, 5 in the first line and j = 4, 5 in the second line combine in the following compact expression:

 X 4 ii+1 4ig



  A6,1 = −     34

45

56

61

i=4,5

×

Ki i

" 3

23

Ki i+1

3

Ki 1

K2 i

+

Ki 2

3 #

3

12 Ki 3 (Ki +p2 )2

. (11.170)

The terms i = 3 in the first line and j = 6 in the second line of eq. (11.168) must be handled carefully. Indeed,   more   they both contain a denominator that vanishes when  η ≡ 2 , due to a factor η2 . In order to calculate these terms, we must leave η unspecified, but such that     η2 ≪ ηj for j 6= 2 , (11.171)   and expand in powers of η2 . After simplifications involving Schouten identity,   the the sum of these two terms is found to be finite when η → 2 and equal to A6,2

   

2  36 14 12 23 4ig4 13 s13 +2(s 12 +s 32 )







    +   +    , = 34

45

56

61

12

32

12

16

23

34

(11.172)

  where we denote sij ≡ ij ij = (pi + pj )2 . Combining all the contributions, the 6-point amplitude reads 4

   A6 ([123]− [456]+ ) =  4ig 34 45 56 61 "  3 3 3 3 # 

 X 12 Ki 3 23 Ki 1 ii+1



  + (Ki +p2 )2 × − K2 i=4,5

2 + 13

Ki i



Ki i+1

Ki 2

i

     36 14 12 23 s13 +2(s +s ) 12 32    . (11.173) +   +    12

32

12

16

23

34

This is the simplest of the 6-point amplitudes with three negative helicities, because the legs with negative helicities are adjacent. The non-adjacent cases lead to more complex expressions, but nevertheless considerably simpler than what one would get from traditional perturbation theory.

11.6.4

General CSW rules

In order to prove the CSW rules in general, we now proceed by induction. i.e. we assume that the CSW rules are applicable to all on-shell amplitudes with up to N − 1

403

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

Figure 11.10: Propagators affected by the shift of eq. (11.174) in a generic amplitude. They correspond to the subgraph obtained by following the momentum that flows from the external lines of negative helicity, assuming that the momenta of the positive helicity ones are held fixed.

b a

negative helicities and we consider an amplitude with N negative helicities. Let us denote N the set  of external lines of negative helicity. We then introduce an auxiliary square spinor η , and we apply the following complex shift to these lines      bı ≡ i + ri z η , bı ≡ i for i ∈ N , (11.174)

where the ri are coefficients chosen in such a way that momentum conservation is preserved at any z. Namely, they must satisfy X  ri i = 0 . (11.175) i∈N

(We further assume that partial sums are not zero, so that the internal propagators connected to the external negative helicities all carry a z-dependent momentum.)

When |z| → ∞, the shifted amplitude goes to zero like |z|1−N (graph by graph), which ensures that there is no boundary term when we integrate over z on the circle at infinity. The propagators that contribute poles to this integral are among those represented in dark in the figure 11.10 (we may assume that they become singular at distinct values of z, so that all the poles are simple). When we assign helicities to these singular propagators, two cases may arise: • The singular propagator is directly connected to one of the external lines of negative helicity (e.g. the propagator labeled “a” in the figure). In this case, only one choice of helicity is allowed, and the amplitude factorizes into an amplitude with 2 negative helicities and one with N − 1 negative helicities. • The singular propagator is an inner propagator, such as the one labeled “b” in the figure. It means that both the left and the right sub-amplitudes contain at least two of the N external lines of negative helicity. In this case, both choices of helicity are valid for the singular propagator, and in both cases the left and right sub-amplitudes have at most N − 1 negative helicities.

404

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

In these two cases, the theorem of residues divides the amplitude into a left and right on-shell sub-amplitudes that have at most N − 1 negative helicities, for which we may use the CSW rules assumed to be valid by the induction hypothesis. After the left and right sub-amplitudes have been replaced by using the CSW rules proven for < N negative helicities, all the poles produce terms that correspond to the same topology of MHV graph, but whose expressions differ because the value of z∗ is different for each pole. How this sum of contributions produces the product of denominators one would obtain by applying directly the CSW rules for amplitudes with N negative helicities requires some clarification. Let us consider a graph with nI internal shifted propagators. The application of the theorem of residues to the shifted amplitude divided by z produces nI terms, whose sum corresponds to the following combination of denominators: Dpoles

nI X 1 Y 1 = . b2 K2 I=1 I J6=I KJ,I

(11.176)

Each term in this sum corresponds to the vanishing of one of the shifted internal b 2 = 0, propagators. The factor K2I comes from the residue of the pole zI for which K I b 2 denotes the value taken by K b 2 at this z . and K I J,I J

❈

From eq. (11.174), we can write the SL(2, ) matrices that represent the shifted internal momenta as follows,  b = K + z η I , K (11.177) I I



where I is a linear combination of the ri i that depends on the precise topology of the MHV graph and of the internal propagator under consideration. This implies that   b 2 = K2 − z η K I , K (11.178) I I I   and the corresponding denominator vanishes at zI = K2I / η KI I . Therefore, we have Dpoles

nI X 1 Y = K2I I=1

20

J6=I

n

I Y 1 1   = . 2 K η K J I=1 I K2J − K2I  J  η K I

(11.179)

I

The last equality shows that the nI terms produced by the theorem of residues combine into an expression which is nothing but the product of unshifted denominators that would appear in the CSW rules. This completes the proof of the general CSW diagrammatic rules: 20 This

may be proven by integrating over a circle at infinity the following function n

f(z) ≡ z−1

I Y

I=1

1   . K2I − z η KI I

405

11. M ODERN TOOLS FOR TREE LEVEL AMPLITUDES

• Draw all the diagrams with the required assignment of external helicities, and such that all vertices have exactly two negative helicities. With N external negative helicities, these graphs all have N − 1 vertices. For instance, the [1234]− [5 · · · n]+ amplitude receives contributions from the following three classes of MHV diagrams: 4

n 1

2

n

-

+

-

3

1

1 n

5

-

+

- 2 3

4

2

5 -

-

3

4

(Only the negative helicities are indicated explicitly.) • The intermediate propagators are scalar propagators i/K2I , with the value of KI determined by momentum conservation at the vertices, • Replace the vertices by MHV amplitudes,  with some off-shell legs for which the angle spinor is defined as KI ≡ η KI , and use the Parke-Taylor formula to obtain their expression. c sileG siocnarF

406

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 12

Worldline formalism In the previous chapter, we have exposed the spinor-helicity language in which the building blocks of scattering amplitudes are expressed in terms of 2-component spinors. As we have seen, when combined with techniques such as on-shell recursion, this leads to great simplifications in the evaluation of on-shell tree amplitudes with physical polarizations. To a large extent, this simplification stems from the fact that the calculation of amplitudes based on these methods bypasses the usual representation in terms of Feynman diagrams. In fact, the spinor-helicity method is not the only one that relegates Feynman diagrams to a minor secondary role. Another approach, that we shall discuss in this chapter, is the worldline formalism. The name comes from the fact that in this approach, Feynman graphs are replaced by a representation in terms of a path integral over a function zµ (τ) (plus additional auxiliary variables in the case of fields with internal degrees of freedom, such as spin or colour), that defines a line embedded in spacetime. This function can be viewed as a parameterization of the whole history of a point-like particle. Historically, this method was first derived by starting from a string theory and by taking the limit of infinite string tension. Subsequently, it was rederived in a more mundane manner, in a first quantized framework. This is the point of view that we shall adopt in this chapter.

12.1 Worldline representation 12.1.1

Heat kernel

In order to illustrate the principles of the worldline representation, consider a scalar field theory of Lagrangian L≡

1 1 (∂µ φ)(∂µ φ) − m2 φ2 − V(φ) . 2 2

407

(12.1)

408

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Let us assume that we wish to obtain the tree-level propagator G(x, y) in a background field ϕ. Up to an irrelevant factor i, this propagator is the inverse of the operator  + m2 + V ′′ (ϕ),
x + m2 + V ′′ (ϕ(x)) G(x, y) = −i δ(x − y) . (12.2)

This equation must be supplemented by boundary conditions that depend on the type of propagator one wishes to obtain (time-ordered, retarded, etc...). Formally, we may write Z∞
−1 2 ′′ dT e−T (+m +V (ϕ)) . (12.3)  + m2 + V ′′ (ϕ) = 0

(The integrand in this formula is sometimes called a heat kernel, by analogy with the propagator of a heat equation.) However, for this integral to make sense in the limit T → ∞, it is necessary that the eigenvalues of the operator  + m2 + V ′′ (ϕ) be positive. The high lying eigenvalues of this operator do not depend on the background field ϕ(x) (assuming that it is smooth enough), and are of the form −gµν kµ kν + m2 .

(12.4)

In order to be positive for any momentum kµ , it is therefore necessary that the metric be Euclidean, with only minus signs. For this reason, we restrict our discussion to an Euclidean field theory from now on, so that we may write −gµν kµ kν = ki ki .

12.1.2

Propagator in a background field

The propagator G(x, y) is obtained by evaluating eq. (12.3) between states of definite position: Z∞ 

2 ′′ G(x, y) = −i dT y e−T (+m +V (ϕ)) x . (12.5) 0

Such a matrix element is quite common in ordinary quantum mechanics, and its representation as a path integral is well known. For a non-relativistic Hamiltonian of the form H≡

P2 + V(Q) , 2M

(12.6)

we have

−i(t −t )H  1 0 x = y e

Z

q(t0 )=x q(t1 )=y



M .2  R t1 Dq(t) ei t0 dt ( 2 q (t)−V(q(t))) .

(12.7)

409

12. W ORLDLINE FORMALISM

Eq. (12.5) can be similarly expressed as a path integral, if we use the following dictionary i(t1 − t0 ) it

τ 1 2 m2 + V ′′ (ϕ(x)) .

→

M

→

V(Q) This leads to G(x, y) = −i

T

→

Z∞ 0

→ dT

Z



z(0)=x z(T )=y

(12.8)

RT 1 .2  2 ′′ Dz(τ) e− 0 dτ ( 4 z (τ)+m +V (ϕ(z(τ)))) , (12.9)

where the dot denotes a derivative with respect to τ. For simplicity, we denote the integration variable z(τ) instead of zµ (τ), although it takes values in a d-dimensional spacetime. This expression is known as the worldline representation of the tree propagator in a background field. Very much like in ordinary quantum mechanics, the function zµ (τ) explores all the paths that start at x and end at y. Note also that the formula contains an integral over the “duration” (we use quotes here because τ is not a physical time) of this evolution. c sileG siocnarF

12.1.3

Alternate derivation

Starting from eq. (12.3), it is possible to follow a slightly different route (that one may view as another derivation of the path integral formulation of quantum mechanics), that has the virtue of providing more control on all the prefactors. Consider first the case of the theory in the vacuum, i.e. with no background field. An important result is Z +∞ z2 2 dz √ (12.10) eT∂ f(x) = e− 4T f(x + z) , 4πT −∞ which may be proven by Fourier transform. In words, this formula means that an operator Gaussian in a derivative is equivalent to a Gaussian smearing. Note here that it is crucial that the squared derivative has a positive prefactor inside the exponential, otherwise in the right hand side we would have a Gaussian with a wrong sign and the result would ill-defined. From this formula, we get Z +∞ z2

−T (+m2 )   dd z − 4T x = e−Tm2 y e y x + z e d/2 | {z } −∞ (4πT ) δ(x+z−y)

−Tm2

=

e (4πT )d/2

(y−x)2 e− 4T

,

(12.11)

410

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

which, apart from the prefactor exp(−Tm2 ), is a Gaussian probability distribution normalized to unity. This Gaussian distribution may be viewed as a Green’s function for the diffusion equation ∂T f(T, x) = −x f(T, x) ,

(12.12)

which highlights the connection that exists between the propagator associated to an elliptic differential operator and diffusion (i.e. Brownian motion). By comparing eqs. (12.9) (without external field) and (12.11), one can obtain the following formula for the absolute normalization of the integral over closed loops in d dimensions Z

z(0)=z(T )



  Dz(τ) exp −

ZT 0

1 1 z2  = = dτ . 4 (4πT )d/2 d=4 (4πT )2 .

(12.13)

The next step is to note that such a Gaussian distribution may be written as the convolution of two similar distributions defined on half the interval: (y−x)2

e− 4T = (4πT )d/2

Z

(z−x)2

(y−z)2

e− 2T e− 2T d z . (2πT )d/2 (2πT )d/2 d

(12.14)

By taking n − 1 of these intermediate points, we arrive at

−T (+m2 )  x = e−Tm2 y e

Z

dd z1 dd z2 · · · dd zn−1 − ǫ Pni=1 e 4 (4πǫ)nd/2

(zi −zi−1 )2 ǫ2

,

(12.15)

where we denote ǫ ≡ T/n, z0 ≡ x and zn ≡ y. In the limit where n → ∞, the argument of the exponential becomes an integral, and we obtain the following path integral

−T (+m2 )  x = e−Tm2 y e

Z

z(0)=x z(T )=y



.2 1 RT  Dz(τ) e− 4 0 dτ z (τ) .

(12.16)

Taking into account the term V ′′ (ϕ) due to a background field poses no difficulty if one breaks the interval [0, T ] into many small intervals. Indeed, even though V ′′ (ϕ(x)) does not commute with x , the Baker-Campbell-Hausdorff formula indicates that the exponential of their sum is equal to the product of their respective exponentials, up to terms of higher order in ǫ = T/n, that do not matter in the limit n → ∞.

12. W ORLDLINE FORMALISM

12.1.4

411

One-loop effective action

A minor modification of this derivation also applies to the quantum effective action at one-loop in the background field ϕ,    + m2 + V ′′ (ϕ) 1 . (12.17) Γ [ϕ] = − Tr ln 2  + m2 The denominator inside the logarithm is not crucial since it is independent of the background field, but it produces an ultraviolet subtraction since the large eigenvalues of the numerator and denominator are almost equal. Firstly, the logarithm may be represented as follows,   Z∞   + m2 + V ′′ (ϕ) dT  −T (+m2 +V ′′ (ϕ)) −T (+m2 ) ln e − e , = −  + m2 T 0 (12.18) which is very similar to eq. (12.3), except for the denominator 1/T . The same restrictions on the signs of the eigenvalues apply here, forcing us to consider an Euclidean theory. The proof of this formula goes along the following lines:   ZB Z∞ Z∞ ZB
A dT −TB dY −TY ln dY = = e − e−TA . (12.19) dT e = B Y T 0 0 A A Then, the trace is obtained as   Z 

Tr · · · = dd x x · · · x .

(12.20)

Therefore, we obtain a path integral representation similar to eq. (12.9), but with a path that starts and ends at the same point: Z Z RT 1 .2   2 ′′ 1 ∞ dT Γ [ϕ] = const + Dz(τ) e− 0 dτ ( 4 z (τ)+m +V (ϕ(z(τ)))) . 2 0 T z(0)=z(T )

(12.21) (We have not written explicitly the term coming from the denominator  + m2 – it is contained in the unspecified additive constant.) In this case, the worldlines are closed, and therefore form loops in spacetime.

12.1.5

Length scales

Let us now discuss some qualitative aspects of eqs. (12.9) and (12.21). Firstly, note that the parameter T has the dimension of an inverse squared mass, T ∼ (mass)−2 .

(12.22)

412

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

On dimensional grounds, one sees that the typical diameter of the loops1 z(τ) that appear in eq. (12.21) is √ (12.23) ∆z ∼ T . In contrast, the perimeter of these loops scales as T . These scaling laws are consistent with a Brownian motion of duration T . c sileG siocnarF

T 1/2

Figure 12.1: Typical worldloop that contributes in eq. (12.21). While its length scales as T , its extent in spacetime only grows like T 1/2 .

The integration measure dT/T corresponds to a uniform distribution of the values of ln(T ). Thus, the loop sizes are uniformly distributed on a log scale. Large loops (i.e. large values of T ) encode the infrared sector of the theory. In a massless theory, loops of arbitrarily high size are allowed. With a non-zero mass, the factor exp(−Tm2 ) suppresses the values T ≫ m2 , i.e. the loops of size larger than the inverse mass (the Compton wavelength of the particle). By suppressing the probability of occurrence of large loops, the mass thus regulates the infrared. Note also how the second derivative V ′′ (ϕ(z)) acts as a position dependent squared mass. On the other hand, small loops encode the ultraviolet behaviour of the theory. When the extent of the loop becomes smaller than the typical scale over which the background field ϕ(x) varies, the loop sees only a constant background field, whose sole effect in eq. (12.21) is an overall rescaling. Therefore, these small loops behave as in the vacuum, and the ultraviolet sector of the theory does not depend on the background field. 1 For

.

a uniform background field, the exponential depends only on the derivative z. As a consequence, this exponential weight constrains the size of the loops, but not the possible location of their barycenter.

12. W ORLDLINE FORMALISM

413

12.2 Quantum electrodynamics 12.2.1

Scalar QED

Let us now consider the case where the background field is an Abelian vector field Aµ (x), while the particle in the loop is a (complex) scalar. The one-loop effective action is now given by   −(∂ − ie A)2 + m2 . (12.24) Γ [A] ≡ − Tr ln  + m2 Note that there is no prefactor 1/2 because the scalar field in this theory is a complex field. Firstly, we obtain Z∞ Z

2 2  dT Γ [A] = const + dd x x e−T (m −(∂−ieA) ) x . (12.25) T 0

Then, note that the exponential contains an operator which is very similar to the Hamiltonian of a charged particle in an external electromagnetic field. We can use this analogy in order to obtain a path integral representation of the matrix element under the integral in the previous equation. This leads to the following worldline representation: Γ [A] = const+

∞ Z 0

dT T

Z



RT . 1 .2  2 Dz(τ) e− 0 dτ ( 4 z (τ)+iez(τ)·A(z(τ))+m ) . (12.26)

z(0)=z(T )

Likewise, the tree-level scalar propagator in a background electromagnetic field is given by Z Z∞ RT . 1 .2   2 Dz(τ) e− 0 dτ ( 4 z (τ)+iez(τ)·A(z(τ))+m ) . (12.27) dT G(x, y) = −i 0

z(0)=x z(T )=y

Under an Abelian gauge transformation of the electromagnetic potential, Aµ (z) .

→

Aµ (z) + ∂µ χ(z) ,

the term in z · A transforms as follows
. . . z · A → z · A + ∂z χ = z · A + ∂τ χ(z(τ)) .

(12.28)

(12.29)

Thus, the gauge transformation modifies this term by the addition of a total derivative with respect to τ, whose integral is ZT dτ ∂τ χ(z(τ)) = χ(z(T )) − χ(z(0)) . (12.30) 0

414

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

In the calculation of the one-loop effective action, the trajectories z(τ) have equal initial and final points, and this shift is therefore zero. Thus, the expression (12.26) of the one-loop effective action is explicitly gauge invariant. If we were considering instead the scalar propagator G(x, y), this term would be ZT dτ ∂τ χ(z(τ)) = χ(y) − χ(x) , (12.31) 0

and as a consequence the propagator transforms as G(x, y)

→

eieχ(x) G(x, y) e−ieχ(y) ,

(12.32)

which is indeed the correct gauge transformation law of the scalar propagator.

12.2.2

Spinor QED

When the particle in the loop is a spin 1/2 fermion, the one-loop effective action involves the Dirac operator   iDµ γµ + m . (12.33) Γ [A] ≡ Tr ln i∂µ γµ + m The first step is to note that

h
i1/2 / = det m − iD / = det m2 +D /2 det m + iD ,

(12.34)

" # / 2 + m2 D 1 Γ [A] ≡ Tr ln 2 . 2 / + m2 ∂

(12.35)

which leads to

Then, we may use /2 D

=

D µ Dν γ µ γ ν

=

D µ Dν

=

−D2 +

=

−D2 − e Fµν Mµν ,




ν ν 1 µ 1 µ 2 {γ , γ } + 2 [γ , γ ] µ ν 1 4 [Dµ , Dν ][γ , γ ]

(12.36)

where Mµν ≡ 4i [γµ , γν ]. (We have assumed an Euclidean metric tensor with only minus signs in the 3rd and 4th lines.) This gives the following representation of the one-loop effective action: 1 Γ [A] = const − tr 2

∞ Z 0

dT −T e T

m2 −D2 −e Fµν Mµν




.

(12.37)

415

12. W ORLDLINE FORMALISM

The term m2 − D2 , identical to the operator encountered in the case of scalar QED, is now supplemented by a potential U(x) ≡ −e Fµν (x) Mµν .

(12.38)

However, because U(x) still contains non-commuting Dirac matrices, the worldline representation of the exponential is now more complicated, and the overall trace applies both to the spacetime dependence and to the Dirac indices. A first possibility is to reproduce the method used in the previous sections, where we introduce a path integral over classical trajectories zµ (τ). When doing this, the matrix Dirac structure inside the exponential is not altered, and is handled by a path ordering: Z  

−T m2 −D2 −e Fµν Mµν
 x = Dz(τ) x e 

× P e−

z(0)=x z(T )=x

RT 0

 . 1 . dτ ( 4 z2 (τ)+iez(τ)·A(z(τ))+m2 +U(z(τ)))

. (12.39)

But it is in fact possible to remove the path ordering by introducing some auxiliary variables. In the procedure that leads to eq. (12.39), one breaks the interval [0, T ] into infinitesimal sub-intervals and one inserts a complete sum of states between each factor. When the evolution operator to be evaluated contains extra internal degrees of freedom (in the present case, the spin degree of freedom encoded in the Dirac matrices), the intermediate states inserted in the expression must contain information about this internal structure, for the matrix elements produced in the process to be c-numbers. Let us define the following operators from the Dirac matrices: c sileG siocnarF

iγ3 ± γ4 iγ1 ± γ2 , c± . (12.40) 2 ≡ 2 2 From the anti-commutation relation obeyed by the Dirac matrices, we have c± 1 ≡

− {c+ r , cs } = δrs ,

+ − − {c+ r , cs } = {cr , cs } = 0 .

(12.41)

c+ r

Therefore the operators are fermionic creation operators (creating independent fermions), and the c− are the corresponding annihilation operators. By inverting i eq. (12.40), − γ1 = −i (c+ 1 + c1 ) , − γ3 = −i (c+ 2 + c2 ) ,

− γ2 = c+ 1 − c1 , − γ4 = c+ 2 − c2 ,

(12.42)

the potential U(x) may be viewed as a Hamiltonian quadratic in fermionic creation and annihilation operators, and a time evolution operator constructed with this Hamiltonian may be written as a Grassmann path integral. In order to see this, let us start from a state 0 which is annihilated by c− 1,2 ,  (12.43) c− 1,2 0 = 0] ,

416

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

and construct populated states by applying c+ 1,2 to it, 

 n1 n2 ≡ c+ n1 c+ n2 0 . 2 1

(12.44)

Consider now a pair of complex Grassmann variables ξ1,2 such that    ξi , ξj = ξi , ξj = ξi , ξj = ξi , ξj = 0 ,   =0. ξi , c ± = ξi , c ± j j 

 A fermionic coherent state ξ may be defined as   ξ ≡ e−ξc+ 0

,

ξc ξ = 0 e .

(12.45)

(12.46)

As with bosonic coherent states, they are eigenstates of the annihilation operators,  

+

, ξ ci = ξi ξ . c− (12.47) i ξ = ξi ξ In addition, the overlap between two such coherent states is given by

 ξ ζ = eξζ ,

(12.48)

and one may construct the identity operator as a superposition of projectors on these coherent states: Z  (12.49) 1 = dξ1 dξ1 dξ2 dξ2 e−ξξ ξ ξ . | {z } ≡ dξdξ

Moreover, if A(c+ , c− ) is a normal ordered operator made of the creation and annihilation operators, then its matrix element between two coherent states is given by 

(12.50) ξ A(c+ , c− ) ζ = eξζ A(ξ, ζ) ,

and the trace over the Dirac indices of an operator A may be written as Z 


tr A = dξdξ e−ξξ − ξ A ξ .

(12.51)

(One may easily check that this gives 4 when A is the identity.) Note that in the calculation of this trace, the coherent states that appear on the left and on the right are defined with opposite Grassmann variables ξ and −ξ. This is a standard property of fermionic traces, whose path integral representation must obey an anti-periodic boundary condition in time. This formalism can be used to transform the Dirac structure in eq. (12.39) into a Grassmann path integral. To achieve this, we follow the standard procedure of

417

12. W ORLDLINE FORMALISM

breaking the interval [0, T ] into N small sub-intervals, and we insert a unit operator given by eq. (12.49) at the boundaries of the sub-intervals. This produces matrix elements of the form 

µν (12.52) ξi+1 e−ǫeFµν (z(τi )) M ξi ,

(ǫ ≡ T/N) that may be evaluated by replacing the Dirac matrices in Mµν by their expression in terms of the operators c± 1,2 and by using the properties of fermionic coherent states. This leads to the following worldline representation for the one-loop effective action in QED, with a spin 1/2 field in the loop: Γ [A] = const −

× e−

RT 0

1 2

Z∞ 0

dT T

Z



z(0)=z(T ) ψ(0)=−ψ(T )

 Dz(τ)Dψ(τ)

. . 1 . 1 dτ ( 4 z2 +iez·A(z)+m2 + 2 ψµ ψµ −ie ψµ Fµν (z)ψν )

, (12.53)

where ψµ is a collection of four Grassmann variables that combine the ξ1,2 , ξ1,2 at each intermediate time. In this formula, the ordering that was necessary to handle the non-commutative nature of the Dirac matrices has now been replaced by a path integral over fermionic internal degrees of freedom.

12.3 Schwinger mechanism Since it provides expressions for propagators and effective actions in a background field, the worldline formalism is well suited to study phenomena that occur in the presence of such an external field, for instance the splitting of a photon into two photons in an external magnetic field (γ → 2γ is forbidden in the vacuum, but becomes possible if an external electromagnetic field provides a fourth photon), or the bremsstrahlung radiation by a charged particle in a magnetic field. c sileG siocnarF

Another interesting process which can be addressed by the worldline formalism is the Schwinger mechanism, which amounts to the spontaneous production of e+ e− pairs by a static and homogeneous electrical field. That this is possible may be understood intuitively as follows. Without any external field, QED has an empty band of states with energy larger than m corresponding to free electrons and anti-electrons, and a filled band of states with energy lower than −m that corresponds to “trapped” particles (the Dirac see). A minimal energy of 2m (the minimal energy of an e+ e− pair) must be provided to move one of these particles from the Dirac see to the positive band of free particles. Consider now an electrical field E in the x direction. If this field is static and homogeneous, we may find a gauge in which it is represented by a vector potential whose only non-zero component is A0 = −Ex, which tilts the boundaries of the band of free states and of the Dirac see, as shown in the figure 12.2. Now, a pair

418

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

E +m −m

x

Figure 12.2: Schematic picture of the tunneling process involved in the Schwinger mechanism. The empty band is the gap between the antielectron Dirac sea and the positive energy electron continuum, tilted by the potential energy eV(x) = −eEx in the presence of an external electrical field E.

from the Dirac see can move to the band of free particles by a tunneling process, that does not require any energy. Standard results of quantum mechanics indicate that the tunneling probability should behave as exp(−const × m2 /(eE)). This expression is non-analytic in the coupling constant e, making it impossible to obtain in a standard perturbative expansion. Although the Schwinger mechanism was computed a long time ago by resummed perturbation theory, the worldline formalism provides a straightforward way to calculate it and offers very interesting new insights about the space-time development of the particle production process. Let us consider the case of scalar QED in order to illustrate this in a simpler setting. The probability of pair production may be inferred from the vacuum-to-vacuum transition amplitude, that can be written as an exponential,

 0out 0in = ei V , (12.54) i V being the sum of all the connected vacuum diagrams. The possibility of particle production is intimately related to the imaginary part of V, since the total probability of producing particles reads  2 Pprod = 1 − 0out 0in = 1 − e−2 Im V .

(12.55)

In scalar QED, the graphs made of one scalar loop embedded in a background electromagnetic field lead to the following contribution to V,
V1 loop = ln det gµν Dµ Dν + m2 , (12.56)

where Dµ is the covariant derivative in the background field. The metric should be Euclidean in order to apply the worldline formalism, i.e. gµν = −δµν . At

419

12. W ORLDLINE FORMALISM

one-loop, the sum of the connected vacuum graphs in the presence of a background electromagnetic field is given by V1 loop =

∞ Z

dT −m2 T e T

0

Z



RT . 1 .2  Dz(τ) e− 0 dτ ( 4 z (τ)+iez(τ)·A(z(τ)) . (12.57)

z(0)=z(T )

This formula involves a double integration: a path integral over all the worldlines z(τ), i.e. closed paths in Euclidean space-time parameterized by the fictitious time τ ∈ [0, T ], and an ordinary integral over the length T of these paths. The sum over all the worldlines can be viewed as a materialization of the quantum fluctuations in space-time, and the prefactor exp(−m2 T ) suppresses the very long worldlines that explore regions of space-time that are much larger than the Compton wavelength of the particles. In eq. (12.57), the path integral can be factored into an integral over the barycenter Z of the worldline and the position ζ(τ) about this barycenter, ZT

z(τ) ≡ Z + ζ(τ) ,

dτ ζ(τ) = 0 .

