Effective Lagrangians for the Standard Model (Theoretical and Mathematical Physics)

Texts and Monographs in Physics Series Editors: R. Balian W. Beiglbock H. Grosse E. H. Lieb N. Reshetikhin H. Spohn W Th...

Author: Antonio Dobado | Angel Gomez-Nicola | Antonio L. Maroto | Jose R. Pelaez

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Texts and Monographs in Physics Series Editors: R. Balian W. Beiglbock H. Grosse E. H. Lieb N. Reshetikhin H. Spohn W Thirring

Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Texts and Monographs in Physics Series Editors: R. Balian W. Beiglbock H. Grosse E. H. Lieb N. Reshetikhin H. Spohn W. Thirring From Microphysics to Macrophysics I + II Methods and Applications of Statistical Physics By R. Balian Variational Methods in Mathematical Physics A Unified Approach By P. Blanchard and E. BrUning Quantum Mechanics: Foundations and Applications 3rd enlarged edition By A. Bohm The Early Universe Facts and Fiction 3rd corrected and enlarged edition By G. Borner Operator Algebras and Quantum Statistical Mechanics I + II 2nd edition By O. Bratteli and D. W. Robinson Geometry of the Standard Model of Elementary Particles By A. Derdzinski Scattering Theory of Classical and Quantum N-Particle Systems By J. Derezinski and C. Gerard Effective Lagrangians for the Standard Model By A. Dobado, A. G6mez-Nicola, A. L. Maroto and J. R. Pelaez Quantum The Quantum Theory of Particles, Fields, and Cosmology By E. Elbaz Quantum Relativity A Synthesis of the Ideas of Einstein and Heisenberg By D. R. Finkelstein Quantum Mechanics I + II By A. Galindo and P. Pascual The Elements of Mechanics By G. Gallavotti Local Quantum Physics Fields, Particles, Algebras 2nd revised and enlarged edition By R. Haag

Supersymmetric Methods in Quantum and Statistical Physics By G. Junker CP Violation Without Strangeness Electric Dipole Moments of Particles, Atoms, and Molecules By I. B. Khriplovich and S. K. Lamoreaux Inverse SchrOdinger Scattering in Three Dimensions By R. G. Newton Scattering Theory of Waves and Particles 2nd edition By R. G. Newton Quantum Entropy and Its Use By M. Ohya and D. Petz Generalized Coherent States and Their Applications By A. Perelomov Essential Relativity Special, General, and Cosmological Revised 2nd edition By W. Rindler Path Integral Approach to Quantum Physics An Introduction 2nd printing By G. Roepstorff Finite Quantum Electrodynamics The Causal Approach 2nd edition By G. Scharf From Electrostatics to Optics A Concise Electrodynamics Course By G. Scharf The Mechanics and Thermodynamics of Continuous Media By M. Silhavy Large Scale Dynamics of Interacting Particles By H. Spohn The Theory of Quark and Gluon Interactions 2nd completely revised and enlarged edition By F. J. Yndurain Relativistic Quantum Mechanics and Introduction to Field Theory By F. 1. Yndurain

A. Dobado A. Gomez-Nicola A. L. Maroto 1. R. Pelaez

Effective Lagrangians for the Standard Model With 33 Figures

Springer

Professor Antonio Dobado Dr. Angel Gomez-Nicola Dr. Antonio L. Maroto Dr. Jose R. Pelaez

Departamento de Ffsica Te6rica Facultad de Ciencias Ffsicas Universidad Complutense E-28040 Madrid, Spain

Editors Roger Balian

Nicolai Reshetikhin

CEA Department of Mathematics Service de Physique Theorique de Saclay University of California F-91191 Gif-sur- Yvette, France Berkeley, CA 94720-3840, USA

Wolf Beiglbock

Herbert Spohn

Institut fur Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 294 0-69120 Heidelberg, Germany

Theoretische Physik Ludwig-Maximilians-Universitat Munchen TheresienstraBe 37 0-80333 Munchen, Germany

Harald Grosse

Walter Thirring

Institut fUr Theoretische Physik Universitat Wien Boltzmanngasse 5 A-1090 Wien, Austria

Institut ftir Theoretische Physik Universitat Wien Boltzmanngasse 5 A-I090 Wien, Austria

Elliott H. Lieb Jadwin Hall Princeton University, P. O. Box 708 Princeton, NJ 08544-0708, USA

Library of Congress Cataloging-in-Publication Data. Effective lagrangians for the standard model 1 Antonio Dobado ... [et al.l. p. cm. - (Texts and monographs in physics) Includes bibliographical references and index. ISBN 3-540-62570-4 (alk. paper) I. Standard model (Nuclear physics)-Mathematics. 2. Quantum field theory. 3. Particles (Nuclear physics)-Chirality. 4. Broken symmetry (Physics) 5. Quantum gravity. I. Dobado, Antonio, 1959-. II. Series. QC794.6.S75E34 1997 539.7'2-dc21 97-1324

ISSN 0172-5998 ISBN 3-540-62570-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and pennission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997

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Preface

This book is devoted to some recently developed techniques in quantum field theory (QFT), as well as to their main applications to different areas of particle physics. All together they are known as the effective or phenomenological Lagrangian formalism. Motivated by the enormous amount of work carried out in this field during the last years, our purpose when writing this book has been to give a modern and pedagogical exposition of the most relevant aspects of the topic. We hope that our efforts will be useful, both for graduated students in the search for a solid theoretical background in modern phenomenology and for more experimented particle physicists willing to learn about this field or to start working on it. Even though we have tried to keep the book as self-contained as possible, it has been written assuming that the reader is familiar, at least, with the most basic concepts and techniques of QFT, gauge theories, the standard model (8M) and differential geometry, at the level of graduate studies. It is therefore possible that senior high-energy physicists may find the book too detailed and so they could probably omit several sections. The book is divided into two main parts and the appendices. In the first part we introduce the fundamentals of the effective Lagrangian formalism and other basic topics such as Ward identities, non-linear sigma models (NL8M), spontaneous symmetry breaking (88B), anomalies, the 8M symmetries, etc. In the second part, we describe their application to different sectors of the 8M; namely, low-energy hadron dynamics, the symmetry breaking sector (8B8) and low-energy gravitational interactions. Finally, in the appendices, we introduce some notation and we cover several topics in geometry and QFT needed in the exposition, but not included in the main text for different reasons. Once more, the advanced reader can avoid reading some of these appendices. The first part consists of five chapters. In Chap. 1 we introduce the nation of the effective Lagrangian and discuss its main properties, which are illustrated with several examples. We also discuss the issue of decoupling in QFT, which has become an important paradigm in modern physics. In simple words, decoupling, when it occurs, allows us to describe a system at some given scale, no matter what the physics at much smaller scales may be. For example, we could study many properties of the atomic spectra without hav-

VI

Preface

ing any knowledge about nuclear structure. In general, decoupling makes it possible to have an independent description of Nature at different scales. In QFT, the concept of decoupling is rigorously formulated in the AppelquistCarazzone theorem. When it can be applied, it is possible to find an effective renormalizable Lagrangian describing the long-distance physics of a system, which otherwise should be described by a more fundamental theory, also valid at shorter distances. However, decoupling is not always present and in fact in most cases it is just a more or less accurate approximation. For example, some tiny hyperfine effects in atomic spectra indeed give information about nuclear structure. In QFT non-decoupling effects are described by non-renormalizable effective Lagrangians, which are the main topic of this book. The parameters appearing in these Lagrangians carry information about the underlying short-distance physics. It is for this reason that the effective formalism has become an important phenomenological tool. In Chap. 2 we review the issue of global symmetries in QFT and how they are realized in terms of Ward identities. Indeed, we study in detail the SSB and the Goldstone theorem, which playa central role in the dynamics of low-energy hadron physics as well as in the generation of masses in the SM. As far as the low-energy behavior of systems displaying SSB is determined just by the symmetry structure, they are naturally well suited for an effective Lagrangian description, namely the NLSM. In Chap. 3 we show in detail the geometrical interpretation and how to build these models. We also discuss the SSB of gauged symmetries in the NLSM (the Higgs mechanism and the generation of massive gauge bosons) as well as the existence and interpretation of topological non-trivial solutions (Skyrmions). The quantization of a theory can spoil the symmetries that would be present at the classical level. The appearance of such anomalies in QFT and, in particular, in effective Lagrangians is discussed in detail in Chap. 4. There we consider axial, gauge, non-perturbative, and NLSM anomalies. The effects of the axial anomalies on the low-energy effective Lagrangians are included in the Wess-ZuminoWitten term, to which we have dedicated a whole section. The trace anomaly, related to renormalization, is briefly discussed at the end of the chapter. Finally, in Chap. 5 we review the SM from the point of view of its symmetries and their realization. We have concentrated on the study of the weak and strong CP violation as well as on the conditions on the hypercharges coming from the cancellation of the gauge anomalies. The latter are needed to preserve the SU(3)c x SU(2)L x U(l)y gauge symmetry at the quantum level. We have briefly reviewed the anomalies in the baryonic and leptonic currents and the running of the couplings with the scale. Finally we summarize all the symmetries, both exact and approximate, that are present in the SM, since many of them will playa central role in the next chapters. The second part of the book starts in Chap. 6. The applicability of effective Lagrangians to low-energy quantum chromodynamics is based on the spontaneous breaking of the chiral symmetry. In Chap. 6 we have followed

