Perspectives in Higgs Physics 981021216X, 9789810212162

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PERSPECTIVES ON HIGGS PHYSICS

ADVANCED SERIES ON DIRECTIONS IN HIGH ENERGY PHYSICS Published Vol. 1 - High Energy Electron-Positron Physics (eds. A. Ali and P. Soding) Vol. 2 - Hadronic Multiparticle Production (ed P. Carruthers) Vol.3 -CPViolation (ed. C. Jarlskog) Vol. 4 - Proton-Antiproton Collider Physics (eds. G. Altarelli andL. Di Leila) Vol.5 - PerturbativeQCD (ed. A. Mueller) Vol. 6 - Quark-Gluon Plasma (ed R. C. Hwa) Vol. 7 - Quantum Electrodynamics (ed. T. Kinoshita) Vol. 9 - Instrumentation in High Energy Physics (ed. F. Sauli) Vol. 10 - Heavy Flavours (eds. A. J. Buras and M. Lindner) Vol. 11 - Quantum Fields on the Computer (ed. M. Creutz) Vol. 13 - Perspectives on Higgs Physics (ed. G. L. Kane) Forthcoming Vol. 8 - Standard Model, Hadron Phenomenology and Weak Decays on the Lattice (ed. G. Martinelli) Vol. 12 - Advances of Accelerator Physics and Technologies (ed. H. Schopper) Vol. 14 - Precision Tests of the Standard Electroweak Model (ed. P. Langacker) Cover artwork by courtesy of Los Alamos National Laboratory. "This work was performed by the University of California, Los Alamos National Laboratory, under the auspices of the United States Department of Energy."

Advanced Series on Directions in High Energy Physics — Vol. 13

PERSPECTIVES ON HIGGS PHYSICS Editor

Gordon L. Kane University of Michigan

V f e World Scientific wfc

• Hong Kong Singapore • New Jersey • London L

Published by World Scientific Publishing Co. Re. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

PERSPECTIVES ON HIGGS PHYSICS Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orby 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 981-02-1216-X ISBN 981-02-1241-0 (pbk)

Printed in Singapore by Stamford Press Pte Ltd

V

CONTENTS

Introduction

xiii

1 The Higgs System by M. Veltman

1

1 Introduction

2

2 Some History

3

3 The 3.1 3.2 3.3

Original Higgs Model Lagrangian and Feynman rules Cosmological constant Unitarity limit, heavy Higgs

4 4 8 9

4 The 4.1 4.2 4.3 4.4 4.5 4.6

cr-Model The model and its symmetries The non-linear (T-model as limit of the (7-model Higgs sector of the standard model Isospin Fermion masses, parity Discussion

12 12 15 17 22 22 25

5 Other Higgs Systems 5.1 Higher representations 5.2 Additional doublets 5.3 Discussion

26 26 29 32

6 Higgs Hunting 6.1 Introduction 6.2 Screening; radiative corrections 6.3 Unitarity limit 6.4 Multi-vector boson production

32 32 33 36 39

References 2 Constraints on Higgs Boson Properties from the Higgs Potential by Marc Sher

42

44

1 Introduction

45

2 The Effective Potential

46

VI

3 Vacuum Stability Bounds in the Standard Model

54

4 Could Our Vacuum Be Unstable? (a) Spontaneous decay (b) Thermal fluctuations (c) Cosmic ray induced vacuum decay (d) Summary

59 59 61 65 68

5 Bounds in Two-Doublet Models (a) Stability bounds (b) Bounds on the ratio of vacuum expectation values

69 69 73

6 Conclusions

75

References

77

3 Higgs Bosons in the Minimal Supersymmetric Model: The Influence of Radiative Corrections by Howard E. Haber

79

1 Introduction

79

2 The Two-Higgs Doublet Model

80

3 The Higgs Sector of the MSSM at Tree Level

87

4 Virtual Higgs Contributions to Precision Electroweak Measurements

88

5 A Theoretical Upper Limit on the Lightest MSSM Higgs Mass . . . 96 6 Radiative Corrections to the MSSM Higgs Masses

98

7 Numerical Results for MSSM Higgs Masses

108

8 Implications of the Radiatively Corrected Higgs Sector

113

Appendix A: Three-Higgs Vertices in the Two-Higgs Doublet Model

117

Appendix B: Renormalization Group Equations

119

Appendix C: MSSM Higgs Sector Contributions to the 5, T, and U Parameters References

122 124

4 Producing the Intermediate Mass Higgs Boson by S. Dawson

129

1 Introduction

130

2 Gluon Fusion of the Higgs Boson

131

vii

(a) Lowest order (b) QCD corrections to the rate (c) QCD corrections to the px distribution 3 Rare Decay Modes of the Higgs Boson

131 135 138 . 142

4 Associated Production with a W Boson

144

5 Associated Production with a Top Quark .

147

6 Conclusions

152

References

.

153

5 Search for Higgs Bosons with Isolated Photons at Large Hadron Colliders by Z. Kunszt

156

1 Introduction

157

2 Branching Ratios

158

3 Production Cross Sections

163

4 Production Rates 5 Conclusions

•••

References 6 Detecting the Supersymmetric Higgs Bosons by John F. Gunion

166 172 176 179

1 Introduction

179

2 Scenarios

180

3 Masses, Coupling Constants and Branching Ratios 3.1 Masses and coupling constants 3.2 Branching ratios

183 183 187

4 The MSSM Higgs Bosons at e+e~ Colliders 4.1 LEP and LEP-II 4.2 The next linear e+e" colliders

191 191 194

5 Detecting MSSM Higgs Boson at Hadron Super Colliders 5.1 Techniques for detecting the SM Higgs boson and application to the MSSM 5.2 SSC/LHC MSSM cross sections 5.3 SSC/LHC MSSM Higgs detection phenomenology 5.4 The no-lose theorem

199 200 201 204 214

VIII

5.5 Alternative detection modes: a brief survey 5.6 Very recent developments 6 Final Remarks References

216 218 218 219

7 W h a t Kind of Higgs Boson Is It? by Gordon L. Kane

223

8 Electroweak Breaking in Supersymmetric Models by Luis E. Ibdnez and Graham G. Ross

229

1 Introduction 1.1 The supermultiplet content of the supersymmetric standard model 1.2 The couplings of the MSSM 1.3 R-parity and discrete symmetries

231 232 235

2 Supersymmetry Breaking and the MSSM Sparticle Masses

238

3 Electroweak Breaking 3.1 One-loop radiatively corrected potential 3.2 Renormalisation group analysis

241 242 245

4 Numerical Analysis 4.1 Unification of gauge couplings 4.2 The electroweak breaking scale 4.3 The fine-tuning problem

252 253 259 260

5 Outlook

230

, . 266

Appendix

267

References

269

9 Addressing the Mysterious with the Obscure — CP Violation via Higgs Dynamics by I. I. Bigi, A. I. Sanda and N. G. Uraltsev

276

1 Introduction

277

2 Implementing CP Violation Through the Higgs Sector 2.1 Manifest vs. spontaneous CP violation 2.2 The question of flavour-changing neutral currents 2.3 Specific models 2.3.1 Models with FCNC