(12.58)

0

After this separation, all the information about the background field contained in eq. (12.57) comes via a Wilson line,    WZ ζ ≡ exp − ie

ZT 0

 . dτ ζ(τ) · A(Z + ζ(τ)) ,

(12.59)

averaged over all closed loop of length T ,

hWZ iT ≡

R

ζ(0)=ζ(T )

R



 R    T Dζ(τ) WZ ζ exp − 0 dτ

ζ(0)=ζ(T )



 R  T Dζ(τ) exp − 0 dτ

.

ζ2 4

.

ζ2 4





.

(12.60)

This path average is dominated by an ensemble of loops localized around the barycenter Z, and hWZ iT encapsulates the local properties of the quantum field theory in the vicinity of Z (roughly up to a distance of order T 1/2 ). In terms of this averaged Wilson loop, the 1-loop Euclidean connected vacuum amplitude reads V1 loop =

1 (4π)2

Z

d4 Z

Z∞ 0

dT −m2 T e hWZ iT . T3

(12.61)

(In this formula, the prefactor and power of T in the measure assume 4 spacetime dimensions.) The imaginary part of V1 loop comes from the existence of poles in

420

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

hWZ iT at real values of the fictitious time T . In terms of these poles, the imaginary part can be written as π Im (V1 loop ) = (4π)2

Z

X e−m2 Tn
d Z Re Res hWZ iTn , Tn . (12.62) Tn3 4

poles Tn

Let us now be more specific and consider a static and uniform electrical field E. Since one can choose a gauge potential which is linear in the coordinates Z, ζ, the path integral that gives the average Wilson loop is Gaussian and can therefore be performed in closed form, leading to hWZ iT =

eET . sin(eET )

(12.63)

(Note that it does not depend on the barycenter Z since the field is constant.) This quantity has an infinite series of single poles along the positive real axis, located at Tn = nπ/(eE) (n = 1, 2, 3, · · · ), that give the following expression for the imaginary part: ∞ X V4 (−1)n−1 −nπm2 /(eE) 2 Im (V1 loop ) = (eE) e . 16π3 n2

(12.64)

n=1

In this formula, V4 is the volume in space-time over which the integration over the barycenter Z is carried out. After exponentiation, this formula gives the vacuum survival probability P0 = exp(−2 Im V). A more detailed study would reveal that the term of index n comes from Bose-Einstein correlations among n produced pairs, while the first pole τ1 only contains information about the uncorrelated part of the spectrum. Given the origin of these terms in the present derivation, as coming from poles Tn that are more distant from T = 0, we see that increasingly intricate (the index n is the number of correlated particles) quantum correlations come from worldlines that explore larger and larger portions of space-time. This supports the intuitive image that quantum fluctuations and correlations are encoded in the fact that the worldlines explore an extended region around the base point Z. c sileG siocnarF

12.4 Calculation of one-loop amplitudes The worldline formalism can also be used in order to derive expressions for one-loop amplitudes. The main difference, compared to the calculation of the Schwinger mechanism, is that in the case of amplitudes the momenta carried by the lines attached to the loop are fixed instead of integrated over. The expected result is therefore a function of N momenta (or coordinates), rather than just a number.

421

12. W ORLDLINE FORMALISM

12.4.1 φ3 scalar field theory As a first simple illustration of the method, let us consider a scalar field theory with a λ cubic coupling V(φ) = 3! φ3 , for which the worldline representation of the one-loop effective action is Z Z RT 1 .2   2 1 ∞ dT Γ [ϕ] = const + Dz(τ) e− 0 dτ ( 4 z +m +λϕ(z)) . (12.65) 2 0 T z(0)=z(T )

Like in the previous section, we first split z(τ) into the barycenter and a deviation about it: z(τ) ≡ Z + ζ(τ) .

(12.66)

In the case of amplitudes, the integration over Z will simply produce the delta function of overall energy-momentum conservation. Using the T -periodicity of the paths over . which we integrate, the term in ζ2 inside the exponential can be integrated by parts, Z Z . 1 T 1 T − dτ ζ2 = dτ ζζ¨ , (12.67) 4 0 4 0

and the the path integral on ζ(τ) involves the inverse G(τ, τ ′ ) of the operator 12 ∂2τ , defined by ∂2τ G(τ, τ ′ ) = 2 δ(τ − τ ′ ) .

(12.68)

This inverse exists thanks to the fact that we have removed the barycenter from z(τ), which amounts to removing the zero mode from ζ(τ). Indeed, a general T -periodic function can be written as X τ (12.69) ζn e2iπn T , ζ(τ) = n∈❩

and excluding the zero mode corresponds to ζ0 = 0. A very useful identity is X 2iπn τ X T =T e δ(τ − nT ) . (12.70) n∈❩

n∈❩

Using this formula, we can check that the propagator G(τ, τ ′ ) is given by G(τ, τ ′ ) = 2T

X

n∈❩∗

(τ−τ 1 2iπn T e (2iπn)2

′

)

.

(12.71)

Note that this function is even in τ − τ ′ and T -periodic. Integrating2 eq. (12.70) twice from 0 to τ − τ ′ , we obtain G(τ, τ ′ ) = |τ − τ ′ | −

(τ − τ ′ )2 T − . T 6

(12.72)

2 adopt a symmetric convention for handling the delta function δ(τ), which amounts to Rτ We 1 ′ ′ 0 dτ δ(τ ) = 2 θ(τ).

422

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

From the quantum effective action, one-particle irreducible amplitudes are obtained by differentiating with respect to the field, as many times as there are external legs, and by setting ϕ ≡ 0 afterwards. Thus, the N-point function is given by ΓN (x1 , · · · , xN ) =

(−λ)N 2 ×

Z∞ 0

N ZT Y

Z

dT −m2 T e T



 Dz(τ)

z(0)=z(T )

dτi δ(z(τi ) − xi ) e−

RT 0

1 . dτ 4 z2

. (12.73)

i=1 0

In this formula, the path integral is over all closed paths that pass at all the coordinates x1 , · · · , xN , in any order, which provides a rather intuitive picture of the worldline representation of the amplitude. Let us now Fourier transform this expression in order to obtain the amplitude in momentum space. The Fourier integrals over the xi are trivial thanks to the N delta functions, and we obtain ΓN (p1 , · · · , pN ) =

(−λ)N 2 ×

Z∞ 0

N ZT Y

dT −m2 T e T

Z

Z

dd Z



 Dζ(τ)

ζ(0)=ζ(T ) RT

dτi eipi ·(Z+ζ(τi )) e

0

1 dτ 4 ζζ¨

,

(12.74)

i=1 0

where we also have separated the barycenter coordinate Z from the deviation ζ and integrated by parts the term in ζ˙ 2 . The integral over Z produces a delta function of the sum of the momenta, and the path integral over ζ is Gaussian, leading to ΓN (p1 , · · · , pN ) =

Z X
∞ dT 2 (−λ)N d/2 (2π) δ e−m T pi 1+d/2 i 21+d/2 T 0 ZT Y N  X  × dτi exp 12 G(τi , τj ) (pi · pj ) .(12.75) 0 i=1

i,j

This is the worldline expression of a one-loop N-point scalar amplitude. One may make a number of remarks about this formula: • In contrast with the formulas obtained from Feynman diagrams, there is no loop momentum. It is replaced by the variable T that measures the length of the worldloops. As we have said earlier, there is loose connection between T and a momentum, since small values of T correspond to the ultraviolet and large values of T to the infrared. c sileG siocnarF

423

12. W ORLDLINE FORMALISM

• The dependence on the external momenta is directly expressed in terms of the Lorentz invariants pi · pj . In a formula that contains a loop momentum k, one would also have all the k · pi . • The integral on T may be divergent at small T , because of the factor 1/T 1+d/2 . However, the integrals of the second line roughly behave as T N (since there are N integrals over an interval of size T , with an integrand of order one). Thus, the overall behaviour of the T integral is dT T N−1−d/2 . This integral is convergent if N − 1 − d/2 > −1, i.e. N > d/2. In four spacetime dimensions, this is N > 2, in agreement with conventional power counting that indicates that all one-loop functions with n > 3 are finite in the φ3 scalar theory. • Each ordering of the fictitious times τi corresponds to a given cyclic ordering of the momenta pi around the loop. The corresponding to one such ordering in the formula (12.75) can be mapped to the expression of the corresponding Feynman diagram. The τi correspond to the N Feynman parameters introduced to combine the N denominators into a single one3 , and T corresponds to the squared momentum that appears in this unique denominator. • The constant term −T/6 in the propagator of eq. (12.72) does not contribute in eq. (12.75). Indeed, its contribution inside the exponential is −

T X  X  T X pi · pj = 0 , pi · pj = − 12 12 i

i,j

(12.76)

j

which is zero thanks to momentum conservation.

12.4.2

Scalar quantum electrodynamics

As a slightly more complicated example of application, let us now derive the expression of the one-loop N-photon amplitude in scalar QED. The starting point is the one-loop quantum effective action in an Abelian background gauge field,

Γ [A] = const +

∞ Z 0

dT −m2 T e T

Z



RT . 1 .2  Dz(τ) e− 0 dτ ( 4 z +iez·A(z)) . (12.77)

z(0)=z(T )

3 Note that there are only N − 1 independent Feynman parameters, since their sum is constrained to be one, but because of the periodicity and translation invariance of the propagators G(τi , τj ), it is possible to choose one of the τi ’s to be equal to zero, hence only N − 1 of them are truly independent.

424

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Differentiating N times with respect to Aµ (x) and setting the background field to zero afterwards, we obtain

µ1 ···µN ΓN (x1 , · · ·

N

, xN ) =

(−ie)

∞ Z

dT −m2 T e T

0

×

ZT Y N

Z



RT  Dz(τ) e− 0 dτ

1 .2 4z

z(0)=z(T )

.

dτi δ(z(τi ) − xi ) zµi (τi ) . (12.78)

0 i=1

Next, we Fourier transform this expression and contract a polarization vector to each external Lorentz index, and we isolate the integral over the barycenter Z,

ΓN (p1 ǫ1 , · · · , pN ǫN ) =

N

d

(−ie) (2π) δ RT

×e

0

X

i

pi

Z N 1 ¨ T Y ζζ

dτ 4

0 i=1




∞ Z

dT −m2 T e T

0

Z



 Dζ(τ)

ζ(0)=ζ(T )

.
dτi eipi ·ζ(τi ) ζ(τi ) · ǫi .

(12.79)

.

The path integral is still Gaussian, but the factors ζ(τi ) · ǫi complicate it significantly compared to the φ3 theory. In particular, the answer will now contain derivatives of the propagator 2 (τ − τ ′ ) ˙ G(τ, τ ′ ) ≡ ∂τ G(τ, τ ′ ) = sign(τ − τ ′ ) − , T 2 ¨ G(τ, τ ′ ) ≡ ∂τ ∂τ ′ G(τ, τ ′ ) = 2 δ(τ − τ ′ ) − . T

(12.80)

Note that the first derivative of the propagator with respect to the second time is the ˙ defined above since the propagator G is even. Note again that the opposite of the G term −T/6 in this propagator will not contribute, thanks to momentum conservation. A convenient trick to perform this integral is to write Y i

.

ζ(τi ) · ǫi = exp

X i

.

ζ(τi ) · ǫi



multi-linear

,

(12.81)

425

12. W ORLDLINE FORMALISM

where the subscript “multi-linear” means that we keep only the term in ǫ1 ǫ2 · · · ǫN in the Taylor expansion of the exponential. This leads to Z X
∞ dT 2 (−ie)N d/2 (2π) δ e−m T p i 1+d/2 i 2d/2 T 0 ZT Y N

X h1 dτi exp G(τi , τj ) (pi · pj ) × 2 0 i=1 i,j i ˙ i , τj ) (pi · ǫj ) + 1 G(τ ¨ i , τj ) (ǫi · ǫj ) +i G(τ . 2 multi-linear (12.82)

ΓN (p1 ǫ1 , · · · , pN ǫN ) =

The expansion of the exponential and extraction of the term that contains each polarization vector exactly once leads to an expression of the form   X

 ˙ G) ¨ exp 1 G(τ , τ ) (p · p ) , (12.83) = PN (G, exp · · · i j i j 2 multi-linear

i,j

where PN is a polynomial in the derivatives of the propagator, with coefficients made of the Lorentz invariants pi · ǫj and ǫi · ǫj . By integration by parts, it is possible ¨ by first derivatives G. ˙ In this operation, the to replace the second derivatives G ˙ polynomial PN is replaced by another polynomial QN that depends only on the G, hence  X

  ˙ exp 1 = QN (G) exp · · · G(τ , τ ) (p · p ) . (12.84) i j i j 2 multi-linear

i,j

˙ corresponds to the combination of numerators that would The polynomial QN (G) appear in the expression of this amplitude from the usual Feynman rules.

12.4.3

Spinor QED

In quantum electrodynamics with spin 1/2 matter fields, the one-loop effective action in a photon background is given by: Z Z   1 ∞ dT Γ [A] = const − Dz(τ)Dψ(τ) 2 0 T z(0)=z(T ) ψ(0)=−ψ(T )

× e−

RT 0

. . 1 . 1 dτ ( 4 z2 +iez·A(z)+m2 + 2 ψµ ψµ −ie ψµ Fµν (z)ψν )

. (12.85)

Now, we have a second path integral, that involves the anti-periodic Grassmann variables ψµ . This additional integral is also Gaussian, and its result can be expressed

426

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

in terms of the inverse of the operator 12 ∂τ over the space of anti-periodic functions4 , whose expression reads X 1 τ−τ ′ 1 S(τ, τ ′ ) = 2 (12.86) e2iπ(n+ 2 ) T = sign (τ − τ ′ ) . 1 2iπ(n + 2 ) n∈❩ One can then in principle follow the same sequence of steps as in the scalar QED case, to obtain an expression of the one-loop N-photon amplitude in spinor QED in ˙ G ¨ and S. In fact, it was shown by Bern and Kosower terms of the propagators G, G, that this expression can be obtained from the corresponding scalar QED amplitude by a simple substitution. Starting from the final scalar QED expression in terms of ˙ (see the eq. (12.84)), one should arrange each term of this the polynomial QN (G) polynomial as a product of cycles of the form ˙ 1 , τ2 )G(τ ˙ 2 , τ3 ) · · · G(τ ˙ c−1 , τc ) . [1, 2, 3, · · · c]G ≡ G(τ

(12.87)

Then, the Bern-Kosower rule states that in order to obtain the analogous spinor QED amplitude, one should perform the following substitution on each such cycle:
(12.88) [1, 2, 3, · · · c]G → −2 [1, 2, 3, · · · c]G − [1, 2, 3, · · · c]S ,

where [· · · ]S is the same cyclic product made of the propagator S defined above ˙ instead of G. c sileG siocnarF

12.4.4

Example: QED polarization tensor

Scalar QED : As an illustration of the calculation of amplitudes in the worldline formalism, let us study the one-loop photon polarization tensor in quantum electrodynamics, starting first from the simpler case of scalar QED. In d dimensions, the polarization tensor is related to the one-particle irreducible two-point function Γ2µ1 µ2 (p1 , p2 ) by Γ2µν (p, q) ≡ (2π)2 δ(p + q) Πµν (p) .

(12.89)

From eq. (12.79), we obtain the following expression in scalar QED Πµν scalar (p)

=

2

−e

∞ Z

dT −m2 T e T

0

×

ZT

Z



 RT Dζ(τ) e 0 dτ

1 ¨ 4 ζζ

ζ(0)=ζ(T )

.

.

dτ1 dτ2 eip·ζ(τ1 ) e−ip·ζ(τ2 ) ζµ (τ1 )ζν (τ2 ) . (12.90)

0

4 Anti-periodic

functions defined over the interval [0, T ] can be written as X 1 τ ψn e2iπ(n+ 2 ) T . ψ(τ) ≡ n∈

❩

When restricted to these functions, the derivative operator ∂τ has no zero mode and is thus invertible.

12. W ORLDLINE FORMALISM

427

Figure 12.3: Feynman graphs contributing to the one-loop photon polarization tensor in scalar QED.

The path integration over ζ leads to Z 1 ¨ . .   RT Dζ(τ) e 0 dτ 4 ζζ eip·ζ(τ1 ) e−ip·ζ(τ2 ) ζµ (τ1 )ζν (τ2 ) ζ(0)=ζ(T )

h i 1 ¨ 1 , τ2 ) − pµ pν G ˙ 2 (τ1 , τ2 ) e−p2 G(τ1 ,τ2 ) gµν G(τ d/2 (4πT )   1 µν 2 µ ν ˙ 2 (τ1 , τ2 ) e−p2 G(τ1 ,τ2 ) , (12.91) = g p − p p G d/2 (4πT ) =

where in the third line we have anticipated an integration by parts on τ1 for the term ¨ We can already see that the polarization tensor is transverse. At this point, it is in G. convenient to use rescaled variables τi ≡ T ϑi . Moreover, thanks to the translation invariance of the integrand in τ1,2 and to its T -periodicity, we are free to set ϑ2 ≡ 0. Having done that, the propagator and its derivative become simple functions of ϑ1 , G(τ1 , τ2 ) = T ϑ1 (1 − ϑ1 ) , ˙ 1 , τ2 ) = 1 − 2 ϑ1 . G(τ

(12.92)

(We have already dropped the constant term in −T/6 from the propagator, since it does not contribute to amplitudes thanks to momentum conservation.) At this point, the polarization tensor reads  Z1 e2  µν 2 µ ν dϑ1 (1 − 2ϑ1 )2 g p − p p Πµν (p) = − scalar (4π)d/2 0 ∞ Z dT 2−d/2 −T (m2 +p2 ϑ1 (1−ϑ1 )) T e × T 0

=

−

 Z1 e2  µν 2 µ ν g p − p p dϑ1 (1 − 2ϑ1 )2 (4π)d/2 0  2 d/2−2 2 d × Γ (2 − 2 ) m + p ϑ1 (1 − ϑ1 ) .

(12.93)

One may check that this expression is identical to the one we would have obtained from the two Feynman diagrams of the figure 12.3, after introducing Feynman parameters and performing the integration over the loop momentum.

428

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Spinor QED : Let us now consider the same quantity in QED with a spin 1/2 fermion. Eq. (12.90) is replaced by Πµν spin 1/2 (p)

=

e2 2

∞ Z

dT −m2 T e T

0

×

Z



. 1 ¨ 1  RT Dζ(τ)Dψ(τ) e 0 dτ ( 4 ζζ− 2 ψ·ψ)

ζ(0)=ζ(T ) ψ(0)=−ψ(T )

ZT

dτ1 dτ2 eip·ζ(τ1 ) e−ip·ζ(τ2 )

0

.
× ζµ (τ1 ) + 2i ψµ (τ1 ) (ψ(τ1 ) · p) .
× ζν (τ2 ) − 2i ψν (τ2 ) (ψ(τ2 ) · p) .

(12.94)

The path integral for the term in ζ˙ µ ζ˙ ν is the same as in eq. (12.91), but we should now multiply the result by5 Z . RT 1   (12.95) Dψ(τ) e− 0 dτ 2 ψ·ψ = 4 . ψ(0)=−ψ(T )

For the terms involving the Grassmann variables, we have Z . RT 1   4 Dψ(τ) e− 0 dτ 2 ψ·ψ ψµ (τ1 ) (ψ(τ1 ) · p)ψν (τ2 ) (ψ(τ2 ) · p) ψ(0)=−ψ(T )

  = − S2 (τ1 , τ2 ) gµν p2 − pµ pν ,

(12.96)

and this result should be multiplied by (4πT )−d/2 to account for the integration over the variable ζ. Thus, we see here an example of the Bern-Kosower substitution rule: ˙ 2 (τ1 , τ2 ) by the spin 1/2 loop can be obtained from the scalar loop, by replacing G 2 2 ˙ G (τ1 , τ2 ) − S (τ1 , τ2 ) and by multiplying by an overall factor −2 (this comes from a −1/2 due to the different prefactors in the scalar and spin 1/2 one-loop effective actions, times the factor 4 from eq. (12.95)). In terms of the variables ϑ1,2 and after setting ϑ2 = 0, we have simply S(τ1 , τ2 ) = 1 ,

(12.97)

˙ 2 becomes and therefore, the factor (1 − 2 ϑ1 )2 from G (1 − 2 ϑ1 )2 − 1 = −4 ϑ1 (1 − ϑ1 ) .

(12.98)

5 This formula may be obtained by ζ function regularization. If we denote A ≡ −∂ (restricted to the τ subspace of anti-periodic functions) and λn = −2iπ(n + 21 ) its eigenvalues, the ζ function of this operator  2 P is ζA (s) ≡ n∈❩ λ−s = n . Since there are four variables ψµ , the value of the path integral is det A ′ (0)). On the other hand, we have ζ (s) = (iπ)−s (1 + eiπs )(1 − 2−s )ζ(s), where ζ(s) is exp(−2ζA A Riemann’s zeta function. This function can be expanded at small s, giving: ζA (s) = − ln(2) s + O(s2 ).

12. W ORLDLINE FORMALISM

429

Therefore, the worldline expression of the one-loop photon polarization tensor in spinor QED reads Πµν spin 1/2 (p) =

 Z1 8 e2  µν 2 µ ν g p − p p dϑ1 ϑ1 (1 − ϑ1 ) (4π)d/2 0  d/2−2 , × Γ (2 − d2 ) m2 + p2 ϑ1 (1 − ϑ1 )

(12.99)

that agrees with the expression obtained from Feynman graphs (only the first topology in the figure 12.3, with the scalar loop replaced by a spinor loop, contributes in this case). Remarkably, all the Dirac algebra usually involved in the calculation of fermion loops is completely avoided in the worldline formalism, since it is encapsulated into the Grassmann functional integration over ψµ .

430

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 13

Lattice field theory We have seen earlier that the running coupling in an SU(N) non-Abelian gauge theories decreases at large energy (provided the number of quark flavours is less than 11N/2). The counterpart of asymptotic freedom is that the coupling increases towards lower energies, precluding the use of perturbation theory to study phenomena in this regime. Among such properties is that of colour confinement, i.e. the fact that coloured states cannot exist as asymptotic states. Instead the quarks and gluons arrange themselves into colour neutral bound states, that can be mesons (e.g. pions, kaons) made of a quark and an antiquark or baryons (e.g. protons, neutrons) made of three quarks1 . A legitimate question would be to determine the mass spectrum of the asymptotic states of QCD from its Lagrangian. Since the perturbative expansion is not applicable for this type of problem, one would like to be able to attack it via some non-perturbative approach. By nonperturbative, we mean a method by which observables would directly be obtained to all orders in the coupling constant, without any expansion. One such method, known as lattice field theory, consists in discretizing space-time in order to evaluate numerically the path integral. The continuous space-time is replaced by a discrete grid of points, the simplest arrangement being a hyper-cubic lattice such as the one shown in the figure 13.1. The distance between nearest neighbor sites is called the lattice spacing, and usually denoted a. The lattice spacing, being the smallest distance that exists in this setup, therefore provides a natural ultraviolet regularization. Indeed, on a lattice of spacing a, the largest conjugate momentum is of order a−1 . Moreover, one usually uses periodic boundary conditions; if the lattice has N spacings in all directions, b ) = φ(x) for bosonic fields and φ(x + N µ b ) = −φ(x) for then we have φ(x + N µ µ is the displacement vector by one lattice spacing in the direction µ fermionic fields (b of spacetime). c sileG siocnarF

1 More

exotic bound states made of four (tetraquarks) or five (pentaquarks) have also been speculated, but the experimental evidence for these states is so far not fully conclusive. Likewise, there may exist bound states without valence quarks, the glueballs.

431

432

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 13.1: Discretization of Euclidean space-time on a hyper-cubic lattice (here shown in three dimensions).

13.1 Discretization of bosonic actions 13.1.1

Scalar field theory

As an illustration of some of the issues involved in the discretization of a quantum field theory, let us consider a simple scalar field theory with a local interaction in φ4 , whose action in continuous space-time is Z 

1 λ S = d4 x − φ(x) ∂µ ∂µ + m2 )φ(x) − φ4 (x) . (13.1) 2 4! A natural choice is to replace the integral over space-time by a discrete sum over the sites of the lattice, weighted by the volume a4 of the elementary cells of the lattice, Z X 4 a → d4 x . (13.2) x∈ lattice

a→0

Then we replace the continuous function φ(x) by a discrete set of real numbers that live on the lattice nodes. For simplicity, we keep denoting φ(x) the value of the field on the lattice site x. The discretization of the mass and interaction terms is trivial, but the discretization of the derivatives that appear in the D’Alembertian operator is not unique. Using only two nearest neighbors, one may define forward or backward finite differences, b ) − f(x) f(x + µ a b) f(x) − f(x − µ µ , ∇B f(x) ≡ a

∇µ f(x) ≡ F

(13.3)

433

13. L ATTICE FIELD THEORY

that both go to the continuum derivative in the limit a → 0. However, unlike the and ∇µ are not anti-adjoint. Instead, assuming periodic continuous derivative, ∇µ F B boundary conditions, we have X

x∈ lattice

   X  µ ∇ f(x) g(x) . f(x) ∇µ g(x) = − B F

(13.4)

x∈ lattice

In other words, ∇µ† = −∇µ . From this, we may construct a self-adjoint discrete F B second derivative as follows: ∇µ ∇µ f(x) = B F

b ) + f(x − µ b ) − 2 f(x) f(x + µ a2

→ f ′′ (x) .

a→0

(13.5)

(There is no summation on µ in the left hand side.) Thus, a self-adjoint discretization of the scalar Lagrangian leads to Slattice = a4

X

x∈ lattice

 1 λ − φ(x) gµν ∇µ ∇ν + m2 )φ(x) − φ4 (x) . (13.6) B F 2 4!

Let us make a few remarks concerning the errors introduced by the discretization. Firstly, the continuous spacetime symmetries (translation and rotation invariance) of the underlying theory are now reduced to the subgroup of the discrete symmetries of a cubic lattice. They are recovered in the limit a → 0. Another source of discrepancy between the continuum and discrete theories is the dispersion relation that relates the energy and momentum of an on-shell particle. In the continuum theory, this relation is of course E2 = p2 + m2 ,

(13.7)

where −p2 is an eigenvalue of the Laplacian. In order to find its counterpart with the above discretization, we must determine the spectrum of the finite difference operator ∇µ ∇µ . On a lattice with N sites and periodic boundary conditions, its eigenfunctions B F are given by kx

φk (x) ≡ e2iπ Na

with k ∈ ❩ ,

−N 2 ≤k≤

N 2

.

(13.8)

The associated eigenvalue is λk ≡


2πk 4 πk 2 cos − 1 = − 2 sin2 . 2 a N a N

(13.9)

Thus, the one dimensional discrete analogue of the continuum p2 + m2 is m2 +

4 πk sin2 . 2 a N

(13.10)

434

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

35

30

Figure 13.2: Discrepancy between the continuous (solid curve) and discrete (points) dispersion relations, on a one-dimensional lattice with N = 40.

25

E

20

15

10

5

0 -20

-15

-10

-5

0

5

10

15

20

k

As long as k ≪ N, this agrees quite well with the continuum dispersion relation, but the agreement is not good for larger values of k. This discrepancy is illustrated in the figure 13.2. This mismatch does not improve by increasing the number of lattice points: only the center of the Brillouin zone has a dispersion relation that agrees with the continuum one. In order to mitigate this problem, one should choose the parameters of the lattice in such a way that the physically relevant scales correspond to values of k for which the distortion of the dispersion curve is small. c sileG siocnarF

13.1.2

Gluons and Wilson action

Non-Abelian gauge theories pose an additional difficulty: since the local gauge invariance plays a central role in their properties, any attempt at discretizing gauge fields should preserve this symmetry. It turns out that there exists a discretization of the Yang-Mills action that goes to the continuum action in the limit where a → 0, and has an exact gauge invariance. The main ingredient in this construction is eq. (4.174), that relates the Wilson loop along a small square, []x;µν ≡ U†ν (x) U†µ (x + ν ^ ) Uν (x + µ ^ ) Uµ (x) ,

(13.11)

to the squared field strength. These elementary lattice Wilson loops are called plaquettes. In the fundamental representation of su(N), we have
g2 a4 µν 6 tr []x;µν = N − Fa (x)Fa µν (x) + O(a ) . 4

(13.12)

Note that, although the first two terms in the right hand side are real valued, the remainder (terms of order a6 and beyond) may be complex. Therefore, it is convenient to take the real part of the trace of the Wilson loop in order to construct a real valued

435

13. L ATTICE FIELD THEORY

discrete action. By summing this equation over all the lattice points x and all the pairs of distinct directions (µ, ν), we obtain a4

X 

x∈ lattice

=

N g2 |

 1 µν Fa (x)Fa µν (x) 4  X X 
N−1 tr Re []x;µν − 1 +O(a2 ) .