Preface

VII

the pioneering ideas of Weinberg and the very complete works by Gasser and Leutwyler, which founded what it is nowadays known as the modern chiral perturbation theory (ChPT). We have discussed the theoretical and phenomenological determinations of the chiral parameters. In addition, we have considered in special detail the problem of unitarity in ChPT and some methods designed to extend the energy applicability range of the standard ChPT, like dispersion relations and the large-N limit, N being the number of Nambu-Goldstone bosons. Another field of application of the effective formalism to the SM is the description of the SU(2)L x U(l)y SSB. At present, it remains as the less known part of the SM since the available experimental information about it is very scarce. The effective Lagrangian description of the SBS, which is not only limited to the minimal SM, turns out to be very appropriate to parametrize our ignorance on this sector. In Chap. 7 we introduce the electroweak chiral Lagrangian, obtaining some of its one-loop predictions, and we review possible ways of determining the effective parameters phenomenologically. We have paid special attention to the equivalence theorem, which relates the longitudinal components of the weak bosons with their corresponding wouldbe Nambu-Goldstone bosons. Therefore it links the physical observables with the SBS and it is extremely useful to simplify the calculations. Finally, we have also studied the unitarity problems mentioned in Chap. 6. In Chap. 8 we consider first the formulation of the SM in presence of weak gravitational fields, i.e. in a curved space-time background. Particular attention has been paid to the baryonic and leptonic anomalies as well as to the cancellation of gauge and gravitational anomalies. Second, we study the effect of the SM matter on the gravitational action. Finally, we analyze the possibility of a phenomenological description of the QFT corresponding to the gravitational interactions in the low-energy limit. This last point is a little more speculative than the rest of the material in the book and hence we have included only the few well-established facts. The book also contains four appendices dealing with basic notions of geometry and QFT. We hope these may help the reader to understand some parts of the main text. In particular, in Appendix A we introduce the notation and give some useful formulae. In Appendix B we review both the Riemannian and the gauge field geometry needed to follow some of the discussions. In Appendix C we briefly review some topics in QFT such as renormalization, perturbative quantization of gauge theories and BRS invariance, as well as some useful methods such as the background field and the heat kernel. Finally, in Appendix D the reader can find an elementary account of the constraints that unitarity imposes on the elastic scattering amplitudes and some formulae for 7m and 7r K scattering, which are too long to be included in Chap. 6. At the end of every chapter and appendix we have included a bibliography that can be useful for the interested reader. Nevertheless, we do not intend

VIII

Preface

that these lists of references are complete, or even that they make justice to the original authors. As may become clear for many readers, none of the authors is a native English speaker. In spite of this, we hope that the exposition is still clear enough for the book to be useful to many students and senior physicists. Another important remark is that we have deliberately omitted some topics more or less related with the issues considered in the book, such as supersymmetry, heavy quark approximations, etc. The main results are proved in detail only when they have some pedagogical interest but, as it is customary in the physics literature, the proofs should be considered more as serious indications of plausibility than as rigorous mathematical proofs. The same applies to the many calculations that have not been shown when they are too long or do not provide further physical insight. In addition, Sects. 4.7, 7.6.2, and 7.6.3, where we give the topological interpretation of the gauge anomalies and the proof of the equivalence theorem, can be omitted if the reader is not interested in the technicalities, without loosing the main discussion. The topics presented in the book may seem a bit heterogeneous to some readers since they range, for instance, from experimental data on pion scattering to the index theorem. However, we have included them since, in our opinion, a solid background on modern high-energy phenomenology does require some understanding of all these subjects. Finally, we would not like to end this introduction without mentioning our early collaborators in this field since our interaction with them has been very stimulating and decisive to clarify many of the issues treated in the book. The list of these people includes Ramon F. Alvarez-Estrada, Domenec Espriu, Marfa J. Herrero, John Morales, Ester Ruiz-Morales, Marfa T. Urdiales and Tran N. Truong. We are specially indebted to Marfa Jose for reading the manuscript and making several suggestions for Chap. 7. We also thank Manuel Mafias, since, even though his patience has been toughly probed by our many discussions, his help and advice have been very useful. Some discussions with L. Alvarez-Gaume, J. F. Donoghue and J. Gasser are also acknowledged. Of course we are the only ones responsible for any misunderstanding or mistake that could be found in the book. Madrid March 1997

A. Dobado A. Gomez-Nicola A.L. Maroto J.R. Pelaez

This work has received partial financial support from the CICYT under contracts TXT-2253 and AEN93-0776.

Contents

1.

The Notion of Effective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction........................................... 1.2 Integration of the Heavy Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Effective Action for the Light Modes. . . . . . . . . . . . 1.2.2 Low Energy Expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Decoupling Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Euler-Heisenberg Lagrangian . . . . . . . . . . . . . .. 1.5 Theories with Spontaneous Symmetry Breaking . . . . . . . . . . .. 1.6 Decoupling of Chiral Fermions . . . . . . . . . . . . . . . . . .. 1.7 References.............................................

2.

Global Symmetries in Quantum Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1 Classical Symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2 Green Functions and the Reduction Formula. . . . . . . . . . . . . .. 2.3 Quantum Symmetries and Ward Identities. . . . . . . . . . . . . . . .. 2.4 Spontaneous Symmetry Breaking and the Goldstone Theorem 2.5 Explicit Symmetry Breaking and the Dashen Conditions 2.6 References.............................................

3.

4.

The Non-linear u Model. . . . . .. . . .. .. . . . . . . . . . . . . . . . .. .. .. 3.1 Introduction........................................... 3.2 The Geometry and the Dynamics of the Non-linear (J Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3 The Quantum Non-linear (J Model. . . . . . . . . . . . . . . . . . . . . . .. 3.4 Reparametrization Invariance of the S-Matrix Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.5 Local Symmetries and the Higgs Mechanism. . . . . . . . . . . . . .. 3.6 Topologically Non-trivial Configurations. . . . . . . . . . . . . . . . . .. 3.7 References............................................. Anomalies................................................ 4.1 Introduction...........................................

1 1 4 4 6 9 11 14 17 21 23 23 28 31 33 37 39 41 41 42 46 48 49 54 57

59 59

X

Contents

4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

4.11 4.12

5.

6.

The Axial Anomaly, Triangle Diagrams and the nO Decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. The Axial Anomaly and the Index Theorem Gauge Anomalies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.1 The Wess-Zumino Consistency Conditions. . . . . . . . . .. Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Ambiguities and Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . .. Topological Interpretation of Non-Abelian Anomalies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Non-perturbative Anomalies. . . . . . . . . . . . . . . . . . . . . . . . . . . .. Non-linear IJ Model Anomalies. . . . . . . . . . . . . . . . . . . . . . . . . .. The Wess-Zumino-Witten Term. . . . . . . ... . . .... . . .. . . . .. 4.10.1 Anomalous Processes in QCD. . . . . . . . . . . . . . . . . . . . .. 4.10.2 The Non-local Anomalous Effective Action. . . . . . . . .. 4.10.3 The WZW Term with Gauge Fields. . . . .. . . . . . . . . . .. 4.10.4 Anomalous Processes and the WZW Term. . .. . . . . . .. 4.10.5 The SU(2) WZW Term. . . . . . . . . . . . . . . . . . . . . . . . . .. The Trace Anomaly References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

60 65 68 72 73 76 78 83 85 86 86 87 89 91 92 93 95

The Symmetries of the Standard Model . . . . . . . . . . . . . . . . .. 5.1 The Elements of the Standard Model 5.1.1 Matter.......................................... 5.1.2 Gauge Fields 5.1.3 The Symmetry Breaking Sector 5.2 The Cabibbo-Kobayashi-Maskawa Matrix and Weak CP Violation 5.3 The Cancellation of Gauge Anomalies in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.4 Baryon and Lepton Number Anomalies in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Evolution of the Coupling Constants 5.6 The Strong CP Problem 5.6.1 The B-Vacuum 5.6.2 The Role of Instantons 5.6.3 The Strong CP Problem 5.7 The Symmetries of the Standard Model 5.8 References

97 97 97 98 101

The 6.1 6.2 6.3 6.4

125 125 128 130 135 135

Effective Lagrangian for QeD The QCD Lagrangian QCD at Low Energies The Chiral Lagrangian at Leading Order The Chiral Lagrangian to Next to Leading Order 6.4.1 The L(4) Lagrangian

103 105 109 111 113 115 117 119 121 123

Contents 6.4.2 One-Loop Renormalization 6.4.3 The Effective Action to One Loop 6.5 The Low-Energy Constants 6.5.1 Phenomenological Estimates 6.5.2 Theoretical Estimates 6.5.3 The N f = 2 Case 6.6 The Problem of Unitarity in ChPT 6.6.1 Unitarity and Dispersion Relations 6.6.2 The Large-N Limit 6.7 References

XI 136 139 142 142 145 149 151 153 161 171

7.

