From Symmetries to Strings : Forty Years of Rochester Conferences, Rochester, New York, 4-5 May 1990 : a symposium to honor Susumu Okubo in his 60th year 9789814540117, 9814540110

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From Symmetries to Strings: Forty Years of Rochester Conferences

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From Symmetries to Strings: Forty Years of Rochester Conferences Rochester, New York 4- 5 May 1990

A Symposium To Honor Susumu Okubo In His 60th Year

Editor

Ashok Oas

b World Scientific 1 II

Singapore· New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. POBox 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Library of Congress Cataloging-in-Publication data is available.

FROM SYMMETRIES TO STRINGS: FORTY YEARS OF ROCHESTER CONFERENCES

Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 9789814540117

Printed in Singapore by Utopia Press.

v

Preface

The Conference "From Symmetries to Strings: Forty Years of Rochester Conferences" - fondly dubbed the "Okubofest" - was held in Rochester on May 4-5, 1990. The Conference had two primary goals - to celebrate the fortieth anniversary of the International Conference on High Energy Physics (Rochester Conference), which originated in Rochester through the efforts of Prof. Robert E. Marshak, as well as to celebrate the sixtieth birthday of Prof. Susumu Okubo, who has long been a valued member at the Department of Physics and Astronomy of the University of Rochester and whose extensive contributions to many areas of high energy and mathematical physics are well known. The Conference was attended by about 150 physicists from around the world and consisted of 21 talks spread over two days. Several speakers as well as chairpersons at the "Okubofest" were also participants at the first Rochester Conference. Furthermore, the scheduling of the talks - which consisted of a blend of experimental and theoretical topics of current interest - and the size of the audience allowed for ample informal interaction between the participants, recreating, in some sense, the atmosphere of the earlier Rochester Conferences. In many of the talks, the influence of Prof. Okubo's work was easy to see. The Conference, therefore, was quite successful from both counts. The Conference was primarily funded by the Department of Energy and the National Science Foundation, to whom I express my sincere gratitude. I would also like to thank the members of the International Advisory Committee for their willingness to help and for their valuable suggestions. It is also a pleasure to thank the members of the local organizing committee for their efforts to make the Conference a success. The actual organization was made easy by the wonderful secretarial staff at the Department of Physics and Astronomy. I would like to thank Betty Cook and Connie Jones for their assistance, and especially Shirley Brignall and Judy Mack for handling all the responsibility with grace, humor and total efficiency. Fi-

vi

nally, it is with fondness that I would like to thank our outgoing undergraduate senior class ('90) as well as the incoming senior class ('91) for undertaking various responsibilities at the Conference site, which made the fest especially memorable. Ashok Das

vii

CONTENTS Mass Formulas and Dynamical Symmetry Breaking Y. Nambu The "Spin" of the Proton and the OZI Limit of QCD G. Veneziano Superconvergence Relations Characterizing Quantum Chromodynamics Kazuhiko Nishijima

1

13

27

Scattering Theory in Quaternionic Quantum Mechanics Stephen L. Adler

37

Understanding the Electric Charge of Quarks and Leptons Rabindra N. Mohapatra

57

Comments on Hadronic Mass Formulae Feza Giirsey

77

The Quantum Envelope of a Classical System E. C. G. Sudarshan

99

Okubofest and the Fortieth Anniversary of the Rochester Conference R. E. Marshak

119

Hadron Colliders: Quest for Violence Leon M. Lederman

125

Looking Beyond the Standard Model Lev Okun

141

New Developments in the s-Channel Theory of Superconductivity T. D. Lee

155

Condensed Matter and High Energy Physics Sergio Fubini

171

viii

Collective Field Theory Applied to the Fractional Quantum Hall Effect B. Sakita and Zhao-bin Su

179

Some Remarks on Quantum Groups - A New Variation on the Theme of Symmetry L. C. Biedenharn and M. A. Lohe

189

From Gell-Mann-Okubo Mass Formula to the Mass of Higgs Boson Sadao Oneda

207

Topological Defects and Anomalies Janos Polonyi

225

The OZI Rule - A Unique Selector of Glueballs and Hadron Spectroscopy S. J. Lindenbaum

241

Baryon and Lepton Number Violation in the Electroweak Theory at Te V Energies Emil Mottola

281

On Okubo Algebras Alberto Elduque and Hyo Chul Myung

299

List of Publications of S. Okubo

311

The Program

325

List of Participants

327

From Symmetries to Strings: Forty Years of Rochester Conferences

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MASS FORMULAS AND DYNAMICAL SYMMETRY BREAKING

Y. Nambu Enrico Fermi Institute and Department of Physics University of Chicago, Chicago, IL 60697

ABSTRACT An overview is given of the BCS mechanism as an agent for dynamical generation of masses in various branches of physics, with emphasis on its universal features like the characteristic mass relations among fermions and collective bosons. 1. Introduction

Understanding the spectrum of energy levels or masses is one of the most obvious tasks in many branches of physics. Very often the spectrum gives us a crucial clue to the underlying physics, or serves as a first test of a theory. A typical example is the Gell-Mann-Okubo formula for hadron masses. 1 It was derived on the basis of SU(3) symmetry considerations, a symmetry that was not perfect, but nevertheless was broken according to a regular pattern. In a second paper dealing with vector mesons,2 Okubo also introduced an additional ansatz, the so-called OZI (Okubo-Zweig-Iizuka) rule, which required a more dynamical explanation. It was eventually supplied by QCD. Unlike the cases of atoms, nuclei, and hadrons, however, one has so far no basic understanding of the mass spectrum of fundamental quarks and leptons. It shows no clear regularities to give one a clue; with the masses of five quarks already known, one is still at a loss to predict the mass of the sixth quark, the top.

2 I would like to discuss dynamical origins of masses and mass spectra, based on the idea of spontaneous symmetry breaking, a topic that has interested me for many years. In particular, the so-called BeS mechanism has turned out to be a very useful concept in diverse phenomena at diverse energy scales. It is my hope that the same concept will shed some light on the riddle of the masses of fundamental fermions, and more generally, the origins of the mass hierarchy one encounters in one's attempts to unify all particles and fields. This article is an attempt to put into perspective my recent work that has been reported in various publications. 3 2. Basics of the BCS Mechanism. First I will begin by giving a characterization of the Bes mechanism. By BSe mechanism I mean a creation of Cooper pair condensate in a medium due to a short range attractive interaction between fermions. In the ideal limit one represents the interaction as a contact 4-fermion type. In conventional superconductors (probably not in the high-To ones), the attraction is due to phonons, but here the physical nature of the interaction need not be specified. The nonvanishing fermion pair correlations < t/Jt/J > and < t/J+t/J+ > defines a complex order parameter characteristic of the condensate. Above the condensate ground state, there are fermionic and bosonic low-lying excitations. The fermionic ones are the Bogoliubov-Valatin quasiparticles, or mixtures of particles and holes of the original fermions, with an energy gap m f. The bosons come in two kinds, the 11'" (Goldstone) and (J (Higgs) modes, corresponding respectively to modulations of the phase and the amplitude of the order parameter. For the time being I will be concerned with nonrelativistic media. energy gap parameter ~ is determined by the BeS gap equation

The

(1) N is the kinematic density of states of fermions at the Fermi surface. The effective potential < V > is given for a Yukawa-type interaction in terms of its coupling constant 9 and mass M, as in the Fermi limit of the W-exchange process. The cut-off parameter A in the logarithm could be equated with M, but here it is left independent. The weak coupling limit, however, means a small factor N < V >, or a large ratio A/~.

In addition to the Bogoliubov-Valatin fermion fields t/J and t/J+ equipped with a mass mf == ~, there are collective bosonic excitations which are derived by examining the propagation of composite fields t/Jt/J and t/J+ t/J+. It is an important feature of the BeS mechanism that these low energy modes satisfy simple mass relations. In most cases they are mlf: mf: m.,.

= 0:

1: 2.

(2)

3

o

SAMPLE M 80 C

E

>-50 >-

v; z

80

w >-

z

B

(

SAMPLE B

E

'\.

G

40 -

60

~. ~O

d

bl}\"7'" ~ ~

-30

20

SAMPLE

\

);{

40

10

G

A~O

I

l/e

J' ·.~,D \

20

.

60

RAMAN SHIFT (em-I)

.1\1

20

60

Raman spectrum oC samples M and B. The lower curve oC each pair [(a)- (d) J Is at 9 K and the upper at 2 K. Raman symmetries [polarlzatlonsJ are E [(xy)J and A [(xx) - ("y)J. C labels CDW modes; G, gap excitations; and I, the Inter layer mode cbaracterlstic oC the 2H polytype.

Figure 1: Data on the

q

mode in superconducting thin films. (Ref. 4).

Except for mll" this is not an exact relation based on symmetry, but is of a dynamical nature valid in the short-range and weak coupling limit, or in the lowest order of a liN expansion. (It can be characterized, however, in terma of what I call quasi-supersymmetry.H) Actually Eq.(2) is a special case of a more general form: 15

(3) where ml and m2 are for a pair of normal modes which are linear combinations of ¢¢ and ¢+ ¢+. For example, the Cooper pairing in 'He occurs in one of the p states, and there are three kinds of complex collective modes corresponding to j = 0, 1, and 2. Some of them are neither Goldstone nor Higgs type, but are subject to Eq.(3). These theoretical relations are almost as old as the BCS theory, but their experimental confirmation is rather recent. Fig. 1 reproduces a Raman map of the excitation spectra of superconducting films exposed to laser beams,4 which was subsequently given the proper theoretical interpretation.' Below the critical temperature one observes a sharp peak at energy w = 2~ corresponding to the sigma mode. The broad enhancement seen both above and below the critical temperature is due to large CDW-type distortions, which modulate the Fermi surface and make the coupling of electromagnetic fields to the collective modes strong.

4

Fig. 2 shows similar data for superfluid sHe-B, obtained from ultrasonic attenuation measurements.6 The pairing occurs in the j = 0 state. The two nonstandard modes (j = 2) have been observed, and they approximately satisfy Eq.(3), with a theoretical ratio m~ : m~ = 2 : 3. 7 The fermionic and bosonic modes have intrinsic interactions among themselves. There is a natural Yukawa coupling between fermions and their bosonic collective modes, but the bosons themselves interact through fermion loops. In the weak coupling limit that led to the mass relations, only Yukawa, 3-boson, and 4-boson couplings are important, and they are controlled by one parameter tl/VN. In other words, the couplings depend on the ratio of two energy scales, the high energy scale charactered by the density of states and the low energy scale of the gap. Alternatively one may say that the low energy phenomenology depends on one mass parameter and one coupling parameter. Knowing these, one can now write down an effective Lagrangian describing the low-lying fermionic and bosonic modes without reference to the original BeS mechanism. It is perhaps not surprising, although not obvious, that one gets in the weak coupling limit precisely a Landau-Ginsburg-Gell-Mann-Higgs type Lagrangian in the broken symmetry phase, which can be converted back to a manifestly symmetric form by shifting the u field. The interaction part of the Hamiltonian is given by

(4) The Higgs vacuum expectation value v, or the shift in the u field, turns out to be v

= VN/2,

mf

= Gv,mu = 2Gv.

(5)

G = mJlv is thus the ratio of the two energy scales as was mentioned above. Table 1. lists various examples of the BeS mechanism and the magnitudes of their parameters. I have included here the standard electroweak theory among them, and will discuss it briefly at the end. Note that the Yukawa couplings come out quite small in general, which implies that the mass relations should be good. One exception is the case of chiral dynamics of the original pion and sigma. Here the magnitudes of two energy scales represented by the constituent quark mass mq and the pion decay constant v are inverted. One would not off hand expect the current analysis to apply literally to QeD, but the ratio mumq = 2 is consistent with the interpretation of the broad enhancement in 7r - 7r scattering as the u.

5

EXCESS ATTENUATION

o

MHZ 12 • 36

··... ·.

P-13.0 BAR

0

60 •

lHe - B

84 '

19 6 BAR

109 •

1515 Mill

133 •

5.15 Mil, ';' J

15 -

-, E

~

10

y-

;:::u

a, 5 E " o ..20,,* _ 0.6

(b)

I

-./ 0.7

....... 0.8

0.9

1.0

T/Tc

Helalive altenuation of 60-MHz sound at 5.3

bars. Arrows show Ule predirled location of the collective-mode and pair-breaking attenuation peaks. The height of the y peak is not known berause of loss of the

Signal for relative attenuaUon >16 em-I,

100

.3

.4

.5

.6

.7 TITe

.8

.9

(a) Attenuation and (b) veloclty of Bound aa a

function of reduced temperature T/Tc. The attenuations at low temperatures should be considered to be

zero. The solld lines are guides to the eye. The dashed Iinc In (hI I. un extr"polallon of the 133-Mllz velocity (lata Lo Its v:lillo In tho "urmnl fluid. ,'hiH Hhould occur

(Ito!. 10) at w=2.Q, whoro pair-breaking commencel.

Figure 2: Data on bosonic modes in 3He-B. (a) Mode m 1=,fI2B~ (from Wheatley, Ref. 6); (b) and (c) Mode mt=.J8i5~ (from Gianeua et aI., and Mast et aI., Ref. 6.)

6

Table 1. Examples of BCS Mechanism Name

v =< u > (ev)

~(ev)

G

superconductor

5.10- 3

10- 3

10- 7

3He

106

10- 6

10- 12

QCD-chiral dynamics

108

109

10

8

6

nuclear pairing

10

10

10- 2

standard model

2.5.1011

106 (1st gen.)

10- 5

> lOll? (top)

> 1/3?

3. Phenomena in Nuclear Physics. I will next discuss the nuclear physics example listed in the Table, which in my opinion has not been well explored yet. To do that, however, I need to add a new conceptual element to the BCS mechanism. I will call it tumbling, after the terminology of Dimopoulos, Raby, and Susskind8 who introduced it as a mechanism for generating a mass hierarchy in unified theories. It is my point here that there already are known examples of it. Suppose the chiral symmetry in a primary theory at a high energy scale is broken, generating massive fermions and composite bosons with a lower mass scale. The Higgs boson will then mediate a new attractive interaction between fermions, and may in turn induce a second symmetry breaking, and so on down to lower and lower energies. This is essentially the idea of tumbling. Consider the usual chiral dynamics of strong interactions. It is generally accepted that QCD leads to quark confinement as well as dynamical generation of quark masses. The latter also generates the pion as a Goldstone boson, and the sigma meson as its chiral partner. The confinement leads to formation of nucleons, and the sigma supplies a strong attractive force between nucleons. This enables many nucleons to bind into nuclei since the forces can add up coherently, and form an average common potential for individual nucleons. The relativistic corrections to the potential give rise to the spin-orbit forces. Besides the common potential, moreover, there are residual attractions between two nucleons, which causes Cooper pair formation out of protons and neutrons. Viewed in this way, the shell model is a consequence of the second generation BCS mechanism. It is easy to set up a BCS equation and make back-of~an-envelope estimates. The necessary parameters in Eq.(l) can be obtained from bulk properties of nuclear matter, namely nuclear density p and work function I, as is illustrated in Fig. 3. One finds N < V >- 1/2, not a very small number. The resulting gap is - 3 MeV compared to the actual value - 1 MeV.

7

1 = N(V) SIj"I( Iltd

(V) = DIP N = 3P1UE F N(V) = 3U/OE F : 1/2 /j :::

3 MeV

Figure 3: Computation of nuclear pairing energy. My next task is to look at the excitation spectra. This was inspired by the interacting boson model (IBM) of Arima and Iachello,9 which was a very successful phenomenological systematization of low lying nuclear levels by grouptheoretical considerations in the manner of the Gell-Mann-Okubo mass formula. Fig. 4 illustrates the degree of success of the IBM. It introduces six (spin 0 and 2) complex self-interacting bosic quanta to represent collective excitations in a nucleus, and assumes that the system has a hierarchy of symmetries along a chain of subgroups starting from U(6). The Hamiltonian is represented as a weighted sum of the Casimir operators of the respective subgroups. In the example shown here, the chain is U(6) -+ 0(6)(~ SU(4)) -+ 0(5)(~ Sp(4)) -+ 0(3). Noteworthy is the fact that the weights of the 0(5) and 0(6) Casimirs are roughly in the ratio 1 : -1, not a small number. It appears that the physical origin of IBM is basically the BCS mechanism.lO In other words, the 0(6) -> 0(5) portion of symmetry breaking chain is of a dynamical nature. To show this, one first observes, following IBM, that the 6 bosons in this case has SU( 4) symmetry because the Cooper pairs at the Fermi surface actually correspond to fermion pairs in the j = 3/2 valence shell, which can be regarded as a 4 by 4 antisymmetric matrix. By forming a Cooper pair in one of the substates (say j = 0), SU(4) naturally breaks to Sp(4). This can be realized in terms of a Higgs potential for the 6 complex bosons. One then computes, after symmetry breaking, the boson-boson interaction in the tree approximation, as in the case of meson-meson scattering in the (J model.

Strictly speaking, the concept of spontanous symmetry breaking does not

8

Exp. (6,0)

(6,1)

Th.

(4,0) (2,0)

(6,0)

(6,1)

(4,0) (2,0)

'L1\ 'p-l,l ,~~\ ilL .l 4!....

It_O-

4-3!...0!....

4~2~

2L 2!.... 0-

o!...

o!...

0(6)

An example of a SpCClrtllll wilh 0(6) symmelry: '~~PIII.' N=6.

Figure 4: IBM formula vs experimental data. (From Arima and Iachello, Ann. Phlls. 123 (1979) 468.) The numbers in parentheses specify an 0(6) representa-

tion and its reduction to 0(5). apply to finite systems like nuclei. The baryon number conservation cannot actually break, so the meaning of the Goldstone bosons accompanying Cooper pair formation becomes obscure. The finite size, on the other hand, has the desired consequence that the scattering interaction causes energy shifts, which go inversely proportional to the volume V. But there is also a problem that the correlation length, or the size of the Cooper pair, turns out to be comparable to the nuclear size. This may be thought of as form factor effects. I wiJI not go into the detail, but basically one takes care of the finite number problem by projecting the results onto a number-diagonal basis. Its effects are to lose distinction between the u and ,.. modes, and eliminate half of the degrees of freedom. The final effective Hamiltonian, or mass formula, expressed in terms of creation and annihilation operators, can be organi:ted into a sum of Casimir operators of 0(6) and 0(5):

H

= C( _N 2 -

0(6)

+ (3/2)0(5)),

C = c/8v 2 V "" 40c/A MeV, (A = mass number).

(6)

Here 0(6) and 0(5) stand for the respective Casimir operators, N is the total number of Cooper pairs in the nucleus, taken to be those above a magic shell. Of the coefficients in front, c represents the form factor suppression effects. Surprisingly, the remaining factors can be expressed solely in terms of the bulk nuclear parameters without involving the pairing energy. A value of"" 1/4 for c gives the right energy scale to fit the data.

9 It looks a bit surprising that the pairing energy and the coupling constant do not figure explicitly in the mass formula. But I would point out that they are functions of bulk nuclear parameters according to the earlier discussion.

4. Bootstrap Symmetry Breaking and the Standard Model. Moving now to the Weinberg-Salam standard model, I will introduce yet another concept: bootstrap, adopting the idea and term promoted by Chew many years ago in hadron dynamics. Applied to the BCS mechanism, it means the following. The tumbling mechanism meant that symmetry breaking cascades down in a sequence, each time creating a new energy scale and new modes. Bootstrap symmetry breaking mechanism, on the other hand, postulates that the Higgs boson itself is the cause of the attractive interaction between fermions that led to symmetry breaking and the formation of Higgs as a fermion pair bound state. This is a theoretical possibility, and there is no a priori reason that it should be happening in electroweak processes or in any other phenomena, but it is at least a possibility that I would like to pursue. l1 In fact it makes sense now that the top quark appears to be quite heavy, which means large Yukawa coupling, hence strong top-top interaction. So I will start with the ansatz, without asking why, that the top quark somehow plays a special role in a BCS mechanism to break the 8(2) x U(l) symmetry a la standard model, but the Higgs field is not an independent entity. This may be called a substandard model because of its frugality. Similar ideas have also been put forward by Miransky et al. 12 and by Bardeen et al. 13 As an exercise, first take the gap equation (1), and substitute for M and 9 the Higgs mass and coupling, and for N the relation (5). One then finds N < V >= 1, so Aim, - 1. In other words, bootstrap is compatible with the (nonrelativistic) gap equation, although not in the weak coupling regime. How can one formulate the bootstrap condition in the real case at hand? Here there can be different approaches. My line of reasoning went as follows. The self-energy of a fermion due to scalar boson exchange consists of two diagrams: a tadpole with a fermion loop, and a conventional self-energy diagram like in QED. The former is the dominant one because it is quadratically divergent, and has the correct sign for mass generation; the other diagram is logarithmically divergent, and has the wrong sign. In the standard model, the mass of a fermion comes from the bare vacuum expectation value Vo in the Higgs potential plus the tadpole and conventional self-energies. The tadpole also contains Higgs boson and gauge boson loops, which are quadratically divergent but with an opposite sign to the fermion loop. Now the notion of bootstrap suggests the following propositions: a) Set Va

= 0, because the masses should generate themselves dynamically.