−

(13.13)

x∈ lattice (µ,ν)

{z Wilson action, denoted

1 S [U] g2 W

}

Note that the error term of order a6 becomes a term of order a2 after summation over the lattice sites, since the number of sites grows like a−4 if the volume is held fixed. Thus, the sum of the traces of the Wilson loops over all the elementary plaquettes of the lattice provides a discretization of the Yang-Mills action. In this discrete formulation, the natural variables are not the gauge potentials Aµ (x) themselves, but the Wilson lines Uµ (x) that live on the edges of the lattice, called link variables. In this notation, x is the starting point and µ the direction of the Wilson line, as illustrated in the left panel of figure 13.3. The Wilson line oriented in the −^ µ direction, i.e.

x+νˆ x

x+µ ˆ Uµ (x)

x

x+µ ˆ

Figure 13.3: Left: link variable. Right: plaquette on an elementary square of the lattice.

starting at the point x + µ ^ and ending at the point x, is simply the Hermitean conjugate of Uµ (x). Under a local gauge transformation, the link variables are changed as follows: Uµ (x)

→

Ω† (x + µ ^ ) Uµ (x) Ω(x) .

(13.14)

The plaquette variable, shown in the right panel of figure 13.3, can then be obtained by multiplying four link variables, as indicated by eq. (13.11), and its trace is obviously invariant under the transformation of eq. (13.14).

436

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

At this stage, the discrete analogue of the path integral that gives the expectation value of a gauge invariant operator reads, hOi =

ZY



NX X 
  . N−1 tr Re []x;µν − 1 dUµ (x) O U exp i 2 g x x,µ (µ,ν)

(13.15)

Since there exists a left- and right-invariant2 group measure dUµ (x), the left hand side of this formula is gauge invariant. Moreover, it goes to the expectation value of the continuum theory in the limit of zero lattice spacing.

13.1.3

Monte-Carlo sampling

Thanks to the discretization, the path integral of the original theory is replaced by an ordinary integral over each of the link variables Uµ (x), whose number is finite. A non-perturbative answer could be obtained if one were able to evaluate these integrals numerically. However, because of the prefactor i inside the exponential in eq. (13.15), the integrand is a strongly oscillating function, whose numerical evaluation is practically impossible except on lattices with a very small number of sites. In order to be amenable to a numerical calculation, this integral must be transformed into an Euclidean one, ZY

N X X 
   −1 N tr Re [] . hOiE = dUµ (x) O U exp x;µν −1 g2 x x,µ (µ,ν)

(13.16)

The exponential under the integral is now real-valued, and thus positive definite. Note that numerical quadratures such as Simpson’s rule, are not practical for this problem, given the huge number of dimensions of the integral to be evaluated. For instance, for the 8-dimensional Lie group SU(3), in 4 space-time dimensions, on a lattice with N4 points, this dimension is 8 × 4 × N4 . For N = 32, the path integral is thus transformed into a 225 -dimensional (225 ∼ 3.107 ) ordinary integral. Instead, one views the exponential of the Wilson action as a probability distribution (up to a normalization constant) for the link variables, that may be sampled by a Monte-Carlo algorithm (e.g. the Metropolis-Hastings algorithm) in order to estimate the integral. In this approach, as long as one is evaluating the expectation value of gauge invariant observables, it is not necessary to fix the gauge in lattice QCD calculations. 2 This

means that:

Z

dU f[U] =

Z

dU f[ΩU] =

Z

dU f[UΩ] .

Such a measure, known as the Haar measure, exists for compact Lie groups, like SU(N).

437

13. L ATTICE FIELD THEORY

Gauge fixing is necessary when calculating non-gauge invariant quantities, such as propagators for instance. The Landau gauge is the most commonly used, because the Landau gauge condition is realized at the extrema of a functional of the link variables, However, the comparison between gauge-fixed lattice calculations and analytical calculations is very delicate, because of the existence of Gribov copies (the problem stems from the fact that the two setups may not select the same Gribov copy). Although considering the Euclidean path-integral instead of the Minkowski one allows for a numerical evaluation by Monte-Carlo sampling, this leads to a serious limitation: only quantities that can be expressed as an Euclidean expectation value are directly calculable. Others could in principle be reached by an analytic continuation from imaginary to real time, but this turns out to be practically impossible numerically. For instance, the masses of hadrons are accessible to lattice QCD calculations (see the section 13.3 for an example), while scattering amplitudes cannot be calculated by this method. c sileG siocnarF

13.2 Fermions 13.2.1

Discretization of the Dirac action

Consider now the Dirac action, whose expression in continuum space reads Z   SD = d4 x ψ(x) i γµ Dµ − m ψ(x) . (13.17)

In the discretization, we assign a spinor ψ(x) to each site of the lattice. Under a gauge transformation Ω(x), these spinors transform in the same way as in the continuous theory, ψ(x) → Ω† (x) ψ(x) ,

ψ(x) → ψ(x) Ω(x) .

(13.18)

The main difficulty in defining a discrete covariant derivative that transforms approb ) transform differently priately under a gauge transformation is that ψ(x) and ψ(x ± µ when Ω(x) depends on space-time. This problem can be remedied by using a link variable between the point x and its neighbors. Like with the ordinary derivatives, one may define forward and backward discrete derivatives, b ) − ψ(x) U†µ (x)ψ(x + µ , a b )ψ(x − µ b) ψ(x) − Uµ (x − µ Dµ ψ(x) ≡ , B a Dµ ψ(x) ≡ F

(13.19)

that both transform like a spinor at the point x, and therefore are valid discretizations of a covariant derivative. However, none of these two operators is anti-adjoint, and

438

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

35

30

Figure 13.4: Discrepancy between the continuous (solid curve) and discrete (points) dispersion curves for fermions, on a onedimensional lattice with N = 40.

25

E

20

15

10

5

0 -20

-15

-10

-5

0

5

10

15

20

k

therefore they would not give
a Hermitean Lagrangian density. This may be achieved + Dµ by using instead 12 Dµ , which corresponds to a symmetric forward-backward F B difference
b ) − Uµ (x − µ b )ψ(x − µ b) 1 µ U†µ (x)ψ(x + µ DF + Dµ . ψ(x) = B 2 2a

13.2.2

(13.20)

Fermion doublers

Let us now study how the dispersion relation of fermions is modified by this discretization. This can easily be done in the vacuum, i.e. by setting all the link
variables to the identity. In this case, the eigenfunctions of the operator 21 Dµ + Dµ are F B ψk (x) = e2iπ

(k+1/2)x Na

with k ∈ ❩ ,

−N 2 ≤k≤

N 2

,

(13.21)

and the corresponding eigenvalue is λk =

2π (k + 1/2) i sin , a N

(13.22)

and the corresponding dispersion relation is E2 = |λk |2 +m2 . This dispersion relation is shown in the figure 13.4. Like in the bosonic case, the discrete dispersion relation agrees with the continuous one only for small enough k. However, the discrepancy at large k is now much more serious, because the discrete dispersion curve has another minimum at the edge of the Brillouin zone. This additional minimum indicates the existence of a second propagating mode of mass m. This spurious mode is called a fermion doubler. In d dimensions, the number of these fermionic modes is 2d , while our goal was to have only one. This problem is quite serious, because it affects all quantities that depend on the number of quark flavours. In particular, this is the case of the running of the coupling constant, whenever quark loops are included.

439

13. L ATTICE FIELD THEORY

35

30

25

20 E

Figure 13.5: Discrepancy between the continuous (solid curve) and discrete (points) dispersion curves in the fermionic case, on a one-dimensional lattice with N = 40, after inclusion of the Wilson term.

15

10

5

0 -20

-15

-10

-5

0

5

10

15

20

k

13.2.3

Wilson term

Various modifications of the discretized Dirac action have been proposed to remedy the problem of fermion doublers. One of these modifications, known as the Wilson term, consists in adding to the Lagrangian the following term (written here for the direction µ), h i 1 b ) + Uµ (x − µ b ) ψ(x − µ b ) − 2ψ(x) , ψ(x) U†µ (x) ψ(x + µ (13.23) − 2a

which is nothing but a D’Alembertian (or a Laplacian in the Euclidean theory) constructed with covariant derivatives. The corresponding operator in the continuum theory is
a ψ Dµ D µ ψ . (13.24) 2

Note that the denominator in eq. (13.23) has a single power of the lattice spacing a, hence the prefactor a in the previous equation. Therefore, this term goes to zero in the limit a → 0, and it should have no effect in the continuum limit. In the absence of gauge field (Uµ (x) ≡ 1), the functions of eq. (13.21) are still eigenfunctions after adding the Wilson term, but with modified eigenvalues, 2π (k + 1/2)  2π (k + 1/2) 1  i 1 − cos . (13.25) + λk = sin a N a N

Thus, the Wilson term does not modify the spectrum at small k, but lifts the spurious minimum that existed at the edge of the Brillouin zone, as shown in the figure 13.5. Roughly speaking, the Wilson term gives a mass of order a−1 to the fermion doublers, making them decouple from the rest of the degrees of freedom when a → 0. However, the Wilson term has an important drawback: there is no Dirac matrix γµ in eqs. (13.23) and (13.24) since the Lorentz indices are contracted directly between

440

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

the two covariant derivatives. Therefore, the Wilson term –like an ordinary mass term– breaks explicitly the chiral symmetry of the Dirac Lagrangian in the case of massless fermions. The fermion doublers are in fact intimately related to chiral symmetry. Without the Wilson term, lattice QCD with massless quarks has an exact chiral symmetry unbroken by the lattice regularization, and therefore there cannot be a chiral anomaly. In fact, this absence of anomaly is precisely due to a cancellation of anomalies among the multiple copies (the doublers) of the fermion modes. This argument is completely general and not specific to the Wilson term: any mechanism that lifts the degeneracy among the doublers will spoil the anomaly cancellation and thus break chiral symmetry. For this reason, the study of phenomena related to chiral symmetry is always delicate in lattice QCD.

13.2.4

Evaluation of the fermion path integral

The path integral representation for fermions uses anti-commuting Grassmann variables. However, such variables are not representable as ordinary numbers in a numerical implementation. To circumvent this difficulty, one exploits the fact that the Dirac action is quadratic in the fermion fields (this remains true after adding the Wilson term to remove the fermion doublers). Therefore, the path integral over the fermion fields can be done exactly. In addition to the fermion fields contained in the action, there may be ψ’s and ψ’s (in equal numbers) in the operator whose expectation value is being evaluated. The result of such a fermionic path integral is given by Z      DψDψ ψ(x1 )ψ(x2 ) eiSD [ψ,ψ] = S(x1 , x2 )×det i γµ Dµ −m ; (13.26)

/ − m between the points x1 where S(x1 , x2 ) is the inverse of the Dirac operator iD and x2 . When there is more than one ψψ pair in the operator, one must sum over all the ways of connecting the ψ’s and the ψ’s by the fermion propagators S(x, y). The same can be done in the lattice formulation. In this case, the Dirac operator is simply a (very large) matrix that depends on the configuration of link variables. Therefore one needs the inverse of this matrix, and its determinant. c sileG siocnarF

In eq. (13.26), the Dirac determinant provides closed quark loops, while the propagator S(x1 , x2 ) connects the external points of the operator under consideration. This observation, illustrated in the figure 13.6, clarifies the meaning of the quenched approximation, in which the determinant of the Dirac operator is replaced by 1. This approximation, motivated primarily by the computational difficulty of evaluating the Dirac determinant, was widely used in lattice QCD computations until advances in algorithms and computer hardware made it unnecessary. Note that, although quark loops are not included in the quenched approximation, gluon loops are present to their full extent. In contrast, lattice QCD calculations that include the Dirac determinant, and thus the effect of quark loops, are said to use dynamical fermions.

441

13. L ATTICE FIELD THEORY

Figure 13.6: Illustration of the two types of quark contributions. In dark: quark propagators (i.e. inverse of the Dirac operator) that connect the ψ’s and ψ’s in the operator being evaluated. In lighter color: quark loops coming from the determinant of the Dirac operator.

13.3 Hadron mass determination on the lattice  Let us consider a hadronic state h . Any operator O that carries same quantum

the numbers as this hadron leads to a non-zero matrix element h O 0 . The vacuum expectation value of the product of two O at different times 0 and T can be rewritten as follows, X   

† 0 O† (0) Ψn Ψn O(T ) 0 0 O (0)O(T ) 0 = n

=

X   0 O† (0) Ψn Ψn O(0) 0 e−Mn T n

=

X  2 Ψn O(0) 0 e−Mn T .

(13.27)

n

In the first equality, we have inserted a complete basis of eigenstates

of the QCD Hamiltonian, and the second equality follows from the fact that Ψn is an eigenstate of rest energy Mn (there is no factor i inside the exponential because of the Euclidean time used in lattice QCD). The sum in the last equality receives non-zero contributions from all the states Ψn that possess the quantum numbers carried by the operator O. However, taking the limit T → ∞ selects the one among these eigenstates that has the smallest mass. This observation can be turned into a method to determine hadron masses in lattice QCD: 1. Choose an operator O that has the quantum numbers of the hadron of interest.

h  The choice of the operator is not crucial, as long as the overlap h O 0 is not zero. However, eq. (13.27) suggests that a better result, i.e. less noisy with limited statistics, may be obtained by trying to maximize this overlap.

442

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 13.7: Hadron mass determination from lattice QCD. Open circles: masses used as input in order to set the lattice parameters. Filled circles: predictions of lattice QCD. Boxes: experimental values.

2. Evaluate the vacuum expectation value of O† (0)O(T ) by Monte-Carlo sampling, as a function of T . 3. Fit the large T tail of this expectation value. The slope of the exponential gives the mass of the lightest hadron that possesses these quantum numbers. The discretized QCD Lagrangian contains several dimensionful parameters: the lattice spacing a and the quark masses mf (one for each quark flavour), whose values need to be fixed before novel predictions can be made. One must choose (at least) an equal number of physical quantities that are known experimentally. Their computed values depend on a, mf , and one should adjust these parameters so that they match the experimental values. After this has been done, quantities computed in lattice QCD do not contain any free parameter anymore and are thus predictions. The figure 13.7 shows hadron masses calculated using lattice QCD.

13.4 Wilson loops and confinement 13.4.1

Strong coupling expansion

While perturbation theory is an expansion in powers of g2 , it is possible to use the lattice formulation of a Yang-Mills theory in order to perform an expansion in powers of the quantity β ≡ g−2 that appears as a prefactor in the Wilson action. This is called a strong coupling expansion, since it becomes exact in the limit of infinite coupling. c sileG siocnarF

This expansion produces integrals over the gauge group such as Z dU Ui1 j1 · · · Uin jn U†k1 l1 · · · U†km lm .

(13.28)

443

13. L ATTICE FIELD THEORY The simplest of these integrals, Z dU = 1

(13.29)

is simply a choice of normalization of the invariant group measure. From the unitarity of the group elements, one then obtains3 Z 1 dU Uij U†kl = δjk δil . (13.30) N In these integrals, the link variables on different edges of the lattice are independent variables, and there is a separate integral for each of them. This is completely general: integrals of the form (13.28) are non-zero only if the integrands contains an equal number of U’s and U† ’s, i.e. for n = m. Therefore, each link variable U that appears in such a group integral must be matched by a corresponding U† . For instance, the group integral of the Wilson loop defined on an elementary plaquette is zero, ZY   (13.31) dUµ (x) tr U†ν (x) U†µ (x + ν ^ ) Uν (x + µ ^ ) Uµ (x) = 0 , {z } | x,µ []x;µν

because the four link variables live on four distinct edges of the lattice. In contrast, the integral of the trace of a plaquette times the trace of the conjugate plaquette is non-zero: ZY    dUµ (x) tr []x;µν (13.32) tr []†x;µν = 1 . x,µ

Using these results, we can calculate to order β the expectation value of the trace of a plaquette:

tr []x;µν RQ ≡



  

P  −1 N tr Re []y;ρσ − 1 dUµ (x) tr []x;µν exp βN x,µ y;ρσ 

RQ P  −1 N tr Re []y;ρσ − 1 dUµ (x) exp βN x,µ

β = + O(β2 ) . 2

y;ρσ

(13.33)

Consider now the trace of a more general Wilson loop along a path γ (planar, to simplify the discussion). Each U and U† in the Wilson loop must be compensated by a link variable coming from the β expansion of the exponential of the Wilson action. The lowest order term in β corresponds to a minimal tiling of the Wilson

444

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 13.8: Tiling of a closed loop by elementary plaquettes.

loop by elementary plaquettes, as illustrated in the figure 13.8. The corresponding contribution is  Area (γ) β htr Wγ i = + ··· , (13.34) 2 where the dots are terms of higher order in β (that can be constructed from nonminimal tilings of the contour γ, such that all the U’s and U† ’s are still paired appropriately).

13.4.2

Heavy quark potential

Let us consider now a rectangular loop, with an extent R in the spatial direction 1 and an extent T in the Euclidean time direction 4. The previous result indicates that the expectation value of the trace of the corresponding Wilson loop has the following form, htr Wγ i ∼ e−σRT + · · · ,

(13.35)

where σ is a constant. Although it is gauge invariant, this expectation value is easier to interpret in an axial gauge where A4 ≡ 0. Indeed, in this gauge, the Wilson loop receives only contributions from gauge links along the spatial direction, as shown in the figure 13.9. Note that the remaining Wilson lines are precisely those that are needed to make a non-local gauge invariant operator with a quark at x = R and an antiquark at x = 0, Oqq (t) ≡ ψ(t, 0) W[0,R] ψ(t, R) ,

(13.36)

3 For SU(2), one may parameterize the group elements in the fundamental representation by U = 2 2 2 2 = 1, and the invariant group measure normalized according to θ0 + 2i θa ta f with θ0 + θ1 + θ2 + θp 3 p eq. (13.29) is dU = dθ1 dθ2 dθ3 /(π2 1 − θ2 ) (with θ0 = 1 − θ2 ). By using this measure and the Fierz identity satisfied by the generators ta f , an explicit calculation leads easily to eq. (13.30).

445

13. L ATTICE FIELD THEORY

R

Figure 13.9: Rectangular Wilson loop in the A4 ≡ 0 gauge.

T

=

t

x

where W[0,R] is a (spatial) Wilson line going from (t, R) to (t, 0). Consider now 

the vacuum expectation value 0 O†qq (0)Oqq (T ) 0 . In this expectation value, the fermionic path integral produces two quark propagators that connect the ψ’s to the ψ’s. However, in the limit of infinite quark mass, the quarks are static and their propagator is just a Wilson line in the temporal direction, that reduces to the identity in the A4 = 0 gauge (represented by the dotted lines in the figure 13.9). Thus, we have 

htr Wγ i ∝ lim 0 O†qq (0)Oqq (T ) 0 . (13.37) M→∞

By inserting a complete basis of eigenstates of the Hamiltonian in the right hand side of eq. (13.37) and by taking the limit T → ∞, we find a result dominated by the quark-antiquark state of lowest energy E0 ,   2

lim 0 O†qq (0)Oqq (T ) 0 = 0 O†qq (0) Ψ0 e−E0 T . (13.38) T →∞

Moreover, in the limit of large mass, the energy E0 of this state is dominated by the potential energy V(R) between the quark and the antiquark (the quark and antiquark are non-relativistic, and their kinetic energy behaves as P2 /2M → 0),  2 

† 0 Oqq (0)Oqq (T ) 0 = 0 O†qq (0) Ψ0 e−V(R) T . lim (13.39) M,T →∞

By comparing this result with that of the strong coupling expansion, eq. (13.35), we conclude that V(R) = σ R .

(13.40)

This linear potential indicates that the force between the quark and antiquark is constant at large distance, in sharp contrast with a Coulomb potential in electrodynamics. This is a consequence of the colour confinement property of QCD. c sileG siocnarF

446

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

13.5 Gauge fixing on the lattice Until now, we have described lattice field theory restricted to the evaluation of the expectation value of gauge invariant operators. One may legitimately argue that this is sufficient as far as the computation of physical observables is concerned. There are however some applications that involve gauge dependent quantities, for which a gauge fixing is necessary. This is the case for instance if one wishes to compare the behaviour of propagators or vertex functions evaluated non-perturbatively on the lattice and in perturbation theory. For practical reasons, it is convenient to consider gauge conditions that may be recasted into the problem of finding the extrema of a functional. This is the reason why the Landau gauge, i.e. the strict covariant gauge ∂µ Aµ = 0, is most often employed in these lattice studies. In the continuum theory, this condition is satisfied at the extrema of the following functional: Z 1 µa d4 x Aa (13.41) FLandau [A, Ω] ≡ Ωµ (x)AΩ (x) , 2 µ where Aµ Ω is the gauge transform of the field configuration A we aim at bringing to Landau gauge. Indeed, if we apply an infinitesimal gauge transformation to the field Aµ Ω , the corresponding variation of FLandau [A, Ω] is

δFLandau [A, Ω] = = =

Z
µa − d4 x DΩ µab θb AΩ Z
− d4 x ∂µ θa − gfcab AcΩµ θb Aµa Ω Z
d4 x θa ∂µ Aµa . Ω

(13.42)

Therefore, if Aµ Ω realizes an extremum of the functional, then this variation must be zero for all possible θa (x), which means that Aµ Ω obeys Landau gauge condition. The discrete analogue of the functional defined in eq. (13.41) reads   XX b) . FLandau [U, Ω] ≡ −2 a2 Re tr Ω(x)Uµ (x)Ω† (x + µ (13.43) x

µ

Finding extrema of such a functional is a rather straightforward task, for instance with the steepest descent algorithm. Due to the existence of Gribov copies, the gauge fixed field configuration is not defined uniquely by the gauge condition, which implies that this functional has more than one extremum corresponding to the various solutions of ∂µ Aµ Ω = 0 along the same gauge orbit (see the figure 13.10). A natural criterion to decide which extrema

447

13. L ATTICE FIELD THEORY

Aµ G(Aµ) = 0

gauge fixed Aµ

gauge orbit

Figure 13.10: Gauge orbit that intersects multiple times the gauge fixing manifold.

448

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

to take into account is to try and reproduce the perturbative Fadeev-Popov procedure. Let us recall here the starting point, which amounts to inserting under the path integral the left hand side of the following equation  a Z   δG µ a DΩ(x) δ[G (AΩ )] det =1, (13.44) δΩ Ga (Aµ )=0 Ω

a

where G (A) = 0 is the gauge fixing condition. However, two conditions must be met for this integral to be really equal to one: • Ga (AΩ ) has a unique zero along each gauge orbit, • The determinant is positive at this zero. However, it was shown by Gribov that the unicity condition is generically not satisfied: the gauge condition has multiple solutions, called Gribov copies. When these conditions are not satisfied, the inserted factor is not one, but instead ZFP

= =

 a  −1  δGa δG det det δΩ i δΩ i zeroes i   a  X δG . sign det δΩ i zeroes i | {z } X



(13.45)

≡ sign(i)

Thus, one may try to mimic the perturbative Fadeev-Popov procedure by the following definition of a gauge fixed operator on the lattice P sign(i) O[AΩi ] extrema P i , (13.46) O[A] ≡ Landau extrema i sign(i)

where the denominator follows from the requirement that gauge invariant operators should remain unaffected by the gauge fixing. However, it was shown by Neuberger that this definition is flawed, because the distribution of Gribov copies is such that both the numerator and the denominator are exactly zero. In order to see this, consider a gauge invariant observable O[U], and let us try to mimic closely the continuum BRST quantization, by introducing ghosts and antighosts, the BRST variation B of the antighost (B ≡ QBRST χ, QBRST B = 0) and a gauge fixing parameter ξ. By doing so, the expectation value of the observable would read Z P P 1 

  − 1 S [U]− 2ξ x χG x BB+QBRST O = Z−1 D(U, χ, χ, B) O[U] e g2 W Z P P 1  − 1 S [U]− 2ξ  x χG x BB+QBRST . (13.47) Z ≡ D(U, χ, χ, B) e g2 W

449

13. L ATTICE FIELD THEORY

(Here, we are assuming a completely generic gauge fixing function G(U).) Note that with a compact gauge group and a finite lattice, all the integrals involved in this formula are finite. Consider now the following quantity: Z P P 1   − 1 S [U]− 2ξ x χG x BB+tQBRST FO (t) ≡ D(U, χ, χ, B) O[U] e g2 W . (13.48)

 The numerator in the gauge fixed definition of O is nothing but FO (1). The derivative of this function is given by dFO dt

=

=

Z



i X h D(U, χ, χ, B) QBRST χG x

−

× O[U] e |

1 g2

1 SW [U]− 2ξ

P

x

BB+tQ

BRST

{z BRST invariant Z i

h X   χG O[U] D(U, χ, χ, B) QBRST

P

×e

1 g2

1 SW [U]− 2ξ

P

x BB+tQ

BRST

P

x

χG

}

x

−

x

χG



=0.

(13.49)

In the last equality, we have used the fact that the integral of a total BRST variation is zero. Thus, we have Z P 1   − 1 S [U]− 2ξ x BB = 0 . (13.50) FO (1) = FO (0) = D(U, χ, χ, B) O[U] e g2 W

This time, the zero follows from the fact that the integrand does not depend on χ or χ, hence the integrals over the ghost and antighost are equal to zero.

The same reasoning applies to the denominator Z in the gauge-fixed definition of O , hence we have an undefined ratio4 :

 O

=

gauge fixed

0 . 0

(13.51)

If we interpret this result in the light of eq. (13.46), we see that these zeroes result from an even number of Gribov copies with alternating signs for the determinant of the Fadeev-Popov operator. One may view this issue as a fundamental obstruction for a non-perturbative definition of gauge fixing by the Fadeev-Popov procedure. Because of this problem, the practical lattice definition of the Landau gauge fixing is simply to pick one of the extrema of the functional (13.43), without any special selection rule. One should be aware of this procedure when comparing with perturbative results, since it is a priori not guaranteed that the solutions of the gauge condition used in the perturbative and in the non-perturbative calculations are the same. c sileG siocnarF

4 The

same conclusion holds if the operator O is not gauge invariant, but simply BRST invariant.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

13.6 Lattice worldline formalism 13.6.1

Discrete analogue of the heat kernel

In the chapter 12, we have exposed the worldline representation for a quantum field theory in a continuous spacetime. However, a similar representation is also possible for propagators in a field theory defined on a discrete spacetime. For the sake of simplicity, let us first consider first a free scalar field theory, defined on a cubic lattice instead of a continuous spacetime. As we have seen earlier in this chapter, the second derivatives ∂µ ∂µ that appear in the inverse propagator are replaced by centered finite differences, and with an Euclidean metric we have ( + m2 ) φ(x) = m2 φ(x) +

X 2 φ(x) − φ(x + µ b ) − φ(x − µ b) a2

µ

. (13.52)

We could in principle introduce a continuous fictitious time T as in eq. (12.3) in order to write a heat-kernel representation of this finite difference operator. But it turns out to be more convenient to introduce a discrete variable here as well, based on the identity A−1 =

∞ X

(1 − A)n .

(13.53)

n=0

13.6.2

Random walks on a cubic lattice

This formula should be applied to a dimensionless operator, for instance a2 ( + m2 ) instead of  +
m2 itself. Since the discrete operator in eq. (13.52) contains a term m2 + 2da−2 φ(x) that acts as the identity, it is in fact more convenient to choose A = (m2 + 2da−2 )−1 ( + m2 ), in order to make 1 − A simpler. Therefore, we may write  + m2 )−1 =

lattice

where

❉

n ∞  X 1 , m2 + 2da−2 2d + m2 a2

(13.54)

n=0

❉ is the discrete operator defined as follows  X ❉ φ(x) ≡ φ(x + µb) + φ(x − µb) .

(13.55)

µ

(In d dimensions, there are 2d terms in the sum of the right hand side.) The operator /(2d + m2 a2 ) realizes one hop from a lattice site to one of its nearest neighbors, with a probability (2d + m2 a2 )−1 for a jump in any given direction. Raised to the

❉

451

13. L ATTICE FIELD THEORY

Figure 13.11: Worldlines on a cubic lattice. The points x and x ′ are materialized by the two little balls, and we have represented three different paths on the lattice connecting these two points.

power n, we get an operator that performs n successive jumps. The probability of a given sequence of n hops is (2d + m2 a2 )−n , and there are (2d)n such sequences, hence a total probability (1 + m2 a2 /2d)−n . This is equal to unity in a massless theory, but suppressed exponentially at large n with a non-zero mass (this observation is the discrete analogue of the fact that long worldlines are suppressed exponentially by a mass in the continuous case). The propagator evaluated between the sites x and x ′ is proportional to the total probability to connect these two sites by a sequence of jumps, regardless of its length (because of the sum on n): 

 + m2 )−1



=

xx ′ lattice

∞ X 1 1 m2 + 2da−2 (2d + m2 a2 )n n=0

X

γ∈Pn

1 , (13.56)

(x,x ′ )

where Pn (x, x ′ ) is the set of all paths of length n drawn on the edges of the lattice that connect x to x ′ (see the figure 13.11). Therefore, the second sum merely counts the number of such paths. This number has an upper bound of (2d)n , which implies trivially the convergence of the sum on n in the massive case.