The Standard Model Symmetry Breaking Sector 175 7.1 The Mass Problem 175 7.2 The Effective Lagrangian for the SM Symmetry Breaking Sector 179 7.3 The O(p4) Lagrangian and One-Loop Renormalization 182 7.3.1 The O(p4) Lagrangian 182 7.3.2 The Covariant Formalism 185 7.3.3 One-Loop Renormalization 186 7.4 The Heavy Higgs and QCD-Like Models 189 7.4.1 The Heavy Higgs Model 189 7.4.2 QCD-Like Models 192 7.5 Phenomenological Determination of the Chiral Parameters 193 7.5.1 Precision Tests of the Standard Model (Oblique Corrections) 194 7.5.2 The Trilinear Gauge Boson Vertex 196 7.5.3 Elastic Gauge Boson Scattering 197 7.6 The Equivalence Theorem 201 7.6.1 Introduction 201 7.6.2 The Slavnov-Taylor Identities 202 7.6.3 The Reduction Formula 208 7.6.4 The Generalized Equivalence Theorem 210 7.6.5 The Equivalence Theorem 211 7.7 The Applicability of the Equivalence Theorem 216 7.8 Gauge Boson Scattering at High Energies 218 7.8.1 Dispersion Relations for the SM Symmetry Breaking Sector 220 7.8.2 The Large-N Limit: The Higgs and the General Case. 221 7.9 References ; 226

8.

Gravity and the Standard Model 8.1 Introduction 8.2 The Standard Model in Curved Space-Time

229 229 231

XII

Contents 236 Anomalies in the Standard Model 8.3.1 The Leptonic and Baryonic Anomalies 237 8.3.2 Gauge Anomalies 238 8.3.3 Gravitational Anomalies 240 242 8.3.4 Charge Quantization in the SM 8.4 The Effect of Matter Fields on Gravitation 243 246 8.5 The Effective Action for Gravity 8.5.1 The Background Field Method in Quantum Gravity .. 246 8.5.2 General Effective Formalism 247 251 8.5.3 Quantum Corrections to the Newton Potential 8.5.4 Perspectives and Other Approaches 254 8.6 References 256 A. Useful Formulae and Notation 259 A.1 Notation in Minkowski Space-Time 259 A.2 Notation in Euclidean Space-Time 261 A.3 Useful Formulae 262 B. Notes on Difterential Geometry 263 B.1 Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 B.2 Homogeneous Spaces 269 B.3 The Geometry of Gauge Fields 270 B.4 References 279 C. Aspects of Quantum Field Theory 279 C.1 Renormalization Group Equations 279 C.2 Quantization of Gauge Theories and BRS Invariance .. 287 C.3 The Background Field Method 293 C.4 The Heat-Kernel Method 298 C.5 References 301 D. Unitarity and Partial Waves 302 D.1 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 D.2 Dispersion Relations 303 NGB Amplitudes to O(p4) 306 D.3 D.4 References 309

8.3

Subject Index

311

1. The Notion of Effective Lagrangian

In this chapter we try to give a motivation for the study of effective Lagrangians, as well as a brief introduction to their main features. We also discuss the important notion of decoupling and analyze the cases in which this property holds.

1.1 Introduction The aim of an effective Lagrangian is to describe the low-energy dynamics of the light modes of some physical system. Although heavier modes will not appear explicitly, their contribution is somehow included through some parameters in the effective theory. At present, this general approach is followed in many contexts of the standard model (8M) and even in more speculative theories like grand unification, supergravity, Kaluza-Klein or superstrings [1]' although in this book we will only consider applications within the 8M.

a)

b) e

e Fig. lola,b. Feynman diagram describing the J1. decay: (a) in the 8M at tree level and (b) in the FFG effective theory

In order to illustrate the use of effective Lagrangians we will start with a simple example which is also interesting for historical reasons. Let us consider the muon decay J.L- -+ e-vevw In the 8M (see Chap. 5) this process is described at lowest order by the Feynman diagram in Fig. 1.1a. In the unitary gauge the corresponding amplitude is given by A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

2

1. The Notion of Effective Lagrangian

M(J-L

_

__ g2 _ gpa _ kPk a IMfv _ -; e vevJ.L) = 2(e"/pPLVe) M2 k2 (VJ.L"/aPLJ-L) ,

w-

(1.1)

where the symbols of the different particles refer to their appropriate spinor fields, 9 is the SU(2)L coupling constant and Mw is the W boson mass. To calculate the decay rate we choose the muon four-momentum as (m, 0, 0, 0) where m is the muon mass. As far as m << Mw it is possible to expand the W propagator in powers of k 2IMfv. At lowest order we find

M(J-L- -; e-vevJ.L) =

g2 M 2 (e,,/pPLve)(VJ.L"/P PLJ-L) . 2 w

(1.2)

The last amplitude could also be obtained by starting, not from the 8M, but from the contact interaction in Fig. LIb which is obtained from the following effective Lagrangian (1.3)

Taking into account that Mw = gv/2, where v is the vacuum expectation value of the Higgs field in the minimal 8M, one finds G F = II (v'2v 2 ). The above Lagrangian, which we owe to Fermi, Feynman and Gell-Mann (FFG) [2]' properly describes the muon decay and from the experimental value of its mean life it is possible to obtain v ::::: 250 GeV. In this example we can observe some of the main features of the effective Lagrangians. First of all, if we have a more fundamental theory (like the 8M), the effective Lagrangian can also be obtained as some low-energy limit of this theory where the heavy modes (the W boson) have been removed. In addition, not only the form of the Lagrangian is determined by the underlying theory but also its parameters (G F in this case) can also be obtained from those of the fundamental theory (g and M w). Nevertheless, even if we did not know a fundamental underlying theory, the effective Lagrangian could be obtained phenomenologically, its form being determined by symmetry considerations based in our limited information about the phenomena we want to describe. The value of the parameters can be obtained by fitting the experimental results (for instance by measuring the muon mean life in our previous example). In fact this was the way followed historically to find the FFG model, which only after the formulation of the 8M was understood as its low-energy limit. Another feature of this model, which is typical of effective Lagrangians, is that it is not renormalizable. Indeed, in the calculation of diagrams with loops one finds divergences that cannot be absorbed by the parameters of the Lagrangian nor by renormalization of the fields (wave function renormalization). If we still want to absorb such divergences, we are forced to introduce new terms of higher dimension, with their corresponding parameters. Let us illustrate the last remark using a simplified model [3] with just one fermion and the following Lagrangian

1.1 Introduction

3

Fig. 1.2. One-loop diagrams appearing in the four-fermion interaction theory

(1.4) which yields again a four fermion interaction, that could be understood as an effective vertex of a more fundamental theory where we had integrated an intermediate boson. Within this model, the diagrams in Fig. 1.2 will give rise to the following divergences

A2 In 2 (41l")2 (2v 2G F) ("l/J,,(J1.P£'lI;)( "ljry J1. PL"l/J)

(1.5)

and 2

1 In 4 A []2 (41l")2(2v2G F ) log Jl2 ("l/J'J1. PL"l/J)("l/J"(J1.PL"l/J) ,

(1.6)

where A is some ultraviolet cutoff. In the context of the effective theory the first divergence can be absorbed in the definition of the renormalized G F, but this is not the case of the second one which needs a new higher dimension operator. Therefore, the effective Lagrangian should contain a new term with the same field structure as (1.6). In that way, the divergence could be absorbed in the definition of the coefficient of the new term. Such parameter or coupling should be found either experimentally or from a more fundamental theory if it were available.

..

--~ -D--~_

..-

Fig. 1.3. One-loop diagrams with internal vector boson lines appearing in the underlying theory

4

1. The Notion of Effective Lagrangian

In our example, we could think of an underlying theory similar to the SM, that is, with massive vector bosons mediating the fermionic interactions. Such theory is renormalizable provided the vector bosons are gauge bosons which obtain their masses through the Higgs mechanism (but not if their mass term is introduced directly in the Lagrangian). Thus, one should calculate the diagrams in Fig. 1.3 which can be properly renormalized, Le. we can eliminate any cutoff dependence. For the full FFG model case the underlying theory is the SM itself. The discussion is quite the same, although some differences appear owing to the existence of different kinds of fermions and the absence of flavor changing neutral currents. To end this informal account of the main properties of effective Lagrangians, it is worth mentioning that usually their predictions are in conflict with unitarity at high energies. Unitarity would be recovered if the heavy modes were included explicitly again in the theory, but that is not always possible or desirable, because either we do not know the fundamental theory or the calculations get too involved. In such cases there are different procedures to restore a good unitarity behavior. We will turn to this interesting point in Chap. 6.

1.2 Integration of the Heavy Modes 1.2.1 The Effective Action for the Light Modes From a formal point of view and assuming that the underlying theory is known, the effective action r eff , which encodes all the information in a quantum field theory (QFT), can be written as follows using the path integral formulation eirerr[tI>z] =

J

[dch]eiS[tI>l ,tI>hl ,

(1. 7)

where .pI and .ph refer to the light and heavy fields respectively and S[.pl, .ph] is the classical action of the underlying theory. Then the effective Lagrangian is defined as (1.8) where dx == d4 x. Note that such a Lagrangian is not necessarily local and, hence, in general it would be a functional of the fields. Indeed, the integration of the heavy modes often produces finite non local terms. This effective action should not be confused with the standard definition of the effective action as the generating functional of the one particle irreducible Green functions. In principle it is possible to compute the effective action reff[.pd, at least formally, using the saddle point technique. One can expand the heavy field .ph around some field configuration ?>h as follows

1.2 Integration of the Heavy Modes

5

In particular, we can choose ~h so that 88[PI, Ph] 8Ph(x)

I

= 0.