278 278 280 280 280

ix

2.3.2 Models with NFC 3 CP 3.1 3.2 3.3

Phenomenology for Light Quark Systems Present constraints Models without NFC Models with NFC 3.3.1 KL decays 3.3.2 The neutron electric dipole moment 3.4 Intermediate resume 3.5 Future CP phenomenology in light quark systems 3.5.1 New contributions to d^ 3.5.2 Electric dipole moments of electrons and atoms and T odd electron-nucleon interactions 3.5.3 K -+ fii/ir 3.6 Conclusions on the CP phenomenology of light quarks

281 283 283 284 285 286 288 290 291 291 292 292 293

4 CP Violation in Heavy Quark Systems 4.1 Models without NFC 4.2 Models with NFC 4.2.1 Beauty decays 4.2.2 Top transitions

294 294 295 295 296

5 Summary and Outlook

296

References 10 Electroweak Baryogenesis by Neil Turok Introduction

298 300 300

1 Sakharov's Proposal and Electroweak Physics

303

2 Electroweak Baryon Number Violation

304

3 B Violation at Finite Temperature

306

4 An Analog Theory in 1+1 Dimensions

310

5 The Electroweak Phase Transition

313

6 Electroweak Baryogenesis in the Two-Doublet Model

319

References 11 Why I Would Be Very Sad If a Higgs Boson Were Discovered by Howard Georgi

328

337

X

12 Strong WW Scattering at the SSC and LHC by Michael S. Chanowitz

343

1 Introduction

343

2 The 2.1 2.2 2.3 2.4

Higgs Mechanism and Its Implications The generic Higgs mechanism The equivalence theorem Low energy theorems Unitarity and the scale of strong WW scattering

347 347 352 353 355

3 Strong WW Scattering Models and Complementarity 3.1 Effective W approximation 3.2 Strong scattering models, 7T7T scattering data and complementarity 3.3 The p chiral Lagrangian and complementarity 3.4 WW scattering models from 7T7T scattering data 3.5 Criterion for observability

358 358

4 Strong Resonances: the "p" Meson

369

5 Strong W+W+ Scattering Signals . . . 5.1 Backgrounds 5.2 The nucleon Q2 scale 5.3 Experimental cuts 5.4 Results 5.5 Complementarity and the "p" chiral Lagrangian

372 373 374 376 379 382

6 Strong Scattering in the ZZ Final State 6.1 The "linear model" for gg -» ZZ 6.2 Results for the "linear moder 6.3 A two-condensate model

385 386 388 390

7 Conclusion

392

References

394

13 Equivalence Theorem and Scattering of Longitudinal Vector Bosons by K. Veltman

359 362 367 368

397

1 The Equivalence Theorem

398

2 Vector Boson Scattering

406

References

413

xi

14 Proposals for Studying TeV Experimentally by C.-P. Yuan

WLWL

—*

WLWL

Interactions 415

1 Introduction

416

2 Signal

416

3 Backgrounds 3.1 W+(-* l+v)W~{-^ qiq2) mode 3.2 W+(— /+i/)Z°(— qq) mode 3.3 W+(-> l+v)W+(-+ l+u) mode

417 417 417 418

4 How to Distinguish the Signal from Background Events 4.1 Global features 4.2 Isolated lepton inW+ ^>l+v 4.3 W -* qxq2

418 418 419 419

5 Various Models 5.1 A TeV scalar resonance 5.2 A TeV vector resonance 5.3 No resonance 5.4 Beam pipe W's 5.5 Conclusions

420 420 422 422 423 424

References

425

15 The Revival of Technicolor Models by Martin B. Einhorn

429

16 Top Quark Condensates by Christopher T. Hill

447

1 Introduction

447

2 Analysis in Fermion Bubble Approximation

450

3 Fully Improved Renormalization Group Solution 3.1 Infrared fixed points 3.2 Sensitivity to irrelevant operators

452 452 454

4 Topcolor: A Gauge Theory That Makes a Top Condensate

457

5 Conclusions

460

Appendix

462

References

466

This page is intentionally left blank

xiii

INTRODUCTION

There have been profound developments in particle physics in the past quarter century, comparable with any discoveries in the history of physics. The minimal requirements for claiming that (a segment of) nature is understood are to know the forces that act, the particles on which the forces act, and the rules for calculating the effect of the forces. The last of these came first, with the discovery of quantum theory. The evidence is increasingly good that the Standard Theory (which used to be called the Standard Model), in which gauge forces mediated by gauge bosons act on quarks and leptons, describes nature's basic forces and particles. As we all know, the Standard Theory is incomplete in several areas. The masses of the top quark and the neutrinos are not yet measured. There are a number of regularities — such as the existence of families, the similarities between quarks and leptons, and the apparent coming together of the gauge couplings at about 1016 GeV — that are not explained or incorporated in the Standard Theory. These guarantee us that more exciting physics is to come. But certainly the central problem of particle physics today is that we do not understand the physics of the Higgs mechanism. In a practical or technical sense the simple Higgs mechanism with one SU(2) doublet of Higgs fields works very well. Many people have tried to construct alternatives to it, and the difficulty of doing so has made many of us appreciate even better how well the simple Higgs mechanism works. It gives mass both to gauge bosons and to fermions with one mechanism, which from the standpoint of any approach involving composite Higgs fields must be viewed as a major accomplishment. It is experimentally testable, though it may not be tested for another decade. As described in several of the chapters in this book, either Higgs bosons will be detected, or the interactions of the longitudinal W bosons will deviate from their expected behavior (appropriately defined) in the Standard Theory. The difficulty of performing these tests has led some people to doubt the existence of Higgs bosons, but that is an inappropriate response. The experimental difficulties with Higgs bosons are largely practical. It is more expensive to make beams of gauge bosons than beams of fermions, and the easiest beams to produce are made of light fermions (e^^u^d) which have very small coupling to Higgs bosons since the coupling is proportional to the mass. In addition, the mass of the Higgs boson expected in the minimal Standard Theory cannot be determined without some measurement from the

xiv Higgs sector; so it is not possible to design an experiment to aim at a particular mass. Thus the first experiments that could have detected a Higgs boson of the natural mass range (say, M^/2 < Mh < v, where v is the Higgs field vacuum expectation value) were those at LEP starting in 1989. By the time this book is published, the single neutral Higgs boson of the one-doublet Standard Theory will have been detected if it is lighter than about 60 GeV, but not if it is heavier than that. Depending entirely on how much funding is put into increasing the energy and intensity of LEP and its detectors, a Higgs boson of mass up to perhaps even 140 GeV could be detected there, possibly by the mid-1990's. That is an important number; recently Espinoza and Quiros have extended the class of models where an upper limit on the mass of the lightest Higgs boson in a supersymmetric theory exists, and they have evaluated it numerically for simple cases, obtaining a value of about 140 GeV. Very recently J. Wells, C. Kolda, and I have been able to build on their work, as well as earlier work of Drees and of Haber and Sher, to get a general limit for any supersymmetric theory. Here "any" means a supersymmetric theory, with any representa­ tions in the Higgs sector, that is perturbatively valid to a mass scale of order 1016 GeV. Since there is phenomenological evidence that some quantities (such as sin2 0\y) can be calculated at the weak scale starting from values at a scale of order 1016 GeV, the perturbative validity to that scale is an appropriate physical requirement and does not imply a belief in a particular form of grand unification. Numerically the limit is not different from that of Espinoza and Quiros. There are some small effects that could lower it a little. Further study of the limit, both numerically and with regard to its underlying assumptions, is in progress. Most likely, LEP will search only up to about Mh < Mz> It has been established that the range above which LEP can reach will be fully covered by the SSC for a Standard Theory Higgs boson, and perhaps for any Higgs boson. More precisely, it will be possible at the SSC to detect Higgs bosons if they exist, and exclude them if they do not, but in order to do so detectors must have certain capabilities. The initial detectors may not have all the needed capabilities, but eventually upgrades can; if upgrades are required it may take longer until we have the data to untangle the physics of the Higgs mechanism. LEP plus the SSC will be sufficient to get the needed data; if LHC runs before SSC or has better detectors it may make the discoveries first. As is well known, unless the approximate coming together of the forces at a scale near 1016 GeV, or alternatively the calculability of sin2 0w to within about 2% in some grand unified theories, is dismissed as an irrelevant accident, a