13.6.3

Scalar electrodynamics

Let us now consider a complex scalar field in a background Abelian gauge field. The D’Alembertian is replaced by the square of the covariant derivative. In order to maintain an exact gauge symmetry in the discrete lattice formulation, the gauge field is represented by link variables Uµ (x) defined on each edge of the lattice, and the

452

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

forward and backward discrete covariant derivatives read b ) − φ(x) U†µ (x)φ(x + µ , a b )φ(x − µ b) φ(x) − Uµ (x − µ . Dµ φ(x) ≡ B a

Dµ φ(x) ≡ F

(13.57)

In order to evaluate the scalar propagator in this gauge background, we can reproduce the derivation of the previous subsection, and the only change will arise in the definition of the operator , whose action now reads

❉

❉ φ(x) ≡

X µ

 b ) + Uµ (x − µ b )φ(x − µ b) . U†µ (x)φ(x + µ

(13.58)

Note that the right hand side transforms like a scalar field at the point x under a gauge transformation. The consequence of this modification is that the jumps performed by the operator are now weighted by U(1) phases corresponding to the link variables that appear in the right hand side of eq. (13.58). The lattice worldline representation of the dressed scalar propagator is

❉



D2 + m2 )−1



xx ′

=

lattice

∞ X 1 1 m2 + 2da−2 (2d + m2 a2 )n

X

Wγ [A] ,

γ∈Pn (x,x ′ )

n=0

(13.59)

where Wγ [A] is the product of all the phases collected along the path γ, which is nothing but the Wilson line defined on this path. This expression transforms as expected for a dressed scalar propagator, thanks to the properties of Wilson lines. A representation similar to eq. (13.59) is also possible for the one-loop quantum effective action in the gauge background, that can be obtained by first noting that ln(A) = ln(1 − (1 − A)) = −

∞ X 1 (1 − A)n . n

(13.60)

n=1

The derivation mimics that of the propagator, and we finally obtain Γ [A] =

lattice

−

∞ XX 1 a4 2 −2 m + 2da 2n(2d + m2 a2 )2n x n=1

X

Wγ [A] . (13.61)

γ∈P2n (x,x)

Note that now the paths γ involved in the sum are the closed paths starting and ending at the point x, which is why only even values of the length are allowed.

453

13. L ATTICE FIELD THEORY

13.6.4

Combinatorics of loop areas on a planar square lattice

In two dimensions, the representation of eq. (13.59) may be used in order to study the properties of charged particles on an atomic lattice, under the influence of an external electromagnetic field. In particular, when this field is purely magnetic and transverse to the plane of the lattice (i.e. the field strength F12 is non-zero), this model is related to the quantum Hall effect. c sileG siocnarF

This relationship may also be exploited in order to derive explicit formulas for the moments of the distribution of areas of random closed loops on a cubic lattice. For this application, it is interesting to consider a 2-dimensional anisotropic lattice, with lattice spacings a1,2 in the two directions. On this lattice, consider the propagator of a massless scalar at equal points in the presence of a transverse magnetic field. It is straightforward to generalize the previous derivation to the anisotropic case, and one obtains the following expression for the propagator G(x, y) =

 n1  n2 ∞ h2 a2 X h1 4 4 4 n1,2 =0

X

Wγ [A] ,

(13.62)

γ∈Pn1 ,n2 (x,y)

where Pn1 ,n2 (x, y) is the set of paths drawn on the edges of the lattice, that connect x to y, and contain n1 jumps in the first direction and n2 jumps in the second direction. In this formula, we have also defined 1 2 1 ≡ 2+ 2 2 a a1 a2

,

h1,2 ≡

a2 . a21,2

(13.63)

If we specialize to a closed path x = y = 0 and to a transverse magnetic field B, the paths γ in the right hand side are closed loops, and we have Wclosed γ [A] = eiΦ Area (γ) ,

(13.64)

where we denote Φ ≡ Ba1 a2 the magnetic flux through an elementary plaquette of the lattice, and where Area (γ) is the area enclosed by the loop γ. Note that this is an algebraic area, whose sign depends on the orientation of the loop, and that accounts for the winding number of the loop. Expanding the exponential, we have G(0, 0) =

 2n1 2n2 X ∞ ∞ a2 X h2 (iΦ)2l h1 4 4 4 (2l)! n1,2 =0

l=0

X


2l Area (γ) .

γ∈P2n1 ,2n2 (0,0)

(13.65)

(Because the area is algebraic, the odd moments are all zero.) In this formula, we have made explicit the fact that a closed path must have an even number of hops in each direction. On the other hand, the propagator can be determined perturbatively order

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

by order in Φ, since this corresponds to a weak field expansion. This calculation requires that one chooses a gauge5 . A convenient choice is provided by the following link variables U1 (x) ≡ 1 ,

U2 (x) ≡ eiΦ i1 ,

(13.66)

P b ). By performwhere i1 the integer that labels the first coordinate (x ≡ µ iµ µ ing the expansion of G(0, 0) and identifying the coefficient of the term of order 1 2n2 Φ2l h2n h2 on both sides of eq. (13.65), one can show that 1 X

2l

(Area (γ))

=

γ∈P2n1 ,2n2 (0,0)

(2(n1 +n2 ))! P2l (n1 , n2 ) , n1 !2 n2 !2

(13.67)

where P2l is a symmetric polynomial in n1 , n2 of degree 2l. Note that the combinatorial factor (2(n1 +n2 ))!/(n1 !2 n2 !2 ) is the number of loops in P2n1 ,2n2 (0, 0). This expansion also provides a semi-explicit form of the polynomial P2l , and the evaluation of the first two terms gives, n1 n2 , 3
n1 n2 7n1 n2 −(n1 +n2 ) P4 (n1 , n2 ) = , 15

P2 (n1 , n2 ) =

(13.68)

All these polynomials satisfy two simple identities: P2l (n1 , 0) = P2l (0, n2 ) = 0 ,

P2l (1, 1) =

1 . 3

(13.69)

The first one is a consequence of the fact that if n1 or n2 is zero, then all the closed paths one can construct have a vanishing area. The second one follows from the fact that for n1 = n2 = 1, all the closed paths have area −1, 0 or +1, and therefore contribute equally to all the even moments. By summing the above results over all n1 + n2 = n, we obtain the moments of the algebraic area of closed loops of length n, with no restriction on the respective number of hops in each direction. Quite generally, these moments can be written as a prefactor (2n)!2 /n!4 (which is the number of closed loops of length 2n on a 2-dimensional lattice) multiplied by a rational fraction in n of degree 2l. For instance, 5 The product of the link variables along a closed loop does not depend on the choice of the gauge, as can be seen from the following gauge transformation formulas for the link variables

b) . Uµ (x) → Ω(x) Uµ (x) Ω∗ (x + µ

455

13. L ATTICE FIELD THEORY the first two moments are X

2

(Area (γ)) =

γ∈P2n (0,0)

X

4

(Area (γ)) =

γ∈P2n (0,0)

(2n)!2 n2 (n − 1) , n!4 6(2n − 1) (2n)!2 n3 (n−1)(7n2 − 18n + 13) . n!4 60(2n − 1)(2n − 3)

(13.70)

456

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 14

Quantum field theory at finite temperature Historically, the main area of developments and applications of Quantum Field Theory has been high energy particle physics. This corresponds to situations where the background is the vacuum, only perturbed by the presence of a few excitations whose interactions one aims at studying. Consequently, most of the QFT tools we have encountered so far are adequate for calculating transition amplitudes between pure states that contain only a few particles. However, there are interesting physical problems that depart from this simple situation. For instance, in the early universe, particles are believed to be in thermal equilibrium and form a hot and dense plasma. The typical energy of a particle in this thermal bath is of the order of the temperature1 , which implies that this surrounding medium may have an influence on all processes whose energy scale is comparable or lower. As a consequence, these problems contain some element of many body physics that was not present in applications of QFT to scattering reactions. Another class of problems where many body effects are important is condensed matter physics. When studied at some sufficiently large distance scale, where the atomic discreteness is no longer important, these problems may be described in terms of (non relativistic) quantum fields where collective effects are usually important.

14.1 Canonical thermal ensemble Usually, the system one would like to study is a little part of a much larger system (this is quite obvious in the case of the early universe, but is also generally true in 1 In this chapter, we extend the natural system of units we have used so far to also set the Boltzmann constant kB = 1. Therefore, the temperature has the dimension of an energy.

457

458

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

condensed matter physics). Thus, its energy and other conserved quantities are not fixed. Instead, they fluctuate due to exchanges with the surroundings, that play the role of a thermal reservoir. The appropriate statistical ensemble for describing this situation is the (grand) canonical ensemble, in which the system is described by the following density operator

 ρ ≡ exp − β H , (14.1)

where β = T −1 is the inverse temperature and H is the Hamiltonian. Given an operator O, one is usually interested in calculating its expectation value in the above statistical ensemble,

 Tr (ρ O) . O ≡ Tr (ρ)

(14.2)

 Let us span the Hilbert space by states n that are eigenstates of H,   H n = En n .

In terms of these states, the trace of ρ O can be represented as follows
X −β En  Tr ρ O = n O n . e

(14.3)

(14.4)

n

From this representation, it is easy to see that the zero temperature limit selects the state of lowest energy, i.e. the ground state of the Hamiltonian. Assuming that this state is non-degenerate, this corresponds to a vacuum expectation value:
 lim Tr ρ O = 0 O 0 . (14.5) T →0

In this sense, eq. (14.2) should be viewed as an extension of the formalism we already know, rather than something entirely different. In this chapter, we discuss various aspects of these thermal averages, starting with the necessary extensions to the formalism in order to perform their perturbative calculation. c sileG siocnarF

14.2 Finite-T perturbation theory 14.2.1

Naive approach

The extension of ordinary perturbation theory to calculate expectation values such as (14.2) is usually called Quantum Field Theory at finite temperature. A first approach for evaluating such an expectation value could be to use the representation of the trace

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

459

provided by eq. (14.4), and a similar formula for the denominator, which would fall back to the perturbative rules we already know (since the temperature and chemical potential appear only in the form of numerical prefactors). however a peculiarity

Note of the matrix elements that appear in eq. (14.4): n and n are identical states since they come from a trace (they are both in-states, since ρ defines the initial state of the system). This is a bit different from the transition amplitudes that enter in scattering cross-sections, where the matrix elements are evaluated between an in-state and an out-state. The perturbative rules to compute these in-in expectation values are provided by the Schwinger-Keldysh formalism introduced in the section 1.16.5. A difficulty with this naive approach is that the number of states that contribute significantly to the sum in eq. (14.4) is large at high temperature, especially when the temperature is large compared to the masses of the fields (and even more so with massless particles  like photons). In fact, it is possible to encapsulate the sum over the eigenstates n and the canonical weight of these states exp(−β En ) directly into the Schwinger-Keldysh rules, by a modest modification of its propagators.

14.2.2

Thermal time contour

To mimic closely the derivation of the Feynman rules at zero temperature, let us consider an observable made of the time-ordered product of elementary fields: O ≡ T φ(x1 ) · · · φ(xn ) .

(14.6)

Each Heisenberg representation field φ can be related to the corresponding field in the interaction representation by φ(x) = U(ti , x0 ) φin (x) U(x0 , ti ) ,

(14.7)

where ti is the time at which the system is prepared in equilibrium, and U is the time evolution operator defined by: U(t2 , t1 ) ≡ T exp i

Z t2

dx0 d3 x LI (φin (x)) ,

(14.8)

t1

with LI the interaction term in the Lagrangian. Thanks to eq. (14.7), we remove all the interactions from the field, and relegate them into the evolution operator where they can easily be Taylor expanded. In the canonical ensemble at non-zero temperature, there is another source of dependence on the interactions, hidden in the Hamiltonian inside the density operator. Indeed, for the system to be in statistical equilibrium, the canonical density operator should be defined with the same Hamiltonian as the one that drives the time evolution,

460

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

i.e. a Hamiltonian that also contains the interactions of the system2 . If we decompose the full Hamiltonian as H ≡ H0 + HI , we have e−βH = e−βH0 T exp i |

Z −ti −iβ −ti

dx0 d3 x LI (φin (x)) . {z }

(14.9)

U(−∞−iβ,−∞)

(This formula in fact does not depend on ti ). It can be proven by noticing that right and left hand sides are equal for β = 0, and by checking that their derivatives with respect to β are also equal (for this, we use the fact that the derivative of the time evolution with respect to its final time is known). From the previous formulas, we can write e−βH T φ(x1 ) · · · φ(xn ) = −βH0

=e

P φin (x1 ) · · · φin (xn ) exp i

Z

dx0 d3 x LI (φin (x)) , C

(14.10)

where the symbol P indicates a path ordering, and where the time integration contour is C = [ti , +∞] ∪ [+∞, ti ] ∪ [ti , ti − iβ]:

ti C =

.

(14.11)

t i − iβ In this contour, ti is the time at which the system is prepared in thermal equilibrium. As we shall see shortly, all observables are independent of this time, which physically means that a system in equilibrium has no memory of when it was put in equilibrium. Note also that in eq. (14.10), the times x01 , · · · , x0n are on the upper branch of the contour (but this constraint can be relaxed shortly).

14.2.3

Generating functional

The time contour (14.11) is very similar to the contour of the figure 1.4, with the addition of a vertical part that captures the interactions hidden in the density operator. Since we had to extend the real time axis into the contour C, it is natural to extend 2 An alternative point of view is to decide that ρ is the density operator of the system at x0 = −∞. There, we may turn off adiabatically the interactions, and therefore use only the free Hamiltonian inside ρ. In this section, we derive the formalism for an initial equilibrium state specified at a finite time x0 = ti .

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

461

also the observable of eq. (14.6) to allow the field operators to be located anywhere on C, with a path ordering instead of a time ordering, O ≡ P φ(x1 ) · · · φ(xn ) .

(14.12)

The expectation values of these operators can be encapsulated in the following generating functional, 

R  Tr ρ P exp i C d4 x j(x)φ(x)
Z[j] ≡ , (14.13) Tr ρ

where the fictitious source j(x) also lives on the contour C. In order to bring this generating functional to a useful form, we can follow very closely the derivation of the section 1.6.2, by first pulling out a factor that contains the interactions, and by rearranging the ordering of the free factor with two successive applications of the Baker-Campbell-Hausdorff formula. This leads to   Z

1Z  δ 4 exp − d4 xd4 y j(x) j(y) G0 (x, y) , Z[j] = exp i d x LI iδj(x) 2 C C (14.14) where the free propagator G0 (x, y), defined on the contour C, is given by   Tr e−β H0 P φin (x)φin (y)
G0 (x, y) ≡ . Tr e−β H0

14.2.4

(14.15)

Expression of the free propagator

In order to calculate the free propagator, we need the free Hamiltonian expressed in terms of creation and annihilation operators3 , Z d3 p H0 = Ep a†p,in ap,in , (14.16) (2π)3 2Ep and the canonical commutation relation of the latter:   ap,in , a†p ′ ,in = (2π)3 2Ep δ(p − p ′ ) .

From this, we get  −βH  0 e , ap,in
Tr e−βH0 ap,in
Tr e−βH0 a†p,in ap ′ ,in
Tr e−βH0

=

(14.17)


e−βH0 1 − e−βEp ap,in

=

0

=

(2π)3 2Ep nB (Ep ) δ(p − p′ ) ,

(14.18)

3 We can drop the zero point energy here. It would simply multiply the density operator by a constant factor, that would be canceled since all expectation values are normalized by the factor 1/Tr (ρ).

462

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where nB (E) is the Bose-Einstein distribution: nB (E) ≡

1 . −1

(14.19)

eβE

This leads to the following formula for the free propagator: Z
d3 p h θc (x0 − y0 ) + nB (Ep ) e−ip·(x−y) G0 (x, y) = 3 (2π) 2Ep i
+ θc (y0 − x0 ) + nB (Ep ) e+ip·(x−y) ,

(14.20)

where θc generalizes the step function to the contour C (i.e. θc (x0 − y0 ) is non-zero if x0 is posterior to y0 according to the contour ordering). This expression of the propagator generalizes to a non-zero temperature the formula (1.119) (the BoseEinstein distribution goes to zero when T → 0). Let us postpone a bit the calculation of the propagator in momentum space. For now, we just note the following rules for the perturbative expansion in coordinate space: 1. Draw all the graphs (with vertices corresponding to the interactions of the theory under consideration) that connect the n points of the observable. Graphs containing disconnected subgraphs should be ignored. Each graph should be weighted by its symmetry factor. c sileG siocnarF

2. Each line of a graph brings a free propagator G0 (x, y). 3. Each vertex brings a factor −iλ. The space-time coordinate of this vertex is integrated out, but the time integration runs over the contour C. Thus, the only differences with the zero temperature Feynman rules are the explicit form of the free propagator, and the fact the time integrations are over the contour C instead of the real axis.

14.2.5

Kubo-Martin-Schwinger symmetry

The canonical density operator exp(−βH) can be viewed as an evolution operator for an imaginary time shift, which implies the following formal identity e−βH φ(x0 −iβ, x) eβH = φ(x0 , x) . Let us now consider the following correlator
G(ti , · · · ) ≡ Tr e−βH P φ(ti , x) · · · ,

(14.21)

(14.22)

463

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

that contains a field whose time argument is the initial time ti (the other fields it contains need not be specified in this discussion). Since ti is the “smallest” time on the contour C, the field operator that carries it is placed to the rightmost position by the path ordering. Thus, we have  
G(ti , · · · ) = Tr e−βH P · · · φ(ti , x) , (14.23)

where the path ordering now applies only to the remaining (unwritten) fields. Using the cyclic invariance of the trace and eq. (14.21), we then get  
G(ti , · · · ) = Tr e−βH φ(ti − iβ, x) P · · ·
= Tr e−βH P φ(ti − iβ, x) · · · =

G(ti − iβ, · · · ) ,

(14.24)

where in the second line we have used the fact that ti − iβ is the “latest” time on the contour C in order to put back the operator carrying it inside the path ordering. This equality is one of the forms of the Kubo-Martin-Schwinger (KMS) symmetry: all bosonic path-ordered correlators take identical values at the two endpoints of the contour C. Note that, although we have singled out the first field in the correlator, this identity applies equally to all the fields. The KMS symmetry is very closely tied to the fact that the system is in thermal equilibrium, since it is satisfied only when the density operator is the canonical equilibrium one. One of its consequences is that all the equilibrium correlation functions are independent of the initial time ti . In order to prove this assertion, let us first note that the free propagator satisfies the KMS symmetry, and does not contain ti explicitly. A generic Feynman graph leads to time integrations that have the following structure: Z G(x1 , · · · , xn ) = dy01 · · · dy0p F(y01 , · · · , y0p | x1 , · · · , xn ) . (14.25) C

(We assume that the integrals over the positions at every vertex have already been performed.) Since the free propagator does not depend on ti , the derivative of the integral with respect to ti comes only from the endpoints of the integration contour, and we can write ∂G(x1 , · · · , xn ) ∂ti

=

p Z Y X i=1 C j6=i

=

0.

h dy0j F(· · · , y0i = ti , · · · | x1 , · · · , xn )

i −F(· · · , y0i = ti − iβ, · · · | x1 , · · · , xn )

(14.26)

The vanishing result follows from the fact that the bracket in the integrand is zero, since it is built from objects that obey the KMS symmetry. The independence with

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

respect to ti merely reflects the fact that, in a system in thermal equilibrium, no measurement can tell at what time the system was prepared in equilibrium. From the analyticity properties of the integrand, the result of the integrations in eq. (14.25) is in fact invariant under all the deformations of the contour C that preserve the spacing −iβ between its endpoints.

14.2.6

Conserved charges

Until now, we have considered only the simplest case of a boson field coupled to a thermal bath. Although energy is conserved, the system under consideration may exchange energy with the environment, which translates into the canonical density operator exp(−β H). Let us now consider a Hermitean operator Q that commutes with the Hamiltonian, i.e. that corresponds to a conserved quantity. A field φ is said to carry a charge q if it obeys the following commutation relation:   Q, φ(x) = −q φ(x) . (14.27) Note that if φ is a real field, then q = −q∗ . Therefore, in order to have a non-zero real valued charge, the field should be complex.

When there are additional conserved quantities such as Q, their conservation constrains in a similar fashion how they may be exchanged with the heat bath. The canonical equilibrium ensemble must be generalized into the grand canonical ensemble, in which the density operator of the subsystem is given by


 ρ ≡ exp − β H + µ Q , (14.28)

where µ is the chemical potential associated to the charge Q. Although we have introduced a single such charge, there could be any number of them, each accompanied by its chemical potential. A first consequence of this generalization is that the KMS symmetry is modified by the conserved charge. Now it reads: G(ti , · · · ) = eβµq G(ti − iβ, · · · ) ,

(14.29)

where q is the charge carried by the field on which the identity applies. Thus, the values of correlation functions at the endpoints are equal up to a twist factor that depends on the chemical potential. c sileG siocnarF

The simplest field that can carry a non trivial charge is a complex scalar field. In the interaction picture, it can be decomposed as follows on a basis of creation and annihilation operators: Z h i d3 p ap,in e−ip·x + b†p,in e+ip·x . φin (x) = 3 (2π) 2Ep

465

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

(This field requires two sets {ap,in , bp,in } of such operators, because it describes a particle which is distinct from its anti-particle.) With this field, it is possible to construct a theory that has a global U(1) symmetry, corresponding to the conservation of the following charge Z

Q≡

 d3 p b†p,in bp,in − a†p,in ap,in . 3 (2π) 2Ep

(14.30)

It is then easy to obtain the following grand-canonical averages: Tr eβ(H0 +µQ) a†p,in ap ′ ,in Tr eβ(H0 +µQ) b†p,in bp ′ ,in







=

(2π)3 2Ep nB (Ep − µq) δ(p − p′ )

=

(2π)3 2Ep nB (Ep + µq) δ(p − p′ ) , (14.31)

and finally obtain the free propagator for a complex scalar carrying the charge q: G0 (x, y) =

Z

d3 p h (θc (x0 − y0 ) + nB (Ep − µq)) e−ip·(x−y) (2π)3 2Ep

i +(θc (y0 − x0 ) + nB (Ep + µq)) e+ip·(x−y) . (14.32)

14.2.7

Fermions

Consider now spin 1/2 fermions, whose interaction picture representation reads ψin (x) =

XZ

s=±



d3 p a†sp,in vs (p)e+ip·x +bsp,in us (p)e+ip·x , (14.33) 3 (2π) 2Ep

where the creation and annihilation operators obey canonical anticommutation relations (see eqs. (1.215)). Because they are anticommuting fields, a minus sign appears in the derivation of the KMS identity: G(ti , · · · ) = −eβµq G(ti − iβ, · · · ) .

(14.34)

Moreover, the eqs. (14.31) are modified into Tr eβ(H0 +µQ) a†p,in ap ′ ,in Tr eβ(H0 +µQ) b†p,in bp ′ ,in







=

(2π)3 2Ep nF (Ep − µq) δ(p − p′ )

=

(2π)3 2Ep nF (Ep + µq) δ(p − p′ ) , (14.35)

466

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where nF (E) is the Fermi-Dirac distribution, nF (E) ≡

1 eβ E

+1

,

(14.36)

and the free propagator reads 0

S (x, y) =

Z


d3 p h / + +m) θc (x0 −y0 )−nF (Ep −µq) e−ip·(x−y) (p (2π)3 2Ep i
/ − +m) θc (y0 −x0 )−nF (Ep +µq) e+ip·(x−y) , +(p

(14.37)

/ ± ≡ ±Ep γ0 − p · γ. with the notation p

14.2.8

Examples of physical observables

Thermodynamical quantities : A central quantity that encapsulates the thermodynamical properties of a system is its partition function, defined in the canonical ensemble as
Z ≡ Tr e−βH . (14.38)

In perturbation theory, the logarithm of Z is obtained as the sum of all the connected vacuum graphs at finite temperature. For instance, for a scalar field, its perturbative expansion starts with the following diagrams:

From Z, one may access various thermodynamical quantities as follows: Energy : Entropy : Free energy :

∂Z , ∂β S = βE + ln(Z) , 1 F = E − TS = − ln(Z) . β E=−

(14.39)

These quantities encode the bulk properties of the system, such as its equation of state or the existence of phase transitions.

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

467

Production rates of weakly coupled particles : In a system at high temperature, it is sometimes interesting to calculate the production rate of a given species of particles. Firstly, note that this quantity is not interesting for particles that are in thermal equilibrium, since by definition they are produced and destroyed in equal amounts, so that their net production rate is zero. The real interest arises for weakly coupled particles that are not in thermal equilibrium with the bulk of the system. For instance, in a hot plasma of quarks and gluons interacting via the strong nuclear force, photons are also produced. However, since they interact only electromagnetically, they may not be thermalized. This is the case for instance when the system size is small compared to the mean free path of photons (i.e. the average distance between two interactions of a photon), because in this situation the produced photons escape without re-interactions. A pedestrian method for calculating a production rate is the following formula: Z dNγ ω ∝ 3 3 unobserved dtd xd p ( particles )

ω 2



n(ω1 ) · · · n(ωn ) , ×(1 ± n(ω′1 )) · · · (1 ± n(ω′p )) (14.40)

where the integration is over the invariant phase-space of the unobserved incoming and outgoing particles, weighted by the appropriate occupation factor (nB or nF for a particle in the initial state, and 1 + nB or 1 − nF for a particle in the final state). In this formula, the gray blob should be calculated with the finite-T Feynman rules. The previous approach becomes rapidly cumbersome as the number of initial and final state particles increase. The bookkeeping may be simplified by using a finite-T generalization of the formula that relates the decay rate of a particle to the imaginary part of its self-energy: ω

dNγ 1 ∝ ω/T Im Πµ µ (ω, p) . | {z } dtd3 xd3 p e −1

(14.41)

photon self-energy

Moreover, there exists a finite-T generalization of the Cutkosky’s cutting rules, in order to organize the perturbative calculation of the imaginary part that appears in the right hand side. c sileG siocnarF

Transport coefficients : Let us now discuss the case of transport coefficients. As their name suggest, these quantities characterize the ability of the system to move certain (locally conserved) quantities. For instance, the electrical conductivity encodes the properties of the system with respect to the transport of electrical charges, the shear viscosity tells us about how the system reacts to a shear stress (this coefficient

468

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

is related to the transport of momentum), etc. Note that in their simplest version, these quantities do not depend on frequency (in fact, they are the zero frequency limit of a 2-point function), and therefore they describe the response of the system to an infinitely slow perturbation. They can be generalized into frequency dependent quantities that also contain information about the response to a dynamical disturbance. The standard approach for evaluating transport coefficients is to use the GreenKubo formula, that relates the transport coefficient to the 2-point correlation function of a current J that couples to the quantity of interest (electrical charge, momentum, etc): h

transport coefficient

i

∼ lim

ω→0

1 Im ω

Z +∞

dtd3 x e−iωt

0



 J(t, x), J(0, 0) .

(14.42)

The physical meaning of this formula is that the system is perturbed at the origin by a current J, and one measures the linear response by evaluating the same current at a generic point (t, x). The transport coefficient is proportional to the Fourier transform of this correlation function at zero energy and momentum. Note that this formula contains the commutator of the two currents, since one wants the two points to be causally connected.

14.2.9

Matsubara formalism

The perturbative rules that we have derived so far are expressed in coordinate space, which is usually not very appropriate for explicit calculations. The standard way of turning them into a set of rules in momentum space is to Fourier transform all the propagators, and to rely on the fact that the Fourier transform of a convolution product is the ordinary product of the Fourier transforms, i.e. symbolically  

FT F ∗ G = FT F × FT G .