(1.10)

'h=;Ph

Note that in this case ~h becomes a functional of the PI, i.e. we could write ~h = ~h(X, Pz] (a function on x and a functional on PI). With this choice for ~h we have eireff[tPd

= eiS[tPt,;Ph]

J

[dPh]

ei

Jdx dyB (tPh(X)-;Ph(X) )A(x,y)(tPh(y)-;Ph (y) )+... } (1.11)

where, by definition, 2

8 8 A(x, y) = 8p (x)8p (y) h

h

I

(1.12)

-

tPh=tPh

Now, by a formal Gaussian integration (see Appendix A) and assuming for simplicity that the heavy field is a boson, we find 00

reff[pd

= L:r(k)[pd = k=O

.

8[PI,~h[pd] + ~TrlogA[Pz] + ... ,

(1.13)

where the above saddle point expansion is in fact an expansion in the number of loops or, in other words, in powers of the Planck constant, so that the first term corresponds to a tree level integration of the heavy field Ph. Note that in the next term of the expansion (the one-loop approximation) we have explicitly written the functional dependence of the A operator on the light field PI. Although the above expansion is quite general and, in principle, it can always be performed, in practice the calculations get very involved in the physically interesting cases. However, as it was previously mentioned, it is possible to get some information about the general form of the effective lowenergy action making use of the symmetry constraints imposed by the underlying theory. This is the way in which the effective Lagrangians have become an extremely useful phenomenological tool.

6

1. The Notion of Effective Lagrangian

1.2.2 Low Energy Expansions As we mentioned above, the calculation of the complete effective action for the light modes after integrating out the heavy ones is not an easy task. However, there is a very general method which in many cases makes it possible to find Leff as an expansion in the number of derivatives of the light fields over some dimensional parameter M.Ifwe consider Green functions in momentum space, this corresponds to an expansion in powers of p/M, where p is any external momentum. Therefore, if one is only interested in the low-energy dynamics of the light modes (which is consistent with integrating out the heavy fields) the derivative expansion offers a useful and systematical way to obtain the low-energy effective action. To illustrate these ideas we will consider a simple model described by the Lagrangian

L(¢, eJ1) =

~aJ.L¢aJ.L¢ - ~m2¢2 -

V(¢)

+~a eJ1aJ.LeJ1 - ~M2eJ12 + ~¢2eJ12 2 J.L

2

(1.14)

2'

whose corresponding action is S[¢, eJ1] = f dxL(¢, eJ1) = S[¢] + S > m in order to obtain the above mentioned derivative expansion of the effective action reff[¢], that we define as eireft!cI>]

=

J

[deJ1]e iS [cI>,
= eiS[cI>]

J

[deJ1]eiS4> [¢,
(1.15)

.

Owing to the simple form of S
eiS[cI>] (det 0)-1/2 , (1.16) 2 where Oxy = (- Ox _M + a¢;)8 xy . Here and in the following we will adopt the notation 8xy = 8(x - y), ¢x = ¢(x), etc. and we will work in the dimensional regularization scheme with dx == dDx where D = 4 - € (see Appendix A). Then we have eireft!cI>] =

r eff [¢] = S[¢]

i

+ 2Trlog 0

(1.17)

,

and now we can write (1.18) From now on, and throughout this chapter, we will forget about the first term in the above expansion, because it does not depend on ¢ and then it only contributes trivially to the effective action. When expanding the logarithm we get .

r eff [¢] = S[¢]- ~

00

1

00

L "kTr[a(O + M 2)-1¢2]k == S[¢] + L r(k)[¢] . (1.19) k=l

k=l

1.2 Integration of the Heavy Modes

7

The operator (0 + M 2)-1 is, by definition, the propagator for the free iP field; that is, if we define G xy == (xl(O + M 2)-lJy) then

J

dz(Ox + M2)DxzGzy = Dxy ,

(1.20)

which can be solved with the usual boundary conditions to give

Gxy -- - Jd-qe -iq(x-y) q2 _ M2 1· + if:

(1.21)

with dij = d D qJ1! /(27r)D where J.L is some arbitrary energy scale.

_0_

Fig. 1.4. One-loop diagrams with internal heavy scalar lines contributing to the two and four points light scalar Green functions

Then, the r(l) [] contribution corresponds to the first diagram in Fig. 1.4, where we have represented the G xx propagator by a loop. It is given by

r(l) []

=

-~a

J

dxGxx ; =

~a

J J ! dx;

dij q2

M2 .

(1.22)

Note that the functional trace that appears in (1.19) has been rewritten as an integration in the space-time variable x. The above momentum integral is divergent, but the divergence can be extracted using (A.32) in Appendix A, so that

J ! dij q2

M2 = i

(~;2 (Nt + 1 -

log

~22)

,

Nt == ~ + log 47r - " and I is the Euler constant. Therefore r(l)[] = - 27:;2 (Nt + 1 -log ~22) Jdx; .

(1.23)

where

(1.24)

Notice that this is a quadratic term in the fields, so that the divergence can be easily absorbed if we renormalize the mass as

2 2 _ 22M2 ( M ) m ~m =m +a(47r)2 N t +l-log J.L2 as a consequence of integrating the virtual iP particles.

(1.25)

8

1. The Notion of Effective Lagrangian

The next term of the expansion in (1.19) is

r(2)[¢] =

_~0'2 j dXdyGxy¢~GyX¢;

(1.26)

,

which corresponds to the second diagram in Fig. 1.4. Thus i

r(2) [¢] = -40'2 introducing k

r(2) [¢] =

2

2)

e-iq(x-y) eik(x-y) dx dydijdk q2 _ M2 ¢~ k2 _ M2 ¢; ,

(1.27)

= p +q

_~0'2 j dx dy¢;¢~ j dpe-ip(y-x) I(p2; M 2)

with

I (p ; M

j

==

j dq_[q2 _ M2][(p1+ q)2 _ M2) .

(1.28)

(1.29)

To calculate this integral we recall (A.33) in Appendix A, so that with the change k == q + p(l - x) and using again (A.32) we get 1

2 I(p2; M ) = (4:)21 dx

[N< -log R)X)]

(1.30)

with R 2(x) = M 2 - x(l - X)p2. Thus

r(2)[",] 'f'

= ~ jdXdYdP-¢2¢2 e -iP(y-X) 4(4'71-)2 x Y x

(N< -log :22 - 1

1

dtlog

[1 - t(l- t) :;2]) .

(1.31)

The last result shows that, as mentioned above, the effective action does not have to be local. However, for p2 << M 2 we can expand the logarithm in this equation so that

(1.32)

1.3 The Decoupling Theorem

9

(1.33) Hence, each term in the derivative expansion is local. Notice that if we had for instance V(rj» = >..rj>4, then the first contribution in the right hand side, which is divergent, could be eliminated by renormalization of >..,

:\ = >.. -

(N< -log ~22) ,

4(~;)2

(1.34)

so that for the effective action we would find

reff[rj>]

=

-~ -

J

dxrj>(O

+ ih?)rj> -

:\

J

dXrj>4

24(4~;2M2 Jdxrj>2 0 rj>2 + 0 (~2) 2

(1.35)

1.3 The Decoupling Theorem One interesting property of the model discussed in the previous section is that the divergent terms of the effective action can be absorbed by renormalization of the rj> field mass and coupling constant. The new terms are not divergent and can be organized as a derivative expansion so that at low energies only some of them are relevant. In addition, we observe that in this case all the new operators vanish in the M --+ 00 limit. When this happens it is usually said that there is decoupling in the effective theory. Intuitively, decoupling means that low-energy physics is blind to highenergy physics. The rigorous formulation of decoupling in QFT is known as the Appelquist-Carazzone theorem [4]. Suppose that we have a theory with light particles and a heavy particle of mass M. In brief, the theorem states that, under some given conditions, the effects of the heavy particle only appear in the light particle physics through corrections proportional to a negative power of M or through renormalization. To be more precise we will consider a theory with a light field rj> and a heavy field 1> with masses m and M, respectively [5]. Let us denote by r(n)(g,m,M,JLikl, ... ,kn) the proper vertex of n light particles with momenta k i , which is derived from the classical action 8[rj>,1>], where 9 stands for the different coupling constants of the model and JL is the renormalization scale. Let us also introduce the action 5'[rj>], which is obtained from 8[rj>, 1>] by omitting the terms that contain any heavy field and replacing the