10

b) The quadratic divergences of the tadpole diagrams should cancel each other, because the bootstrap mechanism should be able to conceal underlying substructure as much as possible. These postulates lead to two equations involving the gauge, Yukawa, and Higgs coupling constants, which can be converted to equations for their masses. The first equation is a quadratic mass sum rule

m~ =

mk/4 + m'tv/2 + m}/4,

(7)

which expresses the absence of quadratic divergences. The second is the gap equation

which is a quartic mass sum rule with In(A/ < m » as a coefficient. The main consequences of them are that the top mass has a lower bound of ~ 70 Ge V from Eq.(7), and one finds from Eq.(8) two allowed regions above it, 70 GeV

~< mt ~


80 GeV,

120 GeV,

mH ~


60 GeV,

200 GeV

(9)

The lower region is probably excluded by experiment. The second region places the top mass in the upper reaches of current theoretical expectation. The Higgsto-top mass ratio is generally less than 2 there. Qualitatively similar results have been obtained by Bardeen et al.13 Some more details are given elsewhere. One of the problems that remain to be studied with respect to the above bootstrap conditions is how to control quadratic divergences in higher orders. The understanding of the entire mass matrix of fermions is a more distant goal. I remark, however, that the large top quark mass poses another problem of hierarchy, a problem that hits us closer to home than the ones known before. The work is supported in part by a grant from the National Science Foundation, PHY 90-00386. References. 1. S. Okubo, Prog. Theor. Phys. 27 (1962) 949. 2. S. Okubo, Phys. Lett. 5 (1963) 165.

11 3. Y. Nambu, Refs. 7, 9, 10, 13; Festi- Val, Festschrift for Val Telegdi, ed. K. Winter (Elsevier Science Publishers B.V., 1988), p. 181; Themes in Contemporary Physics II, Essays in Honor of Julian Schwinger,ed. S. Deser and R. J. Finkelstein (World Scientific, Singapore, 1989), p. 51. 4. R. Sooryakumar and V. Klein, Phys. Rev. Lett. 45 (1980) 660. 5. P. M. Littlewood and C. M. Varma, Phys. Rev. Lett. 47 (1981) 811; Phys. Rev. B26 (1982) 4883. C. A. Balseiro and L. M. Falicov, Phys. Rev. Lett. 45 (1980) 662. 6. J. C. Wheatly, Rev. Mod. Phys. 47 (1975) 415; Progress in Low Tempera-

ture Physics, ed. D. F. Brewer (North-Holland, Amsterdam, 1978), vol. 7. R. W. Giannetta, A. Ahonen, E. Polturak, J. Saunders, E. K. Zeise, R. C. Richardson, and D. M. Lee, Phys. Rev. Lett. 45 (1980) 262. D. B. Mast, Bimal K. Sarma, J. R. Owers-Bradley, I. D. Calder, J. K. Ketterson, and W. P. Halperin, Phys. Rev. Lett. 45 (1980) 266. 7. P. Woelfle, Physica 90B (1977) 96. V. E. Koch and P. Woelfle, Phys. Rev. Lett. 46 (1981) 486. Y. Nambu, Physica 15B (1985) 147. 8. S. Dimopoulos, S. Raby, and L. Susskind Nucl. Phys. B169 (1980) 493. 9. A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069; 40 (1978) 385; Ann. Phys. 99 (1976) 253; 111 (1978) 201; 123 (1979) 436.

10. Y. Nambu and M. Mukerjee, Phys. Lett. 209 (1988) 1. M. Mukerjee and Y. Nambu, Ann. Phys. 191 (1989) 143. 11. Y. Nambu, in New Theories in Physics, Proc. XI Warsaw Symposium on Elementary Particle Physics, ed. Z. A. Ajduk et al. (World Scientific, Singapore, 1989), p. 1; 1988 International Workshop on New Trends in Strong Coupling Gauge Theories, ed. M. Bando et al. (World Scientific, Singapore, 1989), p. 3; Proc. 1989 Workshop on Dynamical Symmetry Breaking, ed. T. Muta and K. Yamawaki (Nagoya University, 1990), p. 2 (U. Chicago preprint EFI 90-36). 12. V. Miransky, M. Tanabashi, and K. Yamawaki, Mod. (1989) 1043; Phys. Lett. B221 (1989) 177. 13. W. Bardeen, C. Hill, and M. Lindner, Phys. Rev.

Phys. Lett. A4

D41 (1990) 1647.

12

14. Y. Nambu, in Rationale of Beings, Festschrift in honor of G. Takeda, Ed. R. Ishikawa et al. (World Scientific, Singapore, 1986)' p. 3. "Supersymmetry and Quasi-Supersymmetry", U. Chicago preprint EFI 89-30, to appear in Murry Gell-Mann Festschrift. 15. Y. Nambu, Ref. 7.

Note added. General discussions concerning the BCS mechanism in various fields of physics are also found in M. Scadron, Ann. Phys. 148 (1983) 257 and 159 (1985) 184. In particular, the second paper deals with nuclear pairing.

13

THE" SPIN" OF THE PROTON AND THE OZI LIMIT OF QeD

G. VENEZIANO

Theory Division, CERN, 1211 Geneva 23, Switzerland

1. A FLORENTINE HONORIS CAUSA I wish to start this talk with a historical digression which has nothing to do

with the spin of the proton but much to do with one of the reasons why we are gathering here. The episode refers to the spring of 1966, when Professor R. Gatto, F. Buccella and myself, working in Florence, were facing some Current Algebra puzzle. Professor Okubo came to visit the University and, one day, shortly after his arrival, Professor Gatto came to see Buccella and myself and announced the news: Professor Okubo had just produced a beautiful argument for the necessity of q-number Schwinger-terms in the commutation relations that were giving us trouble! Buccella and I were shown the extremely elegant proof, which was only assuming the Jacobi identity and some general spectral-function representation. I was stunned; even more so when I found my name appearing in a paper on the subject (Phys. Rev. 149). It is still often quoted, in spite of the fact that, about a year later, in the 1967 Erice Summer School, Coleman was quoting a F.F.F.P. (Four Florentine False Proof, although I am the only genuine Florentine of the lot!): people had found failures of the Jacobi identity in perturbation theory. For a kid (as I was) this was a lot of lessons to be learned: i) nice, elegant proofs can be physically wrong (d. the Coleman-Mandula theorem!) but ii) even then, they can be of use for the development of new important ideas (d. co cycles in today's theoretical physics or supersymmetry for evading the CM theorem).

14

2. OZI-QCD: A REMINDER

The famous OZI rule [1] preceded, of course, and by many years, QeD. Yet, unlike other ideas, it survived QeD or, better, QeD gave it a firm theoretical foundation. I shall recall what the OZI approximation to QeD is by comparing it to three other non-perturbative approximations: the liNe expansion, the topological expansion and the fermion-loop (quenched-QeD) expansion. This is illustrated in Fig. 1. The large Ne limit (at fixed Nf) is the most restrictive approximation (hence, presumably, the most crude but also the easiest) in that it includes only planar diagrams with sources on a single quark line and no extra quark loops (Fig. la). It was introduced by 't Hooft in 1974 [2]. It should provide the easiest checks of confinement and a clear relation between QeD and dual-strings but, unfortunately, it has resisted all attempts at explicit calculations (except in two space-time dimensions ). The quenched and topological approximations improve on the liNe approximation in different directions: the quenched approximation allows non-planar diagrams, but no quark loops (Fig. lb): it corresponds to a small Nf expansion at fixed Ne and is widely used in lattice QeD (because of notorious problems with simulating dynamical fermions). By contrast, the topological expansion, first suggested in dual string theory [3], corresponds, in QeD, to the Ne

--+ 00

limit

at fixed NflNe [4]. It allows any number of quark loops (with sources still on a single quark-line) but keeps, at lowest order, only planar diagrams (quarks and gluons are treated symmetrically, only topology counts, Fig. lc). Its applications are mostly in high-energy soft hadronic collisions treated it la Regge-Gribov: the specific model that embodies it in a fully predictive scheme is the so-called dual parton model [5]. We finally come to the OZI approximation: it combines both the virtues of the topological expansion and those of the quenched approximation since, in OZIQeD non-planar diagrams as well as quark loops are kept (Fig. ld). The only contributions which are neglected in leading order are the so-called hair-pin diagrams (Fig. Ie), in which the external sources (currents) hang on more than one quark-line (OZI rule).

15

fa)

IINc

/ TopoLogicaL

o.uenched

fb)

\

/

(e)

aZI allowed

(d)

(e)

aZI forbidden

Fig. 1: Successively more accurate non-perturbative approximations to QCD. Crosses indicate external sources. a) Typical diagrams of the liNe expansion; b) Typical diagrams of the quenched approximation; c) Diagrams occurring at leading order in the topological expansion; d) OZI-rule-allowed diagrams [besides those of types a), b), c)J; e) OZI-rule forbidden diagrams.

16

I shall spend only a short time recalling the properties that hold true in the OZI limit (and hence, a fortiori, in the other three limits as well).

2.1

M esonic Spectrum

a) nonets rather than octets are predicted; b) no flavour mixing (ideal mixing between octet and singlets). The most famous confirmation of these predictions in the J P

1- sector

where:

¢i ~ ss;

2.2

p,w ~ (uu =f dd); mq,

> mp

~ mw

M esonic couplings, decays

a) strongly coupled systems must share quark-lines (hence flavour). This leads to:

b) decay selection rules, such as the famous:

Things work very nicely with OZI in general (e.g, for JP

= r,2+).

There is,

however, a famous exception: the 0- channel, which is very relevant, as we shall see, for the spin-of-the proton question. The problem is that, in the OZI limit, the Adler-Bell-Jackiw [6] anomaly associated with the U(l) axial current is absent. One thus predicts a full nonet of (pseudo) N ambu-Goldstone bosons

7r,

K, 77 and 77' with the wrong (ideal) mixing

angle. Obviously, corrections must be exceptionally large in this channel. I will show, however, that, in spite of these large violations, the OZI limit is still of great significance for understanding the "spin-of-the-proton" puzzle. But let me first turn momentarily to the experimental findings on which the so-called spin crisis is based.

17

3.

THE (SECOND) EMC (EUROPEAN MUON COLLABORATION) EFFECT

The second EMC effect (after the one on nuclear structure functions) refers to the measurement [7] of the proton polarized structure function called gf. Confirming and extending previous SLAC data [8], EMC finds, in particular:

J

dxgf(x,q2)l q2_11

GeV2

= 0.126

In SU(3), gf is a linear combination of

gi gi 3

),

± 0.010 ± 0.015

8

),

(3.1)

gP). The first of these is given

by Bjorken's sum rule in terms of the usual ,B-decay axial coupling (in order to avoid confusion with g1 we shall use capitals for the axial couplings): (3)

GA while

gi

8

)

= 1.254 ± 0.06,

(3.2)

can be obtained from the known F / D ratio of semi-leptonic hyperon

decays. Modulo various assumptions and extrapolations, one finally arrives at a determination of gP) yielding:

J 1

~(q) ==

dxgP)(x, q2) = 0.10 ± 0.24 at q2

~ 11 GeV 2

(3.3)

o The smallness of this experimental number is what is referred to, improperly, as the "spin crisis". In order to see why the results (3.3) might be embarrassing for QCD let us use the standard operator product expansion (OPE). As shown by Kodaira some time ago [9], the OPE prediction is:

(3.4) where Jp,51q is the flavour-singlet axial current: 3

Jp,5

=L

ifnp,/5ql

(3.5)

1=1 renormalized at the scale q. For simplicity, we shall not write explicitly the renor-

18

malization point from now on. Substituting the definition: 2

-

2

-

(PIJJLsIP) = GA(k )P"(JL"(sP + Gp(k )PkJL,,(sP

(3.6)

(k = momentum transfer between initial and final proton), we finally arrive at the following consequence of the EMC data: GA(O)

= 0.12 ± 0.24

(3.7)

to be compared with the analogous quantity for the octet current. (8) J1'S

+ d-"(JL"(s d == u"(JL"(su

2-8"(1'''(5 8

(3.8)

whch is scale-independent and experimentally given by: (8)

G A (0)

= 0.68 ± 0.08.

(3.9)

Why is the comparison of (3.7) and (3.9) troublesome? It is because it is at variance with the OZI-limit expectation. Indeed, in such a limit, the strange quark part of the currents do not contribute to the proton matrix elements and one expects [10]: (3.10) The so-called "Spin Crisis" is synonimous with the experimental violation of the above relation. It has nothing to do with spin, unless one works with free current quarks as the constituents of a proton, but it has all to do with OZI: it is from this viewpoint that we shall further discuss the implications of the EMC experiment. In fact the EMC data show that the JP

= 1+ channel behaves more like the 0-

than like the 1- ,2+ channels vis-a-vis violations of the OZI rule. Is this surprising? Perhaps not, since 1+ and 0- channels are related, in the case of non-anomalous axial currents, by the famous Goldberger-Treiman (GT) relations. The problem is to extend the GT relations to the case of the flavour-singlet [U(l)] axial current. This can be done, as we shall discuss in a moment.

19

The EMC data have raised a lot of theoretical debate. For brevity, I shall only mention the following ideas: 1. The Skyrmion model has been advocated [11] as a natural framework for suppressing~.

Unfortunately, the authors combined the Skyrmion model

with a large Nc expansion, which, when applied to baryons and to 1+,0mesonic channels, can be treacherous. 2. It has been argued [12] that, within a QCD-parton-model framework, ~ does not just measure the spin of the quarks: it automatically adds a term proportional to the spin of the gluons. Thus, according to these authors: (3.11) where !:l.q is scale-independent and can be identified with the naive (OZI) value, while !:l.f is slowly scale-dependent (because of a one-loop cancellation between the scale dependence of as and !:l.g) and measures the gluonic correction. The reason for the approximate cancellation between the two remains unexplained in this scheme. 3. The authors of [13] have accepted the above picture and have gone further by identifying ways to measure !:l.f in other hard processes. A typical one would consist of looking for two-quark-jet events in the EMC data. 4. Jaffe and Manohar [14] have criticized the claim that !:l.f can be isolated from !:l.q. Their criticism is based on the observation that, in the (OPE)approach, there is only one gauge-invariant dimension-3 operator with the desired quantum numbers that can appear in the relevant channel. Their conclusion is simply that the EMC data point to a large OZI violation in the matrix elements of that operator (Jp 5 of Eq. (3.5)). Forte [15] has stressed the relevance of non-perturbative contributions to the quantity !:l.f. 5. Several authors [16,17] have pointed out the danger of identifying !:l.q and !:l.f with the matrix elements of the soft and anomalous parts of the divergences of axial currents, respectively. It has been shown that such an identification would lead to very large isospin violations in each indivicual spin density and would make, e.g., !:l.f very different for the proton and the neutron.

20 4. THE-TGV LINK I shall now argue that there is something right in each one's claim, the bridge or link between these various points of view being provided by the ... TGV. This is a U(l) extension of the GT relation, which was first "derived" [17] in a rather sloppy way and was later rigorously defined and proved by G. Shore and myself [18]. In our opinion the proof, which is summarized in the Appendix, wipes out all possible claims [19] that the original argument was leaving out an important term. The TGV relation reads (modulo chiral corrections Oem/A)): G A -

(ijq) 2MN

3

-

-

6 f z (ijq,iji5q,N,N,FF) I

6(iji5q)6N6N

__

( 4.1)

FF-O

What is the physical meaning of the TGV formula? Let us recall that the most widely accepted resolution of the U(l) problem in QeD [20] calls for a physical

r/

which is a combination of objects of different U(l) chirality:

r/ = a(iji5q)

+ (J(FF)

( 4.2)

where the first term has chirality ±2 and the second has chirality zero. The fact that objects of different chirality appear in

r/ is a consequence of the anomaly and

of the ghost mechanism, both essential ingredients in any conceivable solution of the U(l) problem. In the OZI limit there is no anomaly, the

r/

is just given by

the (iji5q) term and is massless in the chirallimit. We may call this object 7]OZI since it is the physical

7]'

of the OZI limit.

The TGV relation connects the EMe quantity I; to the coupling of the proton to 7]OZI' making the OZI approximation once more interesting even when inadequate. There is an analogy here with the Ne solution of the U(l) problem: the

7]'

7]'

mass formula which holds in the large

mass is a liNe effect, but can be related

[20] to a quantity (the so-called Yang-Mills topological susceptibility) which is to be computed in the theory without quarks and in leading order in l/Ne. As shown explicitly in Ref. [18], the TGV relation is perfectly consistent with the QeD evolution of I; [9] and one can ask if it is at all possible to break it up into a b.q and a b.f evolving in the way prescribed by the QeD parton model. A simple way to achieve this [18] is to re-express the coupling of 1]~ZI in terms of

21

those of the physical

r/

and of a purely gluonic object. One finds an unambiguous

separation of this kind, which should make Efremov and Teryaev, Altarelli and Ross happy. The price to be paid is that the gluonic component is not the matrix element of a local operator, it has to be defined using the one-particle-irreducible effective action (see Appendix). This solves the isospin violation problems (making the authors of Refs. [16,

17J

happy) and confirms in some sense the Jaffe-Manohar

claim (making them also happy). Finally, unlike the authors of Ref. [l1J we can provide, via TGV, not only an interpretation of the two comments, but also an interpretation of their sum in terms of the 'T/~ZI couplings. This opens the way to a better understanding of which features of the Skyrmion model are essential, and which are not, for explaining the smallness

of~.

In a recent paper [21J (of which I became aware at

this Conference) the coupling of 'T/~Z I to the nucleon is spelled out as one of the two conditions for a non-vanishing

~

in generic Skyrmion-like models. It turns out

that models with just pseudoscalars do not allow such a coupling (the Skyrmion "lives" in the SU(3) X SU(3)/SU(3) coset and has no component in U(l)), while models with additional vector mesons (p,w, ... ) usually do. Thus EMC data will help in finding realistic Skyrmion-like models of the Nucleons, making Brodsky, Ellis and Karliner also happy. So, in conclusion, we seem to make everybody happy.

This is certainly a

nice thing, but it's not my main concern today: what I really hope is to have made Professor Okubo happy, by showing that his pioneering ideas are not only alive and well.. they are also still extremely useful for tackling the most recent challenges that experiments bring, alas more and more rarely, to us!

22

APPENDIX A Derivation of GT and TGV Relations [18] Starting from the usual generating functional of connected correlation functions: (AI) one defines a "Zumino" effective action [22] by

rz(~, V~5) = W ~

JSa~

(A2)

== 8W/8Sa

i.e., by a Legendre transform w.r.t. the "fields" a but not w.r.t. the "currents"

J~s· The choice of the set of fields a is, in general, a matter of convenience. We took a

(i

=

0,1,,,.,8; N(N)

= (ij).iq;

=

ij).i,-YSq, Q =

0'.