(14.43)

However, the main difficulty in doing this at finite temperature is that the time integration in the “convolution product” involves an integration over the complexshaped contour C, which makes it unclear whether we may use the above identity. Two main solutions to this problem have been devised. The first one is the imaginary time formalism, also known as the Matsubara formalism, that we have already presented superficially in the section 2.8.2. The main motivation of this formulation is that the quantities that describe the thermodynamics of a system in thermal equilibrium are time independent. Therefore, one may exploit the freedom to

469

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE deform the contour C in order to simplify it, as shown in the following figure:

ti

0

t i − iβ

−

iβ

It is customary to denote x0 = −iτ, so that the variable τ is real and spans the range [0, β) (the point τ = β should be removed – indeed, because of the KMS symmetry, it is redundant with the point τ = 0). The imaginary time formalism corresponds to the Feynman rules derived earlier, specialized to this purely imaginary time contour. Note that one could in principle use this formalism in order to calculate time-dependent quantities. One would first obtain them as a function of imaginary times τ1 , τ2 , · · · and their dependence upon real times x01 , x02 , · · · may then be obtained by an analytic continuation. From the KMS symmetry, we see that the propagator, and more generally the integrand of any Feynman diagram, is periodic (for bosons) in the variable τ with period β. Therefore, one can go to Fourier space by decomposing the time dependence in the form of a Fourier series and by doing an ordinary Fourier transform in space : +∞ Z X d3 p iωn (τx −τy ) −ip·(x−y) 0 G (τx , x, τy , y) ≡ T e e G (ωn , p) , (2π)3 n=−∞ 0

(14.44)

with ωn ≡ 2πnT . These discrete frequencies are called Matsubara frequencies. Note that for fermions, the propagator is antiperiodic with period β, and the discrete frequencies that appears in the Fourier series are ωn = 2π(n + 12 )T . Moreover, if the line carries a conserved charge q, the Matsubara frequencies are shifted by −iµq, i.e. ωn → ωn − iµq (µ is the chemical potential associated to this conservation law). In the case of scalar fields, an explicit calculation gives the following free bosonic propagator in Fourier space, G0 (ωn , p) =

ω2n

1 1 . ≡ 2 2 2 +p +m P + m2

(14.45)

(For the sake of brevity, we denote P2 ≡ ω2n + p2 .) Note that, up to a factor −i, this propagator is the usual free zero temperature Feynman propagator in which one has substituted p0 → iωn . Let us list here the Feynman rules for perturbative calculations in this formalism: • Propagators :

G0 (ωn , p) =

1 P2

,

470

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS • Vertices : each vertex brings a factor λ. Moreover, the sum of the ωn ’s and of the p’s that enter into each vertex are zero, • Loops : T

Z X X Z d3 p ≡ . (2π)3

n∈❩

P

(The right-hand side of this equation is a frequently used compact notation for the combination of discrete sums and integrals that appear in the Matsubara formalism. This notation includes a factor T that makes its dimension equal to four in four space-time dimensions.) As an illustration of the use of this formalism, let us give two examples of vacuum graphs:

=

Z Z λ XX 1 , 2 2 8 (P + m )(Q2 + m2 ) Q

P

=

Z Z g2 XX 1 . 2 2 2 6 (P +m )(Q +m2 )((P+Q)2 +m2 ) P

(14.46)

Q

The Fourier space version of the Matsubara formalism is structurally very similar to the zero temperature Feynman rules, which makes it quite appealing. There is one caveat however: the continuous integrations over energies are now replaced by discrete sums, which are considerably harder to calculate. Let us expose here two general methods for evaluating these sums. The first one is based on the following representation of the propagator of eq. (14.45): 1 G (ωn , p) = 2Ep 0

Zβ 0

i h dτ e−iωn τ (1+nB (Ep )) e−Ep τ +nB (Ep ) eEp τ , (14.47)

where the integrand in the right hand side is a mixed representation that depends on the momentum p and the imaginary time τ. By replacing each propagator of a given graph by this formula, the discrete sums can be easily performed since they are all of the form X X δ(τ − nβ) . (14.48) eiωn τ = β n∈❩

n∈❩

(The left hand side is obviously periodic in τ with period β, which is ensured in the right hand side by the sum over infinitely many shifted copies of the delta function.)

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

471

At this point, one has to integrate over the τ’s that have been introduced when replacing the propagators by (14.47), but these integrals are straightforward since the dependence on these times is in the form of delta functions and exponential. Moreover, only a finite number of the delta functions that appear in the right hand side of eq. (14.48) actually contribute, due to the constraint that each τ must be in the range [0, β). As an illustration, consider the evaluation of the 1-loop tadpole in a scalar theory with quartic coupling: Z λX 1 = 2 2 P + m2 P Zβ X Z h i d3 p λ −Ep τ Ep τ dτ δ(τ−nβ) (1+n (E ))e +n (E )e = p p B B 2 (2π)3 2Ep 0 n Z  λ d3 p  = 1 + 2 nB (Ep ) 2 (2π)3 2Ep  2  Λ T2 = λ + + ··· , (14.49) 16π2 24 where Λ is an ultraviolet cutoff that restricts the integration range |p| ≤ Λ (the final formula assumes that Λ ≫ T , and we have not written the terms that depend on the mass). The first term is the usual zero temperature ultraviolet divergence, while the term coming from the Bose-Einstein distribution exists only at non-zero temperature. This second term is ultraviolet finite, thanks to the exponential suppression of the Bose-Einstein distribution at large energy. We can already note on this example that the ultraviolet divergences are identical to the zero temperature ones. This is a general property: if the action has already been renormalized at zero temperature, there are no additional ultraviolet divergences at finite temperature. This is quite clear on physical grounds: being at finite temperature means that one has a dense medium in which the average inter-particle distance is T −1 . However, in the ultraviolet limit, one probes distance scales that are much smaller than the inverse temperature, for which the effects of the surrounding medium are irrelevant. An alternate method for evaluating the sums over the discrete Matsubara frequencies is to note that the function P(z) ≡

β eβz − 1

has simple poles of residue 1 at all the z = iωn . Therefore, we can write I X dz f(iωn ) = P(z) f(z) , 2iπ γ n∈❩

(14.50)

(14.51)

where γ is an integration contour made of infinitesimal circles around each pole of P(z), as shown in the left part of the figure 14.1. The second step is to deform the

472

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

γ

Figure 14.1: Successive deformations of the contour in order to calculate the discrete sums over Matsubara frequencies. The cross denotes a pole of the function f(z), while the solid dots on the imaginary axis are the poles of P(z).

contour γ as shown in the middle of the figure 14.1. For this transformation to hold as is, with no extra term, the function f(z) should not have any pole on the imaginary axis, which is usually the case. Finally, a second deformation brings the contour along the real axis. If the function f(z) has poles, the new contour should wrap around these poles, which an additional contribution. Thus, after these transformations, the discrete sum over the Matsubara frequencies has been rewritten as a continuous integral along the real axis (and the weight P(z) becomes an ordinary Bose-Einstein distribution), plus some isolated contributions coming from poles of the summand. c sileG siocnarF

14.2.10 Momentum space Schwinger-Keldysh formalism The imaginary time formalism is particularly well suited to calculate the time-independent thermodynamical properties of a system at finite temperature. However, interesting dynamical information is also contained in time-dependent objects. In principle, one could first evaluate them in the Matsubara formalism in terms of imaginary times τ (or imaginary frequencies iωn ), and then perform an analytic continuation to real times or energies. Beyond 2-point functions (i.e. for functions that depend on more than one energy, taking into account energy conservation), this analytic continuation is usually extremely complicated and for this reason it is desirable to be able to obtain the result directly in terms of real energies. In fact, we may ignore4 the vertical part of the contour C. A heuristic justification is to let the initial time ti go to −∞ and turn off adiabatically the interactions in this limit. Therefore, the canonical density operator becomes exp(−β H0 ) and there is no 4 A more careful treatment of the vertical part of the contour indicates that its effect it to replace the statistical distribution nB (Ep ) by nB (|p0 |) in the equations (14.52). See the discussion after eq. (14.60).

473

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

need for the vertical part of the time contour. Let us call + and − respectively the upper and lower horizontal branches of the contour. We may then break down the free propagator G0 (x, y) into four propagators G0++ .G0−− , G0+− and G0−+ depending on where x, y are located, and Fourier transform each of them separately. For a scalar field, this gives: G0++ (p) = G0+− (p) = G0−− (p) =

i + 2π nB (Ep ) δ(p2 − m2 ) , p2 − m2 + iǫ 2π (θ(−p0 ) + nB (Ep )) δ(p2 − m2 ) ,  0 ∗ G++ (p) , G0−+ (p) = G0+− (−p) .

(14.52)

Note that these propagators are very closely related to those of the Schwinger-Keldysh formalism at zero temperature (see eqs. (1.367)), since we have h i for ǫ, ǫ ′ = ± , G0ǫǫ ′ (p) = G0ǫǫ ′ (p) +2π nB (Ep ) δ(p2 −m2 ) . (14.53) T =0

The rules for the vertices and loops are identical to those of the Schwinger-Keldysh formalism at zero temperature, namely: • One must assign types + and − to the vertices of a diagram in all the possible ways,

• Each vertex of type + brings a factor −iλ and each type − vertex a factor +iλ, • A vertex of type ǫ and a vertex of type ǫ ′ are connected by the free propagator G0ǫǫ ′ , • Each loop momentum must be integrated with the measure d4 p/(2π)4 . Since this formalism is a simple extension of the zero temperature Schwinger-Keldysh formalism (the only difference being the propagators in eq. (14.53) ), it makes the connection with perturbation theory at zero temperature more transparent. In the Matsubara formalism, the KMS symmetry is trivially encoded in the fact that all the objects depend only on the discrete frequencies ωn . In the SchwingerKeldysh formalism, it is somewhat more obfuscated. A generic n-point function Γǫ1 ···ǫn (p1 · · · pn ), amputated of its external legs, obeys the following two identities: X

ǫ1 ···ǫn =±

X

Γǫ1 ···ǫn (k1 , · · · , kn ) = 0 , h Y

ǫ1 ···ǫn =± {i|ǫi =−}

i 0 e−βki Γǫ1 ···ǫn (k1 , · · · , kn ) = 0 .

It is the second of these identities that encodes the KMS symmetry.

(14.54)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Finally, let us note for later use that the four propagators of eqs. (14.52) can be related to the zero temperature Feynman propagator and its complex conjugate by the following formula:   0   0 G++ G0+− GF 0 U (14.55) =U G0−+ G0−− 0 G0F ∗ with p 1 + nB  U(p) ≡  θ(+p0 )+n B √ 1+nB

and

G0F (p) ≡

 √ 1+nB   p 1 + nB

θ(−p0 )+nB

(14.56)

i . p2 − m2 + iǫ

(14.57)

Resummation of a mass : In eq. (14.56), we have voluntarily not written the argument of the Bose-Einstein distribution. Given its origin in the propagators listed in eqs. (14.52), this argument could be Ep or |p0 |. It turns out that the second possibility is the correct one. In order to see this, consider a mass term, but instead of including the mass directly into the propagators (as in eqs. (14.52)) let us start from massless propagators and resum the mass to all orders. In order to simplify this calculation, let us introduce the following compact notations:  0   0  G++ G0+− G++ G0+− , , (14.58) 0 ≡ m ≡ G0−+ G0−− m=0 G0−+ G0−− m

●

●

the 2 × 2 matrix of Schwinger-Keldysh propagators, without and with the mass, and  0   0  0 GF 0 GF , (14.59) 0 ≡ m ≡ 0 G0F ∗ m=0 0 G0F ∗ m

❉

❉

the corresponding diagonal matrices made of the Feynman propagator and its complex conjugate. The massive propagators obtained by explicitly summing the mass term are given by

●m

= = =

●0 U U

∞  X

n=0 ∞ X

❉0 ❉0

− im2 σ3 

n=0 ∞  X n=0

●0

n

− im2 Uσ3 U − im2 σ3

❉0

❉0

n

n

U

U=U

❉mU .

(14.60)

475

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

In the first line, the third Pauli matrix σ3 provides the necessary signs for the vertices of types + and − in the Schwinger-Keldysh formalism. The third line uses the fact that Uσ3 U = σ3 . In the final result, only the matrix is affected by the mass, while the matrix U has remained unchanged. If we use the on-shell energy Ep as the argument of nB in the matrix U, then this argument is |p| since we started from a massless propagator. With this choice, the final result p would be inconsistent, since the poles of the massive propagator are at p0 = ± p2 + m2 (since the matrix m in the middle now contains the mass), but the statistical information contained in the U’s is still massless. In contrast, using |p0 | as the argument of nB ensures that the energy inside nB follows the poles of the propagator, and correctly picks the change due to the mass. We also see that the (incorrect) prescription nB (Ep ) is equivalent to neglecting the vertical path of the contour, since it amounts to keeping the interactions (here, the mass term, treated as an interaction) in the time evolution of the system but not in the density operator.

❉

❉

14.2.11 Retarded basis Change of basis : All the objects that appear in the Schwinger-Keldysh formalism carry indices that take the values + or −. Variants of this formalism may be obtained by performing linear combinations of these two indices, akin to a change of basis. For any n-point function G{ǫi } in the ± basis, we may define G{Xi } (k1 , · · · , kn ) ≡

X

{ǫi =±}

G{ǫi } (k1 , · · · , kn )

n Y

UXi ǫi (ki ) ,

(14.61)

i=1

where U is an invertible “rotation” matrix. The new indices Xi also take two values, that we may denote 1 and 2. For consistency, the vertex functions obtained by amputating Feynman graphs of their external lines must be related by Γ

{Xi }

(k1 , · · · , kn ) ≡

X

Γ

{ǫi }

{ǫi =±}

(k1 , · · · , kn )

n Y

V Xi ǫi (ki ) ,

(14.62)

i=1

where the matrix V is defined by T

V Xǫ (k) ≡ ((U )−1 )Xǫ (−k) .

(14.63)

In particular, this formula gives the expression of the vertices in the new formalism. For instance, in a φ4 scalar theory, we have −iλABCD (k1 , · · · , k4 ) = −iλ

X

ǫ V Aǫ (k1 )V Bǫ (k2 )V Cǫ (k3 )V Dǫ (k4 ) .

ǫ=±

(14.64)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Note that the new vertices may be momentum dependent if the rotation matrix is. Moreover, there could be up to 24 non-zero vertices, while there are only two in the original Schwinger-Keldysh formalism (but we will see shortly that these rotations may reduce the number of non-zero entries for the propagators, which is sometimes an advantage). The n-point functions in the new basis may be obtained directly in perturbation theory, in terms of Feynman diagrams made of the bare propagators and vertices of the new basis. c sileG siocnarF

Retarded-advanced formalism : A convenient choice of rotation consists in exploiting the two relations satisfied by the Schwinger-Keldysh propagators, G++ (p) + G−− (p) = G−+ (p) + G+− (p) , G−+ (p) = ep

0

/T

G+− (p) ,

(14.65)

in order to generate two zero entries in the new propagators. Note that the second of these identities is equivalent to KMS, and is therefore only valid in thermal equilibrium. For bosons, the matrix U that achieves this is   1 a(k0 )a(−k0 ) −a(k0 )a(−k0 ) U(k) = , (14.66) −nB (−k0 ) −nB (k0 ) a(−k0 ) where a(k0 ) is an arbitrary non-vanishing function. A similar transformation exists for fermions. In all cases, it is such that the new propagators become   0 GA (k) GXY (k) = , (14.67) GR (k) 0 where GR,A are the vacuum retarded and advanced propagators. In this formalism, it is customary to denote R and A the values taken by the indices X, Y (therefore, RA the term in the upper right location is G ≡ GA , and the other non-zero term is AR G ≡ GR ). These rotated propagators do not depend on temperature, which is now relegated into the vertices. A convenient choice is a(k0 ) = −nB (k0 ), which leads to the following vertices in a φ4 scalar theory AAAA

λ

RRRR

=λ

=0,

ARRR

λ =λ, AARR λ (k1 , · · · , k4 ) = −λ [1 + nB (k01 ) + nB (k02 )] .

(14.68)

(The vertices we have not written explicitly are obtained by circular permutations.) The general expression of the vertices in the rotated formalism is Q nB (−k0i ) λ{Xi } = λ

i|Xi =A

nB

 P

i|Xi =R

k0i

 .

(14.69)

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14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

(For fermionic lines, we must replace nB by −nF , and shift the argument by −qµ if the line carries a conserved charge.) This formalism, compared to the original Schwinger-Keldysh one, has a number of advantages: • Thanks to eq. (14.69), the Bose-Einstein (or Fermi-Dirac in the case of fermions) functions are conveniently factorized in each Feynman graph, • In this formalism, the two identities (14.54) satisfied by n-point functions take a particularly simple form, Γ

A···A

=Γ

R···R

=0,

(14.70)

which renders immediate the simplifications allowed by these identities. • The retarded-advanced formalism has close connections to the Matsubara formalism, since every R/A n-point function can be obtained as a linear combination of the analytical continuations (iωn → p0 ± i0+ ) of the corresponding function in the Matsubara formalism.

14.3 Large distance effective theories 14.3.1

Infrared divergences

Quantum field theories with massless bosons at non-zero temperature suffer from pathologies in the infrared sector, due to the low energy behaviour of the Bose-Einstein distribution: nB (E) ≈

E≪T

T ≫1. E

(14.71)

As we shall see now, using a massless φ4 scalar field theory as a playground, this leads to loop contributions that exhibit soft divergences. The simplest graph that suffers from this problem is the following 2-loop graph,

, that has two nested tadpoles. Let us assume that the uppermost tadpole has already been combined with the corresponding 1-loop ultraviolet counterterm, so that only the finite part remains, and denote µ2 the finite remainder. From eq. (14.49), its expression is given by µ2 ≡

λ T2 . 24

(14.72)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

(This is the exact result for the temperature dependent part in a massless theory.) With this shorthand, we have

=

=

Z λµ2 X 1 2 (P2 )2 P

Z λ µ2 d3 p nB (p)(1 + nB (p)) 2 eβp − e−βp + 2 (2π)3 4p2 T p | {z } ≈ T4 p≪T p

=

λ µ2 T + infrared finite terms , 4π2 ΛIR

(14.73)

where in the last line we have introduced an infrared cutoff ΛIR in order to prevent a divergence at the lower end of the integration range. A similar calculation would indicate an even worse infrared singularity in the following 3-loop graph:

∼ λT µ

µ3 + infrared finite terms , Λ3IR

(14.74)

and more generally for n insertions of the base tadpole on the main loop,

∼ λT µ



µ ΛIR

2n−1

+ infrared finite terms .

(14.75)

Unlike ultraviolet divergences that can, in a renormalizable theory, be disposed of systematically by a redefinition of the couplings in front of a few local operators in the Lagrangian, it is not possible to handle infrared divergences in this manner because they correspond to long distance phenomena. Fortunately, there is a simple way out in the present case: the series of graphs that we have started evaluating are the first terms of a geometrical series, since the repeated insertions of a tadpole equal to µ2 (after subtraction of the appropriate counterterm) merely amounts to dressing by a mass µ2 an originally massless propagator. Namely, we have

+

=

+ λ 2

Z

+ ··· =

p   d3 p p 1 + 2 nB ( p2 + µ2 ) . 3 2 2 (2π) 2 p + µ

(14.76)

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

479

(The thicker propagator indicates a massive scalar with mass µ2 .) The procedure used here, that consists in summing an infinite subset of (individually divergent) perturbative contributions, is a simple form of resummation. We can readily see that it leads to an infrared finite sum, since now the quantity µ2 plays the role of a cutoff at small momentum. Let us now estimate the √ contribution of the infrared sector to this integral. At weak coupling, we have µ ∼ λ T ≪ T . Therefore, for momenta p ∼ µ, we have p Z Z 2 2 dp p2 T 2 1 + 2 nB ( p + µ ) p λ dp p ∼λ ∼ λT µ ∼ λ3/2 T 2 . (14.77) p2 + µ2 p2 + µ2

This contribution comes in addition to the ultraviolet divergence λΛ2 and the contribution λT 2 that are both contained in the first diagram of the resummed series (these terms come from momenta of order T or above). We observe here an unexpected feature; the appearance of half powers of the coupling constant λ. On the surface, this is quite surprising since the power counting indicates that one power of λ should come with each loop. This oddity is in fact a consequence of the infrared behaviour of the Bose-Einstein distribution, in T/E, combined with the fact that the µ introduced in the resummation is of order λ1/2 . Although the loop expansion generates a series which is analytic in λ, this property may be broken if some parameters in the integrands depend on λ1/2 . c sileG siocnarF

14.3.2

Screened perturbation theory

The resummation of the finite part of the 1-loop tadpole is sufficient in order to screen the infrared divergences in the graphs corresponding to a strict loop expansion. However, since such a resummation amounts to a reorganization of perturbation theory (here, by already including an infinite set of graphs into the propagator), it should be done in a careful way that avoids any double countings, and ensures that we are not modifying the original theory. This can be achieved by a method, called screened perturbation theory, that consists in adding and subtracting a mass term to the Lagrangian, L=



λ 1 1 1 ∂µ φ ∂µ φ − φ4 − µ2 φ2 + µ2 φ2 . 2 4! 2 2

(14.78)

This manipulation clearly ensures that nothing is changed to the original theory. The reorganization of perturbation theory allowed by this trick comes from treating the two mass terms on different footings: the term − 21 µ2 φ2 is treated non-perturbatively by including it directly into the definition of the free propagator, while the term + 21 µ2 φ2 is treated order by order, as a finite counterterm. In this reorganization, the value of µ2 has so far been left unspecified, and it could a priori be chosen arbitrarily. A general rule governing this choice is to include in µ2

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

as much as possible of the large contributions coming from loop corrections to the propagator. The 1-loop contribution in λT 2 is an obvious candidate for including in µ2 , since for momenta p2 . λT 2 this is indeed a large correction to the denominator of the propagator. At small coupling λ ≪ 1, this is the dominant one. However, when the coupling increases, the propagator may receive additional large corrections from higher order loop corrections, and an improved resummation scheme could include these additional corrections. A further improvement, sometimes considered in some applications, is to let µ2 free and to use some reasonable condition to choose an “optimal” value. For instance, this condition may be the minimization of the 1-loop correction, which in a sense would indicate that the resummation has shifted most of this loop contribution into the free propagator. For instance, one may try to achieve

0=

λ + counterterms = 2

ZΛ

p d3 p 1 + 2 nB ( p2 + µ2 ) λ Λ2 p − − µ2 , (2π)3 16π2 2 p2 + µ2 (14.79)

where the two subtractions are respectively the ultraviolet counterterm and the finite counterterm necessary in order not to overcount the mass µ2 . The equation, that provides an implicit definition of the mass µ2 , is called a gap equation5 . Because this equation is non-linear in µ2 , its solution contains all orders in λ, but at small λ it is dominated by the 1-loop result µ2 = λT 2 /24. We show an application of this method to the calculation of the free energy F in the figure 14.2. In this figure, the results obtained at 1-loop and 2-loops in screened perturbation theory are compared to the first two orders (λ and λ3/2 ) of the ordinary perturbative expansion. Firstly, we can see that the latter is quite unstable except at low coupling: the two subsequent orders differ substantially, and even the sign of the correction due to the interactions flips. In contrast, screened perturbation theory leads to a remarkably stable result, with very small changes when going from 1-loop to 2-loops. To a large extent, this success is due to the non trivial coupling dependence of the mass µ2 , acquired by solving the gap equation (14.79) (screened perturbation theory with only the 1-loop mass, would be better than strict perturbation theory, but would encounter some difficulties at large coupling).

14.3.3

Symmetry restoration at high temperature

The thermal correction to the mass µ2 = λT 2 /24 also explains why symmetries that may be spontaneously broken at low temperature are generically restored at high 5 The terminology comes from the fact that the solution of this equation usually shifts the energy of a particle, generating a “gap” in the spectrum, and thus requiring a non-zero energy to create such a particle.

481

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

2.5 2.0 1.5

-F

Figure 14.2: Free energy at non-zero temperature in the φ4 scalar field theory (normalized to the free energy of the non-interacting theory). The horizontal axis is the coupling strength g ≡ λ1/2 . Curves “g2” and “g3”: orders λ and λ3/2 in the original perturbative expansion. Curves “a” and “b”: screened perturbation theory at 1-loop and 2-loops, with the mass µ2 determined as the exact solution of the gap equation (14.79).

1.0 b a g2 g3

0.5 0.0

0

1

2

3

4

5

6

7

8

9

g

100

80

60

V(φ)

Figure 14.3: Evolution of a scalar potential with increasing temperature. Thick dark curve: potential with degenerate non-trivial minima at low temperature, leading to spontaneous symmetry breaking. Thick light curve: potential at the critical temperature.

40

20

0

-10

-5

0 φ

5

10

10

482

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

temperature. Let us consider for instance a scalar theory whose potential at zero temperature is m20 2 λ 4 φ + φ . (14.80) 2 4! Because of the sign of the mass term, this potential has two degenerate minima. The true vacuum of this theory is at a non-zero value of φ, and the discrete symmetry φ → −φ is thus spontaneously broken. When the temperature increases, the thermal fluctuations generate a positive correction to the square of the mass, proportional to λT 2 . Eventually m2 becomes positive, i.e. the potential has a unique minimum at φ = 0, and the symmetry is restored. The critical temperature, that separates the low temperature broken phase and the high temperature symmetric phase, is the point at which m2 = 0. Vφ = −

14.3.4

Hard Thermal Loops

The scalar φ4 theory considered in the previous subsection is rather special because the one-loop tadpole diagram that gives the thermal correction to the mass is momentum independent, and because it is calculable analytically in a massless theory. In gauge theories at finite temperature, there are also important thermal corrections to the propagator of fermions and gauge bosons, but their structure is much richer. As we shall see now, the calculation of the corresponding one-loop self-energies requires an approximation based on the assumption that the loop momentum is of order of T while the external momentum is much smaller, p ≪ T . The resulting self-energies are known as Hard Thermal Loops. Photon hard thermal loop : A simple example of hard thermal loop is that of a photon in QED, for the only graph at is shown on the right of the figure 14.4. In the c sileG siocnarF

K

K P

P

Figure 14.4: Fermion and photon self-energies at 1-loop.

Matsubara formalism, the expression of the photon polarization tensor is Z X / − /K)γνP /) tr (γµ (P µν 2 . Π (K) = e 2 2 P (P − K) P

(14.81)

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

483

(We neglect the fermion mass in this expression.) Let us pause a moment in order to discuss the possible form of this tensor. In QED, Πµν (K) must be symmetric under the exchange of the Lorentz indices, and must obey the following Ward-Takahashi identity: Kµ Πµν (K) = Kν Πµν (K) = 0 .

(14.82)

In the vacuum, this relation is sufficient to fully constrain the tensorial structure of Πµν , up to an overall function of K2 . In the presence of a surrounding thermal bath, the situation is more complicated: besides the metric tensor gµν and the 4-momentum Kµ , this tensor may also contain the 4-velocity Uµ of the thermal bath (with respect to the observer). Let us first introduce V µ ≡ K2 Uµ − (K · U) Kµ .

(14.83)

Then, one may check that Ward-Takahashi identity is satisfied by two symmetric tensors PTµν

≡

PLµν

≡

gµν −

Kµ Kν V µ V ν − , K2 V2

V µV ν . V2

(14.84)

Besides being transverse to Kµ , these two tensors satisfy PTµ ν PTνρ = PTµρ , PLµ ν PLνρ = PLµρ , PTµ µ = 2 , PLµ µ = 1 ,

PTµ ν PLνρ = 0 , (14.85)

which means that they are mutually orthogonal projectors (the values of their traces indicate that PTµν encodes two degrees of freedom, while PLµν contains only one). Moreover, in the rest frame of the thermal bath, we have Uµ = δµ0 , and the first of these tensors reads PT00 = PTi0 = PT0i = 0 ,

PTij = δij −

ki kj k2

.

(14.86)

Therefore, PTµν is a projector orthogonal to the 3-momentum k. In terms of these projectors, the most general photon polarization tensor is of the form: Πµν (K) = PTµν ΠT (K) + PLµν ΠL (K) .

(14.87)

Note that in the presence of a heat bath, the functions ΠT,L (K) may depend on the four components of Kµ separately (in the vacuum, the corresponding function would

484

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

depend only on the Lorentz invariant K2 ). This complication is due to the fact the thermal bath imposes a preferred frame that breaks Lorentz invariance. If the photon self-energy is resummed on the propagator, one obtains the following dressed propagator in a generic covariant gauge: −Dµν (K) =

PTµν 2

K + ΠT (K)

+

PLµν 2

K + ΠL (K)

+ξ

Kµ Kν K2

,

(14.88)

thanks to the orthogonality properties of these projectors (the gauge dependent term in the propagator is not affected by the resummation). The two functions ΠT,L (K) may be obtained from Πµ µ and Π00 by using Πµ µ = 2 ΠT + ΠL

Π00 = −

k2 Π . K2 L

(14.89)

The fully traced polarization tensor, Πµ µ , is the easiest to evaluate: µ

Π

µ (K)

=

Z X /) / − /K)γµP tr (γµ (P

2

e

P2 (P − K)2

P

=

2

4e

Z  X P

K2 P2 (P − K)2

−

2 P2



.