10

1. The Notion of Effective Lagrangian

original light particle mass and couplings by new parameters in and g. The proper vertex of n light particles derived from 5[] will be represented by f(n) (g, in, jl; k l , ... ,kn ). For the sake of simplicity we will assume in the following that the Green functions, couplings and masses are defined in some mass independent renormalization scheme (see Appendix C.l). Then the precise statement of the decoupling theorem is that, to all orders in perturbation theory, r(n) (g, m, M, jl; k l , ... ,kn ) is given by f(n) (g, in, jl; k l , ... , k n ) up to order 11M corrections, where the new coupling constants, mass and scale of the fields (wave-function renormalization) are certain functions g(g, M, jl), m(g, m, M, jl) and Z(g, M, jl), so that we can write r(n) (g, m, M, jl; k l , ... ,kn ) = zn/2(g, M, jl)f(n) (g(g, M, jl), m(g, m, M, jl), jl; k 1, ... , k n )

(1.36)

+ (] (~) ,

where the precise form of the above functions will depend on the details of the renormalization prescriptions. As we commented before, the validity of the decoupling theorem requires some conditions on the original action S[ , ]. In particular, to be sure that the theorem holds, the theory has to be renormalizable, it should not have a spontaneous symmetry breaking nor chiral fermions. The simplest model where the theorem can be used is that considered in the previous section. The examples of greatest physical interest are vector gauge theories (without chiral couplings) when the decoupled fermions constitute a complete multiplet of the gauge group which is not spontaneously broken. Among them, the most relevant are quantum electrodynamics (QED) and quantum chromodynamics (QCD). In the first case it is possible to decouple the fermions (electrons) and obtain an effective theory for low-energy photons (see next section). In QCD we have quarks with different flavors u, d, 5, c, band t, each of them with a different mass m u < md < m s «: me «: mb « mt. For a given flavor, quarks appear with three colors, transforming under the fundamental representation of the color gauge group SU(3). Then one can apply the Appelquist-Carazzone theorem in order to decouple the heaviest quark flavors. The effective low-energy theory will be another QCD-like theory but only with the lightest flavors. This is particularly useful when working at energies smaller than the heavy quark masses. In such a case, the main physical consequence of the decoupling theorem is that it is not possible to predict the masses of the heaviest quarks by means of low energy experiments, as far as only strong interactions are involved. Indeed, the decoupling theorem ensures that the only heavy quark effect which is not suppressed by powers of their masses is just the renormalization of the effective low-energy couplings, masses and wave functions, thus leaving the form of the strong interaction invariant. Intuitively, one would expect that this was the case, since otherwise it would be possible to determine with low-energy experiments the existence of quarks of arbitrarily high

1.4 The Euler-Heisenberg Lagrangian

11

mass. AB a matter of fact, this was the original motivation for the decoupling theorem. A typical example of those decoupling effects is found in the magnitude R which is defined as

R(s) = a(e+e- -+ hadrons) . a(e+e- -+ J.L+J.L-)

(1.37)

At the lowest order in perturbation theory it is given by R( s) ~ N c

L QJ '

(1.38)

flavor

where N c is the number of colors, Qf is the electric charge of the quarks with flavor f and the sum is done over all the flavors whose corresponding quarks can be produced by pairs at the center of mass energy. The above equation is quite simple but gives a crude description of the experimental data. In fact, it predicts a sudden jump of R in the thresholds of new quark production, which is indeed observed, although it fails in giving the details of these thresholds or any resonance. Notice however that the above equation is completely blind to heavier quarks below their thresholds, which means that discovering a new quark actually requires to reach energies above its threshold. That is a simple example of how the decoupling theorem works in QCD (at least at the lowest order in perturbation theory). However, as we have already mentioned, the decoupling theorem does not always hold and indeed there are low-energy experiments that can provide information on higher energy physics. In particular that is the case with the standard electroweak theory, as we will see in Chap. 7. In the following we will study in some detail different examples where it is possible to apply the theorem and where it is not.

1.4 The Euler-Heisenberg Lagrangian Let us see then another example of the applicability of the decoupling theorem, namely, a derivative expansion of the photon effective action obtained by integrating out the electron field in QED [6]. Thus, the validity of this approach will be limited to energies much lower than the electron mass. In this case, we start with the QED action SQED[A IL , 'lj;, 1/J] =

J

= -

dx.cQED(A IL , 'lj;, 1/J)

~

J

dxFILyFILY

+

J

dX1/J( i

f/J -

M)'lj;

(1.39)

with M the electron mass and the usual definitions F lLy = 0IL Ay - oyA IL DIL'lj;=(olL - ieAIL)'lj;·

(1.40)

12

1. The Notion of Effective Lagrangian

Fig. 1.5. One-loop diagrams with internal fermion lines contributing to the two and four points photon Green functions

The photon effective action can be defined as follows

eireff[A"l = j[d'l/J][d1jj]e i J dX.cQEO(A",,p,1/J) = e- i {

JdxF""F"" detO,

(1.41)

where

Oxy = (i l/Jx - M)8 xy ,

(1.42)

and we have taken into account that the integration variables are fermionic, in contrast with the example studied in Sect. 1.2. Therefore

r[AJL] = -~ j dxFJLvFJLV

=-~ j

dxFJLvFJLV

+ i f (_;)k Tr[(i " -

M)-l .¢I]k

k=l

+

f

r(k) [A] .

(1.43)

k=l

It can be shown that those terms with an odd number of photon fields vanish (Furry's theorem [7]), as it could be expected from invariance under charge conjugation. Let us now define G xy

= (02

JIl_ '"

M)-l = jd- -iq(x-y) if + M xy qe q 2 - M2 + 2€ 0

,

(1.44)

which is nothing but the Feynman propagator. Therefore, we can write the first contribution to the effective action (first diagram in Fig. 1.5) as follows

r(2) [A] = ~e2tr j dydxG xy .¢IyG yx .¢Ix i

=

2"e

2

tr

j

_ - e-iq(x-y) eik(x-y) dydxdqdk q2 _ M2 k 2 _ M2 (if + M) .¢Iy(j( + M).¢Ix

With the change of variable k

r(2)[A] =

~e2tr j

(1.45)

= p + q,

°

we can rewrite

dydxdp eip(x-y) A~A~IJLv(p; M) ,

(1.46)

where (1.47)

1.4 The Euler-Heisenberg Lagrangian

13

So that using again the Feynman trick in (A.33), with the change k = q + p(l - x) and finally calculating the Dirac traces in the numerator, we obtain

Jor

1

IfJ.v(p; M) = 4

dx

J-( dk

2kfJ.k v gfJ.V [k2 _ R2j2 - [k2 _ R2]

_ 2x(1 - x) [pJ.LPV -,- p2gfJ.v]) [k 2 _ R 2 j 2 '

(

) 1.48

where we have defined R 2 = +M 2 - x(l - x)p2. Notice that we can replace kfJ.k v by k 2gfJ.V / D in the first term in the right hand side of the above equation. Then, using (A.32) in Appendix A we see that the first two terms in (1.48) cancel each other and finally we are left with IfJ.v(p; M)

r = (47r)2 Jo

1

8i

dx 2x(1 - x)

9

x [(p2 J.LV - PfJ.Pv)

(N. -log ::)]

(1.49)

Therefore, (1.46) is nothing but

J

2

r(2)[A]

-4e = (47r)2

.

dydxdjj etP(x-Y)A~A~I(p; M)(p2gJ.LV - PfJ.Pv),

(1.50)

where I(p;

M) = ~ (N. -log ~22)

1 1

+

dx x(x -1) log

[1 + :;2X(X -1)] .

(1.51)

Once more we have found that r(2) is non-local. We can also perform a derivative expansion, i.e. in powers of p2 / M 2, of the logarithm in the above integral as follows I(p; M) =

Ll

"6 +

£; 00

r

(p2 )k 1 (_l)k+l M2 Jo dx[x(x - l)]k+l k '

(1.52)

where we have defined Ll

M2 == N. - log - 2

(1.53)

.