XV

Standard Model Higgs boson must be lighter than about 180 GeV. Finding all the Higgs bosons of a supersymmetric world (or excluding them if they do not exist), and deciding what kind of Higgs bosons they are, will probably require the next linear collider (NLC), in order to measure difficult branching ratios and overcome background problems. One possibility included in the above is that no Higgs boson is discovered; this is discussed in several chapters. Since the Higgs mechanism has already given rise to the longitudinal W± and Z polarization states (call them WL), it will be possible to learn about the Higgs mechanism by studying WL- Un­ fortunately, the Lorentz invariance of the theory guarantees that nothing can be learned by studying a single WL even though it really is a particle that originates with the Higgs mechanism, because one can transform to the WL rest frame and rotate longitudinal and transverse polarization states into one another. Put differently, it is not possible to define "longitudinal" for a single particle. It is necessary to study the interactions of pairs of WLWL- Even then, symmetry properties of the theory determine the form of the interaction near threshold (basically the value and slope at threshold) so that Mww ^ 2Mw is needed before different physics produces different behavior. It could happen that differences from the behavior of the WLWL interaction if no light Higgs boson exists could be observed for Mww as low as 1 TeV, but the most likely kinds of effects may not be directly observable until Mww approaches several TeV or even higher energies (see the chapter by H. Veltman), depending on the mass scale of the new physics that is required if no light Higgs boson exists; perhaps precise measurements of Mt — M& will be established; such a limit would exclude a significant part of interesting parameter space. In addition, this book contains no chapter about constraints on the ex­ istence or mass of ft from analysis of the precise LEP measurments, for two reasons. First, although some treatments of the data have been interpreted to suggest that the data are better described if a light Higgs boson exists, others do not agree, and in any case such an effect is present only with 1991 and earlier data if one-standard-deviation effects are taken very seriously. Second, once M t o p is measured the analysis of the data will become much more sen­ sitive to the existence of a light ft, but that will not happen before this book is available. However, for perspective it is worth mentioning here what can be expected. There are several high quality analyses of the implications of the precision measurements. The following comments are based on the analysis of F. del Aguila, M. Martinez, and M. Quiros (AMQ), which has several fea­ tures that tend to make it reliable (but the results are essentially the same for all analyses). One point to note is that whether we can learn about the existence of a light Higgs boson after Mt is measured depends on the value of Mt. As Mt gets heavier the sensitivity to it decreases, and once Mt > 160 GeV it will require a major improvement in the errors even beyond presently pro­ jected values to learn anything about M&. On the other hand, if we know that (for example) Mt = 135 GeV the AMQ analysis, including 1991 data, will allow us to conclude (from data!) that M& < 425 GeV at 90% CL, or that Mh < 660 GeV at 95% CL. An important thing to note is that our abil­ ity to learn about Mh is very insensitive to the lower limit on Mt% and very sensitive to the upper limit on it. Thus it is possible that sometime next year we may have the first experimental information (unfortunately, still indirect) about the Higgs sector, from the analysis of precision measurements once Mt is measured.

Gordon L. Kane September 1992

This page is intentionally left blank

1

The Higgs System M. Veltman Randall Laboratory of Physics University of Michigan Ann Arbor, MI 48109-1120

2

1

Introduction

The standard model of weak, electromagnetic and strong interactions stands today virtually unchallenged. The model is renormalizable, thus may be considered internally consistent. In principle observable quantities can be calculated to arbitrary precision, and compared with experiment. While relatively few radiative corrections are accessible to present day experiment, those that have been measured agree with the theory. Yet there is a deep feeling of frustration among many theorists. The standard model accomodates most observable facts, but leaves many unex­ plained. Why three families? Why the particular symmetry structure? Etc. In addition there are many ad hoc parameters, so many that no one seri­ ously believes them all to be fundamental: the particle masses, the coupling constants are all unexplained free parameters. We do not even know if the neutrino mass is fundamentally zero or just very small. To add insult to injury there is a whole segment of the theory that has not even been observed or even directly tested. This is the Higgs sector. In the next decade experimental physics will enter this domain, first indirect, later direct. It will be very difficult but at the same time, in all likelihood, very interesting. It is necessary here to explain what is meant by "interesting". If the Higgs particle turns out to exist as conventionally described, with a reasonably low mass (say less than 200 GeV) then that closes the standard model from a mathematical point of view. It is then quite conceivable that new physics, not contained in the standard model, is way beyond reach of any accelerator imaginable today. Humanity might in that case never get an answer to the questions posed above. This is a distinct possibility. It is not an "interesting" possibility. On the other hand, the Higgs sector might well contain essential clues to the further understanding of matter. To support this view consider the following points. - Within the standard model there are two ways for any particle to obtain mass. One of them is through the interaction with the Higgs field. It now happens that all known particles obtain their mass exclusively through the Higgs system. - If we insist that all particles obtain their mass through one and the same Higgs system then parity non-conservation follows almost automati­ cally. The argument is not airtight, but nonetheless this is the first time that there is at least some argument pointing to violation of parity. - Any theoretical speculation based on taking seriously the vacuum structure as generated by the Higgs system has failed: monopoles, axions are examples.

3

- At the same time the Higgs system conflicts with precisely the weak point of the theory of gravity, namely the observed vanishing of the cosmological constant. Theoretically, the Higgs field functions in the standard model as an ul­ traviolet cut-off. Without that field the Yang-Mills interactions are nonrenormalizable, i.e. uncontrollably divergent. The long range, i.e. low en­ ergy part of the Higgs system brings us in great difficulty (the cosmological constant). It is only natural to guess that somewhere the extrapolation from the ultraviolet to the infrared breaks down. If so, that is what one expects to see when investigating the Higgs sector. This is "extremely interesting''. Most of the above must be classified as daydreaming or speculation. But there is enough in it to stimulate research in the Higgs sector. Such research, for a theorist, tends to be seemingly dull: at this moment because of lack of imagination one cannot do much more then try to calculate effects due to the Higgs system in order to allow comparison with experiment. This con­ frontation turns out to be quite difficult and will require a huge experimental effort. In these notes we will attempt to discuss the Higgs system in a systematic but not necessarily complete way.