(14.90)

The hard thermal loop approximation consists in assuming that the external momentum K is much smaller than the temperature, that controls the typical loop momentum. In this approximation, we have Z X 1 e2 T 2 . (14.91) = Πµ µ (K) = −8e2 HTL 3 P2 |P {z } T2 − 24

The sum-integral in this expression has a very simple tadpole structure, but note that the Matsubara frequencies are the fermionic ones, hence the result −T 2 /24 for its thermal contribution (instead of T 2 /12 in the bosonic case). The 00 component is a bit more complicated, Z X /) / − /K)γ0P tr (γ0 (P 00 2 Π (K) = e 2 2 P (P − K) P   Z X 4 8P0 (P0 − K0 ) 2 − 2 = e HTL P2 (P − K)2 P P

=

HTL

e2 T 2 h k0  k0 + k i 1− . ln 0 3 2k k −k

(14.92)

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

485

In the second line, we have dropped a non-HTL term in K2 /P2 (P − K)2 , and in the last line we have analytically continued the discrete Matsubara frequency to a real energy K0 → ik0 . Therefore, the transverse and longitudinal self-energies of the photon in the HTL approximation read e2 T 2 k0 h k0 1  k2   k0 + k i 1 − 20 ln 0 + 6 k k 2 k k −k e2 T 2  k20 h k0  k0 + k i 1− 2 1− . ln 0 3 k 2k k −k

ΠT (K) = ΠL (K) =

(14.93)

Electron hard thermal loop : A similar approximation can be used for fermions in QED. Due to the breaking of Lorentz invariance caused by the thermal bath, the self-energy may be decomposed as / (K) ≡ α(K) γ0 + β(K) p b·γ, Σ

(14.94)

b ≡ p/|p|. Using the same method as above, one finds where p / (K)) = tr (/K Σ

/ (K)) = tr (γ0Σ

e2 T 2 , HTL 2  e2 T 2 k0 + k  4α = . ln 0 HTL 4k k −k

4 (K0 α + kβ) =

(14.95)

Moreover, the HTL approximation leads to a fermion self-energy that does not depend on the gauge chosen for the photon propagator. After summation of this self-energy to all orders, the fermion propagator becomes S(K) =

b·γ b·γ γ0 + k γ0 − k + , 2(k0 − k − Σ+ ) 2(k0 + k + Σ− )

(14.96)

where Σ± ≡ β ± α. c sileG siocnarF

Non-Abelian gauge theories : In the case of a non-Abelian gauge theory such as QCD, the fermion self-energy is given by the same graph as in QED, the only change a 2 2 being the substitution e2 → g2 ta f tf = g (N − 1)/(2N) (assuming fermions in the fundamental representation of su(N)). Interestingly, although it is given by four graphs (see the second line in the figure 14.5), the gluon self-energy in the HTL approximation has the same form as the photon one, modulo the change e2 → g2 (N + Nf /2) with Nf the number of quark flavours. In addition, there are Hard Thermal Loops specific to non-Abelian gauge theories, both in the n-gluon function, and in the function with a quark-antiquark pair and n − 2 gluons, as shown in the figure 14.5.

486

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 14.5: List of Hard Thermal Loops in QCD.

Quasi-particles : One of the effects of the summation of Hard Thermal Loops on the boson and fermion propagators is to shift their poles, i.e. to modify their dispersion relations (such modified excitations are called quasi-particles). This can be seen from the fact that the self energies ΠT,L and Σ± do not vanish on the original mass-shell, at k0 = ±k. This modification is due to the multiple interactions of a particle with those of the surrounding bath6 , which tend to make it heavier than it would be in the vacuum. In the case of bosons, the dispersion relations of the transverse and longitudinal modes become distinct (except at zero momentum), as shown in the left part of the figure 14.6. Another peculiarity of this change of the dispersion curves is that it does not correspond to a constant mass, but rather to a momentum dependent one (in the figure m2γ ≡ e2 T 2 /9 denotes the mass of long wavelength excitations). Moreover, the residue of the longitudinal pole vanishes exponentially for k ≫ T , which means that this mode decouples at low temperature. This is indeed expected, since the longitudinal mode is unphysical in the vacuum. Thus, the longitudinal mode is a purely collective phenomenon, that exists only in the presence of a dense medium. In contrast, the residue of the transverse pole goes to one at large momentum, and one thus recovers in this limit the in-vacuum gauge boson propagator. Furthermore, the self-energies in eqs. (14.93) are purely real above the light-cone (they can have 6 The

same happens to electrons in a crystal, due to their interactions with phonons.

487

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

Figure 14.6: Gluon (left) and quark (right) dispersion relations.

p0 / m γ

p0 / m f

(+)

(T)

1

1

(L)

p / mγ

1

(−)

1

p / mf

an imaginary part only when the argument of the logarithm is negative), which implies that the shifted poles remain on the real axis. In other words, in the HTL approximation, the gauge boson excitations remain infinitely long-lived. In the case of fermions, there are also two distinct modes, denoted (+) and (−), that merge at zero momentum and k0 = m2f ≡ e2 T 2 /8. The + mode is the analogue of the zero temperature fermion, modified by the surrounding thermal bath (the residue of this pole goes to one when k ≫ T ). In contrast, the − mode is a purely collective mode (the corresponding residue vanishes exponentially at low temperature). Like for bosons, these fermionic modes have an infinite lifetime in the HTL approximation. Debye screening : The Hard Thermal Loop correction to the gauge boson propagator also encodes interesting phenomena in the space-like region. In particular, by taking the zero frequency limit of the photon self-energy, and then its zero momentum limit (in this order), one can determine how the Coulomb potential of a static electrical charge is modified at long distance. Simply recall that the Coulomb potential is given by the Fourier transform of the longitudinal7 term in the propagator, Z 3 eik·r d k . (14.97) A0 (r) ∼ (2π)3 k2 + ΠL (0, k) At large distance, we need the small k behaviour of ΠL (0, k), which is given by lim ΠL (k0 = 0, k) =

k→0

e2 T 2 . 3 } | {z

(14.98)

m2

D

7 The transverse projector does not couple to the electromagnetic current of a static charge, e.g. an infinitely heavy charged particle.

488

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

(The mass mD is called the Debye mass.) The Fourier transform then gives the following Coulomb potential at long distance, A0 (r) ∼

e−mD r , r

(14.99)

which is exponentially attenuated compared to the vacuum Coulomb potential of a point-like charge. The inverse of the Debye mass characterizes the typical distance beyond which this screening is sizeable. Physically, this phenomenon is due to the

−m A0(r) = e r

r

D

r

Figure 14.7: Debye screening in QED.

fact that the test charge polarizes the charged medium surrounding it, by attracting in its vicinity charges of the opposite sign. Because of this, a distant observer sees an effective charge which is much small than the bare charge visible at short distance (see the figure 14.7).

Landau damping : The last collective phenomenon included in Hard Thermal Loops is Landau damping, which manifests itself in the fact that the HTL self-energies have an imaginary part in the space-like region. This imaginary part indicates that a wave propagating in such a dense medium is attenuated over distance scales of order (eT )−1 . In the case of photons, the microscopic mechanism of this damping is the absorption of a photon by the surrounding electrical charges, by a process such as e− γ → e− . c sileG siocnarF

Sum rules : Propagators resummed with HTL self-energies should be used in processes involving soft momenta of order eT or below, in order to capture the main collective effects. However, given the explicit form of the self-energies (recall for instance eqs. (14.93)), the use of these dressed propagators complicates significantly the calculations in which they appear. Nevertheless, some integrals in which these propagators enter can be evaluated in closed form by exploiting their analytical properties.

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

489

Let us return to Minkowski spacetime, and consider the retarded resummed propagators, defined as R

∆T,L (k0 , k) ≡

i . k20 − k2 − ΠT,L (k0 , k) + ik0 0+

(14.100)

This propagator admits the following spectral representation, i = 2 2 k0 − k − ΠT,L (k0 , k) + ik0 0+

+∞ Z

−∞

dω i . ω ρT,L (ω, k) 2 2 2π k0 − ω + ik0 0+ (14.101)

where the spectral function ρT,L is defined by R

ρT,L (k0 , k) ≡ 2 i Im ∆T,L (l0 , l) .

(14.102)

From eq. (14.101), we may derive other useful integrals that contain the spectral functions ρT,L . The starting point is to take the imaginary part of eq. (14.101), by denoting ω ≡ kx et k0 ≡ ky, which gives the following identity +∞ Z

−∞

# " dx 1 x ρT,L (kx, k) P 2 2π y − x2 =

(k2 (y2

k2 (y2 − 1) − Re ΠT,L (ky, k) .(14.103) − 1) − Re ΠT,L (ky, k))2 + (Im ΠT,L (ky, k))2

Various interesting integrals can then be obtained by taking special values of y. With y = 0, we obtain +∞ Z

dx ρT (kx, k) 1 = 2 , 2π x k

−∞

+∞ Z

−∞

1 dx ρL (kx, k) = 2 , 2π x k + m2D

(14.104)

while y = +∞ leads to +∞ Z

1 dx x ρT,L (kx, k) = 2 . 2π k

(14.105)

−∞

Let us also mention another exact integral involving the HTL photon self-energies, Z1 0

i h dx 1 1 2 Im Π(x) , =π − 2 2 x (z + Re Π(x)) + (Im Π(x)) z + Re Π(∞) z + Re Π(0)

490

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS (14.106)

where Π(x) is any of ΠT,L (kx, k) (which does not depend on k since the bosonic HTL self-energies depend only on the ratio k0 /k). The values at x = ∞ and x = 0 of these self-energies that appear in the right-hand side are easily determined from eqs. (14.93). This integral, where the value of x is bounded by one, appears in the scattering cross-section of a hard particle on a particle of the thermal bath, by exchange of a soft photon (the momentum of this photon is space-like, hence |x| ≤ 1). Relevant physical scales : When discussing the physics of a weakly coupled system of particles at high temperature T (much larger than the masses), it is useful to have in mind the following hierarchy of length scales: • ℓ = T −1 . T is the typical momentum of a particle in this system, and its inverse is the typical separation between two neighboring particles. At shorter distance scales, a particle behaves exactly as if it were in the vacuum. This is why ultraviolet renormalization at non-zero temperature can be done with the zero-temperature counterterms. • ℓ = (gT )−1 . This is the typical distance over which a particle “feels” modifications of its dispersion relation. Besides the appearance of a thermal gap in the spectrum of gauge bosons and matter fields, the HTL self-energies also encode Debye screening and Landau damping. c sileG siocnarF

• ℓ = (g2 T )−1 . This is the mean distance between scatterings with a soft colour exchange. These are forward scatterings, since the momentum transfer (of order gT , the scale of the infrared cutoff provided by the dressing of the gluon propagator) is much smaller than the momentum of the incoming particles (typically T ). A gross way to obtain this scale is by estimating the corresponding scattering rate:

Γ

soft collisions

=

2 p⊥

∼

g4 T 3

Z

gT .p⊥

d2 p⊥ p4⊥

∼ g2 T , (14.107)

where p⊥ is the momentum transfer transverse to the momentum of the incoming particles. Although these scatterings do not lead to an appreciable transport of momentum, they reshuffle the colour of the particles and hence contribute to the colour conductivity. • ℓ = (g4 T )−1 . This is the mean distance between scatterings with a momentum transfer of order T , i.e. those that scatter particles at large angles. Estimating

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

491

1

/

g4 T

1

/

g2 T

1 / gT

Figure 14.8: Relevant distance scales in a relativistic plasma at high temperature.

this scale is done as above, but with a lower limit of order T for the momentum transfer: 2 Z d2 p⊥ p⊥ ∼ g4 T . (14.108) Γ hard = ∼ g4 T 3 4 collisions p ⊥ T .p⊥

This scale is usually called the mean free path. This is the relevant scale for all transport phenomena that require significant momentum exchanges, for instance the viscosity. Beyond this scale is the realm of collective effects such as sound waves (on these scales, it is more appropriate to describe the system as a fluid rather than in terms of elementary field excitations). Perturbative and non-perturbative modes : Although it is in principle possible to study any phenomenon at finite temperature in terms of the bare Lagrangian, this becomes increasingly difficult at large distance because of non-trivial in-medium effects. In order to circumvent this difficulty, various resummation schemes and effective descriptions have been devised, one of which is the resummation of HTLs discussed above. Our goal here is not to give a detailed account of these various techniques, but to provide general principles regarding what can and cannot be treated perturbatively,

492

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

focusing on gauge bosons. Let us first recall that a mode is perturbative if its kinetic energy dominates its interaction energy. For a mode of momentum k, the kinetic energy of a gauge field can be estimated as



 K ∼ (∂A)2 ∼ k2 A2 . (14.109) For the interaction energy, we have



2 I ∼ g2 A4 ∼ g2 A2 .

(14.110)

(The second part of the equation is of course not exact, but it gives the correct

2  order 2 2 of magnitude.) Thus, a mode of momentum k is perturbative if k ≫ g A . When

2 discussing the order of magnitude of A , it is useful to distinguish the contribution of the various momentum scales by defining Z κ∗ 3

2 d p n (Ep ) , A κ∗ ∼ Ep B

the contribution of all the thermal modes up to the scale κ∗ . From these considerations, we can now distinguish three types of modes:

 • Hard modes : k ∼ T . For these modes, we have A2 T ∼ T 2 , and K ≫ I. They are therefore fully perturbative.

 • Soft modes : k ∼ gT . For these modes, k2 ∼ g2 A2 T , which implies that the

2soft  modes interact strongly with thehard modes. However, we also have A gT ∼ gT 2 , so that k2 ≫ g2 A2 gT . Thus, the soft modes interact perturbatively among themselves. Consequently, it is possible to describe perturbatively the soft modes, provided one has performed first a resummation of the contribution of the hard modes. Screened perturbation theory is a realization of this idea.

 • Ultrasoft modes : k ∼ g2 T . For these modes, we have A2 g2 T ∼ g2 T 2 , so

 that k2 ∼ g2 A2 g2 T . Therefore, the ultrasoft modes interact non perturbatively among themselves, and there is no way to treat them in a perturbative approach. A non perturbative approach, such as lattice field theory, is necessary for this.

14.4 Out-of-equilibrium systems Until now, we have discussed only systems in equilibrium, whose initial state is described by the canonical density operator ρ ≡ exp(−β H). However, many interesting questions could also be asked for a system which is not initially in thermal equilibrium, the prime of them being to describe its relaxation towards equilibrium. In this section, we discuss a few aspects of the quantum field theory treatment of out-of-equilibrium systems. c sileG siocnarF

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

14.4.1

493

Pathologies of the naive approach

Firstly, let us note that the Matsubara formalism does not seem prone to a simple out-of-equilibrium generalization, since the KMS symmetry (that encodes into the correlation functions the fact that the system is in equilibrium) is in a sense hardwired into the discrete Matsubara frequencies. The Schwinger-Keldysh formalism appears to be a more adequate starting point for such a generalization. Let us first discuss a simple extension that does not work, because the reasons of its failure will teach us a useful lesson. Since in eqs. (14.53), the only reference to the statistical state of the system is contained in the Bose-Einstein distribution nB (Ep ), we may try to replace it by an arbitrary distribution f(p) that describes the particle distribution in an out-of-equilibrium system8 : h i + 2π f(p) δ(p2 − m2 ) . (14.111) ∀ǫ, ǫ ′ = ± , G0ǫǫ ′ (p) = G0ǫǫ ′ (p) T =0

Consider now the insertion of a self-energy Σ on the bare propagator,

Σ

=

X

G0+ǫ (p) Σǫǫ′ (p) G0ǫ′ + (p) .

(14.112)

ǫ,ǫ′ =±

Such an expression is delicate to expand, because it involves products of distributions that are notoriously ill-defined, such as δ2 (p2 − m2 ). Let us first determine which of these products are well defined and which are not. For this, let us write 

iP

1 z

2 + πδ(z)

= = =

   2 1 1 + 2iπδ(z)P π δ (z) − P z z   2  i i d = −i z + i0+ dz z + i0+        ′ 1 1 d − iπδ ′ (z) . iP + πδ(z) = P −i dz z z (14.113) 2 2

From this exercise, we obtain the following two identities:    2 1 π δ (z) − P z 1 2δ(z)P z 2 2

= =

   ′ 1 P z −δ ′ (z) .

(14.114)

8 Note firstly that this would not encompass the most general initial states, only those for which the initial correlations are only 2-point correlations.

494

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Since the derivative of a distribution is well-defined, this indicates that certain products (or combinations of products) of delta functions and principal values are well defined, but not all of them (for instance, the product δ2 (z) makes no sense). Returning now to eq. (14.112) and expanding the propagators, we see that it contains terms that are ill-defined: =

Σ

h well defined i distributions

h i +π2 δ2 (p2 −m2 ) (1+f(p))Σ+− −f(p)Σ−+ , (14.115)

where we have used the first of eqs. (14.54) in order to simplify the combination of self-energies that appear in the square bracket. Note that the square bracket vanishes in equilibrium thanks to the KMS symmetry. We are thus facing a very peculiar pathology, that exists only out-of-equilibrium. We may learn a bit more about this issue by formally resumming the self-energy Σ on the propagator. Let us introduce the following notations:

●

0

≡



G0++ G0−+

G0+− G0−−



,

❉≡



G0F 0

0 G0∗ F



,

❙≡



Σ++ Σ−+

 Σ+− , Σ−− (14.116)

and consider the resummed propagator defined by

●≡

∞ h X

in

●0(−i❙) ●0 .

n=0

A straightforward calculation shows that   e ∗ GF GF ΣG F U =U 0 G∗F

●

(14.117)

(14.118)

where U is the matrix defined in eq. (14.56), but with f(p) instead of the Bose-Einstein distribution, and where we have used the following notations GF (p) ≡

p2

m2

i , − ΣF + iǫ ,

− ΣF ≡ Σ++ + Σ+− h i 1 e≡ (1 + f(p))Σ+− − f(p)Σ−+ . Σ 1 + f(p)

(14.119)

Note that the Feynman propagator and its complex conjugate have mirror poles on each side of the real energy axis. If the self-energy ΣF has no imaginary part, then

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

495

these poles “pinch” the real axis and lead to a singularity (this is in fact a pathology of the same nature as the product δ2 in eq. (14.115)). By performing explicitly the multiplication with the matrix U, we obtain the resummed propagator in the following form: Gǫǫ ′ (p) =

h

i G0ǫǫ ′ (p)

+ 2π f(p) δ(p2 − m2 ) h i + (1 + f(p))Σ+− − f(p)Σ−+ GF (p)G∗F (p) .

T =0

(14.120)

Since it does not depend on the indices ǫǫ ′ , the pathological term (on the second line) appears on the same footing as the second term, that contains the distribution f(p). Thus, the lesson of this calculation is that one may consider hiding this pathology into a redefinition of the distribution f(p). However, the naive formalism that we have tried to use so far is not adequate for doing this consistently, and must be amended in a number of ways: • The initial time ti should not be taken to −∞, as is done when using the Schwinger-Keldysh formalism in momentum space. Indeed, this is the time at which the system was prepared in an out-of-equilibrium state. If it were equal to −∞, the system would have had an infinite amount of time for relaxing to equilibrium at the finite time where a measurement is performed. Note that observables will in general depend on the initial time ti , in contrast with what happens in equilibrium. c sileG siocnarF

• The Schwinger-Keldysh formalism in momentum space assumes that the system is invariant by translation, in particular in the time direction. This is clearly not the case when the system starts out-of-equilibrium, since it is expected to evolve towards equilibrium. Thus, one should stick to the formalism in coordinate space.

14.4.2

Kadanoff-Baym equations

The Kadanoff-Baym equations, that we shall derive now, may be viewed as a kind of quantum kinetic equations. These equations are exact, but contain a self-energy that must be truncated to a manageable number of diagrams in order to be usable in practical applications. In the next subsection, we will show how the traditional kinetic equations can be derived from the Kadanoff-Baym equations. The starting point is the Dyson-Schwinger equation, written in coordinate space,

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

that expresses the resummation of a self-energy on the propagator: G(x, y) = G0 (x, y) + 0

G(x, y) = G (x, y) +

Z

ZC C

  d4 ud4 v G0 (x, u) − iΣ(u, v) G(v, y)

  d4 ud4 v G(x, u) − iΣ(u, v) G0 (v, y) ,

(14.121)

where G0 is the free propagator and G is the resummed one. Note that the time integrations run over the Schwinger-Keldysh contour C. Here, we have written the equation in two ways, depending on whether the self-energy is inserted on the right or on the left of the bare propagator (in the end, the resulting propagator G is the same in both cases). Next, we apply the operator x + m2 on the first equation and y + m2 on the second equation. This eliminates the bare propagators, and we obtain: 2

(x + m )G(x, y) = −iδc (x − y) − (y + m2 )G(x, y) = −iδc (x − y) −

Z

d4 v Σ(x, v) G(v, y) , C

Z

d4 v G(x, v) Σ(v, y) , (14.122) C

where δc (x − y) is the generalization of the delta function to the contour C. This is one of the forms of the Kadanoff-Baym equations.

14.4.3

From QFT to kinetic theory

Kinetic theory is an approximation of the underlying dynamics in terms of a spacetime dependent distribution of particles f(x, p). One may note right away that this is necessarily an approximate description, because it is not possible to define simultaneously the position and momentum of a particle. In the Kadanoff-Baym equations (14.122), the dressed propagator G and the self-energy Σ are in general not invariant under translations, precisely because the system is out-of-equilibrium. Therefore, one may not Fourier transform them in the usual way. Instead, one uses a Wigner transform, defined as follows Z s
s , (14.123) F(X, p) ≡ d4 s eip·s F X + , X − 2 2

where F(x, y) is a generic 2-point function (we use the same symbol for its Wigner transform, since the arguments are sufficient to distinguish them). In other words, the Wigner transform is an usual Fourier transform with respect to the separation s ≡ x − y, and the result still depends on the mid-point X ≡ (x + y)/2. Note that in eq. (14.123), the time integration is over the real axis, not over the contour C. Wigner

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

497

transforms do not share with the Fourier transform their properties with respect to convolution. Given two 2-point functions F and G, let us define: Z H(x, y) ≡ d4 z F(x, z) G(z, y) . (14.124) The Wigner transform of H is given by H(X, p) = F(X, p) exp

i ← → → ←  G(X, p) , ∂X ∂p − ∂X ∂p 2

(14.125)

where the arrows indicate on which side the corresponding derivative acts. The right hand side of this formula reduces to the ordinary product of the transforms when there is no X dependence, i.e. when the functions F and G are translation invariant. The first correction to the translation invariant case is proportional to the Poisson bracket of F and G, H(X, p) = F(X, p)G(X, p) +

i F(X, p), G(X, p) + · · · . 2

(14.126)

The derivatives with respect to x and y that appear in the Kadanoff-Baym equations can be written in terms of derivatives with respect to X and s : 1 1 ∂ X + ∂ s , ∂y = ∂ X − ∂ s 2 2 1 1 x = X + ∂X · ∂s + s , y = X − ∂X · ∂s + s .(14.127) 4 4 ∂x =

In these operators, the Wigner transform just amounts to a substitution ∂s

→

−ip ,

s

→

−p2 .

(14.128)

In order to go from the Kadanoff-Baym equations to kinetic equations, two approximations are necessary: 1. Gradient approximation : p ∼ ∂s ≫ ∂X . The derivatives with respect to the mid-point X characterize the space and time scales over which the properties of the system (e.g. its particle distribution) change significantly. This approximation therefore means that these scales, that characterize the off-equilibriumness of the system, should be much larger than the De Broglie wavelength of the particles. Another way to state this approximation is that the mean free path in the system should be much larger than the wavelength of the particles, which amounts to a certain diluteness of the system. Using this approximation in the two Kadanoff-Baym equations (14.122), taking their

498

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS difference, and breaking it down into its ++, −−, +− and −+ components, one obtains −2ip · ∂X (G+− (X, p) − G−+ (X, p)) = 0 , i  −2ip · ∂X (G+− (X, p) + G−+ (X, p)) = 2 G−+ Σ+− − G+− Σ−+ . (14.129)

2. Quasi-particle approximation : This approximation consists in assuming that the dressed propagators Gǫǫ ′ can be written in terms of a local particle distribution f(X, p) as in eqs. (14.111). This is equivalent to G−+ (X, p) = (1 + f(X, p)) ρ(X, p) , G+− (X, p) = f(X, p) ρ(X, p) ,

(14.130)

where ρ(X, p) ≡ G−+ (X, p) − G+− (X, p). This would be exact for noninteracting, infinitely long-lived, particles. In the presence of interactions, the approximation is justified when the time between two collisions of a particle is large compared to its wavelength. Using eqs. (14.129) and (14.130), we obtain and equation for f(X, p), which is nothing but a Boltzmann equation: h

i ∂t + vp · ∇x f(X, p) =

i i h (1 + f(X, p))Σ+− − f(X, p)Σ−+ , (14.131) 2Ep {z } |

❈

p [f;X]

where vp ≡ p/Ep is the velocity vector for particles of momentum p. Note that the Boltzmann equation is spatially local since all the objects it contains are evaluated at the coordinate X, but its right hand side is non local in momentum. The right hand side, p [f; X], is called the collision term. The combination ∂t + vp · ∇x that appears in the left hand side is called the transport derivative. It is zero on any function whose t and x dependence arise only in the combination x − vp t (this is the case for a distribution of non-interacting particles, that move at the constant velocity vp prescribed by their momentum).

❈

c sileG siocnarF

In order to obtain an explicit expression of the collision term, it is necessary to truncate the self-energies to a certain order (usually, the lowest order that gives a non-zero result) in the loop expansion. In a scalar theory with a φ4 interaction, the self-energies should be evaluated at two-loops, Σ=

.

(14.132)

14. Q UANTUM FIELD THEORY AT FINITE TEMPERATURE

499

Using the Feynman rules of the Schwinger-Keldysh formalism, this diagram leads to the following collision term

❈p [f; X]

=

λ2 4Ep

Z

d3 p1 d3 p2 d3 p3 (2π)4 δ(p−p1 −p2 −p3 ) 3 3 (2π) 2E1 (2π) 2E2 (2π)3 2E3 h × f(X, p1 )f(X, p2 )(1 + f(X, p3 ))(1 + f(X, p))] i −f(X, p3 )f(X, p)(1 + f(X, p1 ))(1 + f(X, p2 )) . (14.133)

The expression describes the rate of change of the particle distribution, under the effect of 2-body elastic collisions. It is the difference between a production rate (coming from the term in which the particle of momentum p is produced, and thus weighted by a factor 1 + f(X, p)) and a destruction rate (from the term in which the particle of momentum p is destroyed, and has a weight f(x, p)). To close this section, let us mention an additional term that arises when the selfenergy contains a local part, i.e. a term proportional to a delta function in space-time: Σ(u, v) = Φ(u)δc (u − v) + Π(u, v)

(14.134)

When such a local term is present, the difference of the two Kadanoff-Baym equations contains Φ(y)G(x, y) − Φ(x)G(x, y), whose Wigner transform at lowest order in the gradient approximation is

i ∂X Φ(X) · ∂p G(X, p) . (14.135) This extra term leads to a somewhat modified Boltzmann equation, h

i i i h 1 ∂X Φ · ∂p f = (1 + f) Π+− − f Π−+ . (14.136) ∂ t + vp · ∇x f + 2Ep 2Ep

In the new term (underlined), one may interpret ∂X Φ as a mean force field acting on the particles. Under the action of this force, the particles accelerate which implies a change of their momentum. The left hand side of the above equation thus describes the change of the distribution of particles under the effect of this mean field, in the absence of any collisions (that are described by the right hand side).

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Chapter 15

Strong fields and semi-classical methods 15.1 Introduction Until now, all our discussion of quantum field theory has been centered on an expansion about the vacuum, i.e. on situations involving a system with few particles. This is also a regime in which the fields are in a certain sense1 small. The connection between the field amplitude and the density of particles in a state may be grasped by writing the LSZ reduction formula that gives the expectation value of the number operator for a system whose initial state is Φin . By mimicking the derivation of the section (1.4), one obtains easily

 Φin a†p,out ap,out Φin

 Φin φ(x)φ(y) Φin

=

=

1 Z

Z

d4 xd4 y eip·(x−y) (x +m2 )(y +m2 ) 

× Φin φ(x)φ(y) Φin Z   Dφ± (z) φ− (x)φ+ (y) ei (S[φ+ ]−S[φ− ]) ,

(15.1)

where in the second line we have sketched the path integral representation of the matrix element that appears in the reduction formula. Note that, since there is no time ordering in this matrix element, the Schwinger-Keldysh formalism must be used here. This formula is only a sketch, because the boundary conditions of the path 1 When we talk of small or large fields, we are referring to the magnitude of the c-number field in a path integral (it does not make sense to apply these qualifiers to the field operator itself).

501

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

integral at the time should be precised in order to properly account for the initial  initial state Φin . However, what we want to illustrate with these formulas is the direct relationship between large particle occupation numbers (the left hand side of the first equation), and large fields in a path integral. Moreover, in the path integral, the magnitude of the fields is controlled by the boundary conditions (this is the only thing that depends on the initial state of the system in the right hand side of the second equation). There is an implicit assumption of weak fields in the perturbative machinery that we have studied so far, which is best viewed in the path integral formalism. For instance, in the second of eqs. (15.1), the perturbative expansion amounts to writing S = S0 + Sint , and to expand the exponentials in powers of Sint . In a scalar field theory with a quartic coupling, the interaction part of the action reads Z λ Sint [φ] = − d4 x φ4 (x) , (15.2) 4! while the free action (that we keep inside the exponential) is given by Z i h 1 S0 [φ] = d4 x (∂µ φ)(∂µ φ) − m2 φ2 . 2

(15.3)

The common justification of the perturbative expansion is that, when the coupling constant λ is small, we have Sint ≪ S0 . However, since S0 [φ] is quadratic in the field while Sint [φ] contains higher powers of φ, this inequality may not be true if the field is large, even at weak coupling. In order to make this statement more precise, we must account for the fact that the field has mass dimension 1. Let us denote by Q the typical momentum scale in the problem under consideration (for simplicity we assume that there is only one), and then we write φ(x) ∼ ϑ Q ,

(15.4)

where ϑ is a dimensionless number that encodes the order of magnitude of the field. Naive dimensional analysis tells us that (∂µ φ)(∂µ φ) ∼ ϑ2 Q4 , λφ4

∼ λ ϑ4 Q4 .