J1-

Then, we can obtain the first term in the expansion in (1.52), which is r(2)[A] =

~~~: x

JdydxdjjeiP(X-Y)A~A~(p2gJ.LV

[~ +}'ot, + 0 (t,)] v

decouples

-PfJ.Pv)

(154)

14

1. The Notion of Effective Lagrangian

and therefore, the r(2) [A] contribution which does not decouple is the following

J () ~:~: ~ J 3~:2)2 J J 4~~: J 0 2

(2) -4e r ND[A] = (47r)2

= =

dydxdp e

tp

2 L1 x-y A~A~(p gp.v - PP.Pv)f;

v V dx[oP. AvOp.A - ovAP.Op.A ]

L1

dxFp.vFp.v .

(1.55)

Hence the decoupling term is 2 r(2) [A] = -4e dydxdp- eip(x-y) [oP. ADo A V - 0 AP. 0 0 AVj D 30(47r)2M2 v P. v P.

= 15(

M2

dxFp.v

Fp.v .

(1.56)

Finally, we get for the effective action the next expression

refrlA] = -

~1

J

dxFp.vFp.v - 3(::)2 L1

J

dxFp.vFp.v

15(4:~2M2 JdxFp.v 0 Fp.v + 0

(1.57)

(:2)2.

Following the same steps with the second diagram in Fig. 1.5, we can also work out the r(4) [A] contribution (these terms are finite since QED is a renormalizable theory): 4

r(4)[A] _ e - 90(47r)2M4

XJdx [(Fp.vFP.V)2 + ~(Fp.vFP.V)2] + (:2)3, 0

(1.58)

where Fp.v = Ep.vo"P Fup /2. Notice that this term also decouples in the M ~ 00 limit. As it can be seen the terms that do not decouple have the same form as those appearing in the original Lagrangian and therefore they can be absorbed in the wave function renormalization. The new structures vanish in the M ~ 00 limit. Therefore, as it was advanced in the previous section, this is another example of the Appelquist-Carazzone decoupling theorem.

1.5 Theories with Spontaneous Symmetry Breaking As we have mentioned in Sect. 1.3, the decoupling theorem does not hold for all cases of physical interest. An important example is given by spontaneously broken gauge theories, that is, when the vacuum does not have the same

1.5 Theories with Spontaneous Symmetry Breaking

15

symmetry as the Lagrangian. If the symmetry is chiral and gauge then an spontaneous symmetry breaking is essential to give masses to the fermions and gauge bosons. Typically, these masses will be obtained as M

~

(1.59)

g(p} ,

where 9 is a gauge coupling when dealing with gauge bosons, or a Yukawa coupling for fermions (see Chap. 5). If we now want to integrate one of those fields in the M - 4 00 limit, we have two possibilities: we can either keep (p) fixed with 9 - 4 00, or just the opposite. Let us illustrate both cases within the 8M, namely, in the interchange of a WJl boson. In that case, 9 is the electroweak coupling constant and v plays the role of (p), respectively. Remember that the mass of the WJl boson is given by

(1.60) Thus, the two possibilities when taking the M w

-4

00

limit are given by

- 9 fixed, v

- 4 00: A typical example of this case is given by the FFG theory obtained from the 8M Lagrangian, as we commented in the introduction. Remember, for instance, that the muon decay was proportional to

1

g2 2 Mw

(1.61 )

~2'

v

which means that the interaction between fermions mediated by the W indeed decouples when v - 4 00. Therefore, at extremely low energies, the interaction of four fermions via one WI' boson disappears. The FFG Lagrangian is just the first term in the p2 / M 2 effective expansion and its coupling G F is therefore proportional to 1/v 2 . - 9 - 4 00, v fixed: In this case there would not be decoupling since in the g2 - 4 00 limit the interaction between fermions mediated by WJl would remain at low energies and then the decoupling theorem does not hold. Another example of the non-applicability of the decoupling theorem in spontaneously broken theories is the linear a model (L8M) described by the Lagrangian

(1.62) where 11"1

P=

p2

ra 1 1I"N

and

= 1I" a 1l"a + a 2

a

=

1. .. N

(1.63)

16

1. The Notion of Effective Lagrangian

(1.64) This Lagrangian is explicitly invariant under O(N + 1) rotations. At tree level, the minimum of the effective potential is given by (1.65) so that there are many possible vacuum choices, all them related by O(N +1) transformations. However, in order to quantize the theory we have to make a specific choice, for instance

(1.66)

so that this minimum is invariant only under O(N), which means that there is a spontaneous symmetry breaking. If we now perform the change (T = H + v, we can rewrite

.c = ~(8 71")2 + ~(8 H)2 - ~M2H2 2 J1. 2 J1. 2 >.

__ (71"2 4

+ H 2)2 _

>.vH(7I"2

+ H 2)

(1.67)

with M 2 = 2>.v 2. From this expression we see that there are N massless 71" particles, called Nambu-Goldstone bosons (see Chap. 2 for a discussion about the Goldstone theorem), associated to the symmetry breaking O(N + 1) --+ O(N) and a massive particle H. If we now take the M --+ 00 limit with >. --+ 00 and v fixed then cjj2 is nothing but v 2 and we get (T = "';v 2 - 71"2. Therefore the Lagrangian of the L8M can be written as 1 1 .c=2"(8J1.7I"a)2 + 2"(8J1.(T)2 =

~(8 7I"a)2 + ~ (7I"a8J1.7I"a)2 2

J1.

2 v 2 - 71"2

1 8J1.71" a8J1. 71" b -_ 2"gab

(1.68)

with

gab = bab

+ V;~7I":2

= bab

+ 7I"~;b

(1 + :: + ...)

.

(1.69)

This Lagrangian describes the so called non-linear (T model. If we compare it with (1.62) we observe that the effect of integrating (at tree level) the H field is the appearance of infinite new interactions that were not present in the original Lagrangian. The decoupling theorem does not hold. In particular,

1.6 Decoupling of Chiral Fermions

17

this model describes for N = 3 the spontaneous symmetry breaking sector of the minimal SM. We want to remark that it is very important that there is no decoupling in this case, since this fact will allow future experiments to obtain information on the symmetry breaking sector in case it is strongly interacting, (see Chap. 7 for details).

1.6 Decoupling of Chiral Fermions When the heavy particles we want to integrate out are chiral fermions, the applicability of the decoupling theorem can be spoilt in different ways. As usual we understand by chiral fermions those whose right-handed and left-handed components transform differently under certain symmetries. For instance, let us consider the SU(2) quark doublet

Q=

[~]

(1.70)

,

where Q = QL + QR with QL,R = PL,RQ (see Appendix A) and the chiral transformations QL

-+

gLQL

QR

-+

gRQR,

(1. 71)

where gL,R E SU(2)L,R. It is important to notice that the usual mass term (1. 72)

is not invariant under the previous two transformations, except for the particular case gR = gL, i.e. it is only SU(2)L+R invariant. In particular, if we have a gauge interaction SU(2)L for Q, as it happens in the 8M, the mass terms are not compatible with gauge invariance. It is well known that in the 8M this problem is solved introducing fermion masses through the spontaneous symmetry breaking mechanism, i.e coupling the fermions through Yukawa terms to scalar fields which acquire a vacuum expectation value (see Chap. 5). For example

Ly = -QLPAQR

+ h.c.

,

(1.73)

with

A=

[~t ~b]'

(1.74)

where At,b are the Yukawa couplings and p(x) is a 2 x 2 matrix which can be parametrized as P = pU with U E SU(2), so that it can be written as U(x) = expiTa¢a(X)/v (T a being the Pauli matrices). Under a SU(2h,R, P transforms as P -+ gLPgi/. If p acquires a vacuum expectation value v then we have

18

1. The Notion of Effective Lagrangian

(1. 75) where the dots stand for interaction terms between fermions and the if> field. Therefore, t and b have acquired masses mt = AtV and mb = AbV respectively. Note that At =1= Ab produces different masses for t and b without destroying the SU(2)L gauge invariance. Let us now take the limit mt -4 00, with v fixed, and consider the effective theory for band if>. For the sake of simplicity we can neglect the SU(2)L gauge couplings (and also the quark mixing described in Sect. 5.2) and make mb = O. Thus the relevant Lagrangian is

.c

Qi ~Q - QLif>AQR

= ti

~t + bi ~b -

+ h.c

p[h, h]U

[~t ~] [~~ ]+ h.c..

(1.76)

Now we can integrate out the t field at tree level as described in previous sections [8]. In the large mt limit the equations of motion can be written as U

[~ ~] [ ~~ ]

[~ ~] ut [ ~~]

0 =

0,

(1. 77)

and then we have the solutions tR=O

[~~] =u [b~]

(1. 78)

Now we introduce these solutions in (1.76) to find the effective Lagrangian for the b quark and the scalars = bi ~b

1-

3

+ -bL/l-'b L 81-'if> +... .

(1.79) v As far as we are considering the limit mt -4 00 with v fixed we can see that we have new terms in the effective Lagrangian which do not decouple. Those terms correspond to tree diagrams with internal t lines that collapse into the new vertices for very large At. In the above example we have studied the case of a chiral fermion doublet interacting with scalar fields in the limit in which one of the fermions is very heavy. Now we are going to consider what happens with the low-energy effective theory for the scalar fields when all the chiral fermions are heavy and degenerated. This example is relevant since we will find again that the decoupling theorem does not hold. Furthermore for the first time we are going to deal with a new structure in the scalar low-energy effective Lagrangian (the Wess-Zumino-Witten term) which, as we will see in Chap. 4, plays an important phenomenological role. In order to find the effective action let us start from the model defined by the Lagrangian

.ceff

1.6 Decoupling of Chiral Fermions

L(U, 'l/J, i/J) = i/JD(U)'l/J + LO(U)

19

(1.80)

with

D(U)

=i

{iJ - M(PRU

+ PLut) ,

(1.81)

and U being an SU(3) field that can be written as

U(x) = exp

(i ir~X))

,

(1.82)