87l'

Tr(FF), N, N)

Nucleon (antinucleon) interpolating fields). All the chiral

Ward-Takahashi identities are contained in the compact equation (A3) where we have dropped the suffix c for simplicity and 8~ are, by the Adler-Bardeen theorem [6], the tree-level chiral variations of the fields: (A4) and 8~N, 8~N need not be specified. Take now 82 / 8N 8N of Eq. (A3) and set all fields to their v.e.v. On one hand:

(A5) Since, by definition (A2), rz is one-Nambu-Goldstone-boson irreducible. On the other hand, when the differential operator acts on the second term in (A3) it gives

23 zero unless

(A6) In the standard picture of spontaneous chiral symmetry breaking, this selects a =

ii'S>..i q with (A7)

Finally, the third term in (A3) gives zero since, in fz,Q,N,N are independent fields. Combining all these results we arrive at:

(AS)

i.e., upon suitable rescaling of the interpolaring fields:

(A9) We have thus derived the same GT-like equation for the non-singlet and for the singlet channels. This would be absurd, were it not for the fact that physical NG boson only for i of the physical

r/

= 1, ... , S while, for i = 0, 'lr

0

~

'lr

i

is the

ii,sq is only one piece

[given by Eq. (4.2)]. It is precisely what we have called "'~ZI in

the text. Thus, for i

= 0, Eq.

(AS) is just Eq. (4.1) of the main text.

24

REFERENCES [1] S. Okubo, Phys. Lett. 5 (1963) 165 ;

G. Zweig, CERN report No. 8419/TH412 (1964); J. Iizuka, Prog. Theor. Phys. Suppl. 37-38 (1966) 21. [2] G. 't Hooft, Nucl. Phys. B72 (1974) 461. [3] G. Veneziano, Phys. Lett. 52B (1974) 220. [4] G. Veneziano, Nucl. Phys. B117 (1976) 519. [5] A. Capella, U. Sukhatme, C.l. Tan and J. Tran Thanh Van, Phys. Lett. 81B (1979) 68; B. Kaidalov and K.A. Ter-Martirosyan, Phys. Lett. 117B (1982) 247. [6] S.L. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuov. Cim. 60 (1969) 47; W. Bardeen, Phys. Rev. 184 (1969) 1848. [7] J. Ashman et al., Phys. Lett. B206 (1988) 364; Nucl. Phys. B328 (1990) 1. [8] G. Baum et al., Phys. Rev. Lett. 51 (1983) 1135. [9] J. Kodaira, Nucl. Phzs. B165 (1980) 129. [10] J. Ellis and R.L. Jaffe, Phys. Rev. D9 (1974) 1444. [11] S. Brodsky, J. Ellis and M. Karliner, Phys. Lett. B206 (1988) 309. [12] A.V. Efremov and O.V. Terayev, Dubna preprint E2-88-287 (1988); G. Altarelli and G. Ross Phys. Lett. B212 (1988) 391. [13] R.D. Carlitz, J.C. Collins and A.H. Mueller, Phys. Lett. B214 (1988) 229; M. Anselmino and E. Leader, Santa Barbara preprint NSF-ITP 88-142 (1988); G. Altarelli and W.J. Stirling, Particle World 1 (1989) 40; G. Altarelli and B. Lampe, CERN preprint TH.5645/90 (1990). [14] R.L. Jaffe and A. Manohar, MIT preprint CTP

# 1706 (1989).

[15] S. Forte, Phys. Lett. B224 (1989) 189; Nucl. Phys. B331 (1990) 1.

25 [16] T.P. Cheng and Ling-Fong Li, Phys. Rev. Lett. 62 (1989) 1441; H.Y. Cheng, Phys. Lett. B219 (1989) 347; G. Lopez Castro and J. Pestieau, Phys. Lett. B212 (1989) 459; T. Hatsuda, Nucl. Phys. B329 (1990) 376. [17] G. Veneziano, Mod. Phys. Lett. A4 (1989) 1605. [18] G. Shore and G. Veneziano, Phys. Lett. B244 (1990) 75. [19] A.V. Efremov, J. Soffer and N.A. Tornquist, Phys. Rev. Lett. 1495.

64 (1990)

[20] E. Witten, Nucl. Phys. B156 (1979) 269; G. Veneziano, Nucl. Phys. B159 (1979) 357; Phys. Lett. 95B (1980) 90. [21] R. Johnson, N.W. Park, J. Schechter, V. Soni and H. Weigel, Syracuse preprint SU-4288-430 (1990). [22] B. Zumino, in "Lectures on Elementary Particles and Quantum Field Theory" (Brandeis 1970, S. Deser et al., editors, MIT Press) Vol. 2, p. 441.

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27 SUPERCONVERGENCE RELATIONS CHARACTERIZING QUANTUM CHROMODYNAMICS

Kazuhiko Nishijima *

Research Institute for Fundamental Physics Kyoto University, Kyoto 606, Japan

ABSTRACT Various fundamental problems in QeD, such as confinement and dynamical breakdown of chiral symmetry, are basically related to low energy properties of QeD. On the other hand, QeD is characterized by asymptotic freedom governing high energy behavior of Green's functions, and the central issue in QeD is concerned with the question of how to deduce low energy properties from those at high energies. This problem is partly resolved by making use of analyticity. It imposes constraints on the behavior of Green's functions in the entire complex energy plane, and consequently their low energy properties are not independent of those at high energies. These constraints are often expressed in the form of superconvergence relations.

L Renormalization Group Equations QCD is characterized by asymptotic freedom so that renormalization group (RG) equations play an important role in exploring the properties of quarks and gluons. In this section, therefore, we shall briefly recapitulate the essential features of RG equations. We start from the Callan-Symanzik equations 1,2) for Green's functions with exponentiated mass-insertion: 3) C (n,m)( x, ... , y ... , z .... ,: }-) \.

=< OIT['IjJ(x) ... ¢(Y) ... (h,(z)." *

exp(iI,

(1.1)

Address after April 1, 1990: Department of Physics, Faculty of Science and Engineering, ChuQ University, Kasuga, Bunkyo-ku, Tokyo 112.

28 where 1/J and lii denote the quark fields and = u(p)u(p).

(1.2)

Here, Ip > denotes a single quark state of momentum p, and u(p) and u(p) are the corresponding Dirac spinors. The RG equation for c(n,m) is given by3)

(V + WYF

+ m,v)c(n,m) = 0,

(1.3)

where

V

a

,v a

a

= m am + {3 ag - 2a aa - [1 + (l-

a

's )K]aK·

(1.4)

IF, IV and IS denote the anomalous dimensions of 1/J,

(2.5)

=, we find

alb> 0 alb < 0

(2.6)

In QeD the inequality alb> 0 is equivalent to Nf < 10, and this superconvergence relation can never be satisfied in perturbation theory. It indicates a delicate balance among all energy regions, and this illustrates the control of the spectral function Pv(p2) at low energies. When alb> 0, we have the celebrated superconvergence relation

(2.7) When this is the case, it is known that color confinement is realized. This can be shown by exploiting BRST invariance 8) of QeD and representations of BRST 9 16 )

algebra, but we shall refer to other publications for its proof.

31

2) Quark Propagator The gluon propagator is closely related to color confinement as we have seen above. By the same token the quark propagator is directly associated with chiral symmetry. The RG equation for the quark propagator is given by

(2.8) In the Landau gauge perturbation theory we have (2.9) Let us decomposed C(2,0) as

C(2,0)(p; m, m, g)

= -ip· 1~1 (p) + m~2(p),

(2.10)

and assume the following spectral representations implied by analyticity:

(i

= 1,2)

(2.11)

Then we can relate the bare mass mo to the physical mass m b/)

:0 ! =

dp,2 p,(p,2)/

!

(2.12)

dp,2Pl(p,2).

The RG method provides us with the following asymptotic behavior of the quark propagator for large values of p2:

!d

!

p,

PI (p,2)

2

p2

+ p,2 _

p2

+ p,2 _

dp,2

ic

p,(p,2)

Z2"l p2'

(2.13)

rv--

rv

ic

const. p2

(lnL) m2

-c/b

'

(2.14)

where c appears in the anomalous dimension I. as

I.(g) = cg 2 + ...

(2.15)

Z2 is the multiplicative renormalization constant of the quark field and is finite due to Eq.(2.9) as is clear from

a

Z2"l

= exp

[!

g2;(;))].

d

(2.16)

g

Again, by multiplying p2 by Eqs.(2.13) and (2.14) and by taking the limit p2 we find

-+ 00,

(2.17)

32 (2.18) The superconvergence relation (2.18) as combined with Eq.(2.12) implies

mo/m

= o.

(2.19)

This result poses a serious doubt on the classical equivalence between chiral symmetry and the vanishing bare mass. Suppose that the physical mass m is finite, then Eq.(2.19) implies mo = 0 and consequently the resulting theory should be chiral symmetric no matter how we choose the physical mass m, if we should insist on the classical equivalence. This sounds very unlikely, however. We must admit, therefore, that the classical equivalence must be abandoned in quantum theory, and we have to look for the proper definition of chiral symmetry expressed in terms of renormalized quantities alone. This is the subject of the next section.

3. Chiral Symmetry In an attempt to define chiral symmetry we propose to define it by the existence of an axial-vector current X).. satisfying the following two conditions: 1) conservation law

(3.1) 2) equal-time commutation relations (ETCRs)

5(xo - yo)[Xo(x), 1jJ(y)] 5(xo - yo)[Xo(x), ~(y)]

==-

'51jJ(y)5 4(x - y), ~(yh54(x - y).

(3.2a) (3.2b)

In what follows we shall look for the condition for the existence of such a current in QCD. In practice it is more convenient to introduce flavor-changing current X).., but we shall consider the flavor-conserving one for an illustrative purpose. Thus, we introduce some unrenormalized expressions bilinear in the quark fields. .1.(0) _ ':-D 0 ) A (O) ).. -t'/-' '~'5'/-'

P

(O) _ .:t..

= 2mB(g)P.

(3.17)

We also find (3.18)

mB(g) = maZs.

In QeD the bare mass ma vanishes as stated before, whereas Zs is divergent, and the I.h.s. of Eq.(3.18) is finite in general. Thus the theory is chiral-symmetric and Eq.(3.1) is satisfied only when mB(g)

= o.

(3.19)

This implies either m = 0 or B(g) = O. In the former case we have a massless theory and the theory is trivially chiral-symmetric. On the other hand, in the latter case chiral symmetry is dynamically broken thereby generating the massless Nambu-Goldstone boson, provided, of course, that m =f. O. Then, by making use of the Ward-Takahashi identity for the axial-vector current, we can study the large p2 behavior of {J., Sp(p)} or ofthe function b. 2(p) in iS

Eq.(2.10) with the help of the RG equation. We shall quote only the results. 1) m

=f. 0,

B(g)

)

=0 (3.20)

2) m

=f.

0,

B(g)

=f. 0

J

dJL2 p2

p,(JL2) - i€

+ JL2

rv

const. p2

(fnL) -c/b m2

•

(3.21)

Thus we may conclude that for

B(g) = 0

for

B(g)

The superconvergence relation (2.18) is valid in both cases.

=f. O.

(3.22)

35 It is my great pleasure and a privilege to dedicate this article to Professor Susumu Okubo on the occasion of his sixtieth birthday. I always cherish my memories of the inspiration and enthusiasm that he has conveyed to me at various opportunities, and I wish him many happy returns.

References 1. C.G. Callan, Phys. Rev. D2 (1970), 1541. 2. K. Symanzik, Comm. Math. Phys. 18 (1970), 227. 3. K. Nishijima and Y. Tomozawa, Prog. Theor. Phys. 57 (1980), 2197. 4. S. Weinberg, Phys. Rev. D8 (1973), 3497. 5. K. Nishijima, Prog. Theor. Phys. 81 (1989), 878. 6. K. Nishijima and M. Okawa, Prog. Theor. Phys. 82 (1989), 775. 7. R. Oehme and W. Zimmermann, Phys. Rev. D21 (1980), 471, 1661. 8. C. Becchi, A. Rouet and R. Stora, Ann. Phys. (NY) 98 (1976), 287; LV. Tyutin, Lebedev Report FIAN No.39 (1975). 9. K. Nishijima, Nucl. Phys. B238 (1984), 60l. 10. K. Nishijima and Y. Okada, Prog. Theor. Phys. 72 (1984), 294. 11. K. Nishijima, Prog. Theor. Phys. 74 (1985), 889. 12. K. Nishijima, Prog. Theor. Phys. 75 (1986), 1221. 13. K. Nishijima, Prog. Theor. Phys. 77 (1987), 1035. 14. K. Nishijima, Prog. Theor. Phys. 80 (1988), 897, 905. 15. R. Oehme, Phys. Lett. B195 (1987), 60. 16. R. Oehme, Phys. Lett. B232 (1989), 498. 17. S. Adler and W.A. Bardeen, Phys. Rev. 182 (1969), 1517. 18. K. Nishijima, preprint RIFP-844, January 1990.

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37 Scattering Theory in Quaternionic Quantum Mechanics

*

Stephen L. Adler Institute for Advanced Study Princeton, NJ 08540 ABSTRACT We discuss scattering theory in quaternionic quantum mechanics, first by use of the symplectic component formalism, and then by developing the quaternionic analog of time-dependent formal scattering theory and the Moller wave operators. We show that generic quaternionic quantum mechanics has a complex C( 1, i) but time-reversal nonconserving S-matrix.

1. Complex Versus Quaternionic Quantum Mechanics Professor S. Okubo has long had an interest in quaternionic and octonionic structures in quantum mechanics and field theory, and with this motivation, I wish to discuss here scattering theory in nonrelativistic quaternionic quantum mechanics. In quantum mechanics the probability Pab for the transition a

->

b is given as the absolute value squared of

the probability amplitude 00

(59)

limit of this state. We do this by using the fact that

If(-)(E', b, t)) :=n(-) Ifo(E', b, t)) (60a) --+ t~oo

Ifo(E', b, t)),

together with completeness of the If(-l) states,

L JdE'lf(-)(E', b, t)) (f(-l(E', b, t)1 00

1= A+

b

0

L JdE'lf(-l(E', b, t)) (fo(E', b, t)ln(-lt. 00

= A+

b

(60b)

0

Multiplying Eq. (60b) from the right by Eq. (59), and using Eq.(57b), we get

If(+l(E, a, t))

L JdE'lf(-)(E', b, t)) (fo(E', b, t)ln(-ltn(+llfo(E, a, t)). 00

=

b

0

(61)

51 We now define the S-matrix by S = n(-)j n(+l,

(62)

which by the intertwining property of Eq.(50) obeys

HoS

= Hon(-ltn(+l = n(-ltHn(+l = n(-ltn(+lHo =

SHo,

(63a)

or in other words,

[Ho, S] = 0.

(63b)

Hence the matrix element on the right-hand side of Eq.(61) is independent of the time t,

(fo(E', b, t)ln( -ltn(+llfo(E, a, t)) (64)

=(fo(E', b, O)ISlfo(E, a, 0)).

So we have

L JdE'lf(-l(E',b,t)) (fo(E',b,O)ISlfo(E,a,O)) 00

If(+l(E,a,t))

=

b

0

J 00

~L b

(65)

dE'lfo(E',b,t)) (fo(E',b,O)ISlfo(E,a,O)).

0

Equation (65) answers the question of what the incident state Ifo(E, a, t)) evolves to at large times: we get a superposition of free particle states, with the superposition coefficient (or probability amplitude) the corresponding free-state matrix element of the S-matrix. From Eqs. (53), (56), (57) and (62), we can easily show that the S-matrix is unitary,

sts = n(+lt n(-l n(-lt n(+l (66a) = n(+lt[1 - A] n(+l = n(+lt n(+l = 1,

and similarly

sst = n(-)j n(+l n(+lt nH (6Gb) = n(-lt[l- A] n(-l = n(-lt n(-l = 1.

52 5. The S-matrix is ql, i) Everything done in Sec. 4 is a direct analog of time-dependent formal scattering theory for the complex case. We now derive a result which is peculiar to the quaternionic case. Let us choose the basis states Ifo(E, a)) to obey the standard ray convention iE,a = i, so that

Holfo(E, a)) = Ifo(E,a))iE,

(67)

and define the S-matrix element Sba(E',E) by

Sba(E', E) = (fo(E', b)ISlfo(E, a)).

(68)

We then proceed to exploit the information contained in Eq. (63b),

HoS = SHo

=}

iE' Sba(E',E) = iE'(fo(E',b)ISlfo(E,a)) (69) = (fo(E',b) IHoSI fo(E,a)) = (fo(E',b) ISHol fo(E,a))

= (fo(E', b) lSI fo(E, a))iE = Sba(E', E) iE. Equating the absolute value of the left- and right-hand sides of Eq. (69), we get

IE' -

EI

ISba(E', E)I = 0,

which implies that Sba(E', E) vanishes if E'

of E.

(70)

Hence the S-matrix for quaternionic scat-

tering leads only to on-energy-shell transitions. Next setting E' = E and dividing Eq, (69) through by E

of

0, we get

(7Ia) which implies that the S-matrix element

Sba(E, E)

=

(fo(E, b)

lSI fo(E, a))

(71b)

53 is complex C(l, i). This gives us an intrinsically quaternionic derivation of the result obtained by elimination of the

f3

symplectic component in Sec. 3.

The derivation just given is a special case of the following general Lemma: Let S be a symmetry operator which commutes with iI,

siI = iIs,

(72)

and let {If(E, a))} be a complete set of energy eigenstates in the standard ray convention,

(73)

iIlf(E, a)) = If(E, a))iE. Then

(i) (f(E,a)ISlf(E',b)) = 0 for E'

# E, (74)

(ii) (f(E,a)ISlf(E,b)) is C(l,i) for E

#0

.

In other words, symmetry operators which commute with the Hamiltonian

III

quater-

nionic quantum mechanics lead to a complex C(l, i), rather than a quaternionic, matrix representation problem.

6.

Time-Reversal Violation in Quaternionic Scattering Let us now return to a more detailed examination of the optical potential of Eq.(33b).

Splitting Vop! into parts even and odd in

V2 , we have (75)

with -

v,;en(E) = VI

+ V-

1

2

Ho odd (E) Vop!

=

+ E + VI

1 Va Ho + E

t. [ -

_

-

V2

+ Va-

1

_ V- 3 , Ho + E + VI

-J

(76)

1 _ V- z - V-2 - Va . + VI Ho + E + VI

Recalling that the time-reversal operation in standard complex quantum mechanics (for spinless systems) is just complex conjugation, we see that v,;;en is time-reversal even and

54 v~ is time-reversal odd. Since

the generic case is time reversal nonconserving, with time-reversal conservation present only when

V2

and

Va

are linearly dependent.

To understand why T-nonconservation arises and is proportional to ~ what happens when

V2

V3 , let us examine

= O. We then have

(7S)

jiI]

=

-iI,

and so defining a time-reversed wave function

1/JT

by

(79a) we see that

1/JT

obeys the time-reversed Schrodinger equation

(79b) Similarly, if

V3

= 0, the time-reversed wave function

(SOa) obeys

81/JT _ 8( -t) -

-H1/JT,

(SOb)

and more generally, by a rotation of axes in the j-k plane, we can construct a time-reversed wave function

1/JT

whenever ~ and

11;

are linearly dependent.

On the other hand, in the generic case with linearly independent use the method of Eqs.

V2

and

11;,

we cannot

(7S)-(SO) to find a unitary time reversal operation. Moreover,

55 we cannot use the quaternion conjugation operation to give an antiunitary time-reversal operation. Although it is true that the Hamiltonian

iI of Eq.(29) obeys

if = -iI,

(81)

because quaternion conjugation reverses the order of factors in a product, the conjugate of

8t/J 8t

=

-iIt/J

(82a)

IS

81/;

--;::;

8(-t) = t/JH =

(82b)

-1/;iI,

which, since iI now stands to the right of 1/;, does not have the form of a time-reversed Schriidinger equation. The above results concerning time-reversal symmetry in quaternionic quantum mechanics have an interesting interpretation in terms of the classic Wigner theorem on realizations of symmetry operations in quantum mechanics, as extended to the quaternionic case by Bargmann. According to Wigner's theorem in complex quantum mechanics, a correspondence between transition probabilities

l(allb')1 = l(alb)1

(83)

can, with suitable phase choices for the states, always be realized in one of two ways: by a unitary transformation

la') = Ula),

(84a)

la') = (Ula))'.