(15.5)

For the interaction term to be small compared to the kinetic term, we must have λ ϑ2 ≪ 1 ,

(15.6)

which is slightly different from the usual criterion of small λ, since this condition depends on the field magnitude via ϑ. The purpose of this chapter is to explore situations of weak coupling (i.e. λ ≪ 1) where the inequality (15.6) is not satisfied because of strong fields. We call this the strong field regime of quantum field theory. We will discuss two main situations where strong fields may occur:

503

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS • The initial state is a highly occupied state, such as a coherent state. c sileG siocnarF

• The initial state is the ground state, but the system is driven by a strong external source. As we shall see, since the coupling constant is assumed to be small, there is nevertheless a loop expansion, but each loop order (including the tree level approximation) is non-perturbative in a sense that we will clarify in the rest of the chapter.

15.2 Expectation values in a coherent state In the section 1.16.5, we have presented the Schwinger-Keldysh formalism, that allows

theevaluation of expectation values of an observable in the in- vacuum state, 0in O 0in . In the previous chapter, we have generalized this technique to expectation values in a thermal state, i.e. a mixed state whose density matrix is the canonical equilibrium one, ρ ≡ exp(−β H). Another generalization, that we shall consider in this section, is to consider an expectation value in a coherent state, which may be defined from the perturbative in-vacuum as follows

Z    d3 k † χin ≡ Nχ exp 0in , (15.7) χ(k) a k,in (2π)3 2Ek

where χ(k)

is afunction of 3-momentum and Nχ a normalization constant adjusted so that χin χin = 1. From the canonical commutation relation   (15.8) ap,in , a†q,in = (2π)3 2Ep δ(p − q) , it is easy to check the following identity

  ap,in χin = χ(p) χin ,

Z 2  2 d3 k Nχ = exp − . χ(k) (2π)3 2Ek

(15.9)

 The first equation tells us that χin is an eigenstate of annihilation operators, which is another definition of coherent states, and the second one provides the value of the normalization constant. The occupation number in the initial state is closely related to the function χ(k). Indeed, we have 

† 2 (15.10) χin ap,in ap,in χin = |χ(p)| .

In other words, the number of particles in the mode of momentum p is the squared modulus of the function χ(p). A large χ thus corresponds to a highly occupied initial state (at the opposite, χ(p) ≡ 0 corresponds to the vacuum).

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Consider now the generating functional for the extension of the Schwinger-Keldysh formalism in this coherent state,

Zχ [j] ≡



=

χin P exp i

χin P exp i

Z

ZC

C

 d4 x j(x)φ(x) χin

h i  d4 x Lint (φin (x)) + j(x)φin (x) χin ,

(15.11)

where j(x) is a fictitious source that lives on the closed-time contour C introduced in the figure 1.4. As usual, the first step is to factor out the interactions as follows: Z Z  δ   4 χin P exp i d4 x j(x)φin (x) χin . (15.12) Zχ [j] = exp i d x Lint iδj(x) C C | {z } Zχ0 [j]

A first application of the Baker-Campbell-Hausdorff formula enables one to remove the path ordering, which gives Z  Zχ0 [j] = χin exp i d4 x j(x)φin (x) χin C

1Z   4 × exp − d xd4 y j(x)j(y) θc (x0 − y0 ) φin (x), φin (y) , 2 C (15.13)

where θc (x0 − y0 ) generalizes the step function to the ordered contour C. Note that the factor on the second line is a commuting number and thus can be removed from the expectation value. A second application of the Baker-Campbell-Hausdorff formula allows to normal-order the first factor. Decomposing the in-field as follows, φin (x) ≡

Z d3 k d3 k −ik·x + a e a† e+ik·x , k,in 3 (2π) 2Ek (2π)3 2Ek k,in {z } | {z } |

Z

(−)

φin

(x)

(+)

φin

(15.14)

(x)

we obtain the following expression for the free generating functional

Z 

Z  

(+) (−) Zχ0 [j] = χin exp i d4 x j(x)φin (x) exp i d4 y j(y)φin (y) χin C C

1Z   (+) (−) × exp + d4 xd4 y j(x)j(y) φin (x), φin (y) 2 C

1Z   × exp − d4 xd4 y j(x)j(y) θc (x0 − y0 ) φin (x), φin (y) . (15.15) 2 C

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

505

The factor of the first line can be evaluated by using the fact that the coherent state is an eigenstate of annihilation operators:

Z 

Z  

(+) (−) 4 χin exp i d x j(x)φin (x) exp i d4 y j(y)φin (y) χin C C Z  

Z 3 d k −ik·x ∗ +ik·x χ(k)e + χ (k)e . = exp i d4 x j(x) (2π)3 2Ek C | {z } Φχ (x)

(15.16)

We denote Φχ (x) the field obtained by substituting the creation and annihilation operators of the in-field by χ∗ (k) and χ(k) respectively. Note that this is no longer an operator, but a (real valued) c-number field. Moreover, because it is a linear superposition of plane waves, this field is a free field: (x + m2 ) Φχ (x) = 0 .

(15.17)

The second and third factors of eq. (15.15) are commuting numbers, provided we do not attempt to disassemble the commutators. Using the decomposition of the infield in terms of creation and annihilation operators, and the canonical commutation relation of the latter, we obtain    (+)  (−) θc (x0 − y0 ) φin (x), φin (y) − φin (x), φin (y) Z d3 k e−ik·(x−y) = θc (x0 − y0 ) (2π)3 2Ek Z d3 k 0 0 + θc (y − x ) e+ik·(x−y) , (2π)3 2Ek | {z } G0 c (x,y)

(15.18)

which is nothing but the usual bare path-ordered propagator G0c (x, y). Collecting all the factors, the generating functional for path-ordered Green’s functions in the Schwinger-Keldysh formalism with an initial coherent state reads

Z  δ 

Z  Zχ [j] = exp i d4 x Lint exp i d4 x j(x) Φχ (x) iδj(x) C C

1Z  × exp − d4 xd4 y j(x)j(y) G0c (x, y) . (15.19) 2 C It differs from the corresponding functional with the perturbative vacuum2 as initial state only by the second factor, that we have underlined. This generating functional is 2 The

vacuum initial state corresponds to the function χ(k) ≡ 0, i.e. to Φχ (x) = 0.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

also equal to3 Zχ [j] =

Z 

Z  δ  exp i d4 x j(x) Φχ (x) exp i d4 x Lint Φχ (x) + iδj(x) C C

1Z  × exp − d4 xd4 y j(x)j(y) G0c (x, y) . (15.20) 2 C

The first factor has the effect of shifting the fields by Φχ (x). The simplest way to see this is to write φ ≡ Φχ + ζ .

(15.21)

In the definition (15.11), this leads to Z

Z   4 Zχ [j] = exp i d j(x) Φχ (x) χin P exp i d4 x j(x) ζ(x) χin , (15.22) C

C

where the second factor in the right hand side is the generating functional for correlators of ζ. Comparing with eq. (15.20), we see that the generating functional for ζ is identical to the vacuum one, except that the argument φ of the interaction Lagrangian is replaced by Φχ + ζ: Lint (φ)

→

Lint (Φχ + ζ) .

(15.23)

In other words, the field ζ appears to be coupled to a background field Φχ . For instance, for a φ4 interaction term, we have Lint (Φχ + ζ) = −λ

ζ4 4!

+

ζ3 Φχ ζ2 Φ2χ ζΦ3χ Φ4χ  . + + + 8 4 8 4!

(15.24)

The first term, in ζ4 , gives the usual four-leg vertex in the Feynman rules, and the following terms describe the interactions of ζ with the background field Φχ . The last term plays no role since it does not contain the quantum field ζ. Except for the appearance of these new vertices that involve a background field, the Feynman rules are the same as in the Schwinger-Keldysh formalism for a vacuum initial state, with + and − vertices, and bare propagators G0++ , G0+− , G0−+ and G0−− to connect them. In summary, replacing the vacuum initial state by a coherent state amounts to extend the usual Schwinger-Keldysh formalism with a background field Φχ . c sileG siocnarF

As in eq. (15.4), let us assume for the purpose of power counting that Φχ ∼ ϑQ , 3 In

this transformation, we use the functional analogue of F(∂x ) eαx G(x) = eαx F(α + ∂x ) G(x) .

(15.25)

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15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

Figure 15.1: Vertices that appear in the perturbative expansion for the calculation of expectation values with a coherent initial state. The circled cross denotes the field Φχ .

and consider a connected graph G made of nE external lines, nI internal lines, n1 vertices ζΦ3χ , n2 vertices ζ2 Φ2χ , n3 vertices ζ3 Φχ , n4 vertices ζ4 , and nL loops. These parameters are related by the following two identities: nE + 2nI = 4n4 + 3n3 + 2n2 + n1 , nL = nI − (n1 + n2 + n3 + n4 ) + 1 .

(15.26)

Then, the order in λ and ϑ of this graph is given by G

∼ λn1 +n2 +n3 +n4 ϑ3n1 +2n2 +n3 √
3n1 +2n2 +n3 λϑ . ∼ λnL −1+nE /2

(15.27)

The first factor is nothing but the usual order in λ of a connected graph with nE external lines and nL loops. The second factor counts the number of insertions (3n1 + 2n2 + √ n3 ) of the background field Φχ . Interestingly, it involves only the combination λ ϑ, that appears also in the inequality (15.6) that delineates the strong field regime. From eq. (15.27), we can draw the following conclusions: • When λϑ2 ≪ 1, i.e. in the weak field regime, we can make a double perturbative expansion in λ and in ϑ (i.e. in the occupation of the initial coherent state). Leading order results correspond to tree diagrams with zero (or the minimal number necessary for the observable under consideration to be non-zero) insertions of the background field. • When λϑ2 & 1, i.e. in the strong field regime, the expansion in powers of λ is still possible (and is organized by the number of loops in the graphs). But the expansion in powers of the background field becomes illegitimate, and one should instead treat Φχ to all orders. As we shall see now, this leads to important modifications in the calculation of observables in the strong field regime.

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Note that for a system prepared in a coherent initial state, it is the function χ(k) that defines the coherent that determines whether we are in the weak or strong field regime. In order to illustrate the changes to the perturbative expansion in the strong field regime, let us consider a very simple observable, the expectation value of the field operator,  



Φ(x) ≡ χin φ(x) χin = Φχ (x) + χin ζ(x) χin . (15.28)

The beginning of the diagrammatic representation of Φ(x) at tree level reads:

Φ(x)

=

+

+

+

+

+...

tree

(15.29)

In fact, at tree level, Φ(x) is the sum of all the tree diagrams (weighted by the appropriate symmetry factor) whose root is the point x and whose leaves are the coherent field Φχ . This infinite set of trees can be generated recursively by the following integral representation: Z h i λ  − Φ3 (y) . (15.30) Φ(x) = Φχ (x) + i d4 x G0++ (x, y) − G0+− (x, y) {z } | | 6 {z } G0 (x,y) R

U ′ (Φ(y))

Interestingly, after one has summed over the + and − indices carried by the vertices, the propagators G0++ and G0+− of the Schwinger-Keldysh diagrammatic rules always appear via their difference, which is nothing but the bare retarded propagator: G0++ (x, y) − G0+− (x, y) = G0−+ (x, y) − G0−− (x, y) = G0R (x, y) .

(15.31)

Since this propagator obeys (x + m2 ) G0R (x, y) = −i δ(x − y) ,

G0R (x, y) = 0 if x0 < y0 , (15.32)

the expectation value Φ(x) at tree level satisfies (x + m2 ) Φ(x) + U ′ (Φ(x)) = 0 , lim

x0 →−∞

Φ(x) = Φχ (x) .

(15.33)

In other words, at tree level, the field expectation value obeys the classical field equation of motion, with the boundary value Φχ (x) at the initial time. The nonlinearity of this equation of motion is crucial in the strong field regime, and all the

509

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

terms of the series (15.29) have the same magnitude when λ ϑ2 ∼ 1. Nevertheless, the representation of this series as the solution of the classical field equation of motion with a retarded boundary condition is very useful, since it turns the problem of summing an infinite series of Feynman graphs into the much simpler (at least numerically) problem of solving a partial differential equation. c sileG siocnarF

This result for the expectation value of φ(x) generalizes to the expectation value of any observable built from the field operator: at tree level, its expectation value is obtained by replacing the operator φ(x) by the c-number classical field Φ(x) inside the observable:



= O Φ(x) . (15.34) χin O φ(x) χin tree level

We will defer the study of loop corrections to these expectation values until the section 15.4, because this discussion will be common with another strong field situation that we shall discuss first, namely the case of quantum field theories coupled to a strong external source.

15.3 Quantum field theory with external sources Let us now consider a second way to reach the large field regime. This time, the initial state of the system is the vacuum, but the field is coupled to an external source that drives the system away from the ground state. When the external source is large, the field expectation value will eventually become large itself, and the system will again be in the strong field regime. Let us consider a scalar field theory with quartic interaction coupled to a source J, whose Lagrangian is L≡

1 1 λ (∂µ φ)(∂µ φ) − m2 φ2 − φ4 +Jφ . 2 2 4! | {z }

(15.35)

U(φ)

Although we consider here the example of a φ4 interaction term, we will often write the equations for a generic potential U(φ), and sometimes diagrammatic illustrations will be given for a cubic interaction for simplicity. These more general interactions terms will be defined as λ−1+n/2 Qn−4 φn , where Q is an object of mass dimension 1. The Feynman rules for this theory are the usual ones, with the addition of a special rule for the external current J. In momentum space, a source j attached to the end ˜ (where J˜ is the Fourier of a propagator of momentum p contributes a factor iJ(p) transform of J). The source J(x) is a given function of space-time, fixed once for all. As we shall see shortly, the strong field regime corresponds to large sources J ∼ λ−1/2 – we all call this situation the strong source dense. In contrast, the situation where

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 15.2: Generic connected graph in the strong source regime. In this example, nE = 5, nI = 11, nJ = 4, nL = 1, n3 = 5 and n4 = 2.

the external source J is small is called the weak source dilute. Consider a simply connected diagram (see figure 15.2), with nE external legs, nI internal lines, nL (4) independent loops, nJ sources, and n3 cubic vertices, n4 quartic vertices, etc... These parameters are not all independent. First, the number of propagator endpoints should match the available sites to which they can be attached. This leads to a first identity, nE + 2nI = nJ + 3n3 + 4n4 + 5n5 + · · ·

(15.36)

A second identity expresses the number of independent loops in terms of the other parameters, nL = nI − (n3 + n4 + n5 + · · · ) − nJ + 1 .

(15.37)

Thanks to these two relations, the order of a diagram G can be written as 1

3

G ∼ JnJ λ 2 n3 +n4 + 2 n5 +··· = λnL −1+NE /2

√


n λJ J .

(15.38)

This formula is very similar to eq. (15.27). First, it does not depend on the number of vertices and on the number of internal lines; only the number of external legs, the number of loops and the number of sources appear in the result. The strong source regime√is the regime where it is not legitimate to expand in powers of J because the factor λ J is not small. In this case, the order of a diagram does not depend on its number of sources, and an infinite number of diagrams –with fixed nE and nL but arbitrary nJ – contribute at each order.

15.4 Observables at LO and NLO Leading order : Let us consider an observable O(φ), possibly non-local but with fields only at the same time tf (the discussion could be generalized to fields with only space-like separations). At leading order in the strong field regime. As we

511

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

Figure 15.3: The two contributions to observables at NLO in the strong field regime.

x

x

y

have seen in the previous sections, this can be achieved by the presence of strong external sources, or by starting from a highly occupied coherent state. In both case, the calculation of expectation values is done with the Schwinger-Keldysh formalism. Note that since the field operators in the observable are taken at equal times, they commute and the result does not depend on the + or − assignments for those fields. But it is crucial to sum over all the ± indices in the internal vertices of the graphs. At leading order in λ, its expectation value is obtained by simply replacing the field operator φ by the solution Φ of the classical equations of motion,

 O(φ) LO = O(Φ) , (15.39)

with

(x + m2 )Φ + U ′ (Φ) = J , lim Φ(x) = Φχ (x) .

x0 →ti

(15.40)

(We have combined in a single description the two situations, with an external source  J and starting from a non-trivial coherent state χin .) Note that it is the internal sums over the ± indices of the Schwinger-Keldysh formalism that lead to retarded boundary conditions, by virtue of eq. (15.31). Next-to-leading order : For such an observable, the corresponding next-to-leading order correction can be formally written as follows, Z Z

 δO(Φ) 1 δ2 O(Φ) O(φ) NLO = d3 x δΦ(x) + , d3 xd3 y G(x, y) δΦ(x) 2 tf δΦ(x)δΦ(y) tf (15.41) where δΦ is the 1-loop correction to the classical field Φ, and G(x, y) is the propagator dressed by the background field Φ. The two contributions of eq. (15.41) are illustrated in the figure 15.3. Since the fields operators in the observable O(φ) are all separated by space-like intervals, it is not necessary to indicate the ± indices in δΦ and G, and we have in fact: c sileG siocnarF

δΦ+ (x) = δΦ− (x) , G++ (x, y) = G−− (x, y) = G−+ (x, y) = G+− (x, y) if (x − y)2 < 0 . (15.42)

512

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Let us start with δΦ± . The propagators in the diagram on the left of the figure 15.3 are the Schwinger-Keldysh propagators in the presence of a background field Φ, i.e. the propagators Gǫǫ ′ . For a generic interaction potential, we can write δΦ± (x) as follows: Z i X (15.43) δΦǫ (x) = − d4 z ǫ′ Gǫǫ′ (x, z) U′′′ (Φ(z)) Gǫ′ ǫ′ (z, z) . 2 ′ ǫ =±

In this formula, the 1/2 is a symmetry factor, the factor ǫ′ in the integrand takes into account the fact that vertices of type − have an opposite sign in the SchwingerKeldysh formalism, and the factor −i U′′′ (Φ(z)) is the general form of the 3-particle vertex in the presence of an external field (for an arbitrary interaction potential U). Thus, we have reduced the calculation to that of the 2-point functions G±± . These four propagators are defined recursively by the following equations : X Z η d4 z G0ǫη (x, z) U′′ (Φ(z)) Gηǫ′ (z, y) . (15.44) Gǫǫ′ (x, y) = G0ǫǫ′ (x, y)−i η=±

Here, −i U′′ (Φ(z)) is the general form for the insertion of a background field on a propagator in a theory with potential U(Φ). From these equations, we obtain the following equations : 

 x +m2 +U′′ (Φ(x)) G+− (x, y) =   x +m2 +U′′ (Φ(x)) G−+ (x, y) =



 y +m2 +U′′ (Φ(y)) G+− (x, y) = 0 ,   y +m2 +U′′ (Φ(y)) G−+ (x, y) = 0 . (15.45)

In addition to these equations of motion, these propagators must become equal to their free counterparts G0+− and G0−+ when x0 , y0 → −∞. From the definition of the various components of the Schwinger-Keldysh propagators, G++ and G−− are given in terms of G+− and G−+ by the following expressions: G++ (x, y) =

θ(x0 − y0 ) G−+ (x, y) + θ(y0 − x0 ) G+− (x, y) ,

G−− (x, y) =

θ(x0 − y0 ) G+− (x, y) + θ(y0 − x0 ) G−+ (x, y) .

(15.46)

The above conditions determine G+− and G−+ uniquely. In order to find these propagators, let us recall the following representation of their bare counterparts : G0+− (x, y) = G0−+ (x, y) =

Z

Z

d3 p a−p (x)a+p (y) , (2π)3 2Ep d3 p a+p (x)a−p (y) , (2π)3 2Ep

(15.47)

513

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS where (x +m2 ) a±p (x) = 0 ,

lim

x0 →−∞

a±p (x) = e∓ip·x .

(15.48)

It is trivial to generalize this representation of the off-diagonal propagators to the case of a non zero background field, by writing G+− (x, y) = G−+ (x, y) =

Z

Z

d3 p a−p (x)a+p (y) , (2π)3 2Ep d3 p a+p (x)a−p (y) , (2π)3 2Ep

(15.49)

with 

 x +m2 +U′′ (Φ(x)) a±p (x) = 0 ,

lim

x0 →−∞

a±p (x) = e∓ip·x . (15.50)

By construction, these expressions of G+− and G−+ obey the appropriate equations of motion, and go to the correct limit in the remote past. The functions a±p (x) are sometimes called mode functions. They provide a complete basis for the linear space of solutions of the equation (15.50), i.e. the space of linearized perturbations to the classical solution of the field equation of motion. Relationship between LO and NLO : At this point, we have all the building blocks in order to obtain the single inclusive spectrum at NLO. One can go further and obtain a formal relationship between the LO and NLO inclusive spectra. A key observation for this is that the functions ak that appear in the dressed propagators G±∓ can be obtained from the classical field Φ as follows:

❚±k Φ(x) , where the operator ❚±k is defined by a±k (x) =

❚±k · · ·

≡

Z

(15.51)

d3 u e∓ik·u

u0 =−∞

×

❚

h

δ δΦini (u)

∓ iEk

i δ . (15.52) · · · δ(∂0 Φini (u)) Φini ≡Φχ

In words, the operator ±k in eq. (15.51) differentiates the classical field Φ with respect to its initial condition Φini , and replaces it by the initial condition of a±k . Since a±k is a linear perturbation to Φ, this indeed gives the correct result. c sileG siocnarF

514

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS Thus, the propagator G+− (x, y) that enters at NLO can be written as Z h ih i d3 k Φ(x) Φ(y) . G+− (x, y) = −k +k (2π)3 2Ek

❚

❚

(15.53)

In the rest of our NLO calculation, we only need this propagator for a space-like separation between x and y, which implies that G+− (x, y) = G−+ (x, y). In this case, we can symmetrize the expression of the propagator as follows: G+− (x, y) =

Z

h ih i 1 d3 k Φ(x) Φ(y) −k +k 2 (2π)3 2Ek h ih i + +k Φ(x) . −k Φ(y)

❚

❚

❚

❚

(15.54)

As we shall see now, a similar expression can be obtained for δΦ± . Let us start from eq. (15.43). Since the propagators G++ and G−− are equal when the two endpoints are evaluated at equal times, we have i − 2

δΦǫ (x) =

Z

h i d3 k 4 z G (x, z) − G (x, z) d ǫ+ ǫ− | {z } (2π)3 2Ek GR (x, z) ×U′′′ (Φ(z)) a−k (z)a+k (z) ,

(15.55)

where GR is the retarded propagator in the presence of the background field Φ. By writing more explicitly the interactions with the background field, Z h δΦǫ (x) = −i d4 y G0R (x, y) U′′ (Φ(y)) δΦǫ (y) Z i d3 k 1 ′′′ + U (Φ(y)) a∗k (y)ak (y) , (15.56) 3 2 (2π) 2Ek (with G0R the bare retarded propagator), one may prove that δΦǫ (x) =

1 2

Z

d3 k (2π)3 2Ek

❚+k ❚−k Φ(x) .

(15.57)

By inserting this expression, as well as eq. (15.54), in eq. (15.41), we can write the NLO expectation value as follows, " Z #



 1 d3 k O NLO = O LO . (15.58) +k −k 3 2 (2π) 2Ek

❚ ❚

This central result is illustrated in the figure 15.4. Some remarks should be made about this formula:

515

quantum

classical

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

Figure 15.4: Illustration of eq. (15.58). The open squares represent the operator k (u) −k (v). Their action is to remove two instances of the initial classical field (the open circles), and to connect them with the light colored link to form a loop.

❚

❚

i. In this formula, the LO observable that appears in the right hand side must be considered as a functional of the initial classical field. ii. The LO and NLO observables cannot be obtained in closed analytical form, because they contain the classical field Φ – retarded solution of a non-linear partial differential equation that cannot be solved analytically in general. Nevertheless, eq. (15.58) is an exact relationship between the two.

Why is the NLO “nearly classical”? : In a sense, eq. (15.58) indicates that observables at NLO in the strong field regime are almost classical, since they can be obtained from the LO result (that depends only on the classical field Φ) by acting with the operators ±k (i.e. derivatives with respect to the initial value of the classical field). If one had kept track of the powers of h, ¯ the h ¯ that comes at NLO would just be an overall prefactor (the prefactor 1/2 in eq. (15.58) would become h/2), ¯ but all the rest of the formula would not contain any h. ¯

❚

This is in fact not specific to the strong field regime nor to quantum field theory, but is a general property of quantum mechanics. To see this, consider a generic quantum system of Hamiltonian H and density operator ρt . The latter evolves according to the Liouville-von Neumann equation: ih ¯

 ∂ρt  = H, ρt . ∂t

(15.59)

516

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

The next step is to introduce the Wigner transforms of the density operator: Z

s s Wt (x, p) ≡ ds eip·s x + ρt x − . (15.60) 2 2

The Wigner transform of an operator is a Fourier transform of the matrix elements of the operator in the position basis with respect to the difference of coordinates. The function Wt (x, p) may be viewed in a loose sense4 as a probability distribution in the classical phase-space of the system (x and p are classical variables, not operators). Note that the Wigner transform of the Hamiltonian operator H is the classical Hamiltonian H. On may show that the Liouville-von Neumann equation is equivalent to c sileG siocnarF

∂Wt ∂τ

= =

  ← →
ih ¯ ← → 2 H(x, p) sin Wt (x, p) ∂p ∂x − ∂x ∂p ih ¯ 2  + O(¯h2 ) H, Wt | {z } Poisson bracket

(15.61) (15.62)

The first line is an exact equation, known as the Moyal-Groenewold equation. In the second line, we have performed an expansion in powers of h, ¯ and one can readily ¯ is nothing but the classical Liouville equation (it thus see that the order zero in h describes a system whose time evolution is classical). The first quantum correction to the time evolution arises only at the order h ¯ 2 . Therefore, at the order h ¯ (i.e. NLO in the language of quantum field theory), the time evolution of the system remains purely classical. This does not mean that there are no quantum corrections of order h, ¯ but that these corrections can only come from the initial state of the system (in particular, from the fact that a quantum system cannot have well defined x and p at the same time, and the Wigner distribution Wt (x, p) must have a width of order h ¯ at least). The effect of the operator in +k −k that acts on the LO in eq. (15.58) is precisely to restore this quantum width of the initial state.

❚ ❚

15.5 Green’s formulas Eq. (15.51), that formally relates a small field perturbation to the background field on top of which it propagates, plays a crucial role in discussing many questions related to strong fields. A standard proof of this formula relies on Green’s formulas, that we shall discuss in this section. 4 W is not a bona fide probability distribution, because it is not positive definite in general. But the t regions of phase-space where it is negative are small, typically of order h. ¯ After being integrated either over x or over p, it becomes a genuine probability distribution for the expectation values of p or x, respectively.

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

15.5.1

517

Green’s formula for a retarded classical scalar field

Consider the following partial differential equation5 , (x + m2 )Φ(x) + U′ (Φ(x)) = j(x) .

(15.63)

Since this equation contains time derivatives up to second order, it is necessary to specify the initial value of Φ itself as well as that of its first time derivative. Let us assume that we know these values on the surface t = 0. We wish to obtain a formula for Φ(x) at a time x0 > 0 in terms of this initial data. In order to do this, we must introduce the retarded Green’s function of the operator x + m2 , defined by (x + m2 ) G0R (x, y) = −iδ(x − y) , G0R (x, y) = 0 if x0 < y0 .

(15.64)

(The superscript 0 is a reminder of the fact that this is a free Green’s function, that does not depend on the interaction potential U(Φ).) Note that G0R (x, y) obeys the same equation if acted upon with y + m2 instead. From the equations obeyed by Φ and by G0R , we obtain h i
→ G0R (x, y) y +m2 Φ(y) = G0R (x, y) j(y) − U′ (Φ(y)) ,
← G0R (x, y) y +m2 Φ(y) = −iδ(x − y)Φ(y) , (15.65)

where the arrows on the d’Alembertian operators indicate on which side they act. By integrating these equations over y above the initial surface t = 0, and by subtracting them, we get the following relation Z h← i → Φ(x) = i d4 y G0R (x, y) (y − y )Φ(y) + j(y) − U′ (Φ(y)) . (15.66) y0 >0

The last step is to show that the term that involves the difference between the two d’Alembertian operators is in fact a boundary term that depends only on the initial conditions. Note first the following identity, ←

←

→

→

A( − )B = ∂µ A( ∂ µ − ∂ µ )B ,

(15.67)

where the leftmost ∂µ acts on everything on its right. In other words, the left hand side is a total derivative, and its integral over d4 y can be rewritten as a surface integral thanks to Stokes’ theorem. The integration domain defined by y0 > 0 has three boundaries: 5 This

equation is the classical equation of motion in the scalar field theory of Lagrangian L≡

1 1 (∂µ φ)(∂µ φ) − m2 φ2 − U(φ) + jφ . 2 2

518

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Figure 15.5: Typical contribution to Φ(x) in the diagrammatic representation of eq. (15.68), in the case of cubic interactions. The solid dots represent the sources j, the open circles represent the initial value of the field or field derivatives on the surface y0 = 0. The lines are retarded propagators G0R .

x

y0 = 0

i. y0 = +∞ : this boundary at infinite time does not contribute, since the retarded propagator obeys G0R (x, y) = 0 if y0 > x0 . ii. y0 = 0 : this boundary gives a non zero contribution, that depends only on the initial conditions for the field Φ. iii. Boundary at spatial infinity : this boundary does not contribute if we assume that the field vanishes when |x| → ∞, or for a finite volume with periodic boundary conditions in the spatial directions. Therefore, we obtain Φ(x) =

i

Z

y0 >0

+i

Z

h i d4 y G0R (x, y) j(y) − U′ (Φ(y)) →

←

d3 y G0R (x, y)( ∂ y0 − ∂ y0 )Φ(y) .