where ir(x) = 1r a (X)A a , 1r a (x) are scalar fields, Aa are the Cell-Mann matrices and v is some constant with energy dimensions. In addition, M is the fermion mass as it can be seen by expanding U and ut in powers of the ir(x) fields and LO(U) is the part of the Lagrangian depending only on the scalar fields. Among others, it should contain their kinetic terms, although the specific form of LO(U) will not be relevant for the discussion. The model defined above is invariant under the global transformation

SU(3)L x SU(3)R 'l/JL -> gL'l/JL 'l/JR->gR'l/JR U ->glUg R ,

(1.83)

where gL,R E SU(3)L,R, provided LO(U) is also SU(3)L x SU(3)R invariant. As a consequence, a mass term is not allowed for the scalar fields. Thus, according to previous discussions, the effective action for the scalar fields, which will only be useful at energies well below M, is given formally by

ei r [1r] = j[d'l/J][di/J]e i J dx(1[JD(U),p+.Co(U)) = ei J dx.c.o(U) det(D(U)). (1.84) Then, following the same method explained in previous sections, it is possible to write 00 ( )k 00 r[1rj = j dXLo(U) +iL: ~ Tr[(i {iJ- M)-lOjk = L:r(k)[1rj, (1.85) k=l k=O where the 0 operator is defined by

D(U) = i {iJ - M

+ O(U) .

(1.86)

Notice that if we write

U=I+Ll(1r)

(1.87)

then the 0 operator is nothing but

Oxy

= -AV[PRLl(1rx ) + PLLlt(1rx )]Dxy = -iA-lirxD xy + ... ,

where A is defined so that M = AV.

(1.88)

20

1. The Notion of Effective Lagrangian

There will be many terms appearing in the scalar effective action in (1.85), but in the following we will concentrate in the one that contains five fields, namely, we will only consider

r

(5)

[71"]

_

. [ 1

t

-"5 Tr

]5 -"5.Tr [1 ]5 i ~ _ M(t)..'Y 71") + ... , _

i ~_ MO

t

.

5,

(1.89)

which can also be written as

r (5)[ 71"1 = _ )..55 Tr

[i ~+M -0-

M2 ('Y 5,)]5 71" + ...

(1.90)

If we now replace the value of the gamma-matrix trace, we find r(5)

[71"] = i4)" 5 M EJ1.vpu X

8J1. , 8v , 8P 8u , 1 ,] , Tr [ 271" 71" 71" 71" 71" - 0 - M - 0 - M2 - 0 -M2 - 0 - M2 - 0 - M2

+ ... , (1.91 ) which can be simplified by using the antisymmetry properties of the EJ1.vpu symbol. In momentum space we arrive at . 5 r (5) [71"] = -t8)" M EJ1.vpu J

pT , ' J dp- (p2 pJ1. dx tr8 v'71"8P'71"8u'7I"8T 7I"7I" _ M2)6

+ .... (1.92)

This momentum integral can be calculated using the well known formulae of dimensional regularization (see Appendix A) and it yields

pJ1. pv gJ1.V J _ p2 dp (p2 _ M2)6 = D dp (p2 _ M2)6 J Therefore we finally have

.

gJ1.V

= - t (471")25!M6

(1.93)

1 2 5 EJ1.vpu Jd X tr 8J1.'8 ' , + ... (1.94) 71" V71"'8 P71"'8 u 71"71" 24071" v This part of the scalar effective action is referred to as the Wess-ZuminoWitten (WZW) term and it appears in several contexts in different effective Lagrangians [9]. Such a term will be reobtained in Chap. 4 using other techniques. Notice that when taking the M --+ 00 limit with v fixed, the term above does not decouple. Once more, the decoupling theorem does not apply. Thus, even if the fermions are very heavy, the scalar fields will feel their presence at low energies through this WZW term. To summarize, in this chapter we have seen how it is sometimes possible to obtain an effective Lagrangian describing the low-energy dynamics of the light particles of a much more complicated system. In case the underlying theory was known and manageable, the terms in this Lagrangian could be obtained theoretically. However, if we are not able to perform such a calculation, symmetries are very useful to determine which terms are acceptable, i.e,

r (5)[ 71"] =

1.7 References

21

invariant (excluding the anomaly problem that will be treated in Chap. 4). In general, these terms will contain certain coefficients which have to be obtained phenomenologically. In turn, the measurement of these parameters will provide information about the underlying theory in case it was unknown. Moreover, there is another reason which magnifies the importance of symmetry in the effective Lagrangians, which is the spontaneous symmetry breaking that takes place in many physical systems. In QFT the symmetry breaking gives rise to the appearance of Nambu-Goldstone bosons, which are massless particles and, therefore, the natural candidates to constitute the light degrees of freedom of an effective Lagrangian. According to the enormous importance of the symmetry considerations in the study of effective Lagrangians we will spend the next chapter recalling the most important notions of symmetry in QFT, together with the Goldstone theorem.

1.7 References [1]

[2] [3] [4] [5] [6] [7] [8] [9]

R.N. Mohapatra, Unification and Supersymmetry, Springer Verlag, 1986 M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Cambridge University Press, 1987 D. Bailin and A. Love, Supersymmetric Gauge Field Theory and String Theory, lOP Publishing Ltd, 1994 D. Bailin and A. Love, Rep. Prog. Phys. 50 (1987) 1087 E. Fermi, Z. Phys. 88 (1934) 161 R.P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193 M.K. Gaillard, Lectures presented at the Cargese 1987 School on Particle Physics T. Appelquist and J. Carazzone, Phys. Rev. Dll (1975) 2856 E. Witten, Nucl. Phys. B104 (1976) 445 W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714 J. Schwinger, Phys. Rev. 82 (1951) 714 W.H. Furry, Phys. Rev. 51 (1937) 125 F. Feruglio, L. Maiani and A. Masiero, Nucl. Phys. B387 (1992) 523 E. D'Hoker and E. Farhi, Nucl. Phys. B248 (1984) 59, 77 1.J.R. Aitchison and C.M. Fraser, Phys. Rev. D31 (1985) 2605

2. Global Symmetries in Quantum Field Theory

Symmetries play an important role in the determination of the structure of the different terms appearing in an effective Lagrangian. For this reason we briefly review in this chapter the basic notions related with global symmetries in quantum field theories (QFT). Starting from the Noether theorem for classical systems we derive the Ward identities for the Green functions. We also review the spontaneous symmetry breaking mechanism, the Goldstone theorem and the Dashen conditions for explicit symmetry breaking.

2.1 Classical Symmetries Let iP: M --+ S be some classical field, i.e. a mapping from the space-time M into some space S. That field may contain both internal (such as spin or isospin) and space-time degrees of freedom. For the sake of simplicity in the notation we will consider iP as an scalar field, i.e. only internal degrees of freedom (represented by the index i) will be explicitly considered. Anyway, the results are easy to extend to the general case. Now we will assume that iP transforms under a given representation of some Lie group G so that, under a small G transformation it behaves as (2.1) (or just iP

--+

iP' = iP + fJiP in a more condensed notation) where

(2.2) Here ea are some infinitesimal parameters and the Ta are the hermitian G generators in the given representation, satisfying the commutation relations [T a , T b] = irbcye (2.3) and normalized so that trTaT b = fafJ ab /2 with fa = ±. Whenever the group is compact fa = +. For instance, the SU(2) generators in the fundamental representation can be chosen as Ta = T a /2, T a being the Pauli matrices. Note that the ea parameters are space-time independent, since here we are only considering global or rigid transformations (see Appendix B.3 for the case of local or gauge fields). A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

24

2. Global Symmetries in Quantum Field Theory

The classical evolution of the system is obtained from the action 5[P] =

l

(2.4)

dx£(P,8Il P),

where dx = dxdt represents the space-time volume element and for simplicity we assume that the Lagrangian is only a function of the field and its first derivative. More general cases will be considered in the next chapter. Note that the action is an integral functional on the field P, where the integral is taken over a given region n of the space-time. The precise definition of the action also requires knowing the functional space in which it is defined, or in other words, the field boundary conditions on 8n. As usual we will consider here that P and 81l P vanish on 8n but this will not necessarily be the case, for instance, when dealing with gauge fields (see Appendix B.3). The equations of motion are obtained from the variational principle

85[P] =

o.

(2.5)

where 8 is an arbitrary variation, not to be confused with the variation under the G group 8, as given in (2.2). By means of the Gauss theorem and the above mentioned boundary conditions, (2.5) leads to the well known EulerLagrange equations

8£

8£

8p = 81l 8(8 P)

(2.6)

Il

Now we will assume 5[P] to be invariant under the G group. Therefore we have 5[P] = 5[P'] = 5[P

+ 8P]

.

(2.7)

Here we are only considering internal symmetries (i.e. not affecting the space-time coordinates) and then the above equation is equivalent to

8£ =

o.

(2.8)

In this case, and using the equation of motion, it is not difficult to find 81l

(8(~~P/P)

=0.

(2.9)

If we define a Noether current as 8£ Tan. J Ila = 8(8 P) '¥,

(2.10)

Il

we have

8 1l

r

Il

=0'

(2.11)

which is the well known Noether theorem relating symmetries with conserved currents (see [1] for a detailed review). Moreover, one can define the charges

Qa(t)

=

J

dxJg

(2.12)

2.1 Classical Symmetries

25

integrating the time-like J8 component only over the space volume element dx. Then, by using (2.11) and the Gauss theorem with the above mentioned boundary conditions we obtain

dQa(t) = 0 (2.13) dt or in other words, the Qa charges are constants of motion. In summary, we have shown that the invariance of the action under G transformations (2.7) leads to the conservation of the corresponding Noether currents on shell, that is, provided the evolution of the fields satisfies the classical equations of motion. When considering transformations involving not only internal degrees of freedom but also space-time coordinates the statement of the theorem is slightly more involved. In this case the transformations can be written as x-+x'=x+8x