(84b)

or by an anti-unitary transformation

56 Bargmann has shown that in the extension of this result to quaternionic quantum mechanics (for Hilbert spaces of dimension 2': 3), a correspondence between transition probabilities as in Eq.(83) can only have a unitary realization, as in Eq.(84a); there is no anti-unitary case in quaternionic quantum mechanics. Hence, as we have seen, time-reversal symmetry in quaternionic quantum mechanics cannot be realized using the quaternion conjugation operation; if a time-reversal invariance is present, it must have a unitary realization, which is possible only when 7.

V2

and

V3

are linearly dependent.

Conclusion We conclude that generic quaternionic quantum mechanics has a complex C(l, i) but

time-reversal nonconserving S-matrix. For further details, references, and a discussion of the possible relevance of these results to elementary particle physics, the reader is referred to the author's forthcoming book Quaternionic Quantum Mechanics (to be published by Oxford University Press). This work was supported by the U.S. Department of Energy under Grant No.DE-FG0290ER40542.

57

Understanding the Electric Charge of Quarks and Leptons* Rabindra N. Mohapatra t Department of Physics and Astronomy University of Maryland College Park, MD 20742

April 1990

Abstract I discuss ways to understand the quantization of electric charges of quarks and leptons as well as the vector-like nature of electromagnetism in the context of unified gauge theories. In particular, I note that in the context of the standard model, vanishing of neutrino electric charge combined with freedom from axial anomalies explains both the vectorlike nature of QED as well as quantization of electric charge. As a corollary, it follows that, if neutrino has a tiny charge, it implies parity violation in quantum electrodynamics and breakdown of electric charge conservation.

I. Introduction It is a particular honor for me to have this opportunity to pay tribute to

Prof. Susumu Okubo on his sixtieth birthday since I had the good fortune to be a student of his at Rochester from 1966 - 1969. Okubo is well-known for his contributions to particle physics such as the Gell-Mann-Okubo mass formula, Okubo-Zweig-Izuka rule, etc. But his contributions spanned many other areas of particle physics as well. In the early sixties, after Gell-Mann and Zweig proposed the quark model to explain the observed hadron spectra, many variants of this idea appeared in the literature - notable among them were the Han-Nambu threetriplet model and Maki-Hara model which postulated intercharged quarks unlike *Invited talk presented at the symposium held in Rochester, N.Y., May 4-5, 1990 to honor Prof. Susumu Okubo on his sixtieth birthday and to celebrate forty years of the Rochester Conferences. Work supported by a grant from Natioual Scieuce Foundatiou. To be published in the proceedings. tWork supported by the National Science Foundation.

58

the original Gell-Mann-Zweig model, which postulated fractional charges for the quarks. An important. question in t.hose days was how t.o dist.inguish between the different models. Okubo was the first person to realize in 1965 that 'frO -+ 2, decay provides a way to distinguish between t.he various quark models. He continued this work, continually refining it until 1969 when he used the axial anomalies for this purpose. Today, it is widely accepted that the fundamental constituents of matter the quarks and leptons have the following electric charges:

where the numbers within the bracket gives the electric charges in units of the positron charge. This has introduced remarkable simplification into physics. Along with these values of electric charge, it is of course well-known that electromagnetic interactions conserved parity and no evidence exists for any parity violating component of quantum electrodynamics. The question I would like to discuss in this article is that while we accept the above two facts regarding electro-magnetism i.e. (i) quantization of electric charge of quarks and leptons as well as the observed pattern of charges; and (ii) vector nature of QED, how well do we understand it from deeper principles. I will analyze this question within the framework of unified gauge theories and isolate the conditions that enable us to answer this question. The conclusion will be that both these properties of QED are connected with properties of the neutrino. 1 ,2 This article has been organized as follows: in sec. II I summarize the evidence in favor of charge quantization and charge conservation and review earlier attempts to understand charge quantization; in secs. III, IV, V, VI, VII, VIII I discuss our proposap,2 to understand these properties of E&M using the constraints that follow from the absence of axial anomalies and properties of the neutrinos; in sec. IX I comment on the fact that grandunified theories other than SU(5) do not in general lead to observed quantized values of quark and lepton charges; in sec. X and XI, I discuss the implications of a non-vanishing neutrino charge in the standard model; I then conclude with comments on the situation in models with extended gauge or fermionic sector.

59

2. Evidence for Electric Charge Quantization and Conservation: As not.ed earlier, t.hree outst.anding properties of electric charge are: (i) its conservation; (ii) Quantization and (iii) its vector-like character. In this section, we will discuss t.he experimental observations that enable us t.o arrive at the above conclusions.

(i) Charge Conservation: If electric charge were not conserved, one would expect many new decay processes t.o take place in atoms and nuclei

(ref. 3) (ref. 4) n---+p+, ---+p+v+ii ---+

p

+ e- + e+

(ref. 5) .

There exist stringent limits on the above decay processes from various experiments, which look for spontaneous X-rays or ,-ray emission from atoms 6 and nuclei 7 : we list below some of these limits:

r(e- ---+ v + ,):::: 10 25 yrs. r(n ---+ p+ neutral$) GeV1.9 x 10 18 yrs. r(G a ---+ G e + neutrals):::: 2.3 x 10 23 yrs. 1.9 x 10 21 yrs. (K- Capture in Iodine)

rAQicO ::::

(A vignone et al., ref. 6) (Norman et al., ref. 7) (Barabanov et al., ref. 7) (Holjevic et al., ref. 7).

One can also derive a limit on the characteristic transition time for efrom the first limit (i. e. r( e- --) v + ,)) to be 4

(1) r+

e+

(2) There has also been a great deal of theoret.ical discussion of possible charge non-conserving processes occurring in nature 3 ,4,8,9 One of the inevitable consequences of electric charge non-conservation is a non-vanishing mass of the photon, m-y (although a non-vanishing photon mass does not necessarily imply electric

60 charge non-conservation.) There exist limits on the photon mass from the existing geomagnetic data as well as magnetic fields near the Jupiter:10 m"( ~

10- 22 MeV.

(4)

There also exist other bounds on the photon mass from other considerations: Galactic Magnetic Fields m"( ~ Alfren waves: m"( ~ Gauss Law Tests: ~ 1.0 Satellite Data: ~ 6 x

3 x 10- 33 MeV 7.3 x 10- 22 Me V x 10- 20 Me V 10- 21 MeV

(ref· 11) (ref. 12) (ref. lS) (ref. 14) .

(5)

Let us now discuss the evidence for quantization of electric charge. We will discuss upper limits on residual charges of atoms, neutrons and neutrinos: The electric charge on an atom is given by Q(Atom) = N Qn

+ (Qp + Qe) Z

(6)

(in units of the positron charge). By looking for the effect of electric fields on intense beams of reactor neutrons, a group from ILL, Grenoble 15 has obtained the most stringent limit on the charge of the neutron:

(7) Investigations into the neutrality of atoms goes back many years and was initiated by Piccard and Kessler,16 who obtained an upper limit on the residual charge on CO 2 :

(8) Since then, experiments have been carried out on a large number of gases as well as molecular beams with the following set of limits:

± 6) x 10- 20 Q(Ar) ~ (4 ± 4) x 10- 20 Q(He) = (4 ± 2) x 10- 20 Q(H 2 ) = (-2 ..5 ± 1..5) x 10- 20 Q( C s ) = (1.3 ± 5.6) x 10- 17 Q(K) = (3.8 ± 11.8) x 10- 17 Q(N2 ) ~ (6

ref. 17 ref. 17 ref. 18 ref. 18 ref. 19 ref. 19.

(9)

61

Finally, from the observed time spread 20 in the neutrino signal from SN 1987 A, one can obtain a limit on the electric charge of the neut.rino to be 21

(10) The idea here is that. if neutrino had an electric charge, the slower neutrinos would bend more in the intergalactic magnetic field than the faster moving neutrinos and travel longer distances. As a result, they would arrive at the earth later causing a time spread between the faster and slower neutrinos. If the bound in eqn. (10) was violated, the time spread between neutrinos would be more than what is observed. It is worth noting at this point that if indeed the proton, neutron, electron and neutrino have tiny amounts of residual electric charges in addition to their usually assumed charges, assumption of charge conservation in beta decay would provide an extra constraint on the charges residing in their left-handed chiral components:

(11) Under the assumption of SU(3)c color invariance and t.he usual quark model according to which p = uud and n = udd, eqn. (11) translates to

(12) If we writ.e QUL = ~ + €UL' eqn. (12) implies that

QdL

=

-~

+ €dL'

QeL

= -1 + EeL

and Q"

= €",

then

(13) If we further assume that (u, d) and (v, e) transform as isodoublets under an SU(2) gauge symmetry, then we get only two free parameters €"L and €h = €UL' The key theoretical question that we will discuss in the ensuing section is: "are there

theoretical considerations that imply €"L = €UL = 0 as well as QUR = ~, QdR - ~ , QeR = -1, thereby leading to an understanding of charge quantization. Finally, we discuss a related matter, which concerns the possible existence of mini charged particles. Such particles have been discussed in various theoretical models. 8,9,n If such particles are light (me :c:: 10 Me V), they would effect (9 2) of electron and muon, Lambshift., energy loss from stars and supernova, etc. Limits on these electric charges of light mini charged particles have been obtained recently:23,24,25 SN 1987 A: 7 X 10- 10 :c:: Qe :c:: 10- 7 me:C:: 10 Me V (ref· 24) (ref. 23) Qe :c:: 10- 2 , me:C:: 10 Me V (9 - 2)e,1' 13 Cooling of Red giant Qe:C:: 10- , me:C:: 10 Ke V (ref. 25)

(14)

62

3. Theoretical Understanding of Electric Charge Quantization and Vector-like QED: In the past there have been two celebrated attempts to understand the quantization of electric charge: (i) Dirac's hypothesis 26 of magnetic monopoles and (ii) Grandunification of all forces of nature in a gauge theory framework. 27 According to the Dirac's hypothesis, if there exists a monopole of magnetic charge 9 , single-valuedness of quantum mechanical wavefunction involving a monopole and an electric charge e implies the quantization condition: eg = 27rnn .

(15)

Thus, the existence of a single monopole implies the discreteness of all electric charges in the universe. The unsatisfactory aspect of this approach is that it does not explain why Qu = -~ Qe = -2Qd = ~. Turning now to grand unified theories, since one uses only simple groups or products of simple groups to describe the interaction of elementary particles, it is obvious that the ratios of electric charges in these theories such as Qu/Qe, etc. are rational numbers; but again as in the case of monopoles, it does not explain why Qu = -~Qe = -2Qd = ~. To make these detailed predictions one needs additional assumptions concerning the nature of symmetry breaking from the grandunification scale down to the standard model symmetry. As we will show later on in this article, different choices of Higgs boson can lead to different values for the electric charges of quarks and leptons, in grand unified theories. Since we will conduct our discussion within the framework of unified gauge theories, let us briefly review the gauge theory framework for particle interactions. For each generation of fermions, the minimal gauge group is SU(3)c x SU(2)L x U(I)y and it will be assumed to break down to U(1 )em' Of course, the gauge group could be bigger and in general there are two distinct classes of possibilities:

(i) Gauge group G

=

Go

X

U(1)Y:

Here, Go could simply be SU(3)c x SU(2)L or could be a more general nonabelian group that contains SU(3)c x SU(2)L' In either case, since we assume unbroken SU(3)c , the electric charge will be a linear combination of nonabelian diagonal generators T; of Go and Y as follows: r-2

Q =2: UiTi+ Y i::;::l

(16)

63 where r is the rank of the group and the sum goes only upto r - 2 because unbroken SU(3)c has rank 2 and its diagonal generators do not. conhihl1t.p t.o Q. We can redefine electric charge to set at = 1. At this stage, we can see the problems with both charge quantization and vectorlike nature of QED. The number of free parameters giving electric charges of particles are nj + r - 3 where nj is the number of irreducible fermion multiplets and each fermion multiplet has an arbitrary continuous hypercharge Y,. Since apriori Yt's are continuous parameters, electric charge is not quantized. Apriori the vectorlike nature of QED also does not follow.

(ii) Non-Abelian Gauge Symmetry G ::) SU(3)c

X

SU(2)L x U(I)y:

Obviously, in this case, there are no arbitrary hypercharge quantum numbers to worry us; but the electric charge formula is given apriori as follows: r-2

Q=

Lai T;.

(17)

i=l

Again as before, by redefining electric charge, we can set at

=

1 leaving us with

r-3 parameters, which we must fix using other constraints. Since Ti are generators of a non-abelian group, the values of electric charge are not continuous but they will not in general lead to desired values, without additional assumptions. The only exception is the SU(5) model which has r = 4 and therefore, leads to unique values for quark charges under the assumption that SU(5) breaks down to SU(3)c x SU(2)L x U(I)y. Thus, it is important. to stress that there is a problem with understanding the values of electric charge even in general grandunified theories. In order to improve our understanding of the electric charges, let us use the axial anomaly constraints on the parameters of the theory. As is well-known in general gauge theories with chiral fermions one can have Adler-Bell-J ackiw anomalies, which will destroy the conservation of gauge current at the one-loop level. These anomalies arise from triangle diagrams with three gauge bosons (or currents) attached to the three vert.ices. Therefore, unless we rest.ore the current conservation, the renormalizability of the t.heory, which relies on the assumption of current conservation will be spoilt. Thus, renormalizability of gauge theory requires t.hat the chiral generators of the gauge group satisfy the constraint:

(18) In the next section, we will explore the role of the anomaly constraints in

64

fixing the electric charges of elementary particles.

4. Anomaly Constraints and Electric Charge Quantization: Let us first note that in the case of a completely non-abelian gauge symmetry, anomaly constraints only fix the allowed fermion representations; but unfortunately, they do not shed any light on a;'s in eqn. (17) - thus, arbitrariness of the electric charge in grandunified theories persists at this stage. The situation, however, is very different in theories with U(l)y group, where renormalizability demands that the following constraints be satisfied by the different hyper charge parameters:

(19)

where df = Tr {T;, Tj

}

Irermwn rep. D'

• ,

T; being the generators of the nonabelian

part of the group. The first constraint follows from consideration of triangle diagrams involving three U( 1) gauge currents, second follows from the diagrams with one U(l) gauge current and two non-abelian gauge currents and the third from one U (1 )-gauge current and two gravitons. The case for the freedom from mixed gauge-gravity anomaly is perhaps not as strong as the other two but, eventually, in a fully unified theory, one will have to demand it for consistency. To see how it restricts the arbitrariness of electric charges, let us first apply these considerations to the standard model and the left-right symmetric model. Consider gauge models of the type SU(3)c x Gw x U(l)y. There are then the following types of anomalies to be considered, if we assume that Gw is SU(2) or a product of SU(2) groups. [This is easily generalized if Gw is other than SU(2) .]

(A) Tr U(l)y [SU(3)C]2 (B) Tr U(l)y [G W ]2 (C) Tr [U(1)y]3 (D) U(l)y x Gravitational anomaly.

(20)

65 One could also consider the global Witten anomaly. Since this does not involve the hyper charge quantum number, we shall not concern ourselves with it in this paper. Let us now consider the various cases.

(i) G w = SU(~)£: Depending on whet.her we include the right-handed neutrino, l/R, in the fermionic spectrum or not., we will have two possibilities. Let us first consider the standard model spectrum (i.e., no l/R)' We take the following assignments of the quarks and leptons keeping the hypercharge Y as free parameters. Q£ UR

dR l£ eR

(3,2,Yq) (3,1, Y u ) (3,1,Yd) (1,2,Yi) (l,l,Y.).

(21)

We consider only one generation of fermions for the moment. The vanishing of the four anomaly co-efficients implies the following relations between the Y's.

(A) (B) (C) (D)

2Yq -Yu -Yd =O 3Yq - Yi = 0 6Y,/ - 3Y; - 3Y] + 2Y? - 1';,3 = 0 6Yq + 2Yi - 3Yu - 3Yd - Yo = 0 .

(22)

These constraints imply either28

Yd

Yo

=

Yl

= Y2 = 0

and

=

2

"3 Yi;

Ye

Y u = -Yd'

=

2Yi .

(23a)

(23b)

In eqn. (23a) there is an overall arbitrariness in the hypercharges and therefore, there is no charge quantization. Of course, one might argue that couldn't we simply redefine the U(l)y gauge coupling to choose Yi = -1, thereby implying charge quantization? The point is, however, more subtle since to determine what the unbroken U(l) generator that corresponds to electromagnetism is, we must break the gauge symmetry via the Higgs mechanism, i.e. by giving a non-zero vacuum expectation value (vev) to a Higgs field with hypercharge Y. If we choose an 8U(2)£ doublet· Higgs field and give a vev to its f3 = f3 component,

66 the electric charge formula is

(24) Note that if we have already rescaled the U(l)y gauge coupling to choose YI = -1, we have no more freedom to fix Y¢ , which, being arbitrary does not lead to charge quantization. Turning now to the eqn. (23b), this is a solution which allows a color nonsinglet baryon violating mass term30 involving URand dR. We will see how to eliminate this solution in the next section. Let us now consider the extension of the standard model with VR included. Eqs. (22C) and (22D) receive new contributions and become (denoting Y of VR by Yv )

(C) 6yq3 - 3Yu3 - 3Y} + 21,? - Y,,3 - Yv3 = 0 (D) 6Yq - 2Y1 - 3Yu - 3Yd - Yo - Y,:, = 0 .

(25)

Eqn. (3) with Yv = 0 is of course a solution for this case, but it is not the only solution and again electric charge remains arbitrary.31

(ii) SU(2)L X SU(2)R :32 In this case, the right-handed neutrino is automatically included as an isopartner of eR and again assigning fermions to doublets, we have QL : (3,2,1,YqL) QR : (3,1,2, YqR ) lL : (1,2,1, Yi L ) lR : (1,1,2,Yi R ) ·

(26)

As before, the anomaly constraints on Yare

(A) YqL - YqR = 0 (B) 3YqL - YlL = 0; 3YqR

+ YlR =

0.

(27)

The above equations imply YqL = YqR and YiL = Yl R , i.e., the U(l)y is vectorlike. As a result, the constraints (C) and (D) are automatically satisfied. It is amusing to see that parity conservation of weak interactions is implied by anomaly cancellation. Eq. (27B) then implies

1 Yq=--Y/. 3

(28)

67 Again, even though there are fewer parameters to start with, there is no charge quantization. It is dear that we need additional constraints to understand charge quantization.

5. Massive Charged Fermions and Vector-like QED: Any realistic gauge model of electroweak interactions must have a mechanism to explain the non-vanishing masses of quarks and charged leptons. The usual procedure is to write down the gauge-invariant Yukawa coupling between fermions and the Higgs scalar responsible for spontaneous symmetry breaking. Once the Higgs scalar acquires non-zero vev and reduces the gauge symmetry down to U(l )em , it will also generate non-zero masses for the charged fermions. The gauge invariance of the Yukawa couplings relates the hyper charges of the Higgs boson to those of quarks and leptons. As explained before, the hypercharge of the Higgs boson is what determines the electric charge; one might, therefore, hope that non-vanishing masses of quarks and leptons will provide enough constraints to lead to electric charge quantization. To see what really happens, let us again analyze the case of the standard model. As is well known, the Higgs boson that triggers gauge symmetry breaking as well as fermion masses is an SU(2) doublet with hypercharge Yci>' Let us redefine the U (1) gauge coupling to make Yci> = +1. The electric charge formula following from Eq. (24) is then (choosing I3ci> = -1/2)

Q = In

Y

+"2 .

(29)

The gauge invariance of the Yukawa Lagrangian

(30) where ~

= iT2¢>* , then implies Yu =Yq+1 Yd =Yq -1 Ye =Yi+1.

(31)

It then follows from Eqs. (23), (29) and (31) that electric charge is quantized. This point could have also been seen without any reference to Higgs bosons 33 as

68 follows. Electric charge conservation implies that for mass terms to appear the left- and right-handed helicity component.s of each charged fermion must have the same electric charge. It is easy to see using eqn. (29) that this implies eqn. (31). A relevant point to recognize here is that demanding non-vanishing fermion masses is equivalent to requiring electrodynamics to be vector like since left- and right-handed helicities pair up for each value of electric charge. In the context of standard model, this implies electric charge quantization. 6. Constraints of Vanishing electric charge of the Neutrinos: It has been pointed out by Babu and this author22 in a recent paper that an alternative viewpoint to take is not to use mass constraints but to demand that the neutrino is electrically neutral. It then follows from eqn. (29) that Y( = 1, which leads to Qu = ~, Qd = -~ and Q, = -1 as well as vectorlike QED.