2

Some history

The idea of mass generation through interaction with a non-empty vacuum appears to have been mentioned explicitly for the first time in a fundamental paper by Schwinger [1] in 1957. This paper describes what we now call the linear (7-model. That model will be described amply further on. Here we will quote a few fines from this paper. On page 416: "... a coupling of the form illustrated by ... will produce effective mass terms for each field through the action of the vacuum fluctu­ ations of the other fields". On page 423: "As a field which is a scalar under all operations in the three-dimensional isotopic space and in space time, ^(0) has a nonvanishing expectation value in the vacuum. Although unable to affix the value implied by the strong interactions with heavy fermions, one could at least anticipate that < would have the magnitude of nucleon masses, and thus a suitable /i-meson mass constant might emerge from g^ < without requiring a particularly large coupling constant gj*. On page 424: "... in which we again use the ')2

The relation 5 + H « = 4M 2 holds. The variable 5 is related to the total energy in the centre of mass system; s = l?cm = 4^oThe three diagrams can be evaluated readily. One finds:

+

where t' and u' are different from < and u: t' =

i ( l - c o s ^ ) s = -< + 0(A/2)

v! =

i ( l + c o s ^ ) 5 = - « + 0(M 2 )

For fixed 6,t,u the amplitude goes to zero for large Higgs mass m. However, consider now the case that m » Af, but fixed, and then take the limit s —► oo. To clearly see what happens we specialize to the case of scattering in the forward direction, i.e. cos0 = 1. Then t = 0 and u = — s 4- 4M 2 . In the limit M2 < < rn2 < < 5 one finds: Ampl(0) =

2

m2 ^

11

If m is sufficiently large compared to M(i.e. g2m2/M2of order 1) than the am­ plitude is larger than allowed by unitarity. This limit, including the method shown here is very much analogous to that usualy quoted in connection with the standard model [14]. As is clear from the above the limit of large Higgs mass is quite delicate. The longitudinal four-photon amplitude remains small for energies below the Higgs mass, but for higher energies the amplitude reaches a maximum that grows with m2. What is the limit really? More precisely, what happens with the one-loop radiative corrections? Such corrections can roughly be un­ derstood as the square of the tree amplitude integrated over all intermediate energies. We then have two effects that tend to cancel each other (asTO—► oo the region where the amplitude is substantial decreases, while the amplitude increases in that region). A complete calculation in the limit M2 « s,t,u « m2 shows that a result remains. For the longitudinal photon-photon scattering amplitude the one loop radiative corrections grow quadratically with s and t:

Ampl(l) = ij£(st + s2 + 1. Even if the tree amplitude is well within bounds (in fact it is vanishingly small if m 2 —* oo), the one loop radiative corrections violate the unitarity bound. The above expression for this amplitude suggest that perhaps a partial summation of diagrams in the s, t and u channels take care of the problem. That however is pure speculation. In the limit of heavy Higgs the most obvious area to study is longitudinal vector-boson scattering. In this context the equivalence theorem [15,16] and its application to vector-boson scattering is of great interest. This is not discussed here.

12

4 4.1

The a-model The model and its symmetries

Schwinger's = m ($Lx/;R + ^ i fy L ) , thus connecting t/>L and \j>R as expected. To generate such a mass term via the Higgs mechanism we must introduce a coupling between xpL^R and $. This coupling must be S£/(2)-left invariant (the G-transformations), which is the gauge symmetry of the model. One cannot make both i\)L and i\>R to behave as doublets under 5C/(2)-left because one cannot make an invariant coupling with three doublets (tpL^R and $ from the left). Thus necessarily either x/>L or tpR must in fact be invariant under SU(2)-\eft. We will take this t o b e %I>R.

Adding electromagnetism, i.e. invariance under Aj, is simple. Making tpR a doublet with respect to SU(2)-right it is then trivial to write a coupling that is both SU(2)-\eit and SU(2)-nght invariant:

Since the full SU(2)-right is respected this coupling also respects isospin. If all fermion mass terms were of this form then in fact the masses within a doublet would be equal. This becomes obvious substituting for $ its vacuum expectation value fo$abL

fm -> -9ffo ($ifl>i + tfy?)

+ h.c.

24

The mass generated is determined by the arbitrary parameter gf and the a vacuum expectation value f0. However, full invariance under global SU(2)-right is not a requirement. Only the gauged 5£/(2)-left, and the gauged rotation around the third axis of 5(7(2)-right must be rigorously respected. Since

expQA3r3) commutes with r$ it follows that the fermion mass term Lfm = ~9 ft a* ah (l + V*) J*?

+ h.C.

is acceptable. The parameter rj is arbitrary. If tf is non-zero isospin is broken and the masses within a doublet are different, as can be seen by substituting /o for $:

£ / » - - gtfotfii + iT*)^ The above reasoning suggest that under the U(l) invariance of the stan­ dard model, which in the Higgs sector of the standard model relates to the transformation

the left handed fermion tpL is invariant while x/>R transforms as a rotation around the third axis: ^ e x p ^ ' A V ) ^ . However, the above mass term has another manifest symmetry, whereby all %1>R and t/>L obtain the same phase factor: V>L -» exp ( - ^ A r 0 ) xl>L ^^exp(-iAr°)vR One can actually identify the U(l) invariance with any addition of the above, for example (infinitesimal form):

^

l

_» ^ _ 1 A A V * Z»

Zt

-g'A°T34>R

25 with arbitrary A. It is interesting to check the coupling of the photon field A^ to the fermions. The invariant Lagrangian is:

-$LD^L

-

D^L

$RD^R

= d,xl>L + gb^L-%-\Blxl>L

D^R =

d^-^Bl^-ig'By^*

The e.m. coupling emerges if we write

K = -\sAs3

K =cA*

with 5 and c as before, i.e. sine and cosine of the weak mixing angle. Inserting this: D^L

:

-^(gsT3

+ \c)rJ>L

D^R

:

- ^ ( < 7 ' C T 3 + AC)V>*

Remembering that g' = gsjc we see that ipL and il>R couple identically to the photon field. As a consequence parity is conserved in e.m. interactions. If we define the electric charge e as e = gs = g sin $w then the above expressions show that the two members of the isospin doublet differ by e in charge. The central charge of the doublet is determined by A, a free parameter.

4.6

Discussion

Making the following basic assumptions: i the gauge symmetry is SU(2) x U(l); ii the Higgs sector is as the linear 2 then the integral behaves like (m2)~*. In other words, if the integral is convergent without the Higgs propagators one may make the approximation q2 + m2 ~ rn2. If j = 2 the integral will be logarithmically divergent if no Higgs propagators were present; the leading behaviour is like (m2)~k£n(m2).If j = 1 the behaviour is like (m 2 )~* +1 with possibly a factor £n(m2). If j = 0 one has (using dimensional regularization):

/

d

n9

!

_ I

( 9 2 + m2)3 ~ 2

35

The general rule is this: consider the dimensionality of the integral. That determines the highest power of m2 that can appear. Thus integrals with the dimension of a (mass)2 behave maximally like m2. These are quadratically divergent integrals (j = 0, k = 1 in the above). Dimensionless integrals (logarithmically divergent) behave like ln(m2). The rest goes to zero as m 2 —> oo. For completeness here the result for j = k = 1: 27T2

{- d b + l + T ^ W (M2^(M2) - - 2 M™ 2 ))}

The conclusion is quite simple. For any expression involving Higgs propaga­ tors consider only logarithmic and quadratic divergent parts. They behave like £n(m2) and m2. Since m 2 and £n(m2) occur associated with (unobservable) divergencies it is not obvious that these terms can be seen. The obvious place to look is for differences (ratio's) of such expressions where the divergencies cancel out but hopefully some m2 or £n(rn2) remains. The only quadratically divergent integrals occur for vector boson self energies. Therefore the /^-parameter, essentially the ratio of vector boson masses, is the obvious candidate for large Higgs mass dependence. Unfortunately, while rn2 terms appear in the vector boson self-energies they cancel out in the ratio. Only £n(m2) remaines. That is then the final part of the screening statement: no observable rn2 dependence in any one-loop radiative correction. The £n(m2) correction to the /^-parameter can be calculated. It is [25]: , P = 1

3GM2 s2

.

m2

2 £n

W2

" ~8^~ T

where M = charged vector boson mass and G is the fermi coupling constant 02 x 1( _ 1.02 10"5 2 G= j=—-— \/2m

mp proton mass.