(15.68)

y0 =0

In this Green’s formula, the first term in the right hand side provides the dependence on the source j, and on the interactions, while the second term tells us how Φ(x) depends on the initial values of Φ(x) and of its first time derivative. c sileG siocnarF

Except in the trivial case where the potential U(Φ) is zero, eq. (15.68) does not provide an explicit result for Φ(x), since the right hand side depends on Φ(y) at points above the initial surface. Despite this limitation, this is a very useful tool in order to perform formal manipulations involving retarded solutions of eq. (15.63). To end this section, let us mention a diagrammatic interpretation of eq. (15.68), illustrated in figure 15.5. One can expand the right hand side of eq. (15.68) in powers of the interactions. The starting point is the zeroth order approximation, obtained by setting

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

519

the potential to U = 0, and then by proceeding recursively in order to keep higher orders in U. The outcome of this expansion is an infinite series of terms that have a tree structure. The root of this tree is the point x where the field is evaluated, and its leaves are either sources j (if there are any above the surface y0 = 0) or the initial data on the surface y0 = 0. In particular, if the source j(x) vanishes at y0 > 0, then all the j dependence of the classical field is implicitly hidden in the Φ(y) that appears in the boundary term. Extension to a generic initial surface : In eq. (15.68), the initial conditions for the field Φ have been set on the surface of constant time y0 = 0. However, there are many situations in which this initial data is known on a different initial surface. Let us consider a generic surface Σ, on which the field Φ and its derivatives are known. As before, we wish to obtain a formula that expresses Φ(x) at some point x above Σ in terms of these initial conditions on Σ. Most of the derivation is identical to the case of a constant time initial surface, with all the integrals over the domain y0 > 0 replaced by integrals over the domain Ω located above Σ. The only significant change occurs when we apply Stokes’ theorem in order to transform the 4-dimensional integral of a total derivative into an integral over the boundary of Ω. Like in the previous case, the boundaries at infinite time, and at infinity in the spatial directions do not contribute, and we have only a contribution from the surface Σ. Stokes’ theorem can then be written as Z Z 4 µ d y ∂µ F (y) = − d3 Sy nµ Fµ (y) , (15.69) Ω

Σ

where d3 Sy is the measure on the surface Σ, and nµ is a 4-vector normal to the surface Σ at the point y, pointing above the surface Σ. In the important case where the initial surface is invariant by translation in the transverse directions, the proper normalization for nµ and d3 Sy can be obtained as follows. Parameterize an arbitrary displacement dyµ on the surface Σ about the point y as dyµ = (βdy3 , dy1 , dy2 , dy3 ), where β is the local slope of the surface Σ in the (y3 , y0 ) plane. Then, we have: nµ dyµ = 0 , nµ nµ = 1 , n 0 > 0 , p d3 Sy = 1 − β2 dy1 dy2 dy3 .

(15.70)

The second and third conditions require to have β < 1 in order to make sense. This implies that the surface Σ must be locally space-like. Physically, this means that a signal emitted from a point of the surface Σ cannot reach the surface again in the future. The relations (15.70) are illustrated in figure 15.6. Note that the orthogonality defined by nµ dyµ = 0 does not correspond to the Euclidean concept of orthogonality.

520

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

nµ

Ω

dyµ

Σ

y

Figure 15.6: Illustration of eqs. (15.70).

Thanks to eq. (15.69), it is possible to write the Green’s formula for an arbitrary initial surface Σ as Z h i Φ(x) = i d4 y G0R (x, y) j(y) − U′ (Φ(y)) Ω

Z → ← +i d3 Sy G0R (x, y)(n· ∂ y −n· ∂ y )Φ(y) .

(15.71)

Σ

For an arbitrary surface Σ, the second term in the right hand side of this formula tells us explicitly what information about Φ we must provide on the initial surface in order to determine it uniquely above the surface: at every point y ∈ Σ, one must specify the values of the field Φ(y) and of its normal derivative n · ∂y Φ(y).

15.5.2

Green’s formula for small field perturbations

Consider now a small perturbation a(x) to the classical field, and assume that a(x) ≪ Φ(x). Therefore, one can linearize the equation of motion of a(x), and we get h i x + m2 + U′′ (Φ(x)) a(x) = 0 . (15.72)

Treating the term U′′ (Φ(x))a(x) as an interaction, we can easily derive a Green’s formula that expresses the field fluctuation a(x) in terms of its initial conditions on a surface Σ, Z h i a(x) = i d4 y G0R (x, y) − U′′ (Φ(y))a(y) Ω

Z

→

←

+i d3 Sy G0R (x, y)(n· ∂ y −n· ∂ y )a(y) .

(15.73)

Σ

Eq. (15.73) is illustrated in the figure 15.7. Every diagram contributing to a(x) has

521

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

Figure 15.7: Typical contribution to a(x) in the diagrammatic representation of eq. (15.73), in the case of cubic interactions. The solid dots represent the sources j, the open circles represent the initial data for Φ(y) on the surface y0 = 0, and the open square the initial data for a(y). The lines are retarded propagators G0R . The dashed line is the retarded propagator of the fluctuation in the background Φ, i.e. an inverse of the operator  + U′′ (Φ).

x

y0 = 0

exactly one instance of the initial value of a(y) (represented by an open square in the figure) on the initial surface. Indeed, it is easy to see from eq. (15.73) that a(x) depends linearly on its value a(y) on the initial surface. This is a consequence of the fact that equation of motion for a small fluctuation is a linear equation. c sileG siocnarF

By comparing the figures 15.5 and 15.7, one sees that they differ only by the fact that one instance of the field Φ(y) has been replaced by the small fluctuation a(y) on the initial surface. Therefore, we expect a linear relationship between a(x) and Φ(x), of the form a(x) =

❚

❚a Φ(x) ,

(15.74)

where a is a linear operator that substitutes one power of Φ(y) by a(y) on Σ (i.e. an operator that involves first derivatives with respect to the initial conditions on Σ). It is easy to prove this relation by using eqs. (15.71) and (15.73). In order to do so and at the same time determine the form of the operator a , let us apply a to the Green’s formula that gives Φ(x). We get6

❚

❚a Φ(x)

=

i

Z

Ω

+i

h d4 y G0R (x, y) − U′′ (Φ(y))

❚a

Z

→

i

❚a Φ(y) ←

d3 Sy G0R (x, y)(n· ∂ y −n· ∂ y )Φ(y) .

Σ 6 Since

❚

❚a acts only on the initial fields on Σ, we have ❚a j = 0.

(15.75)

522

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

If the boundary term in this formula can be made identical to the boundary term in the Green’s formula for a(x), then this equation will be identical to the Green’s formula for a(x) and we will have proven the announced relationship between a(x) and Φ(x). This is the case if the operator a is chosen as   Z δ δ 3 , (15.76) + (n · ∂a(y)) a ≡ d Sy a(y) δΦ(y) δ(n · ∂Φ(y))

❚

❚

Σ

which is nothing but the operator that substitutes a(y) to Φ(y) on the initial surface Σ, as announced (this definition generalizes eq. (15.52) to a generic initial surface and to generic initial conditions for the perturbation).

❚

Note that a is an operator that performs an infinitesimal translation (by an amount a(y)) to the initial condition of the classical field. By exponentiation, it may be promoted into an operator that performs a finite shift of the initial condition. In particular, if we denote by Φ[Φ0 ] the classical field whose initial value on Σ is Φ0 , then we have e❚a Φ[Φ0 ] = Φ[Φ0 + a] .

15.5.3

(15.77)

Schwinger-Keldysh formalism

It is often useful to obtain Green’s formulas for fields in the Schwinger-Keldysh formalism. In this case, one has a pair of fields Φ± (x), that are both solutions of the classical equation of motion (x + m2 )Φ± (x) + U′ (Φ± (x)) = j(x) .

(15.78)

(For simplicity we take the same source j(x) for the two fields, but this limitation is easily circumvented if necessary.) Since the Schwinger-Keldysh propagator G0++ is also a Green’s function of the operator x + m2 , we can reproduce the previous derivation of Green’s formula, which leads to7 Φ+ (x) =

Z h i i d4 y G0++ (x, y) j(y) − U′ (Φ+ (y)) Z iy0 =+∞ h → ← +i d3 y G0++ (x, y)( ∂ y0 − ∂ y0 )Φ+ (y) , (15.79) y0 =−∞

0

y =b where we used the notation [f(y0 )]y 0 =a ≡ f(b) − f(a). The only difference with the Green’s formula derived with retarded propagators is the boundary term: since 7 Here also, the prefactors i follow from our convention for the propagators of the Schwinger-Keldysh formalism (see eqs. (1.367)).

523

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

G0++ (x, y) does not vanish when y0 > x0 , there is also a non-zero contribution from the boundary at y0 = +∞. Then, by using the fact that (y + m2 )G0+− (x, y) = 0, we obtain in a similar way : Z h i 0 = i d4 y G0+− (x, y) j(y) − U′ (Φ− (y)) Z iy0 =+∞ h → ← +i d3 y G0+− (x, y)( ∂ y0 − ∂ y0 )Φ− (y) . (15.80) y0 =−∞

Subtracting this equation from eq. (15.79), we obtain Φ+ (x) Z h i h i = i d4 y G0++ (x, y) j(y)−U′ (Φ+ (y)) − G0+− (x, y) j(y)−U′ (Φ− (y)) Z h iy0 =+∞ ↔ ↔ , −i d3 y G0++ (x, y) ∂ y0 Φ+ (y) − G0+− (x, y) ∂ y0 Φ− (y) y0 =−∞

(15.81)

↔

→

←

where A ∂ y0 B ≡ A( ∂ y0 − ∂ y0 )B. Similarly, we obtain for Φ− (x) : Φ− (x) Z h i h i = i d4 y G0−+ (x, y) j(y)−U′ (Φ+ (y)) − G0−− (x, y) j(y)−U′ (Φ− (y)) Z iy0 =+∞ h ↔ ↔ . −i d3 y G0−+ (x, y) ∂ y0 Φ+ (y) − G0−− (x, y) ∂ y0 Φ− (y) 0 y =−∞

(15.82)

At this point, these formulas are rather formal, and it is not clear why we have gone through the trouble of subtracting the quantity given by eq. (15.80), since it is identically zero. This will become transparent in the next section, where we show that these formulas enable one to sum series of tree diagrams encountered in the Schwinger-Keldysh formalism. Note also that the only property of the propagators G0−+ and G0+− that we have used in this derivation is the fact that they are annihilated by the operator y . Therefore, the equations (15.81) and (15.82) remain valid if we replace these propagators by any other pair of propagators sharing the same property. For instance, one can replace the propagators G0+− and G0−+ of eqs. (1.367) by the following objects 0

G+− (x, y) = 0

G−+ (x, y) =

Z

d4 p −ip·(x−y) e u(p) 2πθ(−p0 )δ(p2 ) , (2π)4 Z 4 d p −ip·(x−y) e v(p) 2πθ(+p0 )δ(p2 ) , (2π)4

(15.83)

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

where u(p) and v(p) are some arbitrary functions of the momentum p, without altering any of the formulas in this section. We will make use of this freedom in the next section. c sileG siocnarF

15.5.4

Summing tree diagrams using Green’s formulas

Many problems involving strong fields require that one sums infinite series of tree diagrams. These sums of diagrams can in general be expressed in terms of solutions of the classical equations of motion. However, in order to determine them uniquely, one must know the boundary conditions obeyed by these classical solutions. The strategy in order to obtain them is to write the sum of tree diagrams as a recursive integral equation. Then, by comparing this integral equation with a Green’s formula such as eq. (15.68), one can read off the boundary conditions easily. Sum of retarded trees : Let us illustrate this first in the simplest case, where one must sum all the tree diagrams built with retarded propagators, and whose leaves are a source j(x). Let us call Φ(x) the sum of all such tree diagrams. Given the recursive structure of such trees, one can write immediately : Z h i Φ(x) = i d4 y G0R (x, y) j(y) − U′ (Φ(y)) , (15.84) where the integration over d4 y is extended to the entire8 space-time. Therefore, we see that this formula is identical to the Green’s formula (15.68), with the initial surface at y0 = −∞ instead of y0 = 0, and where the boundary term is be identically zero. This means that the sum of these tree diagrams is a retarded solution of the classical equation of motion with a null boundary condition in the remote past (x + m2 )Φ(x) + U′ (Φ(x)) = j(x) , lim

y0 →−∞

Φ(y0 , y) = 0 ,

lim

y0 →−∞

∂0 Φ(y0 , y) = 0 .

(15.85)

Sum of trees in the Schwinger-Keldysh formalism : Consider now a more complicated example, in which one must sum tree diagrams in the Schwinger-Keldysh formalism. Now, each vertex is carrying an index ǫ = ±. For simplicity, we assume that the + and − sources are identical, so that we still have a single source j(x). Because this is necessary in certain applications, we are going to use the modified 0 0 propagators G+− and G−+ defined in eqs. (15.83), instead of the propagators G0+− 0 and G−+ defined in eqs. (1.367) (the propagators G0++ and G0−− are kept unchanged). In addition to summing over all the possible trees, we sum over all the combinations of 8 In

the Feynman rules the integration at each vertex is extended to the full space-time

❘4 .

525

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

± indices at every internal vertex. Firstly, the sum of these trees can be written in the form of two coupled integral equations (there are now two fields Φ± (x) depending on the index carried by the root of the tree) : Z h i Φ+ (x) = i d4 y G0++ (x, y) j(y) − U′ (Φ+ (y)) Z h i 0 −i d4 y G+− (x, y) j(y) − U′ (Φ− (y)) , Z h i 0 Φ− (x) = i d4 y G−+ (x, y) j(y) − U′ (Φ+ (y)) Z h i −i d4 y G0−− (x, y) j(y) − U′ (Φ− (y)) . (15.86) At this point, we recognize that the right hand side of these equations is identical to the first term in the right hand side of eqs. (15.81) and (15.82). From this observation, we conclude that Φ+ (x) and Φ− (x) are solutions of the classical equation of motion, (x + m2 )Φ± (x) + U′ (Φ± (x)) = j(x) ,

(15.87)

and that they obey the following boundary conditions Z

Z

h iy0 =+∞ ↔ ↔ 0 d3 y G0++ (x, y) ∂ y0 Φ+ (y) − G+− (x, y) ∂ y0 Φ− (y) =0, 0 y =−∞

h

0

↔

↔

d3 y G−+ (x, y) ∂ y0 Φ+ (y) − G0−− (x, y) ∂ y0

iy0 =+∞ Φ− (y) =0. y0 =−∞

(15.88)

We have now coupled boundary conditions for the fields Φ+ and Φ− , that involve the value of the fields both at y0 = −∞ and at y0 = +∞. In addition, these boundary conditions are non-local in coordinate space, since they involve integrals over d3 y on the surfaces y0 = ±∞. However, they can be simplified considerably if one uses the following Fourier representations for the propagators Z

i d3 p h 0 0 −ip·(x−y) 0 0 +ip·(x−y) θ(x − y )e +θ(y − x )e , (2π)3 2Ep Z i d3 p h 0 0 +ip·(x−y) 0 0 −ip·(x−y) θ(x − y )e +θ(y − x )e , G0−− (x, y) = (2π)3 2Ep Z d3 p 0 u(p) e+ip·(x−y) , G+− (x, y) = (2π)3 2Ep Z d3 p 0 v(p) e−ip·(x−y) . (15.89) G−+ (x, y) = (2π)3 2Ep G0++ (x, y)

=

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F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Compared to the expressions of these propagators in eqs. (1.367) and (15.83), we have performed explicitly the integration over p0 in order to obtain these formulas. Thus, whenever p0 appears in these expressions, it should be replaced by the positive on-shell value p0 = |p|. The other ingredient we need in order to simplify the boundary conditions is a Fourier representation for the fields Φ± (y), Φǫ (y) ≡

Z

i h d3 p (+) 0 −ip·y (−) 0 +ip·y . (15.90) f (y , p) e + f (y , p) e ǫ (2π)3 2Ep ǫ

The superscripts (±) on the Fourier coefficients serve to distinguish the positive and negative frequency modes. Note that because the fields Φ± (y) are not free fields, these Fourier coefficients are time dependent. In practice, one may assume that the (±) interactions are switched off at y0 = ±∞, so that the coefficients fǫ (y0 , p) tend to constants when y0 → ±∞. However, these limiting values are different at y0 = +∞ and at y0 = −∞, and we must keep the y0 argument to distinguish them. Using the identity Z ↔ ′ ′ d3 y eiǫp·(x−y) ∂ y0 eiǫ p ·y = iδǫǫ′ eiǫp·x (2π)3 2Ep δ(p − p′ ) , (15.91) (valid for ǫ, ǫ′ = ±) it is easy to rewrite the boundary conditions (15.88) as a set of separate conditions for each Fourier mode p: (+)

(−)

f+ (−∞, p) = f− (−∞, p) = 0 , (−)

(−)

(+)

(+)

f+ (+∞, p) = u(p) f− (+∞, p) , f− (+∞, p) = v(p) f+ (+∞, p) .

(15.92)

The boundary conditions have a very compact expression in terms of the Fourier coefficients of the fields Φ± . At y0 = −∞, Φ+ (y) has no positive energy modes and Φ− (y) has no negative energy modes. At y0 = +∞, the negative energy modes of Φ+ (y) and Φ− (y) are proportional (with a proportionality relation that involves the function u(p)). A similar relation, that involves the function v(p), holds between their positive energy modes at y0 = +∞. Eqs. (15.92), together with the equations of motion (15.87), determine uniquely the fields Φ± (x) and therefore provide the solution to our original problem of summing tree diagrams in the Schwinger-Keldysh formalism. One should however keep in mind that this solution is somewhat formal, because it is in general extremely difficult to solve a non-linear field equation of motion with boundary conditions specified both at y0 = −∞ and y0 = +∞. Let us also mention that these boundary conditions become considerably simpler in the case where u(p) = v(p) ≡ 1. Indeed, from the second and third of eqs. (15.92), we see that the fields Φ± (y) have identical Fourier coefficients at y0 = +∞. Therefore, the two fields must be equal in the limit y0 → +∞. Then, by solving their

527

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

equation of motion backwards in time, one sees trivially that they are equal at all times (since they obey identical equations of motion), if u(p) = v(p) ≡ 1 ,

Φ+ (x) = Φ− (x) ,

for all x ∈

❘4 .

(15.93)

Finally, the first of eqs. (15.92) tells us that if u(p) = v(p) ≡ 1 ,

lim

x0 →−∞

Φ± (x) = 0 .

(15.94)

To summarize, when u(p) = v(p) ≡ 1, the two fields Φ± (x) are equal to the retarded field that vanishes when x0 → −∞. This result could in fact have been obtained by a much more elementary argument. Indeed, when u(p) = v(p) ≡ 1, the summation over the ± indices at the vertices of tree diagrams always leads to the following combinations of propagators, G0++ − G0+− = G0−+ − G0−− = G0R .

(15.95)

In other words, summing over these indices amounts to replacing all the propagators in a given tree by retarded propagators, and one is thus led to the problem discussed in section 15.5.4. c sileG siocnarF

15.6 Mode functions 15.6.1

Propagators in a background field

We have introduced in eqs. (15.50) a set of small perturbations on top of a background field Φ, as a way to express propagators dressed by this background. We shall discuss further properties of these functions in this section. However, before we do so, let us propose an alternative derivation of the dressed propagators. In eqs. (15.49), this problem was solved by writing the equations of motion obeyed by the various propagators, as well as their boundary conditions, and by exhibiting an expression that fulfills both. The method proposed here simply amounts to performing explicitly the resummation of the background field insertions. As one will see, this approach is arguably more tedious, but is in a sense much more elementary. The starting point is eq. (15.44), that performs the resummation of the background field. In this form, the equation is fairly complicated to solve because the four components of the Schwinger-Keldysh propagator get mixed already after the first insertion of the background field. However, there is a simple way to simplify these equations. It is based on the observation that the four propagators are not independent, but satisfy a linear relation, G0++ + G0−− G++ + G−−

= =

G0+− + G0−+ , G+− + G−+ ,

(15.96)

528

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

which follows immediately from their definition as path-ordered products of two fields, and from the identity θ(x) + θ(−x) = 1. It is possible to exploit this relation as follows: perform a rotation on the matrix made of the four propagators so that one component of the rotated matrix becomes zero. Having a zero in the matrix of propagators makes the resummation of the background field considerably simpler. Therefore, let us define X (15.97) Ωαǫ Ωβǫ′ Gǫǫ′ . αβ ≡

●

ǫ,ǫ′ =±

(The same rotation is applied to the free propagators.) There is not a unique choice of the matrix Ωαǫ that gives a zero component in αβ , but the following choice is convenient:   1 −1 Ωαǫ ≡ . (15.98) 1/2 1/2

●

The rotated propagators read   0 G0A 0 , αβ = G0R G0S

●

●αβ =



0 GR

GA GS



,

(15.99)

where we have introduced G0R = G0++ − G0+−

,

GR = G++ − G0+− ,

G0A = G0++ − G0−+

,

GA = G++ − G0−+ ,

G0S = G0++ + G0−−

,

GS = G++ + G0−− .

(15.100)

(The subscripts R, A and S stand respectively for retarded, advanced and symmetric.) After having performed this rotation, eq. (15.44) is transformed into XZ 0 (x, y) = (x, y) − i d4 z 0αδ (x, z) U′′ (Φ(z)) σδγ γβ (z, y) , αβ αβ

●

●

●

●

δ,γ

(15.101)

where we denote   0 1 σ≡ . 1 0

(15.102)

In order to make the notations more compact, let us introduce the following shorthand, h

i

❆ ◦ ❇ αβ(x, y) ≡ −i

XZ δ,γ

d4 z

❆αδ(x, z) U′′(Φ(z)) σδγ ❇γβ(z, y) . (15.103)

529

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS With this notation, eq. (15.101) takes a very compact form,

● = ●0 + ●0 ◦ ● ,

(15.104)

and its solution is

●=

∞ h X

n=0

●0

i◦n

,

with

❆◦n ≡ ❆| ◦ ·{z· · ◦ ❆} .

(15.105)

n times

What makes the calculation of this infinite sum easy after the rotation we have performed is the fact that the elementary object 0 σ is the sum of a diagonal and a nilpotent matrix:  0    0 0 0 GA 0 σ= + , ≡ , ≡ . (15.106) G0S 0 0 G0R

●

●

❉ ◆

❉

◆

◆

●

One has 2 = 0, which simplifies a lot the calculation of the n-th power of 0 σ. From this observation, it is easy to obtain !  0 ⋆(n+1) h i◦(n+1) 0 GA 0 =  ⋆(n+1) Pn  ⋆i  ⋆(n−i) , (15.107) 0 ⋆ G0S ⋆ G0A G0R i=0 GR

●

with the notation 

Z



A ⋆ B (x, y) ≡ −i d4 z A(x, z) U′′ (Φ(z)) B(z, y) ,

(15.108)

(and an obvious definition for the ⋆-exponentiation.) The summation of the offdiagonal components of eq. (15.107) is trivial since these terms do not mix. Moreover, the resummed GS propagator has a simple expression in terms of the resummed retarded and advanced propagators. These results can be summarized by GR =

∞ h X

n=0

G0R

i⋆n

,

GA =

∞ h X

G0A

n=0 0 −1

GS = GR (G0R )−1 G0S (GA )

i⋆n

,

GA .

(15.109)

At this stage, we know all the components of the resummed propagator in the rotated basis. In order to obtain them in the original basis, we just have to invert the rotation of eq. (15.97), which gives G−+

=

GR (G0R )−1 G0−+ (G0A )−1 GA ,

G+−

=

GR (G0R )−1 G0+− (G0A )−1 GA .

(15.110)

It is easy to check that these equations are equivalent to eqs. (15.49). c sileG siocnarF

530

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

15.6.2

Basis of retarded small fluctuations

Since small perturbations obey a linear equation of motion, it is always possible to write them as a linear superposition of small fluctuations that obey retarded boundary conditions. Let us introduce a±k (x), defined by the equation of motion h i x + m2 + U′′ (Φ(x)) a±k (x) = 0 , (15.111) and the retarded boundary condition lim

x0 →−∞

a±k (x) = e∓ik·x .

(15.112)

Note that for a real potential U(Φ), a±k (x) are mutual complex conjugates. Any solution of the equation of motion for small fluctuations can be written as Z h i d3 k k a(x) = αk (15.113) + a+k (x) + α− a−k (x) , 3 (2π) 2Ek where the αk ± are constant coefficients that depend on the boundary conditions (the boundary conditions in general lead to a set of linear equations for the coefficients).

15.6.3

Completeness relations

The set of small fluctuations {a+k (x), a−k (x)} obey some useful relations, that are a consequence of unitarity. Consider first two generic solutions a1 (x) and a2 (x) of the equation of small fluctuations. In order to make the notations more compact in the rest of this section, it is useful to introduce the following notations  

a ≡ a(x) , a ≡ a∗ (x) a˙ ∗ (x) σ2 , (15.114) ˙a(x)

where the dot denotes a time derivative and σ2 is the second Pauli matrix. Thanks to the fact that the background potential U′′ (Φ(x)) is real, one can construct from a1 and a2 an inner product which is an invariant of the time evolution of the two perturbations. This quantity is reminiscent of the Wronskian for two solutions of a second order ordinary differential equation, and it is defined as follows Z h i
a1 a2 ≡ i d3 x a˙ ∗1 (x) a2 (x) − a∗1 (x) a˙ 2 (x) . (15.115)


Although a1 a2 could in principle depend on time (since one integrates only over space in its definition), it is immediate to verify that
∂ a1 a2 = 0 . 0 ∂x

(15.116)

531

15. S TRONG FIELDS AND SEMI - CLASSICAL METHODS

Since it is a constant in time, one can compute this inner product from the value of the field fluctuations in the remote past. This is particularly handy when the fluctuations under consideration are specified by retarded boundary conditions, as is the case for a±k (x). One finds
a+k a+l
a−k a−l
a+k a−l

=

(2π)3 2Ek δ(k − l) ,

=

−(2π)3 2Ek δ(k − l) ,
a−k a+l = 0 .

=

(15.117)

Consider now a generic solution a(x) of eq. (15.111). Since the a±k a basis of the linear space of solutions, one can write a(x) as a linear superposition
a =

Z


i
d3 k h k , α+ a+k + αk − a−k 3 (2π) 2k

(15.118)

where the coefficients αk ± do not depend
on time or space. By using the orthogonality relations obeyed by the vectors a±k , one gets easily
αk + = a+k a

,


αk − = − a−k a .

(15.119)

By inserting these relations back into eq. (15.118), and by using the fact that it is valid for any small fluctuation a(x) solution of eq. (15.111), we obtain the following identity Z

i
d3 k h
ak ak − a−k a−k = 1 . 3 (2π) 2k

(15.120)

This identity is valid at all times over the space of solutions of eq. (15.111). It is a manifestation of the fact that, when the background
field is real, the time evolution preserves the completeness of the set of states a±k .

15.7 Multi-point correlation functions at tree level 15.7.1

Generating functional for local measurements

Definition : In the previous section, we have studied a generic observable at leading and next-to-leading orders in λ, and we have established a general functional relationship that relates them. In a sense, this relationship reflects the fact the first h ¯ correction in a quantum theory is not fully quantum: at this order only the initial state contains quantum effects, but the time evolution of the system is still classical. c sileG siocnarF

532

F. G ELIS – A S TROLL T HROUGH Q UANTUM F IELDS

Let us consider now an observable involving multiple points x1 , · · · , xn , corresponding to n measurements. For simplicity, we assume that the points xi where the measurements are performed lie on the same surface of constant time x0 = tf , but the final results are valid for any locally space-like surface (this ensures that there is no causal relation between the points xi , and also that the ordering between the operators in the correlator does not matter). In this case, the leading order is a completely disconnected contribution made of n separate factors, that does not contain any correlation between the n measurements. However, the physically interesting information lies in the correlation between these measurements,

 C{1···n} ≡ O(x1 ) · · · O(xn ) c , (15.121)

where the subscript c indicates that we retain only the connected part of the correlator. From the generic power counting arguments developed in the previous sections, these connected correlators are all of order λ−1 in the strong field regime. It is also important to realize that the connected part of these correlators is subleading compared to their fully disconnected part, since

 O(x1 ) · · · O(xn )

=





 O(x1 ) · · · O(xn ) | {z }

+

X

i