While we recognize that the neutrality of the neutrino may be considered an adhoc requirement, we wish to stress that the neutrino is the only unpaired chiral fermion in the standard model and a non-zero neutrino charge may therefore imply pathologies in quantum electrodynamics such as non-decoupling of longitudinal photon, etc. 34 A second point of interest is more historical in the sense that /5invariance of a massless neutrino was used by Marshak and Sudarshan to argue in favor of V-A theory of weak interactions; our assertion is that charge neutrality of the neutrino may tell us why QED is vectorlike. Thus, our understanding of the Lorentz transformation properties of electroweak interactions may be tied to two properties of the neutrino i.e. m" = 0 and Q" = O. Conversely, if the neutrino has a tiny non-vanishing electric charge, it would imply a small admixture of parity violating component in QED. We will elaborate on this possibility later on. The situation can be illustrated as follows:

//

Ivector

QED I

Electric Charge Quantization

7. Models with Right-handed neutrino (VR): There is immense physical interest for considering models with right-handed neutrino since there exist some hint.s from experimental results such as the solar

69 neutrino puzzle, cosmological scenarios such as the role of dark matter in galaxy formation, etc. that neutrinos may be massive. Most theories for massive ne11trinos contain a right-handed neutrino for each generation. We saw in sec. 4 that in the presence of the right-handed neutrino, there is an additional hypercharge parameter entering our discussion; therefore anomaly constraints leave two parameters free Yl and Y". The question now is whether the additional charged fermion mass constraint (or equivalently the vector-like nature of QED) can help us to eliminate this ambiguity leading to charge quantization. We will consider both the models with VR discussed earlier: (i) Standard Model with VR: In this case, a Dirac mass term which connects VL to VR can be written: hJL¢VR' In addition to Eq. (31), we have the following new constraint involving the hypercharges: (32) (Note that Eq. (32) also follows from Eqs. (22) and (25) even in the absence of ihvR mass term.) Suddenly the situation becomes very different. Equations (22A),

(22B), (22C) and (22D) are satisfied by (33) This leaves the hypercharges as well as the electric charges arbitrary. Thus, if neutrinos are found to have mass, and they are Dirac particles, one looses charge quantization.

(ii) Left-Right Symmetric Model: Before discussing the mass constraints, let us note that, in general, in leftright symmetric models the electric charge is given by Q

Y

= In + bI3R + c 2

.

(34)

If we want the charged fermions to be massive that electric charge conservat.ion implies that b = 1 or QED is vectorlike. But there is no new constraint on the hypercharges. Thus Dirac mass constraint again does not lead to electric charge quantization. Therefore, the electric charges of the neutrino are left-free although Q"L = Q"R as would be required for their Dirac masses. 8. Majorana Neutrinos and Electric Charge Quantization:

70 It was suggested in ref. 1 that requiring neutrino to be a Majorana particle leads to quantization of electric charge. This can be seen very easily becanse Majorana VR mass term is of the form V~C-IVR' which means, from gauge invariance that Q"L = Q"R = 0 or II = -1 both in the standard as well as the left-right model. Applying this to eqn. (23), we predict that in the standard model Yq = ~ j Yu = ~ and Yd = - ~ , which leads to correct values for the quark charges. Turning to the left-right model, again Majorana v implies from eqn. (34) that, c = -#;. Using eqn. (28), we can then conclude that Qu = ~ and Qd = -~.

This is physically interesting since gauge theories apriori did not have any way to distinguish between Dirac and Majorana neutrinos but understanding charge quantization provides a way and choosese the Majorana alternative.

9. Situation in Grand-unified Theories: As discussed in sec. 3, in the general class of theories without U(l)'s and in particular in grandunified theories, the electric charge formula is given by the formula in eqn. (17), where ai's are unknown parameters. The constraints of anomalies do not provide any additional constraints. Therefore, except for the simplest grandunified theory based on the SU(5) group, the correct values of quarks and lepton changes do not follow without making additional assumptions. This is related to the fact that when we demand that a grand unified symmetry breakdown to SU(3)c x SU(2)L x U(l)y, the Y-values of fermions depend on which Higgs multiplet is used to break the gauge symmetry. We illustrate this by using the simple example of SU(2h x SU(2)R x SU(4)c gauge symmetry. As mentioned earlier, the electric charge formula in this case is given by

Q

= J3L + bJ3R + eFts

(35)

Let us consider two choices for the Higgs bosons that break this symmetry down to SU(3)c x SU(2)L x U(l)y:

(a) Higgs Multiplet (1, 3, i-o)

==

~R:

Since we want to preserve SU(3)c , we choose the color-neutral component of ~R which has J3R = -1 and F IS = to acquire a non-zero vev. This implies,

VI

71

c=

+If b leading to the charge formula (36)

Demanding vectorlike nature of QED implies b = 1 giving the usual formula,32 leading to Q" = 0, etc.

(b) Higgs multiplet (1,2,10) = ER On the other hand, if we choose ~R to break the symmetry down to the standard model, we find, under the assumption of vectorlike QED,

(37)

This leads to Q"R = Q"L = ~ , Qu = {2' Qd = -{2 , etc. which values are of course quantized but totally unphysical. This kind of situation persists in all grandunified theories with r > 4. 35

10. Charged Neutrino and Parity Violation in Quantum Electrodynamics We saw in sec. 6 that a zero electric charge of the neutrino implies parity conservation by QED. We could turn this argument around to conclude that if the neutrino had a tiny electric charge in the standard model, via anomaly equations, this would imply parity violation in the interactions of photon with charged particles such as the electron and the proton. In fact, the anomaly equations (22a-d) imply that, if Q" = f/2, then from the electric charge formula Q = 13L + ~ , Yi=-l+f

Yq

1

= -

3

(1 - 10)

72

Yd

2 (1 - £) 3

= --

(38)

leading to QUL - QUR = £/2, Qd L - QdR = -£/2, QeL - QeR = -£/2. This results in a parity violating component of QED given by

(39)

Since the most stringent bound on the electric charge of the neutrino is obtained from SN1987 A observations 21 to be Q" :::; 10- 17 e, we conclude that £ :::; 10- 17 • This means that the strength of the parity violating QED interactions is:::; ex 10- 17 • In the non-relativistic limit, Eq. (39) leads to spin dependent interactions of the form 'ljJ t ii'ljJ . i as well as 'ljJ t ii . V'ljJ¢ , where AI' = ('ljJ, A). Also, we the atoms will not

f/2

be neutral; in such a case, the stability of galaxies requires £ :::; (G N . m;/lO = 10- 20 • Furthermore, in our model Q( e) + Q(p) = 0 whereas Qn = -Q" = -£/2. (This applies both to the vector and axial vector components.) When Qp + Qe = o, a more stringent bound on £ can be obtained from the following considerations. The earth will be electrically charged due to its neutron content and will have a radial electric field E ~ £ X 10 30 volts/meters. If we assume that E should not exceed 100 volts/meters, this implies e :::; 10- 28 e which is the most stringent observational bound on the neutrino charge in the context of the SU(2)L x U(l)y model. 11. Fermion Masses and Breakdown of Electric Charge Conservation

In order to study further implications of a nonvanishing neutrino charge, we note that once the fermions acquire mass, there will be breakdown of electric charge conservation since a Dirac mass connects the left and the right-handed chirality components. To show this explicitly, we first realize that the Higgs doublet which defined the electric charge by causing breakdown of SU(2)L x U(l)y to U(l )em had its hypercharge Y", scaled to Y", = 1; it therefore has no gauge invariant Yukawa couplings. let us therefore introduce a second Higgs doublet 'P2 with Y 2 = 1 - £. Its components have electric charges Q (Qt1/2) = 1 - £/2 and Q ('P~1/2) = -£/2. It has gauge invariant Yukawa couplings to the fermions. To get fermion masses we have to give non-zero v.e.v. to ('P~1/2) = V2 . This not only breaks electric

73 charge conservation 9 but also gives photon a mass: e

m-y

VI V2

= J2 € JVl + Vl

( 40)

Due to Z - I mlxmg in the mass matrix [the mlxmg angle is given by 8 = (M-y I M z ) (V2 IVI ) 1, additional couplings (both parity-violating and parity-conserving) of the photon to the fermions proportional to € are induced. These new couplings also obey Qe + Qp = 0 and Qn + Q" = 0, so that the bounds on € discussed in sec. 4 are still valid. Since 2.7 X 10- 4 e Vi (ii) V2 < 2.7 X 10- 4 e V. The value 2.7

74 10- 4 e V corresponds simple to the present black-body radiation temperature. The significance of t.his number is that. in case (i), U(l )em symmetry is already broken now and the limit on photon mass discussed in eqn. (.5) applies; on the other hand, in case (ii), U(l)em symmetry is exact in the present universe since the present universe is hotter than the electric charge violating vev, V2 • Therefore, photon mass is zero now and the bounds in eqn. (5) are irrelevant. 36 An intriguing possibility is that V2 is slightly less than 2.7° K so that, in only a "few" years, the universe cools below V2 and photon suddenly acquires mass after that "date." The value of Tn" in that case could be as large as 10- 4 e V (corresponding to a range :::::: 10- 1 em, for electromagnetic forces which would have debilitating consequences for electronic modes of telecommunication, light propagation, etc. 37 X

To summarize, in this article, we have attempted to understand the electric charges of quarks and leptons as well as the vector-like nature of QED in the unified gauge theories. It appears that a proper and complete understanding of these questions requires in addition to the theoretical constraint of anomalies, also some propert.ies of the neutrino such as its electrical neutrality or equivalently its Majorana character. We also present some speculations on parity violation in QED and electric charge non-conservation in the context of these models. It is also worth remarking that the situation is in general more complicated in models with extra fermions such as technicolor38 or mirror fermion models and electric charge quantization does not follow.

References

1.

K.S. Babu and R.N. Mohapatra, Phys. Rev. Lett. 64,938 (1989) and Phys. Rev. D41, 271 (1990).

2.

K.S. Babu and R.N. Mohapatra, Univ. of Maryland Preprint U.MD-90-217 (1990).

3.

L.B. Okun and Ya. B. Zeldovich, Phys. Lett. 78B, 597 (1978); L.B. Okun and M.B. Voloshin, Pizma Zh. Eksp . . Theor. Fiz. 28, 156 (1978).

4.

R.N. Mohapatra, Phys. Rev. Lett. 59,1510 (1987).

5.

J.N. Bahcall, Rev. Mod. Phys. 50,881 (1978).

6.

M.K. Moe and F. Reines, Phys. Rev. 140B, 992 (1965); R.I. Steinberg et ai., Phys. Rev. D 12, 2582 (1975); E. Belloti et ai., Phys. Lett. 124B,

75 435 (1983); A.A. Pomansky, Proceedings of the International Neutrino Conference, Aachen, ed. H. Faissner et al. (Vieweg, Braunschewig, Germany), p. 641; F. Avignone et al., Phys. Rev. D34, 97 (1985). 7.

S. Holjevic, B.A. Logan and A. Ljubicic, Phys. Rev. D35, 341 (1987); E.B. Norman and Alan G. Seamster, Phys. Rev. Lett. 43, 1226 (1979); I. Barabanov et al., Pizma Zh. Eksp. Theor. Fiz. 32, 384 (1980); S.C. Vaidya et al., Phys. Rev. D27, 486 (1983).

8.

A.Y. Ignatiev, V.A. Kuzmin and M.E. Shaposnikov, Phys. Lett.B84, 315 (1979).

9.

S. Nussinov, Phys. Rev. Lett.59, 2401 (1987); M. Tsypin, ITEP preprint 8588 (1988); M. Suzuki, Phys. Rev. D 38, 1544 (19 ); L.B. Okun, Comments in Nucl. and Part. Physics 19, 99 (1989).

10.

A.S. Goldhaber and M.M. Nieto, Rev. Mod. Phys. 43,277 (1971); L. Davis, A.S. Goldhaber and M.M. Nieto, Phys. Rev. Lett. 35, 1402 (1975).

11.

G.V. Chibisov, Usp. Fiz. Nauk, 119,551 (1976).

12.

R. Hollweg, Phys. Rev. lett. 32, 961 (1974).

13.

R. Williams, R. Faller and D. Hill, Phys. Rev. Lett. 26,726 (1971).

14.

Gintsburg, Sov. Astr. AJ7, 536 (1964); (1965).

15.

W. Mampe et al., Nucl. Instr. and Methods A 284,130 (1989).

16.

A. Piccard and E. Kessler, Archives des Sciences Physique et Naturelles (Geneve) 5me periode, vol. 7 (Geneva, 1925), p. 340.

17.

A.M. Hillas and T.E. Cranshaw, Nature 184,892 (1959); 186,459 (1960).

18.

J.G. King, Phys. Rev. Lett. 5,562 (1960).

19.

J.C. Zorn, G.E. Chamberlain and V.W. Hughes, Phys. (1963).

20.

A. Bionta et al., Phys. Rev. lett. 58, 1494 (1987); K. Hirata et al., Phys. Rev. Lett. 58, 1490 (1987).

21.

G. Barbiellini and G. Cocconi, Nature 329, 21 (1987).

. Patel, Phys. Lett.14, 105

Rev.

129, 2566

76

22.

K.S. Babu and R.N. Mohapatra, Univ. of Md. Preprint 90-217, (1990).

23.

M.1. Dobroliubov and A. Yu. Ignatiev, Phys. Left. B206, 346 (1988); Kyoto preprint, RIFP-837 (1990).

24.

R.N. Mohapatra and I. Rothstein, Univ. of Md. Preprint, 90-227 (1990).

25.

M. Suzuki, ref. 9.

26. /P.A.M. Dirac, Pmc. Roy. Soc. A133, 60 (1931). 27.

J.C. Pati and A. Salam, Phys. Rev. DI0, 275 (1974); H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32,438 (1974).

28.

C.Q. Geng and R.E. Marshak, Phys. Rev. D39, 693 (1989).

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J.Minahan, P. Ramond and R. Warner, Phys. Rev. Dll, 715 (1990).

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C.Q. Geng, Phys. Rev. D41, 1292 (1990).

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32.

J .C. Pati and A. Salam, Phys. Rev. D 10, 27.5 (1974); R.N. Mohapatra and J.C. Pati, Phys. Rev. Dll, 566, 2558 (1975); G. Senjanovic and R.N. Mohapatra, Phys. Rev. D 12, 1502 (1975).

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35.

See, for instance, L.B. Okun, M. Voloshin and V. Zakharov, Phys. Lett. 138B, 115 (1984).

36.

J. Primack and M. Sher, Nature 288,680 (1980).

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38.

X.G. He et al., Phys. Rev. D40, 3140 (1989).

77

COMMENTS ON HADRONIC MASS FORMULAE

*t

FEZAGURSEY Center for Theoretical Physics, Yale University New Haven, CT 06511

ABSTRACT The evolution ofhadronic mass formulae since Okubo's pioneering contributions is briefly reviewed with special emphasis on group theoretical descriptions and supersymmetry suggested by QCD and based on diquark-antiquark symmetry.

I. Introduction

1 In 1962 the Gell-Mann-Okubo mass fonnula illuminated the low-lying hadronic mass spectrum. It was based on the flavor SU(3) and its breaking into its SU(2)xU(1) maximal subgroup of isospin and hypercharge. It led to the pseudoscalar octet mass fortnula

[

2 2 11 = 110 + a I (I + 1) -

2]

Y "4

(1.1)

where I is the total isospin and Y the hypercharge. A more refined mass fonnula 2 depending also on 13 would describe electromagnetic mass differences which we ignore in this talk. The baryon masses for the octet and the decuplet are also well described by the linear mass fonnula m = mo + a. [ I (I + 1).- :2] +

~ Y.

*Research supported in part by the U. S. DOE contract No. DE-AC-02-76 and ER03074 and 3075 tlnvited lecture for the Okubo Fest, Rochester, May 1990

(1.2)

78 These lead to the well-known sum rules m 2 = (4m 2 11 K

m 2 ) /3

(1.3)

1t

for the pseudoscalar octet, (1.4)

for the baryon octet, and (1.5)

for the baryon decuplet. The latter fonnula led to the discovery of ~r by Samios and 3 collaborators just as the pseudoscalar octet fonnula had led to a successful search for 4 the 11 meson earlier . The mass fonnula for the vector mesons presented a more delicate problem since the isospin singlet members of the nine vector mesons, namely the physical ro and were mixtures of SU(3) octet and singlet states, involving a mixing angle ev as a new parameter. In 1963 Okubo proposed a modelS for the detennination of this mixing angle by requiring the nine vector mesons to fit in the 3x3 matrix K*+ ) K*

(1.6)

o

The octet and singlet SU(3) multiplets are given respectively by the traceless part of V and the trace of V with proper nonnalization. If we regard the elements of V as qq states, this means that is an ss state while ro and p are composed of u and d, where u, d and s are the three lightest quarks. This hypothesis fixes ev to satisfy 1 sin e = v

giving the ideal mixing angle

ev - 35.3°.

(1.7)

(3

Experimentally

ev is 36° for the linear mass ,

fonnula and 39° for the quadratic one. Since there is a ninth pseudoscalar 11, Okubo's analysis of the vector nonet led to a reexamination of the octet-singlet mixing for pseudoscalars. In that case the mixing angle e is small and around 10° for the s quadratic mass fonnula.

79 6 The group theoretic interpretation of ideal mixing followed soon after with the enlargement of SU(3) to SU(3) x SU(3)_ , one SU(3) being associated with the q q quarks and the other SU(3) with the antiquarks that are the constituents of the vector mesons. Then, the nonet corresponds to the representation (3,3) of this group. Since the u, d quarks are much lighter than the strange quark s, the SU(2) x SU(2) subgroup is not badly broken, so that we must decompose with respect to the subgroup SU(2)q x U(l)q x SU(2)q x U(l)q by using the (I,Y) labels for each SU(2)xU(I). We find

(~ ~ ;~ ~ ) ,

,-

:,

(ro, p)

(1.8a)

K*

(1.8b)

i(*

(1.8c)

C2' 1 2)

- -·0 - . 3' , 3 .

(0,- 3;2 2'-3 1 1) : (0, -

~; 0, ~ ) :

( 1.8d)

With respect to the diagonal SU(3) subgroup the hypercharge Y is the sum of 1 Yq and Yq while the isospin I is zero for and , one for p and 2 for K* and K*.

ro

Now, Okubo's octet breaking hypothesis involves the octet-singlet mixture given by

y2 K=I(I+ 1) - - . 4

(1.9)

For the nonet the energy breaking requires the combination E = Eo + a (Kq + K_) q +bK

(1.10)

80 We find the following assignments K_ q

K q

q

13

13

18

18

13

13

13

18

18

9

9

11

9

1

13

11

9

18

18

- -

- -

2

-

18

i(* :

0

-

13

K* :

K

q

13

CO

P

K +K_

1

1

-

-

9

2

18

9

2

2

(1.11)

0

-

9

Note that the sum K + K_ of the two octet breakings gives equal spacing for the

q

q

energy levels and degeneracy for co and p. Now it was shown in Ref.(5) that the rest energy breaking formula leads to a quadratic mass formula when the energy differences are large with respect to the mean energy, as in the case of the pseudoscalar mesons and to a linear mass formula when the ratio of the energy splittings to the mean energy is small as in the case of baryons. The vector mesons being nearer in mass to the baryons than the pseudoscalar mesons we can use the linear mass formula as also suggested by the value of the mixing angle being nearer perfect mixing in this case. Then we get

26

(1.12a)

+ - a 18 '

26 1l+- a + 2b

(1.12b)

18

m K*

=

mi(*

m

=

11

11 + 11 -

18 4

18

1 a + 2 b,

(l.12c)

a,

(1.12d)

leading to the mass sum rule ( -

K*)

1 (K* -co) + 2 (co -

p)

(1.13)

81

which gives deviations of less than one percent with the choice Jl = 988 MeV, a = - 144 MeV, b = - 6 MeV.