As usual, s and c are the sine and cosine of the weak mixing angle. Even with m ~ 3000 GeV the correction is only 0.4%. Since at this point no other mass ratio's are predicted by the standard model there is no other possibility for m2 dependence. Other divergent integrals occurring in the standard model arise in vertex type diagrams. Since all vertices basically involve the same coupling con­ stant a £n(rn2) dependence may be observed by comparing vertices. This indeed occurs, and comparing the three vector boson vertex with a vector boson- fermion vertex (as occurring for example in /j-decay) shows an ob­ servable £n(m2) correction. Such effects are part of the radiative corrections to e+e~ -» WW [24]. Analyzing two loop integrals is much more complicated than the one loop effects. As a general rule it appears that if £n(m2) occurs at the one loop

36

level for some observable quantity, then m2 dependence occurs in two loop effects [17]. If m is of order of 1000 GeV such two-loop effects are of the same order as the one-loop effects. Perturbation theory becomes useless. Hence the statement that one has strong interactions if the Higgs is heavier than 1 TeV = 1000 GeV. At the same time strong interactions of a somewhat different type occur in vector boson scattering. That is the subject of the next section.

6.3

Unitarity limit

In a renormalizable theory amplitudes must not grow indefinitly as a function of the momenta of the particles. The standard model including the Higgs system is renormalizable, and amplitudes behave as should be. Without the Higgs the theory is non-renormalizable, and indeed certain amplitudes, at the tree level, do not behave correctly in this sense. For a finite but large Higgs mass such amplitudes behave badly up till energies of the order of that mass. For still higher energies behaviour as in a renormalizable theory re-appears. Thus certain amplitudes reach a maximum for energies of the order of the Higgs mass. If, at the tree level, such amplitudes become of order one then that generally implies a breakdown of perturbation theory. One loop diagrams, that can sloppily be understood as the product of two tree diagrams, become of the same order of magnitude as the tree diagrams. The elastic scattering amplitude for longitudinally polarized vector bosons is perhaps the most instructive, and also most practical case. Considering the dependence on s, the centre off mass energy squared, the following happens: - Individual diagrams grow as s2. - The Yang-Mills structure of the tree and four vector boson vertices leads to certain cancellations, and growth as s results. - Diagrams involving Higgs exchange give further cancellations, and the total behaves as a constant. It is interesting to note that absence of a Yang-Mills structure can be observed quite before a missing Higgs. That aspect will not be discussed any further here. First the tree-level situation will be considered. We will restrict our­ selves to the simplified model with the Feynman rules as in appendix A. Also no fermions are considered. Then isospin invariance holds, which sim­ plifies things considerably. Thus consider two ingoing longitudinally polarized vector bosons in the centre of mass. The momenta and polarization vectors are taken to be: k = (0,0, kiy ik0) p = (0,0, -kt, ik0) «(*) = i ( 0 , 0 ,ft*,ikt) e(p) = £ ( 0 , 0 , -fc, ik€)

37

There are two outgoing longitudinally polarized vector bosons. By convention momenta are always taken to be ingoing and we write: k* = — (kt sin 0,0, kt cos 0, ik0) p' = — (—&/ sin 0,0, —ki cos 0, iko) e(k') = — — (ko sin 0,0,fcocos 0, ifc/) e(p') = — — (—A?o sin 0,0, —fc0cos 0, ikt) The angle 0 is the scattering angle in the centre of mass system. The coor­ dinate system has been chosen such that the incoming vector bosons have momenta along the third axis while the outgoing bosons are in the 1-3 plane. Obviously the individual components of the polarization vectors can be­ come very large for large s = 4&Q. Roughly speaking the polarization vectors behave like y/s. Consider now the Feynman rules. The WWWW vertex is momentum independent, as is the WWH vertex, while the WWW vertex is linear in the momenta. A propagator behaves as 1/6 (or \jt or 1/w). It is easily seen then that

behaves as s2

ii

W exchange, behaves a s s 2

iii

Higgs exchange, behaves as s

Since the total must (and indeed does) behave as a constant for large s we deduce that the s2 behaviour of diagrams i and ii must cancel, while the remaining behaviour proportional to s must cancel against that of diagrams

iii.

38

This is precisely what happens. Denoting the isospin states of the vector boson with momentumfcby a (simarly b to p, c to p' and d tofc')the leading behaviour of diagram i is: i:

S^S^st-^

+ ^/M'

+ Ois)

There are terms proportional to 6ac6M and £a D u t those are not shown. A factor g2(2ir)4i has been omitted. Diagrams i and ii combined show the behaviour: b b 1 + u: ^ cd-^p Finally Higgs exchange: s2 1 ill: Sabred AM2 ' - H m 2 Developing the Higgs propagator: 1 _ _l_m^ —3 + m2 s s2 and noting that there is no other m2 dependence the total result obtains 2

i 4- ii + iii = ~i(2ir)4g26ab&cd TTTi + ( m ~ independent constant) The unitarity limit of ref. [14] is based on this constant. Roughly speaking, if 2 2 m a ~ 1 9 AM2 then higher order effects become important. Considerations concerning the unitarity limit show the precise moment where the tree level amplitude is larger than allowed by unitarity. But in general no one takes the trouble to say by how much, i.e. how large higher order effects must be in order to restore unitarity. Here we refer to the fact that the complete theory is unitary by construction. Furthermore, even if tree level unitarity is not violated it still might be that higher order corrections are sizable. Therefore a calculation of the one loop radiative corrections to this amplitude is of interest. This calculation has been done in the limit M2 < s,t,u < m 2 and the result is [26]: One loop: _ ^ | - _ - _ + _

32

s2 , s 32 m 2

(st \96

t2\ 48/

- (s+SW*)} +

*oe*M { * - » « , t -» S, U -+ t}

+

Sadfoc {*—* t, t —» U, t l - t j }

m

t m2

39 with Bo = w/y/3 - 2. The terms involving £n(m2) have been reported before [27], [28]. In this last ref. also other approximations then the one considered here are discussed. The ratio of the one-loop result to the tree level amplitude may now be considered. Keeping only terms containing logarithms one finds for the case t = — s (thus u ~ 0): » _... ~

a

*>3

U

1

^ 3 nM '24m2 2

for the 6^6^ channel. In here aw = g2/4ir ~ 1/30. Again, this result has been deduced in the approximation M2 41.7 GeV. This is the most model independent bound and assumes only that the H^ decays dominantly into r + i/ r , cs and cb. The LEP limits on the masses of h° and A0 are obtained by searching simultaneously for Z —► h°ff and Z —> h°A° [13,14]. The ZZh° and Zh°A° couplings that govern these two decay rates are proportional to sin(/? — a) and cos(/# — a ) , respectively. Thus, one can use the LEP data to deduce limits on m^o and m^o as a function of sin(/? — a). Stronger limits can be obtained in the MSSM where sin(/? — a) is fixed by other model parameters. The present limits as summarized by the Particle Data Group [12] are m^o > 29 GeV and ra^o > 12 GeV based on supersymmetric tree-level relations among Higgs parameters, but with no assumption for the value of tan 0. If leading log radiative corrections are incorporated and tan/? > 1 is assumed, then recent results of the ALEPH Collaboration [14] yield mho > 41 GeV and mAo > 20 GeV (at 95% CL). However, the limit onra^omay be substantially weaker if large squark mixing is permitted [15].