(1.14)

(Eq. 1.13) which we have derived here using purely the SU(3)xSU(3) group theoretical assignments of (1.8) is usually written as the mass square formula

(1.15) for which the deviations are much larger than the linear mass formula. It is also important to note that the quantum numbers of (1.8) which forbid the

into pions, or more generally of the s s system into systems involving u and d is consistent with the OZI rule also due to Okubo7 and independently to others. This rule must be violated in QCD through gluonic intermediate states and yet it is surprisingly well verified, reinforcing the validity of the symmetry breaking chain that gives eq. (1.10). decay of the

The next step came from an attempt to put pseudoscalar mesons and vector mesons in a single multiplet. This is natural in a quark model since the lightest quarks 1 u d s have each spin -, giving 6 states as a representation of the group SU(6). Then, 2 the group SU(6) xSU(6)_ has a SU(6) diagonal subgroup that generalizes the SU(3) of

q

q

Gell-Mann and Ne'eman through the incorporation of the quark spin associated with SU(2). However, spin is conserved independently of total relativistic angular momentum only for free particles, so that spin can only be combined with flavor quantum numbers like isospin if quarks inside the meson or the baryon behave like approximately free fermions. This is a posteriori justified by the asymptotic freedom 8 of the QCD theory. It was one of the mysteries of the SU(6) model . The low lying mesons, now fit in the singlet and 35 - dimensional adjoint representations of SU(6) with TJ being the singlet and the eight pseudoscalars and the three s=1 states of the vector nonet completing the 35. Assuming again symmetry breaking to arise from a mixture of SU(3) singlet and the 8th member of the octet in the symmetric 35 x 35 product which contains the I, 35, 189 and 405 representations of SU(6), one finds the new general energy formula

y2

E

Eo + a [I (I + 1) -

4 ]

For baryons, E is the mass, for mesons

~

+

~

y + 'Y s (s +1).

(1.16)

vanishes and one can approximate E

by Jl2 since pseudoscalar mesons are involved. This gives the breaking of the (56)

82 1 3 baryons (s=- octet together with s=- decuplet) as well as the (35) mesons.

2

2

For

instance one obtains the mass formula

- mp2

- m1t2

(1.17)

relating vector and pseudoscalar meson mass splittings and hence showing approximate spin independence of the binding forces in addition to flavor independence implied by SU(3) symmetry. II. From Spin Independence to Supersymmetry

At this point, the baryons which are three quark (qqq) fermionic systems are treated separately from the (qeD bosonic mesons. But if the quark binding forces are approximately spin independent they should also be blind to the distinction between fermions with half odd-integer spin and bosons with integer spin. It is this idea that led Miyazawa9, 10 in the years 1966-68 to propose a fermion-boson symmetry in the hadronic spectrum which later became known as supersymmetry. What are the indications for a broken supersymmetry for hadrons? First of all the low lying me sonic 1 3 spectrum for s=O,l extends from 0.14 GeV to 1.02 GeV while the s=2' 2 baryons cover the range 0.94 GeV to 1.67 GeV, so that the two spectra overlap. The mass splittings between adjacent isotopic multiplets are also similar, being of the order of 0.1 GeV in both cases. Hence all low lying hadrons seem to be members of a single 1 3 supermultiplet with the spin taking the values 0, -, 1, -. Another experimental

2

2

evidence is provided by excited hadronic states with s ~ 2. If such states are represented by points on a Chew-Frautschi plot of m2 versus s, then they fall on parallel linear Regge trajectories. The astounding fact is that the slopes of the baryon and meson trajectories are nearly equal. This is a manifestation of a deep and unexpected supersymmetry between excited (qeD and (qqq) states. The mathematical expression of supersymmetry arises through a generalization of Lie algebras to superalgebras. When the Lie algebra is su(n) it can be extended to a graded algebra (superalgebra) su(n/m) with even and odd generators, the even generators being paired with commuting (bosonic) parameters and the odd generator with Grassmann (fermionic) parameters. The algebra can then be exponentiated to the supergroup SU(rn/n). This was done by Miyazawa who derived the correct commutation and anticommutation relations for such a superalgebra as well as the generalized Jacobi identity. This discovery antidates the supersymmetry in dual resonance models 11 or supersymmetry in quantum field theories 12 invariant under the super-Poincare group13 that generalizes special relativity. Miyazawa looked for a 10 supergroup that would contain SU(6) and settled on a broken SU(6/21). He showed that an SU(3) singlet-octet breaking of this supergroup leads to a new kind of mass formula relating fermionic and bosonic mass splittings. An example for non strange hadrons is

83

ml\

2

(2.1)

The emergence of the group SU(6/21) can be understood on the basis of the quark model. The quarks (u,d,s) are associated with the six dimensional representation of SU(6). Assuming that two quarks can form a bound state (this being justified by QCD), the diquarks (qq) belong to representations 15 or 21. Actually, the SU(3) color coupled with the Pauli principle and the SU(3) singlet nature of (qqq) baryon states gives 21 for the diquark bosonic multiplet. The diquarks and quarks can then combine to give baryonic states which are in the (56) representation of SU(6). It is now clear that the 6 fermionic q-states and the 21 bosonic qq states together form the 27-dimensional fundamental representation of the supergroup SU(6/21). The reason . we must take the anti diquark to be in the same multiplet as the quark is given by QCD based on color SU(3)c. Each of the 6 colored quark states belongs to the triplet representation of SU(3)c. On the other hand, the diquark is in the 3' representation present in 3x3, so that the antidiquark has the color group representation 3. QCD also gives an attractive force between two quarks (half in strength of the qq force) due to one gluon exchange, leading to the formation of a qq bound state. Besides the 27-dimensional representation ~ of SU(6/21) there is also the complex conjugate 27 representation consistiQg of antiquarks and diquarks. The adjoint representation arises from the product 27x27 and contains the mesons qq (l + 35), the baryons qqq (56 + 70), the antibaryons qqq and the exotic mesons qqqq which can be regarded as diquark-antidiquark bound states. Thus, QCD provides a basis for formation of a supermultiplet that contains baryons and mesons, the starting point of Miyazawa's 14 model. . The other manifestation of hadronic supersymmetry, namely parallel Regge trajectories for all hadrons is more difficult to relate to group theory. It arises naturally from the string theory15 of Nambu and Goto which is associated with the infinite parameter Virasoro algebra 16 rather than a Lie algebra. The parallel Regge trajectories arise from the flavor and spin independence of the qq or qq forces. QCD is certainly flavor independent. As to approximate spin independence, there is mounting evidence 17 that the confining potential is a relativistic scalar rather than the fourth component of a vector potential, although this conclusion has been challenged by some authors 18 from a discussion of heavy meson spectra. Now assuming that the confining potential is a relativistic scalar, we know from lattice QCD .that in its static form it is proportional to the distance r between the quarks. Such a potential for the relativistic two body problem has two consequences. Firstly, as shown by Eguchi, and also by Johnson and Thorn 19, for high rotational excitation the three-quark system tends to a quark-diquark two body system. Secondly, the squared mass of the two-body system with a linear potential becomes proportional to the angular momentum J of the system. The first property tells us that the excited baryon can be treated as a two body q-D syst~m (D=qq), just like the meson which is a q-q system. Now, both D and q are in the 3 color representation, so that the q-D potential is the same as the q-q potential provided we neglect the spin dependence of the forces. The short range force, due to a gluon exchange, is obtained from a vector Coulomb-like potential and is spin

84 dependent. But for high excitation the two constituents have a large separation and the spin independent confining force takes over, resulting in the approximate equivalence of the Hamiltonians for the qq and q-D systems. The slope of the mother Regge trajectory depends only on the parameter of the confining potential, resulting in parallel linear Regge trajectories for baryons and mesons. The Hamiltonian is also approximately invariant under the transformation of q into D, which is a supersymmetry transformation belonging to the supergroup SU(6/21) for the low lying hadrons. The string approximation to QCD gives therefore a new type of mass formula m2 = a'-1 J + C

=

a'-1 (J - J ),

(2.2)

o

, valid for both baryons and mesons. Here, the Regge slope a is of the order of 1 (GeVf2.

Contrast this equation with Eq. (1.16) in which for large J the mass is Eq. (2.2) gives proportional to

P.

a'-1 ~ J

(2.3)

as for ~ J = 1 we obtain (2.4)

, both for baryons and mesons. Now a is the same for the also the same for the N and ~ trajectories, giving

1t

and p trajectories. It is

(2.5)

which is the same as Miyazawa's sum rule obtained from supersymmetry. Here the relation is also valid for any two pairs of points on the same trajectories provided ~ J is unity. In short the Miyazawa hadronic supersymmetry for the low lying hadrons is extended through QCD to the rotationally excited hadronic levels. Breaking of this supersymmetry has two origins. First, the q and qq mass differences as well as mass differences among quarks. This results in different values of the constant Join Eq. (2.2), leading to different intercepts for parallel Regge trajectories. The second breaking comes from the contribution to the potential from one gluon exchange. This potential is a 4-vector and is spin dependent. Since the I quark and antiquark have s=- and the diquark has s = 0 or 1, the spin dependent part 2 of the q - qq potential is different from that of q-q, causing supersymmetry breaking. Another consequence is the deviation of the Regge trajectories from linearity for low spin, since the potential is no longer proportional to the distance. Examples

of

effective

Hamiltonians

obtained

from

a

two-body

85 Schrooinger-Dirac approximation to the quark-QCD system after elimination of the gluon degrees of freedom will be presented in the next sections. They exhibit an approximate SU(6) symmetry and SU(6/21) supersymmetry with explicit symmetry breaking terms.

III. Derivation of the Effective Hamiltonian Consider a quark q(1) at point xl' and an antiquark q(2) at point x represented 2 by the charge conjugate spinor qC(2). The q(1)q(2) system for a meson has 16 space components that are the elements of the 4x4 matrix (12) = q (1) q (2)

(3.1)

Elimination of gluon degrees of freedom generates a potential between these constituents that in the static approximation depends on the relative distance r. It will be approximated by a scalar potential S(r) and the fourth component of a vector potential VCr) given by 4 S(r) = b r, V (r) = -- a 3 sr '

(3.2)

where as is the strong coupling constant at an energy scale corresponding to low-lying meson masses. In this approximation the relativistic Hamiltonian can be written as

(3.3a) with (3.3b)

where i = 1,2; pi

-i V' (i) .

(3.3c)

In the center of mass system

0,

(3.4a)

86 -+

a ar

-+

p =- i V

-i---=J'

(3.4b)

-+

(3.4c)

r

Now, since Y).1 (1) and Y ).1(2) commute we can represent them by left and right multiplication of cI> by Dirac matrices so that

(3.5) We shall make one more approximation by replacing the scalar potential by its expectation value in a given state for the qq system (3.6)

where ro is the Bohr radius for the relevant state. In the next section an estimate for ro will be given. Now we can make a unitary non-local transformation similar to the Foldy-Wouthuysen transformation by defining a new two-body wave-function (3.7a) where

W.

1

[

m.1 + 1/2 S0 +

. -+ (i) -+ (i) 1

Y P

r

(3.7b)

and

1.

(3.7c)

The transformed Hamiltonian H involves a transformed vector potential V (r), where

87 H

=

WHW- 1 , V(r)

=

WV(r) W- 1 .

(3.8)

In the approximation that S(r) is spin-independent we have S(r)

-1

=

W S(r) W

V(r)

V(r) +

'" S(r)

=

S(r ) o

+( S(r)

-

S(r )) , o

(3.9)

(3.10)

20 so that V (r) differs from V(r) by a spin-dependent Breit term . The new two-body equation now reads

H

E

= -V (r) -

+'yAI (m1 +V2

2 2 S(r)) -V

(3.11)

This equation is invariant under the spin transformation

........

-1/2 i e

0" .

(02 (3.12)

since

(3.13)

provided the Breit term is neglected. The potential being flavor independent, flavor independence is only broken by the flavor-dependent mass differences m - m2. This 1 leads to an approximate SU(6)xSU(6) symmetry of H broken by mass differences and spin dependent Breit terms. Note that the original Hamiltonian H does not display this invariance. The invariance of H does not conflict with relativity since it is obtained by a non-local transformation.

88

Iteration of eq. (3.11) gives the "Krolikowski" type second order equation

which was successfully used by LiChtenberg and his collaborators

22

21

to calculate

meson masses in the approximation Y = Y. For a q-D system q(2) is replaced by a wave function with four spin components associated with s=1 and s=O. In addition there are flavor quantum numbers. For the low-lying hadrons D has 21 degrees of freedom as mentioned in the previous section. Then Y (r)

=

Y (r) +

K

(3.15)

with m and m being respectively the quark and diquark masses, SO) the s = 1/2 1 2 4

Pauli matrices and S (2) being either zero or the 3x3 s=1 matrices. Otherwise the two-body equation is the same as for the meson case. If m - m and the 1 2 spin-dependent terms are neglected the two-body equation for the meson-baryon system in which the wave function is

'if =($qq 'PDq

~qD)

(3.16)

qD

becomes invariant under the transformation

'P -> U 'P

yt

(3.17)

where U and Y are elements of the Miyazawa supergroup SU(6/21). In a relativistic treatment we can represent the diquark for each flavor by a 16-component wave function satisfying a linear Kemmer type equation. 10 components are for s=1 (F /.1v and A,.) and 5 are for s=o (/.1 and 10 GeV / c and W ~ Iv) should give information on the W structure. Sensitivity also exists to anomalous WW production, e.g. such as the production of "W sprays" or boson pair resonances.

132

There are the indications of new physics which may appear and help guide both theory and the SSC research directions.... Supersymmetry can be searched in the mass ~ange up to 300 GeV, heavy gauge bosons to -1 TeV /c 2 , and there are the new things that no one has yet proposed!

5.

HISTORY OF THE FAR FUTURE

(2000~)

This history has gotten off to a lively start. SSC was "conceived" in the late 1970 ICFA studies but brought to a sharp focus as a national plan in 1982. By July, 1983, it was embraced by the DOE and there began a serious design study under Tigner at the LBL headquarters of the Design Group. The energy is 20 TeV in each beam yielding a splendidly violent 40 TeV in the CM with a collision rate of loB/sec. Magnet R&D aimed at SSC was diversified to three laboratories (LBL, FNAL, BNL) and did not break speed records. In 1987, sse became U.S. policy and the site was selected and the SSC Laboratory founded under Roy Schwitters. As of current writing, the cost estimate for the SSC "hovers" between $7.8 billion and $11.8 billion and I digress from the physics account to explain this to the theoretical friends of Professor Okubo who may be confused. Once upon a time, say when Fermilab was built, it was enough to estimate the cost of the machine and its accompanying laboratory. Contingency was a matter of conscience, inflation didn't exist and detectors were something to be discussed later, in the context of the operating budget. And so Fermilab "cost" $250 million which in today's dollars, is about $1 billion. Detectors, preoperating costs, inflation and contingencies would have doubled this. In 1983 at the first workshop held at Cornell and devoted to examining the cost and technical feasibility, one guestimated that the machine and laboratory would cost $2.7 billion. This is about 20% less than the June, 1990, cost estimate of the machine (in 1990 dollars) without contingency, equipment, etc. Thus, when comparing cost estimates, one must make an effort to find out what is and is not induded. In fact, the

133

differences are almost entirely in the contingency and in the amount of preoperating and detector funds. So is it worth it? We start with the list that any Congressman is completely familiar with: 1. Higgs! [In the more sophisticated circles around Waxahachee, it is known as electroweak symmetry breaking.] In particular, H ~ ZOZO ~ 4jl or 4e which, for a mass of ~800 GeV, yields only a few tens of events a year at the design luminosity of 1033 cm- 2 sec I ! 2. Z's, W's 3. Top physics 4. SUSY searches (it is possible to kill, KILL SUSY?) 5. Compositeness 6. Strong WL WL scattering 7. B physics There is an incredible literature on the physics at SSC, and/ or its European version, LHC. At this time, "expressions of interest" running to hundreds of pages have been received. This confirms the notion that interest is worldwide. About 5 or 6 propose to build generic detectors which are modelled on the DO and CDF or UA I, 2 style of "41t" do-everything detectors. One detector proposed by 837.5 authors is a 61t detector. Other expressions of interest vary from the ubiquitous logs physics, to fixed-target beauty research. So far, only one, perhaps two, seem to be based upon totally new technologies. The next few years will see a refinement of these expressions-of-interest. One of the more challenging aspects of SSC experimentation has to do with the collision rate. The design luminosity would yield 108 interactions/ sec, each interaction generates -100 particles requiring -10 7 bytes to describe. This data rate requires all kinds of new techniques, radiation-hard detectors and up-close electronics, a refined mechanism for selecting the interesting events, etc. Whereas very few experts would claim that this problem is now completely solved, there is nevertheless considerable pressure to go to 10 times this rate or even more! From the

134

theoretical physics point of view it is clear that this would help the higgs problem, but from the experimental point of view, it is not at all clear that 1990-1992 technology can deal with these kinds of data rates.

6.

HISTORY OF THE FAR, FAR FUTURE (2020

~)

*It is the 60th anniversary of the Rochester Conference *Samios has his 80th Festshrift So by now we can also invent sse results, e.g.

*H~

= 422 GeV found at SSC in 2004

o

*H 2 == 699 but only 3a Indications exist that there is a higgs sector with a rich higgsian spectroscopy. To study this, we obviously need higher energy. SSC may instead discover a new class of strong interactions which may, in the words of Steven Weinberg, revive the physics of our youth; dispersion relations, Regge poles, sum rules, all at a much higher energy. Again we'll need a machine appropriate to the energy. To decide the state of hadron colliders, we are fortunate to have the well-tested Livingston Chart (Fig. 2). This predicts that by 2030, we will have 1000 TeV in the CM. In order not to violate this schedule, we must start in 2020. Now all of this was foreseen by the well-known futurist but obscure accelerator designer J. D. Bjorken back in 1985 and published by Professor Thomas Ferbel. (See his Techniques and Concepts of HEP, Vol. II.) The story is told about Gorbachev visiting Finland and asking to be shown the Tomb of the Unknown Soldier for which he had brought a special wreath. His hosts exchanged furtive and puzzled looks when one smiled and led the caravan to the Sibelius memorial. Gorbachev, the European intellectual, exclaimed, "Why that's not the Unknown Soldier, that's Sibelius the famous composer!" "Yes," said his host, "as a composer he is famous, but as a soldier, he is completely unknown."

135

Bjorken designed this pp collider as a pedagogical exercise with the following table of specifications: 500 TeV Energy /beam Peak Field lOT Radius 170 km Circumference 1260 km 7m Dipole length 170,000 No. of dipoles 16,000 No. of quads Luminosity 1033 200 MW RF power As we can seE!, all are very reasonable parameters, although the graph (Fig. 3) indicates that a field of 20T can perhaps be more optimal and substantially decrease the tunnel size. One must realize that there is a worldwide peace dividend which is estimated at 1 trillion dollars per year that must be put to good use. It is poetic if these funds were invested in the violence of hadron colliders.

6.

CONCLUSION

My final comment is perhaps to raise a controversial question from a biased, subjective point of view. So far, we cheerfully dealt with hadron colliders and ignored brand X. This goes against a wide consensus that the next accelerator (following SSC) will be a linear collider accelerating electrons. A very able community of creative accelerator experts are working hard on these problems and it may well be that their efforts will produce a breakthrough in technology, making the next energy domain one explored by electron-positron collisions. However, this is very far from clear in 1990. There is a segment of devotees of e+e- collisions that seem to hold to a belief that the next machine "belongs" to electrons and this is as sensible as if the experts on Geiger counters would insist that they be employed on the

136

next detector. The point is that we are all driven by physics. Electron machines were powerful in the 1970's and LEP's contribution to ZO physics, especially the width, is clear. The virtue of electrons, their clean initial state, may however count for less and less as the violence increases. Very narrow resonances like the ZO, strongly ocupled to electromagnetism, is one of the few states that strongly favor e+e- machines and these may be a vanishing breed at post sse energies. If hadron colliders can solve the rate problems and the messiness of the spectator partons, its relative economy in dollars per GeV and its large variety of initial states may win over e+e- colliders in the next round. A strong indication of this does not have to wait for sse results in the 2000's but will be guided by 1032 luminosity in the upgraded Tevatron in the mid-1990's. If constituent collisions continue to be as clearly discernible at these rates, it will be a strong indicator that a 500 TeV x 500 TeV pp machine can be the 2020 machine, rather than the equivalent 10 TeV x 10 TeV e+e-. We must keep our minds open and weigh the physics potential of these two approaches. Both have formidable challenges, the former is largely in cost reduction. In the e+e- case, the technical challenges are so daunting that it is likely that the only sensible approach is an iterative, learning process, through a, e.g., 200 GeV x 200 GeV collider, then a 1 TeV x 1 TeV, etc. Each process is in the billion dollar category and probably requires of the order of five to ten years.