85 The experimental information on the parameter tan/? is quite meager. For definiteness, let us assume that the Higgs-fermion couplings are specified as in the MSSM. In the Standard Model, the Higgs coupling to top quarks is proportional to gmt/2m\v, and is therefore the strongest of all Higgs-fermion couplings. For tan/9 < 1, the Higgs couplings to top-quarks in the two-Higgs-doublet model dis­ cussed above are further enhanced by a factor of 1/tan/?. As a result, some weak experimental limits on tan /? exist based on the non-observation of virtual effects involving the H~tb coupling. Clearly, such limits depend both on m#± and tan /?. For example, for ra#± ~ raw, limits from the analysis of B°-B° mixing imply that tan/? ^ 0.5 [16]. No comparable limits exist based on top-quark couplings to neutral Higgs bosons. Theoretical constraints on tan /? are also useful. If tan /? becomes too small, then the Higgs coupling to top quarks becomes strong. In this case, the treeunit arity of processes involving the Higgs-top quark Yukawa coupling is violated. Perhaps this should not be regarded as a theoretical defect, although it does render any perturbative analysis unreliable. A rough lower bound advocated by ref. 16, tan /? ^ mt/600 GeV, corresponds to a Higgs-top quark coupling in the perturba­ tive region. A similar argument involving the Higgs-bottom quark coupling would yield tan /? ^ 120. A more solid theoretical constraint is based on the requirement that Higgs-fermion couplings remain finite when running from the electroweak scale to some large energy scale A [17-19]. Beyond A, one assumes that new physics enters. The limits on tan/? depend on mt and the choice of the high en­ ergy scale A. Using the renormalization group equations given in Appendix B, we integrate from the electroweak scale to A (allowing for the possible existence of a supersymmetry-breaking scale, mz < -^SUSY — ^ ) , anc * determine the region of tan/3-mt parameter space in which the Higgs-fermion Yukawa couplings remain finite. (The t, b and r are all included in the analysis.) The results are shown in figs. 1 and 2 for two different choices of A [19]. The allowed region of param­ eter space lies below the curves shown. For example, if there is no new physics (other than perhaps minimal supersymmetry) below the grand unification scale of 10 16 GeV, then based on the CDF limit [20] of mt > 91 GeV, one would conclude that 0.5 ^ tan/? ^ 50. The lower limit on tan/? becomes even sharper if the top-quark mass is heavier. Remarkably, the limits on tan/? do not get substan­ tially weaker for A as low as 100 TeV. Finally, it is interesting to note that the limits on tan /? shown in fig. 2 are not very different from those that emerge from models of low-energy supersymmetry based on supergravity which strongly favor t a n / ? > 1 [21].

86

260

-T

1—i—r

200 h-

>

^■"-" 1 1 11 11

O

■//

ALLOWED REGION Msuw- A Msusr^STeV

( (

MMT-ITW

(

MsusY = 250GeV M

0.5

1

SUSY-

m

) ) )

(

Z

(

5

10

) )

50

tan0 Fig. 1. The region of tan/?-mt parameter space in which all running Higgs-fermion Yukawa couplings remain finite at all energy scales, fi, from mz to A = 10 16 GeV [19]. Non-supersymmetric twoHiggs-doublet (one-loop) renormalization group equations (RGEs) are used for mz < /i < AfSUSY and the RGEs of the minimal supersymmetric model are used for M S U S Y < /i < A (see Appendix B). Five different values of M S U S Y are shown; the allowed parameter space lies below the respective curves.

~l—I—» I I I 11

350 h-

i

|

A=100 TeV

111111 ALLOWED REGION A ( 5 TeV ( 1 TeV ( 250 GeV ( mz ( 10 1

) ) ) ) ) 10 2

tan/3 Fig. 2. The region of tan /?-mt parameter space in which all running Higgs-fermion Yukawa couplings remain finite at all energy scales from mz to A = 100 TeV. See caption to fig. 1.

87

3 . T h e H i g g s Sector of t h e M S S M at Tree Level The Higgs sector of the MSSM is a CP-conserving two-Higgs-doublet model, with a Higgs potential whose dimension-four terms respect supersymmetry and with restricted Higgs-fermion couplings in which $ i couples exclusively to downtype fermions while $2 couples exclusively to up-type fermions [8]. Using the notation of eq. (1), the quartic couplings A,- are given by A i = A 2 = ±( mz) on electroweak observables. In this case, since q2 is of order m2z, one only makes an error of 0(m2z/M2) by neglecting the q2 dependence of the Fij. Then, one can show that the oblique corrections to electroweak observables due to heavy physics can be expressed in terms of three particular combinations of the A,j(0) and Fij aT

^ - 5

_ Aww(P) _ Azz(0) _ 2*wAZt(0) m 2w

. Fzz{m2z)

m\

cw

- F„(mi) + (

m\

^

)

i^(m|)

(28)

2

^ ( 5 + U) = Fww(m2w)

- F„(m2w)

-

^FZy(m2w).

Note that the A,j(0) and Fij in the above formulae are divergent quantities. Nev­ ertheless, if one includes a complete set of contributions from a gauge invariant sector, then 5 , T, and U will be finite constants. The Higgs sector by itself does not constitute a gauge invariant sector in this regard, so one must include the vector boson sector as well to obtain a non-divergent result for S, T and U. Alternatively, in order to obtain finite quantities that solely reflect the influence of heavy Higgs physics, one can define 6S, ST and SU relative to some reference Standard Model where rn^o is fixed to a convenient value. For example, if we choose a RSM with m^o = mz, then the change in S due to a fourth generation of fermions U and D (with electric charges eD + 1 and eD respectively) and a heavy Higgs boson of

X In addition, the sum of heavy particle contributions to Azy(0) also vanishes exactly. Only gauge boson loops can produce nonzero contributions to J4Z 7 (0) (in the standard i2-gauge).

91 mass rrifo is given by

6ir

3

^ODKK^f)- *^

1 + (1 + 2e z

107 6

,

(29)

where m^, m^, m^o >• mz has been assumed. Once again, the non-decoupling effects of the heavy physics are apparent. Radiative corrections (in the oblique approximation) to electroweak observables can be expressed in terms of S, T and U. The /^-parameter discussed above is one such example. A second example is the W mass prediction. The one-loop prediction is obtained by solving the following equation for the W mass rn2w\__(

( ^>2 mw

V

where 7ca/y/2GF

ml)

™

V

\V2GJ

1

(30)

1-Ar

= (37.2802 GeV) 2 and

8?r

S-2c2wT

+

&?)']

(31)

Other examples can be found in refs. 33-35. Thus the effects of heavy physics on numerous electroweak observables are immediately known once the corresponding contributions to S, T and U have been computed. In order to compute the contributions of the Higgs sector of the MSSM to 5, T and U one must first define the RSM. Then, S — ^RSM + ^ > T

= ?RSM +

6T

i

(32)

where the MSSM Higgs sector contributions to £5, ST and SU are obtained from eq. (28) by computing the MSSM Higgs loops contributing to Aij(0) and Fij (in­ cluding diagrams with one virtual Higgs boson and one virtual gauge boson) and subtracting off the corresponding Higgs loops to the RSM. In nearly all cases of interest, one finds that SU < SS, ST, so I shall focus on SS and ST below. In the present case, it is most convenient to define the RSM to be the Standard Model with the Standard Model Higgs boson mass set equal to the mass of the lightest CP-even Higgs boson of the MSSM. In addition, until ra< is known, the definitions

92 of SS and ST will depend on the value of m* chosen for the RSM. Typically, one chooses mt = mz (equal to the present experimental CDF lower bound [20]) in order to obtain conservative limits on the possible new physics contributions to S and T. Consider now the specific contributions of the MSSM Higgs bosons to SS and ST. As indicated above, the sum of these contributions is finite after subtracting out the contribution of the Standard Model Higgs boson with m^o = m^o. In contrast to the Standard Model where the Higgs contributions to S and T grow logarithmically with Higgs mass, the contributions of the MSSM Higgs bosons smoothly decouple as the Higgs masses become large. This behavior is easy to understand. According to the results of section 3, the mass of h° cannot be arbi­ trarily large—it is bounded at tree level by mz* AH other Higgs masses can become large by taking m^p >• mz- In this limit, we see that mjj± ~ mjjo ~ m^o and ra^o ~ mz\ cos2/3|. However, in this limit, the large Higgs masses are due to the large value of the mass parameter mu [see eq. (18)] rather than a large Higgs selfcoupling (which is the case in the large Higgs mass limit of the Standard Model). In particular, the Higgs self-couplings in the MSSM are gauge couplings which can never become large. As a result, the decoupling theorem applies, and one must find that SS and ST approach zero quadratically as m^o —► oo. This result can be generalized to all other sectors of the MSSM! All MSSM contributions to 5, T and U vanish in the limit of large supersymmetry-breaking mass parameters [37,38]. In this limit, the effects of the supersymmetric particles (and all Higgs bosons beyond h°) smoothly decouple; the resulting low-energy effective theory at the scale mz is precisely that of the Standard Model. The results of an exact one-loop computation of the MSSM Higgs contributions to S and T are given in Appendix C [38]. (See refs. [25,39-41] for previous work on radiative corrections in two-Higgs doublet models.) Numerical results are shown in figs. 3 and 4. To understand why the numerical values for the MSSM-Higgs con­ tributions to SS and ST are so small, it is instructive to evaluate the corresponding expressions of Appendix C in the limit of large m^o. I find*

. / w , m f TT. x mifsin 2 23 - 2 cos 2 0\y) £S(MSSM-Higgs) - - ^ -f—. ^ , z47rm^0

, , (33)

^(MSSM-Higgs) * nXsff^W

(34)

f

.

487rm^0 sin* 0\v • Asymptotically, 6U(MSSM-Higgs) = 0(m%lm\),

which is completely negligible.

93

-0.004 CO

-0.008 h -

tan^-l ( tan0 = 2 ( tan£=10 ( tan0=lO [asympt] (

/

-0.008

/

/ . / .

-0.010

200

100

300

) ) ) )

400

500

mAo (GeV) Fig. 3. The contribution to the S parameter from the MSSM Higgs sector relative to the Standard Model with Higgs mass set equal to m^o, as a function of m^o. Three curves corresponding to tan/? = 1, 2, and 10 are shown. For comparison, the dotted curve depicts the asymptotic prediction [eq. (33)] for tan/? = 10. "i—i—|—i—|—i—i—i—i—I—i—I—i—i—|—I—i—i—i—I—i—i—i—r-

0.04

tan0 = l ( tan/? = 2 ( tan£ = 10 ( asymptotic (

\ "\ '\ I- I

I \

) ) ) )

0.02

0.00

-0.02

_i—i—ii—i—I—i—i—i—i—I—i—i

100

200

i

i

I

300

i

i

i

i_

500

mAo (GeV) Fig. 4. The contribution to the T parameter from the MSSM Higgs sector relative to the Standard Model with Higgs mass set equal to m^o, as a function of m^o. Three curves corresponding to tan/? = 1, 2, and 10 are shown. For comparison, the two dotted curves depict the asymptotic predictions [eq. (34)] for tan/? = 1 and 10 respectively. The p parameter is related to T via dp = aST, where a is the fine structure constant.

94 A recent analysis of 5, T and U based on LEP data (assuming a RSM where mt = m^> = mz) reported in ref. 30 yields: SS = - 0 . 9 7 ±0.67, ST = - 0 . 1 8 ±0.51 and SU = 0.07 ± 0.97. It is hard to imagine that the these quantities could ever be measured to an accuracy better than 0.1 One must also consider the possibility of other contributions to SS and ST. As long as m* is not well known, there will be mt dependence in these quantities (entering through the mt choice of the RSM). Moreover, in the supersymmetric model, SS and ST would also acquire contributions from other MSSM sectors which, although small, are nearly always larger than the MSSM-Higgs contributions shown above [38]. Thus, I conclude that virtual effects of the MSSM Higgs sector will never be detected via its oblique radiative corrections. One class of processes for which virtual Higgs effects could be important are those involving external b or t quarks. Here, I shall briefly focus on processes in­ volving ^-mesons. In such cases, vertex corrections and box diagrams that involve an intermediate t-quark and charged Higgs boson can be substantial, because gjj-tl contains a piece proportional to mt cot f3/m\y [see eq. (16)]. Thus the impact of such contributions can be significant for small tan/? and m#±. Three examples of relevant processes studied in the literature are: (i) charged Higgs box diagram con­ tributions to B°-B° mixing [42-45, 16] (briefly mentioned at the end of section 2); (ii) the charged Higgs vertex correction to Z —► bb [46]; and (iii) the charged Higgs vertex corrections to various rare 6-decays [47,43-45] such as b —* 37, 6 —+ si+i", b —> sg and b —* svv. Of course, if the two-Higgs-doublet model is a piece of the MSSM, then there will also be one-loop supersymmetric particle contributions to all of the processes mentioned above. Some of these contributions (e.g., loops con­ taining top-squarks) could dominate over the virtual charged Higgs effects [44,46]. Among the rare 6-decays, the charged Higgs contribution to b —* 57 is perhaps the most promising. The theoretical prediction for this rate in the Standard Model is BR(B -> K~f + X) ~ 3.6 x 10~ 4 (4.1 x l 0 ~ 4 ) , for mt = 150 (200) GeV, where leading log QCD corrections have been included. Incorporating the charged Higgs contribution [assuming an H"tb coupling specified in eq. (16)] yields results for this branching ratio shown in fig. 5. These results correspond to an enhancement over the Standard Model expectation as shown in fig. 6. Whether effects of the charged Higgs boson can be detected in this way (or interesting limits set) depends on the reliability of the Standard Model prediction. At present, this prediction is reliable to within a factor of two [47]. Improved theoretical analysis as well as more i?-decay data will be required before definite conclusions can be drawn. In summary, the only potentially important virtual Higgs boson corrections to Standard Model processes arise either through oblique radiative corrections (i.e., Higgs loop corrections to vector boson propagators), or through charged Higgs vertex and box diagram corrections to processes with external 6 and/or t quarks. I

95

0.0050

.0.0010

H-

t

s 0.0005 m

m t =150 GeV taitf-0.5 ( tan^»l ( tan^-20 ( 0.0001

m t = 200 GeV tan^-0.5 ( tan0«l ( tan/?-20 (

) ) )

) ) ) _l

100

200

500

200

1000 100

I I l—L

500

1000

mH+ (GeV) Fig. 5. The branching ratio for B —► Ky + X in the two-Higgs-doublet model as a function of the charged Higgs mass for m< = 150 and 200 GeV and various choices for tan/?, assuming an H~tb coupling given by eq. (16). This graph is based on calculations of ref. 45.