137

Standard Model Lego Version

t

Figure 1.

A "lego" plot of the standard model.

138 I

I

I

I

I

I,'

1012 _ I

I

,/ I

_

/ GUTS Level /(l014GeV) _ I

I-

/ by 2140 I

-

I 1

I

91000TeVinCM

10

~---------------7

/

/ I

I

-

1

/

>

I

/

Q)

I

~

,1

-

I

1

6 10 ~

1

-

I

I 1 1 1

-

-

1

~,'SSC /

,

,,

I

-

1

1

3·' TeV 1-'

10

-

1

I

2000

I

I

I

2020

2040

I

I

2060

Year

Figure 2.

Upper reaches of the Livingston Plot whose absolute validity is established in the off-side years 1930-1990.

139

600r-----------~----------_r----------~

400

>

Q)

-

~

a.

200

---H=5T

-_SSC_--.. -

O~~-------------~--------------~------------~

200

L(

Figure 3.

400

K

600

m)

Synchrotron radiation constraints. L is the proton accelerator length, p the particle momentum, H is the mean field taken to be 90% of the actual peak dipole field. The solid curve with dashing bounds the region in which synchrotron radiation loss is less than 10% of the LEP II case.

This page is intentionally left blank

141

LOOKING BEYOND THE STANDARD MODEL ( An account of a talk)

Lev Okun ITEP, Moscow,

117259, USSR

When Ashok Das invited me to the Okubofest and to

give

a

talk

"Beyond the Standard

him that this was the title of my

talk

asked

Model",

I

to

given

be

warned in

August at the XXV Rochester Conference in Singapore. So he proposed to add the word "Looking",

I agreed and spoke

at

the Okubofest. My main topics were: the problem of higgses, the

re-

cent theoretical speculations about multiple production of higgses, W- and Z- bosons, the possible time variations of solar neutrinos observed by R.Davis

and

the

anomalously

large neutrino magnetic moment. Now, at the end of July, the deadline for the submission of the text to the Proceedings of the

Okubofest,

is

nearing. And I have realized that I could not write simultaneously two different texts on the same subject. At same time I did not want to "self-plagiarize". So I ded to subtract from my Okubo

talk

my

future

the deci-

Singapore

talk. What is left is essentially my heartly

greetings

Susumu Okubo on his sixtieth birthday and xerox

copies

to of

142 several pages from two papers which I have used

as

tran-

sparencies in my talk in Rochester. The papers were

writ-

ten by A.D.Sakharov and B.M.Pontecorvo. The paper by A.D.Sakharov "Violation of CP

invarian-

ce, the C asymmetry and baryon asymmetry of the

universe"

(Pis'ma ~,

Lett.

ZhETF,~,

32

24 (1967»

(1967>,

English

translation

JETP

opened a new chapter in cosmology. The

starting point of the paper is

the

observation

1958 by S.Okubo that CP-violation leads to the

made

in

difference

in partial widths of charge conjugated decays. At that time there were no reprints or

offprints

of

Pis'ma ZhETF, so authors received from the editor a certain number of copies of the Journal. On

February

Andrej Dmitrievich gave a copy with

rhyme

a

Lvovich Feinberg (and a copy without rhyme

to

but

misprints corrected to me). Before coming to

3,

1967

Evgeniy with

all

Okubofest

I

asked Professor E.L.Feinberg to give me a !{erox of his copy, and he kindly complied with my request. I

appologize

for

non-poetic

translation

of

the

inscription: From the effect of S.Okubo At high temperature A furcoat is tailored for the Universe To fit its skew figure. For the convenience of the reader a xerox from ters is also added

(with misprints not

JETP

corrected

Let-

in

the

English translation>. The paper by

Bruno

Pontecorvo

(Chalk River report PD-205, 20

"Inverse

November,

1946)

to-process" was

the

first in which a method of detection of neutrinos (and

in

particular of solar

In

neutrinos)

has

been

suggested.

January 1988 the paper was typed afresh and was with a foreword by

W.F.Davidson,

who

justly

published called

it

1~

prophetic. Two pages of this report are reproduced here. I conclude this short note

by

expressing

again

my

best wishes to Susumu Okubo, whose work played such an important role in the creation of the Standard Model and the efforts to go beyond it.

in

144

HAPYIlIEHI1E CP-I1HBAPI1AHTHOCTI1, C-ACI1MMETPH51 H BAPHOHHA51 ACHMMETPH51 BCEJlEHHOH A./{.Caxap08 Teoplili paCUlIIpli JOll\ellcli bceJ!eHHoll. npeAnO.lara JOluali CBepXnJIOTHOe 'laJIbHoe COCTOllHlle Bell\eCTBa. nO-BIIAIiMOMY.

IICKJIIG'l~T

Ha-

B03MO)!(HOCTb MaK-

pOCKOnll'leCKOrO pa3Ae.1eHlI1I Bell\eCTBa II aHTIiBell\eCT9a; n03TOMY CJIeAyeT npllHlITb. 'lTO B nplipOAe OTCYTCTBYJOT TeJIa 113 aHTIiBelUeCTBa. T.e. BCeJIeHHall aCIiMMeTplI'lHa B OTHOllieHlI1I 'lIlCJIa 'laCTliu II aHTII'laCTIiU

B 4aCTHOCTIi.

pllll).

(C -aCI1MMeT-

OTCYTCTBlie aHTII6apliOHOB II npeAITOJIaraeMOe OTCYTCT-

Bile Hell3BeCTHblX 6apIIOHHblX HellTpliHO 03Ha'laeT OTJlII'llie OT HYJIli 6ap1IOHHoro 3apllAa (6aplIOHHali aCIiMMeTplIll).

Mbl XOTIIM YKa3aTb Ha B03MO)!(Hoe

06bllCHeHlle C-aCIiMMeTplIlI B rOPll'lell MOAeJIIi paClIIllpliJOll\elicli Bce.leHHoli (CM. [

[2]).

1])

C nplIBJle'leHlleM 3cpcpeKToB HapYllieHlIlI

CP-IIHBapliaHTHocTIi

(CM.

,l\JIli o6bllcHeHII1I 6aplloHHoli aCIiMMeTplI1I AOnOJIHIITe.lbHO npeAnOJIi'.ra-

eM npll6JlIIll(eHHblli xapaKTep 3aKOHa coxpaHeHlIlI 6apIiOHOB. f\lIIHIIMaeM. 'lTO 33.KOHbI coxpaHeHlIlI 6apliOHOB II MIOOHOB He llBJllllOTCll a6COJllOTHblMli II AOJlll(Hbl 6b1Tb 06beAIIHeHbl B 3aKOH coxpaHeHlIll pOBaHHoro "

6 aplloH- MIOOHHoro

3apllAa n K =

3n

B -

nf'.

"KOMOIIHII-

nOJlOll(eHO:

3

145

AIITlI~IIOOllhl Il_

\IIOOllbl

f)ap~lollhl

~ ~ ~ lie: nil = - I,

Il+

~

= +1

"'II ~ lie: nil = +1, n K =-1

~I

N:

n 11 = + I,

AIITlI6apIIOllhi

p~

N:

P

nK

n K = +

1

nll~-I, nK~-3 KBapKa~1

TaKaH q,OpMa :lamlCII C13H3aHa C npCIll:Tal3J1eHlleM 0 KBapKaX;

~

n,

npHlIIlCblHaeM n K

= +1.

npelle6pClKilMYlO PO,lb npOueCl:OIJ HapYWCHIIH JlOBIIHX

-I.

0, realizing total CPT symmetry of the Universe. All < 0 are assumed in this hypothesis to be CPT reflections of the phenomena

decay with an excess of quarks when t the phenomena at t at t

> 0.

We note that in the cold model CPT reflection is impossible and only T and TP re-

flections are kinematically possible.

TP reflection was considered by Milne, and T reflection

by the author; according to modern notions, such a reflection is

~arnically

impossible be-

cause of violation of TP and T invariance. We regard rnaxirnons as particles whose energy per particle €/n depends implicitly on the If we assume that €/n - n- 1/ 3 , then €/n is proportional to the

average particle density n.

interaction energy of two "neighboring" maximons (€/n)2n 1/3 (cf. the arguments in [6]). Then n2/ 3 and R~ - (€ + 3p) = 0, i.e., the average distance between maximons is n- 1/" - t.

E -

Such dynamics are in good agreement with the concept of CPT reflection at the point t = 0. We are unable at present to estimate theoretically the magnitude of the C asymmetry, Which apparently (for the neutrino) amounts to about [(v - v)/(v + v)] - 10- 8 _ 10- 10 • The strong violation of the baryon charge during the superdense state and the fact that the baryons are stable in practice do not contradict each other. model.

Let us consider a concrete

We introduce interactions of two types. 1.

An interaction between the quark-muon transformation current and the vector boson

field aia' to which we ascribe a fractional electric charge ~ = ±1/3, ±2/3, ±4/3 and a mass ma "" (10 - 10 3 ) mp.

This interaction produces reactions q

-+

a +

il,

q + ~

-+

a, etc.

The in-

teraction of the first type conserves the fractional part of the electric charge and therefore the actual number of quarks minus the number of antiquarks (= 3~) is conserved in pro-

150 cesses that include the a-boson only virtually.

We estimate the constant of this interaction at ga siderations:

137- 3 / 2 , from the following con-

=

The vector interaction of the a-boson with the IJ.-neutrino leads to the presence

The upper bound of the maSS ~O is estimated in [7J on

of a certain rest mass in the latter.

the basis of cosmological considerations.

If we assume a flat cosmological model of the

Universe and assume that the greater part of its density p - 1.2

ascribed to J..l ' then the rest mass of O then from the hypothetical formula

m

~O

J..l

O

is close to 30 eVe

X

10- 29 g/cm3 should be

The given value of ga follows

g2

a

m:-=~'V

We note that the presence in the Universe of a large number of

J..l

O

with finite rest mass should

lead to a number of very important cosmological consequences. 2.

The baryon charge is violated if the interaction described in Item 1 is supplemented

with a three-boson interaction leading to virtual processes of the type aa1 +

a~2

+ aa3

~

o.

At the advice of B. L. Ioffe, I. Yu. Kobzarev, and L. B. Okun', the Lagrangian of this inter-

action is assumed to be dependent on the derivatives of the a-field, for example,

Inasmuch as L2 vanishes when two tensors coincide, in this concrete form of the theory we should aSSume the presence of several types of a-fields.

Assuming g2 = l/~ and

f"o

= 2

X

10- 5

g, we have strong interaction at n ..... 10 98 cm- 3 and very weak interaction under laboratory conditions.

'!he figure shows a proton-decay diagram including three vertices of the first type, one vertex of the second, and the vertex of proton decay into quarks, which we assume to contain the factor

J£.

q~J

p.

f;r

q~J

'-10 g.

g.

1/p~

(due, for example, to the pro-

pagator of the "diquark" boson binding the quarks

)i.

in the baryon).

ii,

divergence at pq =

Cutting off the logarithmic ~,

we find the decay pro-

bability

The lifetime of the proton turns out to be very large (more than 10 50 years), albeit finite. The author is grateful to Ya. B. Zel'dovich, B. Ya. Zel'dovich, B. L. Ioffe, 1. Yu.

Kobzarev, L. B. Oknn', and I. E. Tamm for discussions and advice. [lJ

Ya. B. Zel'dovich, UFN 89, 647 (1966) (Review), Soviet Phys. Uspekhi

[2J

L. B. Okun', UFN 89, 603 (1966) (Review), Soviet Phys. Uspekhi

2

2

(1967), in press.

(1967), in press.

[3J

M. A. Markov, JETP 51, 878 (1966), Soviet Phys. JETP ~ (1967), in press.

[4 J

A. D. Sakharov, JETP Letters

[5J

Ya. B. Zel'dovich and S. S. Gershtein, JETP Letters~, 174 (1966), trans1. p. 120.

2,

439 (1966), trans1. p. 288.

151

NATIONAL RESEARCH COUNCIL OF CANADA DIVISION OF ATOMIC ENERGY

INVERSE

~

PROCESS

by B. Pontecorvo

Chalk River, Ontario 20 November, 1946

152

v

v

+ C~37 ~18- + All

Br 71 ,'1

+

~

8-

Br 11 , •

+ Kr ' I , ' l

Al7 ~ C~37

Kr 11

(34 days; K capture)

(34 hrs.j emission of

~

, • 1

1

positrons of 0.4 MeV) The experiment with Chlorine, irradiating

with

neutrinos

a

large

for example, would consist in

volume

of

Chlorine

or

Carbon

Tetra-Chloride,

for a time of the order of one month, and extracting

the radioactive

Al7

from such volume by boiling.

The radioactive argon

would be introduced inside a small counter; the counting efficiency is close to 100%, because of the high Auger electron yield. 2,

3, 4, are reasonably fulfilled in this example.

also

that

condition

5,

implying

a

relatively

Conditions I,

It can be shovn

low

background,

is

fulfilled. Causes other than inverse

8

processes capable of producing

the radioelement looked for are: (a)

(n,p)

processes and Nuclear Explosions.

The production of

background by (n,p) process against the nucleus bombarded is zero,

if

the particular inverse 8 process selected involves the emission of a negatron rather than the emission of a positron. the

inverse

arguments

8

show

process

which

that

"cosmic

background of A37 from Cl 3 ' . fact that (b) through

Kl7

would produce ray

stars"

A31

This is the case in frOID Cl J

cannot

Similar

1 •

produce

a

direct

As for (n,p) processes in impurities. the

does not exist in nature rules out this possibility.

(n.1)

Process.

impurities.

In

This

effect

principle

at

addition of neutron absorbing material.

can

produce

least,

background

it can be

only

reduced

by

In the case cons idered, AJ

1

153

Such a value of the neutrino flux, though extremely high, is not too far from what could be obtained with present day facilities.

Sources The neutrino neutrinos/cmZ/sec. very energetic.

flux

from

the

sun is of the

order of 10 10

The neutrinos emitted by the sun, however, are not

The use of high intensity piles permits two possible

strong neutrino sources:

1.

The neutrino source is the pile itself, during operation.

In

this case, neutrinos must be utilized beyond the usual pile shie ld. The advantage of such an arrangement is the possibility of using high energy

neutrinos

fragments. 2.

emitted

by

all

the

very

short

period

fission

Probably this is the most convenient neutrino source.

The neutrino source is the "hot" uranium metal extracted from

a pile, or the fission fragment concentrate frOID "hot" uranium metal. In this case, neutrinos can be utilized near to the surface of the source,

but the high energy neutrinos

emitted by the

short period

fragments are not present. In the case

of

the

investigation of

inverse

~

processes

produced by electrons of 1-rays of high energy, the best source is a betatron or a synchrotron.

Chalk River Laboratories Chalk River, Ontario

November 13, 1946 B. Pontecorvo

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155

NEW DEVELOPMENTS IN THE s-CHANNEL THEORY OF SUPERCONDUCTIVITY

T. D. Lee Columbia University, New York, NY. 10027

ABSTRACT The essential features and new developments of the s-channel theory of superconductivity are discussed.

1. Introduction

One of the important differences between the recently discovered high temperature superconductors1 ,2 and the usual lower temperature superconductors is the smallness of the coherence length ~. For the former,3,4 ~ is ~ lOA; for the latter, ~ is much longer, typically ~ 10 4 A for type I and ~ 102 A for type II. In addition, it

is known that in all these superconductors the magnetic flux carried by each vortex filament is 27rnc/2e, showing the existence of a pairing state. The observation of such a small coherence length ~ in the high Tc superconductors indicates that the pairing between electrons, or holes, is reasonably localized in the coordinate space. Hence, the pair-state can be well approximated by a phenomenological local boson field ¢ (f), whose mass M is ~ 2me and whose elementary charge unit is 2e, where me and e are the mass and charge of an electron. It follows then that the transition

2e

-+

¢

-+

2e

(1.1)

156

must occur,

In

which e denotes either an electron or a hole; furthermore, the

localization of 1> implies that phenomena at distances larger than the physical extension of 1> (which is


. Since ~ is of

the same order as the scale of a lattice unit cell, it becomes possible to develop a phenomenological theory of superconductivity based only on the local character of 1>. Of course, physics at large does depend on several overall properties: the spin of 1>, the stability of an individual 1>-quantum, the isotropicity and homogeneity (or their absence) of the space containing 1> and so on. The situation is analogous to that in particle physics: the smallness of the radii of pions, p-mesons, kaons, ... makes it possible for us to handle much of the dynamics without any reference to their internal structure, such as quark-antiquark pairs or bag models. Hence, the origin of their formation becomes a problem separate from the description of their mechanics. An important ingredient in this type of phenomenological approach is the selection of the basic interaction Hamiltonian that describes the underlying dominant process. In the usual low-temperature superconductors, the large ~-value makes the corresponding pairing state 1> too extended and ill-defined in the coordinate space; therefore, (1.1) does not play an important role. Instead, the BCS theory of superconductivity 5 is based on the emission and absorption of phonons, 2e

-+

2e

+ phonon

-+

2e.

(1.2)

In the language of particle physics, (1.1) is an s-channel process, while (1.2) is tchannel. The BCS theory may be called the t-channel theory, and the model that is based on (1.1) the s-channel theory.6,1 The use of a boson field for the superfluidity of Liquid H ell has had a long history. However, there are some major differences in the following application to (high temperature) superconductors: 1. The 1>-quantum is charged, carrying 2e, while the helium atom is neutral. 2.

We assume each individual 1>-quantum to be unstable, with 2v as its

excitation energy. In any microscopic attempt to construct 1> out of 2e, because of the short-range Coulomb repulsion it is very difficult to have 1> stable. The explicit assumption of instability bypasses this difficulty; it also makes the present boson-fermion model different from the theory of Schafroth8 ,9 and others.

157

In the rest frame of a single cP-quantum (in isolation), the decay

cP

--t

(1.3)

2e

occurs, in which each e carries an energy = v.

Consequently, in a large system, there are macroscopic numbers of both bosons (the cP-quanta) and fermions (electrons or holes), distributed according to the principles of statistical mechanics. At temperature T


..T / d)2 would be logarithmically 00. However, the cupric superconductors are three-dimensional structures, made of parallel layers of CU02 planes with spacing e. Even without a definite theoretical idea, one may approach the problem heuristically by writing

(AT/d)2 ~ constant

+ Inc/I!

where I! is a characteristic two-dimensional distance. When the spacing e

AT

-+ 00

(2.9) -+ 00,

and consequently Tc = 0 for a two-dimensional system. We recall that for

an ideal three-dimensional boson system AT/d = (2.612)! = 1.377 and for liquid He II AT/d ~ 1.65. Here, because of the log term in (2.9), it seems reasonable

160

that the ratio AT/d for cupric superconductors could be somewhat larger ~ 2.B. (See Table 1). In the BCS theory, Tc depends sensitively on the interaction between electrons and phonons (or other excitations). In the Bose-Einstein condensation, Tc is determined by AT ,...., d, which is of statistical origin and therefore can be much higher

(Tc exists even without interaction). In the boson picture, on account of (2.7), we have (2.10) "Empirically", this product varies by only a factor less than, or ,...., 2, from ideal boson to He, and to cupric superconductors. For He, d ~ 3.5BA, Tc ~ 2.20 K and

M

~

BOOOm e , whereas M,...., 2me for cupric superconductors; i.e., a mass change

by a factor ,...., 4000. Thus, if one could have smaller d, then Tc would increase accordingly. Of course, d must not be too small; otherwise, the pair-states overlap, and the boson approximation breaks down (as in the case of cold superconductors).

Arid

Ideal bosons

He II

1.377

1.65

High To S uperco nductor 2.8

Table 1

3. A Prototype s-Channel Model As a prototype of the s-channel theory of superconductivity, we assume ¢ to be of spin 0 and that the space containing ¢ is a three-dimensional homogeneous and isotropic continuum. For realistic applications, as emphasized before, a more appropriate approximation of the latter would be the product of a two-dimensional

x, y-continuum (simulating the CU02 plane) and a discrete lattice of spacing c along the z-direction. The two-dimensional layer character of CU02 planes helps in the localization of the pair state in the z-direction, making the ¢-quantum disc-

161

shaped. The space that 4> moves in becomes a three-dimensional continuum when c

-t

0, but two-dimensional when c

- t 00.

This interesting case, plus the general-

ization to higher spin, will be discussed elsewhere. Here we consider an idealized system consisting of the local scalar field 4> of mass M and the electron (or hole) field 1/;" of mass m, with a = the spin. The Hamiltonian is

i

or

!

denoting

(1i = 1) H

=

(3.1)

Ho+HI

in which the free Hamiltonian is

with the repeated spin index a summed over and

t

denoting the hermitian conju-

gate. The interaction HI can be simply

(3.3) Both 4> and 1/;" are the usual quantized field operators whose equal-time commutator and anticommutator are

and

The total particle number operator is defined to be

(3.4) which commutes with H and is therefore conserved. Expand the field operators in Fourier components inside a volume periodic boundary conditions:

L k

n-I 1

"k-a-k,,, e'"r

n

with

162

and

(3.5)

L

n- t

b"k ei"k.r

k

8-:k k-:,8 uu ' and the commutator

t} where the anticommutator {a-k,u' ak',UI

[ b"k' b1, I = 8k"k"

Equation (3.3) can then be written as

In (3.2), 2vo is the "bare" excitation energy of . Because of the interaction, the "physical" (i.e., renormalized) excitation energy 2v in reaction (1.3) is given by 2v = 2vo +

g2

n2 L k

1 p-v - Wk

(3.7)

where P denotes the principal value and

(3.8) The decay width

r

of a -quantum (in vacuum) is given by

(3.9)

4. Gap Energy For T < T c , the zero-momentum occupation number b~ bo of the boson field becomes macroscopic; hence, we may replace the operator bo by a macroscopic constant. Set in (3.5)

n-t bo

B

c. number

and write

(4.1)

163

where

. Equations (6.3)-(6.4) also follow from general arguments: (i) At zero momentum k = 0, as in the Higgs mechanism,ll the energies of the three spin-components of the massive vector field V become the same; i.e., they are all equal to the rest mass m v , given by

167

mV

(ii) When e with

Wt

=k

= 0,

we have mv

=0

=

\ -1

AL

(6.6)

.

and the transverse mode is the usual photon

(since the velocity of light c is 1). On the other hand, the longitudinal

mode describes the Goldstone-Nambu boson 12 which, for e = 0, corresponds to the vibration of ¢J, propagating with the sound velocity v < < 1 (i.e., Wi -+ kv as k -+ 0). (iii) For very large k, the excitation of ¢J approaches the free boson spectrum k 2 /2M,

k »2Mv

for

and

(6.7)

For e =f 0, the Goldstone-Nambu boson joins with the transverse photon to form a massive vector field V, which leads to the above formulas for Wi and Wt, consistent with (i)-(iii). For the coherence length ~, we may set wICk) = 0 and k becomes complex, which gives a boson-amplitude, say exp (ikx), that decreases exponentially with distance (e.g., along the radius of a vortex filament). The decay rate in x determines ~.

From (6.4), the root

k

= iv'2l1±

for

(6.8)

satisfies

(6.9) The amplitude exp (ikx) becomes, then, exp (-v'2I1±X). To conform to the usual definition, the coherence length ~ is given by [Re(I1_) ]-1, which is always ~

[Re(I1+)]-l. (1) For v 2

> (MAL)-l, 11+ and 11- are real and (6.10)

(2) For v 2

< (MAL)-l, 11±

are complex with

(6.11) where

cos2a

(6.12)

168 and

(6.13)

sin 20: correspondingly,

[f;

e = VAi sec 0: .

(6.14)

A complex /-l± implies the condensate amplitude inside a vortex filament also contains an oscillatory component, which may lead to new observational possiblities. In the case v 2 < (MAL)-1 , according to (6.12) cos 20: varies from 0 to 1; therefore, cos 0: is between ~ and 1. Hence

(6.15) (Recall that Xr: 2 = (2eB)2IMc2. The product AL times the Compton wavelength hiM c is independent of c, the velocity of light.) Assume a boson condensate density B2 (at T

< < Tc) between 1020

-

1021 cm- 3

•

On account of (6.5),

M ~ 2m. and e /47r = 1/137, the London length is 2

for and AL '" 3800A for B2 '" 1020 cm- 3 '" 2

X

10-

3

A,

.

(6.16) Since the Compton wavelength M- 1 is

we see that in case (2), (6.14)-(6.15) give

e '" few A.

(6.17)

Case (1) holds only if v is larger than (MAL)-' '" 10- 3 times the velocity of light; hence, depending on v, (M V)-1 < few A, or ..j2 VAL < < AL. In either case, the theory predicts a very small consistent with experimental observations.

e'"

e,

e'"

e, the s-channel theory gives, in general, a type II superconductor. The s-channel theory is based on the observation that the coherence length e

Because AL

»

is small for all recently discovered high Tc superconductors. It is indeed satisfying that based on the s-channel theory, the smallness of

e can in turn be calculated.

169

REFERENCES This research was supported in part by the U.5. Department of Energy. [1]. J. C. Bednorz and K. A. Muller, Z.Phys. B64, 189 (1986). [2]. M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang and C. W. Chu, Phys.Rev.Lett. 58,908 (1987). Z. X. Zhao, L.Chen, Q. Yang, Y. Huang, G. Chen, R. Tang, G. Liu, C. Cui, L. Wang, S. Guo, S. Lin and J. Bi, Kexue Tongbao 6, 412 (1987). [3]. P. Chaudhari et al., Phys.Rev. B36, 8903 (1987). [4]. T. K. Worthington, W. J. Gallagher and T. R. Dinger, Phys.Rev.Lett. 59, 1160 (1987). [5]. J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys.Rev. 106,162 (1957); 108, 1175 (1957). [6]. T.D. Lee, "s-Channel Theory of Superconductivty," in Symmetry in Nature (Pisa, Scuola Normale Superiore, 1989), Vol. 2. R. Friedberg and T.D. Lee, Phys.Lett. 138A, 423 (1989). [7]. R. Friedberg and T.D. Lee, Phys.Rev. B40, 6745 (1989). [8]. M.R. Schafroth, Phys.Rev. 100,463 (1955). [9]. R. Friedberg, T.D. Lee and H.C. Ren, "A Correction to Schafroth's Superconductivity Solution of an Ideal Charged Boson System," Preprint CU- TP-460. [10]. Y.J. Uemura et aI., Phys.Rev.Lett. 62, 2317 (1989).

[11]. PW. Higgs, Phys.Lett. 12, 132 (1964). [12]. Y. Nambu, Phys.Rev.Lett. 4, 350 (1960). J. Goldstone, Nuovo Cimento 19, 154 (1961).

This page is intentionally left blank

171

Condensed matter and high energy physics* Sergio Fubini CERN - Geneva - Switzerland and Dipartimento di FiJica Teorica Torino - Italy

I wish to dedicate this short essay to my friend Susumu Okubo on his sixtieth

birthday. I wish to convey to him my appreciation for his important contribution to theoret-

ical physics together with my best wishes for the future.

*

Invited talk at RocheJter Conference - Mall 5, 1990

172 1)

Condensed matter and high energy physics deal with two very far apart corners

of the energy spectrum. In spite of that, there are deep analogies between the ideas and the methods employed in both fields. Cross-fertilization between them has been essential to their development.

It is instructive to recall the situation in the fifties when field theory has been applied both to high energy physics and to superconductivity. The successful use of gauge theory in electroweak theory had to wait until high energy physicists learned the concept of spontaneous broken symmetry which has been successfully applied in the theory of superconductivity. Cross fertilization went of course the other way, when the renormalization group became an essential element of modern theories of phase transitions. I wish to argue that we might now head to a similar situation. It is likely that the beautiful new developments in Hall effect and, possibly, in high Tc superconductivity can be nicely expressed in the string language, familiar to high energy physicists. The link is the concept of "anyons", i.e. of objects wih fractional spin and statistics which appear naturally in two dimensional theory. In what follows I am giving a few indications on how this connection comes about.

2) Let me first introduce a few well known properties of the Landau levels. We study a charged particle in a constant magnetic field jj oriented along the z axis. The problem is essentially two dimensional (everything interesting takes place in the (xy) plane). The classical particle rotates with the Larmor frequency

qB

(1)

w=-

mc

In the quantized problems we have energy levels

(2) It is well known that each level is infinitely degenerate. We have indeed two non commuting constants of the motion

(3) where 1I'i is the kinetic momentum

(4)

1I'i = Pi -

~Ai(X) C

173 The non commutativity of WI and

W2 :

(5) is the cause of the infinite degeneracy of each Landau level. In the symmetric gauge

(6) the wave function for the lowest level is

-,p(:!:, Y) = f(:!: + iy)e _~w

(7) where f(z) (z =

:!:

+ iy)

(z'+y')

can be chosen as any normalizable analytic function of z.

The arbitrariness in the choice of f( z) is indeed the expression of the degeneracy of the first Landau level. The theory of the quantum Hall effect implies the study of a large number of electrons in a constant magnetic field. In the case of very large magnetic field and low temperature, all electrons will indeed lie in the lowest Landau level. The fundamental work of R. Laughlin plays fundamental role in all investigations of quantum Hall effect. I wish now to show that Laughlin ansatz and its relations to anyons can be simply understood in the language of dual resonance model developed about twenty years ago as a starting point to the string theory 3) Let me now recall a little string mathematics in a language which is appropriate to the problem at hand. The vertex operator Vk(Z) introduced by G. Veneziano and the present author in the study of the dual resonance model (DRM) plays the fundamental role in all our discussions. Let us define the creation and destruction operators an and a~

(8) and the zero mode operators, p and q

(9)

[q, p]

=

i

The FV vertex operator is given by the ordered product

(10)

174 where

V.+ = exp (k L V~ = exp (k[iq

(10')

V.- = exp

~zn )

+ plog z])

(-k L ~z-n)

Introducing the well known Vir as oro operators L n , we have the fundamental algebra

(11) We shall be particularly interested in the action of the L_l operator (traslation) and of the Lo operator (rotation) We have

(12)

We wish to introduce a new operator W.(z) transforming as a free field under translations. Defining

*

(13) we have (14)

(15)

d

[Ll' Wk(z)] = dz Wk(z) [Lo, W.(z)j = zdd W.(z) Z

+ ~k2W.(z) 2

We already see an analogy with what happens in Wilczek definition of anyons. The single valued operator V.( z), which does not transform under translations as a free operator is changed through eq. (13) in the operator W.(z) which (see eq. (14» transforms under

*

The W.(z) operator can aiJo be ezpreJJed aJ an ordered product

W: = V/

j

W.- = V.-

W~ = exp (ikq) . exp (kp log

z)

175 translation as a free, but (for k 2 non integer) transforms under rotations as a non integer spin operator. As a consequence Wk(z) is multivaluen since (see eq. (15))

(16)

So, after a rotation of 27r, Wk(z) is multiplied by a phase factor eiwk '. The algebraic relation

(17)

where () = arg z shows that the non integer spin operator obeys a corresponding non integer statistic. Equations (16) and (17) show that Wk(z) is the natural candidate for an anyon operator. Let us now study a slight extension of the usual (DRM) procedure to compute the matrix elements of those operators 4) Let us consider the matrix element

where

< p" I and

IPb

> lie in the ground state of all oscillators

< p"la~ = 0

(18)

bu t belong to different eigenvalues p" and Pb of the zero mode operator p. Because of the zero mode conservation law we have

(19)

p" -Pb =

Lki

We wish both initial and final states to be translation invariant. Since the translation operator L-l is given by

(20)

L-l =

paL+ ...

we have always

(21)

< p"IL_ 1 = 0

176 but we have (22)

L-1lpb >= 0

only for Pb = O. We shall thus always choose Pb = 0

(23)

{ Pa = Lki

In order to simplify the notation we shall simply indicate:

when eqs. (23) are satisfied.

It is now a simple exercise to prove that (24)

< IWk1 (Zl)'" Wk.(zn)1 >= IT(z; - Zj)k;ki i=

2 + ... 0

< ILo =< 1 (2: k i )2

2 Using the commutator (15) together with eqs. (25) one can simply verify that the r.h.s. of eq. (24) is a homogeneus function of the Zi with dimensionality d = 2: k;k j • We are now ready to use the DRM formalism to the many electron system within the theoretical framework developed by R. Laughlin. 5) Let us first consider the indipendent particle model. The wave function for the lowest spin state is given, apart from the standard gaussian {actor by the Slater determinant 1

(26)

ffree

=

1

1

177 which can be easily expressed as

frr •• =

(27)

= 1,

for k

k2

II(zi - Zj) =< IWk(Zl)'" Wk(zn)1 i

= 1.

Looking back to equations (16) and (17) we see that for k = 1, Wk(Z) is the lowest

fermion operator, it changes sign for a 271" rotation and it does anticommute.

It has been the fundamental contribution of R. Laughlin to give strong argument to show that, because of the Coulomb interactions, other "ground states" responsible for fractional Hall effect can be obtained by choosing the higher fermionic representations of Wk i.e. by writing

(28) with k 2 = 3,5,7 .... Until now we still deal with fermionic operators. Anyons enter the game as soon as we consider excited states above the Laughlin ground state. For an excitation at the point y the wave function is

!B(y, Zl,'" zn) =

(29)

II(y -

z;)f(zl"" zn)

=< IWt(y)Wk(Zl)",Wk(zn)1 > We see that the operator Wl(y) for the excitation is indeed an anyonic with fractional spin

• -b = t, L ~ ... and a corresponding fractional statistic.

As a consequence the wave function corresponding to many excitations will be

f(Yl, ... y"zl""zn) =

< IWt(Yl)'" Wt(y.)Wk(zd··· Wk(zn)1 >=

(30)

= II(Yr -

Y.)~(Yr - Zi)(Zi - Zjt

One can proceed further and interpret the hyerarchical model as the appearance of

W k operators labelled by higher fractions, but now I think it is time to stop. I have constructed a narrow bridge between two equally exciting fields of research: string theory which deals with ultra high energy physics and with gravitation and quantum Hall effect related with extremely low temperature properties of matter. I sincerely hope that the ensuing cross fertilization will be profitable for the development of both fields.

178 Bibliography For a review on dual models

M. Green, J. Schwarz, E. Witten, "Superstring Theory", Cambridge University Press, 1987. on Hall effect

R. Prange, S. Girvin, "The Quantum Hall Effect", Springer Verlag 1990.

179

COLLECTIVE FIELD THEORY APPLIED TO THE FRACTIONAL QUANTUM HALL EFFECT B. Sakita Department of Physics, City College of the City Vniversity of l\ew York, New York NY tOO31 Zhao-bin Su Institute of Theoretical Physics, Academia Sineca, P.O.Box 2735, Beijing, China

ABSTRAcr

We present an application of lolkrtive field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the nofree-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin's wave function with and without quasi-hole excitations is also given. 1. Introduction At present we have a fairly good understanding of the fractional quantum Hall effect (FQHE) [I] baed on the celebrated Laughlin's wave function [2]. This approach is quite intuitive and pragmatic. but it is based on a rather ad hoc though brilliant choice of wave function. A question was raised by Girvin and by Girvin and MacDonald [3] and pursued by them as to whether or not there exists an effective field theory of FQHE, just like the LandauGinzburg theory of superconductivity. This work was followed by the recent attempts at using a 2+ I dimensional Schr(Xjinger Bose wave field coupled with a Chern-Simon gauge field as an effective field theory of FQHE [4] [5] [6]. Although these are interesting and beautiful. the connection of these to the Laughlin's wave function approach is not entirely clear to us. In particular, it is not obvious in these theories whether the constraint for the electrons being in the lowest Landau level is correctly imposed or not. In this paper we report some of our field theoretical studies of FQHE. in particular the use of the collective field theory. which wa0

(2.2)

and He is the Coulomb interaction: He

= L. _ a >b

I Xa

e2 -

Xi, I

- 7J

(2.3)

where 7J is the contribution from the uniform background charge which neutralizes the system. We choose such that He = 0 when the charge distribution of the electrons is uniform. The commutation relations for the 1T'S are

-z,

(2.4) where (2.5)

>.. being the radius of classical cyclotron motion. Nex t one defines the guiding center coordinates [8} >..2 (2.6) X i = X i - 2 Ei] 1T] f1

Then the following commutation relations hold. [ Xi, X

J ]

= i 'A 2 Eij

[

XI

,1T J ]

=0

(2.7)

If we neglect the Coulomb force, the Hamiltonian (2.3) is equivalent to that of a harmonic oscillator and consequently forms Landau levels. Because of the second commutation relation of (2.7) the X 's are constants of motion so that the electron states are degenerate in energy. The degeneracy is easily computed by using (2.7) and one obtains the following expression for the I

181

maximum density allowed for each Landau level: 1

eB

(28)

Po = 211')..2 = 211'flc

Under a strong magnetic field one may assume that the electrons are all in the lowest Landau level (LLL). This occurs when the average electron density p is less than Po- We define the filling factor by -,

v = P Po =

2 11'/\\ 2211'fIc P = --eJ3"P

(29)

Since the energy of the Landau levels is given by

.!..2 fl6lo(l + 2n ) = .!..fI eB (1 + 2n ) 2 me

(210)

which is inversely proportional to the mass of electron, the LLL condition is equivalently achieved by taking the zero mass limit. We first define the Hamiltonian by subtracting the LLL energy: 1 H = (21) - 21't (r)e N = -2 r.(11'aX

+ i 11'IX11'ax -

i 11'1)

+ He

(211)

In a

We recall that the SchrOdinger equation is derived from the energy functional by a variation: =

(212)

If we demand that the system has finite energy even for the zero electron mass limit, we must have (11'aX

-

i 11'1) 'l(XI,x2.···.xiv)

=a

a

= 1. 2•...• N

(213)

which we call the LLL constraint equation. (2.12) is now = E

=

(214)

where W is a totally anti-symmetric many-body wave function. The quantum mechanics of the system is defined by (213) and (2.14). The LLL condition (2.14) is a set of constraints on the state, and in the symmetric gauge, (2.15) it is given by

(216) where z=x+iy

(2.17)

182

3. Collective Field Method and Laughlin Wave Function Let us first bosonize the wave function: ,TI

~ ~

~)

= II

'¥\X I'X 2' ••• ,XN

(za - Zb)m

nJ ~

~

m 'i'\X I'X

a>b 'Za -

~)

2, ••• ,XN

Zb ,

-im Llmln(z. - zb )

= e

~X\,X' 2' •••

a,.h

.xiv )

(3.1)

where m is an odd integer. is now a totally symmetric wave function. The constraint equation is now given by 1 a [-aZa- + -4>t2 Za

-

m

-

2

~

InJ~ ~ ... ,~x N ) -- 0 'i'\X I'X 2'

1

Lb "'" Za -

(3.2)

Zb

Since the wave function is a totally symmetric function of the x's , one may apply the collective field method [7], a method of change of variables from the particle coordinates to the density variables, which are the most general symmetric combinations of the x 's: N

pCX')

= L.

8(X' -

x;,)

(3.3)

a = I

One first regards as a functional of p and uses the chain rule of differentiation to convert the derivatives. The number of degrees of freedom of the x 's and p(x ) is in general different so that the transformation is not a genuine point canonical transformation. But in the high density limit it would tend to a canonical transformation. The constraint equation is then expressed as

1~_8_ + az8p(X')

_l_z _ !!!:.2 4>t

2

rJ

JdX" z -1z ' p(X") I x0- ... 7 0 V of OZ -opex) 2

(4.4)

and consequently -Pv

I + --, 2Tf>"~

m p

=0

(4.5)

Because of vortices the functional Fourier transform of