10

2 CO

5h

1

taxtf-0.5 ( tan^-1 ( tan^-20 ( B-*K7+X

) ) )

r

-V .

>. PQ

a i

>»

u

2h

1—r- i

1

tan£~20 ( >v

—) -) -) -

-

\^ N.

X

N

\

\\ ^ \ '^ v s \ '^ N s, \ X

_

'^ v N X

-

-v

*^ \ '*> \

m t -200 GeV

mt-150 GeV

i

100

» f

B-»Ky+X

- ^*>•' Ns

00

1

tan^-0.5 ( tanfi=l (

200

500

1000 100

200

500

i

i

1000

mH+ (GeV) Fig. 6. The ratio of BR(B -► Ky + X) in the two-Higgs-doublet model relative to its predicted value in the Standard Model (SM) as a function of the charged Higgs mass for mt = 150 and 200 GeV and various choices for tan/?, assuming an H~tb coupling given by eq. (16). This graph is based on calculations of ref. 45.

96 have shown above that in the MSSM, Higgs-mediated oblique radiative corrections are too small to be observed. This leaves heavy quark processes as the only possible arena for observable Higgs-mediated radiative corrections. 5. A Theoretical U p p e r Limit on t h e Lightest M S S M Higgs Mass The tree-level Higgs mass predictions of section 3 have important phenomenological consequences. For example, the bound m^o < mz, if reliable, would have significant implications for future experiments at LEP-II. In principle, experiments running at LEP-II operating at y/s = 200 GeV and design luminosity would either discover the Higgs boson (via e + e ~ —► h°Z) or rule out the MSSM. (Whether this is possible to do in practice depends on whether Higgs bosons with m^o « mz can be detected [48].) However, m^o < mz need not be respected when radiative corrections are incorporated. In the radiative corrections to the neutral CP-even Higgs squared-mass matrix, the 22-element is shifted by a term proportional to {92fntlmw) M ^ f / m < ) [49-51]. Such a term arises from an incomplete cancel­ lation between top-quark and top-squark loop contributions to the neutral Higgs boson self-energy. If m* is large, this term significantly alters the tree-level predic­ tions. Hempfling and I computed the exact one-loop expression for the light Higgs mass bound, as a function of all the relevant supersymmetric parameters [49], This bound is saturated in the formal limit where tan/? —► oo (with all downtype fermions masses set equal to zero) and m^p > mz, m^o. The expression we obtained is quite cumbersome, although straightforward to evaluate numerically. However, it is useful to display an approximate expression, valid for a certain range of supersymmetric parameters. If all supersymmetric mass parameters are roughly of order AfSUSY and if mz < m% h° < ™>Z + A m u ,

(36)

which defines the quantity Am&. A numerical calculation of Am& is displayed in fig. 7. As advertised, the dominant correction to the tree-level formula increases as the fourth power of m*, and therefore can be quite large. Nevertheless, for values of mt & 250 GeV, the perturbative one-loop calculation is reliable. This can be verified by estimating the largest two-loop contributions to Arn^ and showing that the one-loop result is stable [52]. It is also evident from eq. (35) that the dependence of the squared Higgs mass shift on Af£ is logarithmic. Thus, even if Afg is significantly smaller than 1 TeV, Ara& can be appreciable if mt is sufficiently large. This is illustrated in fig. 8, where Am& is plotted as a function of Mx for m* = 100, 150 and 200 GeV. These results are based on an exact numerical one-loop computation; the approximate formula given in eq. (35) is unreliable for values of Mx approaching mt*

6. R a d i a t i v e Corrections t o t h e M S S M Higgs Masses One can also compute radiative corrections to the full CP-even Higgs masssquared matrix [51,53-58]. Full one-loop computations can be found in refs. 53 and 57. Here, I will present the results based on a calculation of the mass-squared ma­ trix in which all leading logarithmic terms are included (see ref. 54 for details). We take the supersymmetry breaking scale ( M S U S Y ) to be somewhat larger than the electroweak scale. For simplicity, we assume that the masses of all supersymmetric particles (squarks, sleptons, neutralinos and charginos) are roughly degenerate and of order M S U S Y . This means that various soft-supersymmetry breaking parameters such as the diagonal squark mass parameter, M g , and the gaugino Majorana mass terms, as well as the supersymmetric Higgs mass parameter are all roughly equal to M S U S Y . Admittedly, this is a crude approximation. However, deviations from this assumption will lead to non-leading logarithmic corrections which tend to be small if the supersymmetric particles are not widely split in mass. Moreover, the procedure outlined below can be modified to incorporate the largest non-leading logarithmic contributions that arise in the case of multiple supersymmetric particle thresholds and/or large squark mixing. The leading logarithmic expressions for Higgs masses are obtained from eqs. (4) and (5) by treating the A,- as running parameters evaluated at the electroweak scale,

99

Afweak. In addition, we identify the W and Z masses by

rnw = \g2{v\ + vl), 2

1/ 2 ,

™>Z = i\9

I2\(

+9

2 i

2\

^

'

) K + VI)>

where the running gauge couplings are also evaluated at A/ weak . Of course, the gauge couplings, g and g1 are known from experimental measurements which are performed at the scale M wcaIc . The Ai(Af^eak) are determined from supersymmetry. Namely, if supersymmetry were unbroken, then the A,- would be fixed according to eq. (17). Since supersymmetry is broken, we regard eq. (17) as boundary conditions for the running parameters, valid at (and above) the energy scale AfSUsY- That is, we take A i ( M | U S Y ) = A 2 (M| U S Y ) = & 2 ( M j U S Y ) + mw by de­ coupling the (t, b) weak doublet from the low-energy theory for scales below rat. The terms in eq. (41) that are proportional to m\ and/or ml arise from selfenergy diagrams containing a tb loop. Thus, such a term should not be present for ™>W < P < fnt. In addition, we recognize the term in eq. (41) proportional to the number of generations Ng as arising from the contributions to the self-energy dia­ grams containing either quark or lepton loops (and their supersymmetric partners). To identify the contribution of the tb loop to this term, simply write Ng = \Ng(Nc + 1) = \NC + \[Nc(Ng - 1) + Ng],

(42)

where Nc = 3 colors. Thus, we identify \NC as the piece of the term proportional to Ng that is due to the tb loop. The rest of this term is then attributed to the

101 lighter quarks and leptons. Finally, the remaining terms in eq. (41) are due to the contributions from the gauge and Higgs boson sector. The final result is [59] Urn2x

Nc94

1„2

l (m*2t ™?\ m\\ 2ml, \A * A)

1 3

m\m\

lnMusY ■^SUSY

~ 96^ {[jVc(JVy ~^ + N' + *N* ~ 10]