(2.14)

where 8
+ 8J.L
(2.15)

and L\
= 8[
.

(2.16)

Notice that we have introduced 8' since the definition of 8 includes the spacetime measure and therefore a Jacobian factor is present whenever the coordinates are involved in the transformation. That is why the explicit functional form of the action changes from 8 to 8'. Again, and following similar steps as in the previous case, it is possible to show that the invariance of the action represented in (2.16) gives rise to the conservation of both the Noether current (8J.L J~ = 0) and their charges, provided J~ is defined as JJ.La =

8.c (8
(2.17)

where xa and ya are defined as follows: 8x J.L=XJ.L aea 8
(2.18)

At this point, it is important to remark that the conserved currents can be conveniently rescaled and shifted by adding a total divergence of any antisymmetric tensor f~v, since with the definition J~a = + 8 v f:v (2.19)

U:

26

2. Global Symmetries in Quantum Field Theory

we also obtain a conserved current 81-' J~a = 81-'

J; =

0.

(2.20)

Once we have reviewed the internal symmetries, let us study the most important transformation groups in QFT which act on both the fields and the coordinates. They are the following: a) The Lorentz Group. It is the group of Lorentz transformations, that we can write as (2.21 ) where L E 0(1,3), that is, the orthogonal transformations in a space-time with metric gl-'v = diag(1, -1, -1, -1). Notice that this group is not compact. It is made of four disconnected sheets. The one connected with the identity (Lg 2: 1, det L = 1) is called the proper orthochronous Lorentz subgroup and it is equivalent to 5L(2, C) j7L. 2 . The other sheets can be reached from this one by applying the discrete parity (P) and time reversal (T) transformations. Thus it will be enough to study the proper orthochronous subgroup, and so we will do in the following. For this group the index a becomes a couple of Lorentz indices so that XI-'a ~ XI-'Vp and yia ~ yil-'v. Then it is not difficult to find that 1 XI-'VP=_(gI-'VxP _ gl-'PXV) 2

Y il-'v=_~El-'vpi 2 tJ '

(2.22)

where the spin symbols Eijv are defined in terms of the corresponding Lorentz group representation for the eJ> field. For instance EI-'v = 0 for scalar fields, EI-'v = i['yI-','Y v ]j4 for spinor fields and EI-'v = MI-'v for vector fields, where MI-'v are the 0(1,3) generators in the fundamental representation. Notice that i, j, ... refer to Dirac matrices components for the spinor case and to space-time indices for the vector representation. Therefore, the Noether current is given by MI-'vp = III-' (88eJ>X 8V P +

~EVPeJ>)

- £XI-'VP

(2.23)

with

I-' _

8£

IIi - 8( 81-'eJ>i)

(

2.24

)

b) The Poincare Group. It is defined as the Lorentz group transformations in (2.21) plus translations defined as X'I-' = xl-' +~I-',

(2.25)

thus, for pure translations we have XI-'v=gl-'V yl-'v=O,

(2.26)

2.1 Classical Symmetries and the corresponding Noether current is TlJ.v = IIr8 vepi _ £gIJ.V .

27

(2.27)

Therefore 8IJ.TIJ.v = 0 in the classical evolution of the system for translationally invariant actions. TlJ.v is called the canonical energy-momentum tensor. The associated conserved charge is just the total momentum

PIJ. =

J

dxTolJ. ,

(2.28)

which is the translation generator. Notice that the Noether current for the pure Lorentz group, defined in (2.23), can be written in terms of TlJ.v as

MlJ.vp = ~ (xVTIJ.P _ xPTIJ.V + SIJ.VP) ,

(2.29)

where SIJ.VP = IIr Evpepi. Hence, introducing PIJ. = TolJ. the Lorentz charges (2.30) generate the Lorentz transformations. Therefore, for Lorentz invariant actions we find that 8IJ.MIJ.vp = O. Using the redefinition of the Noether currents in (2.19), it is possible to consider other energy-momentum tensors. For instance, by adding to the canonical one in (2.27) a total derivative of the form 8u C UIJ.V, with CUIJ.V antisymmetric in the (TJl indices. All these tensors would be conserved in Poincare invariant systems, although, in general, they are not symmetric. However, it is possible to define a symmetric and conserved energy-momentum tensor TlJ.v [2]' by choosing TlJ.v =TlJ.v + 8 C UIJ.V u

CUIJ.V = _~ (SUIJ.V 2

+ SIJ.VU _ SVUIJ.) ,

(2.31)

if there is Poincare invariance. c) Scale Transformations. Dilatations or scale transformations can be seen as a one parameter group, since they are defined as follows

x'lJ. = eO;xlJ.

~

(1

+ a)xlJ.

(2.32)

and then

Y=-dep ,

(2.33)

where d is the energy dimension of the ep field, i.e. d = 1,3/2,1 for scalars, spinors and vectors fields, respectively. The scale invariance of the action requires that no dimensional parameters appear in the Lagrangian. In particular there cannot be any mass term. The corresponding Noether current is given by

28

2. Global Symmetries in Quantum Field Theory DI-'

= III-'(8vcPx v + dcP)

- .cxj1, .

(2.34)

Using the freedom given in (2.19), it is possible in many cases (which include renormalizable theories [3]) to define both a dilatation current and a conserved and symmetric energy-momentum tensor such that (2.35) Thus one has 81-'Dj1, = T:

(2.36)

and therefore

T:

= 0

(2.37)

i.e. a traceless energy-momentum tensor can always be found when the action is scale and Poincare invariant. We remark that the above TI-'v tensor is not the same TI-'v tensor previously defined. d) The Conformal Group. The conformal group can be defined as the most general transformation group preserving patterns (which means for instance angles). Clearly it should contain rotations, translations, boosts and dilatations. In addition to the ten Poincare group generators plus that of dilatations, it also contains four more generators associated to the so called special conformal transformations, which are defined as xl-' - bl-'x 2 (2.38) x ' j1, = 1 _ 2bx + b2x2 ' where the new four transformation parameters are just the b vector components. Thus the conformal group has fifteen parameters (in four space-time dimensions). Conformal invariance trivially includes Poincare invariance and scale invariance. In contrast, Poincare invariance and scale invariance only implies conformal invariance provided there is a symmetric, conserved and traceless energy-momentum tensor satisfying (2.35), (2.36) and (2.37). However, this is not true in general and it is possible to find examples of scale invariant field theories that are not conformally invariant. The conserved charges corresponding to the special conformal transformations are given by KI-' =

J

dx(x 2TI-'O - 2x l-'x V T2) .

(2.39)

Finally it is not difficult to see that the conformal group is just the 50(2,4) group which obviously contains the orthochronous Lorentz 50(1,3) group.

2.2 Green Functions and the Reduction Formula As we have seen in the previous section, the Noether theorem states that whenever an action is invariant under a given symmetry, there are some associated conserved currents in the classical evolution of the system. However,

2.2 Green Functions and the Reduction Formula

29

at the quantum level the consequence of a symmetry becomes much more complicated owing to the fact that in QFT the fundamental objects are the Green functions, which are defined as (2.40) where 10) is the true vacuum of the theory (not to be confused with the Fock space vacuum), T is the time-order operator and If>(x) is the field operator in the Heisenberg representation. Note that we use the same symbols for the classical fields and their operators. The difference will become clear from the context. The Green functions encode all the information contained in the theory and, in particular, from them we can calculate all the physical observables. The Green functions can be obtained as functional derivatives n n 1 8 W[J] (2.41 ) C (Xl,X2, ... ,X n ) = in 8J(Xl)8J(X2) ... 8J(x n ) J=O

I

from the generating functional W[J], which in the path integral formalism is written as W[J]

= eiZ[J] =

J

(2.42)

[dlf>]ei(S[cf>]+<Jcf»} ,

where S[If>] is the classical action and < JIf> >= J dxJi(x)lf>i(X). The functional measure [dlf>] is normalized so that W[O] = 1. The above equation is formal in the sense that the integral is well defined only in the Euclidean space-time. Then the Euclidean Green functions (which are analytical) should be analytically continued to the Minkowski space-time where, in general, they have to be understood as distributions. Note that in the W[J] definition we have also introduced the Z[J] functional, which can be shown to generate the connected Green functions. Indeed n

Cc (Xl, X2,···, x n )

=

1

in-l

n

8 Z[J]

I

8J(Xl)8J(X2)'" 8J(x n ) J=O .

(2.43)

As it is customary we mostly use the Fourier transform of the different Green functions. That is, C n (k l ,k2, ... ,kn ) =

J

dXldx2 ... dxncn(Xl,X2, ... ,xn)eiL::'=lxiki,

(2.44) which, assuming translational invariance of the action (i.e. total momentum conservation), can always be written as C n (k l ,k2, ... ,kn ) 4 -n = (27r) 8(k l + k2 + ... + kn)C (k l , k2, ... , kn ) . (2.45)

30

2. Global Symmetries in Quantum Field Theory

en

Simple poles occur in whenever kf = M 2 , M being the physical mass of the particles appearing in the spectrum of the theory with the same quantum numbers as the field cf>. It is also posf?ible to define more general Green functions by considering also composite operators like (cf>(x))m, or even more complicated functions of the fields and their space-time derivatives (for instance, Noether currents). In this case, the corresponding functions have poles too. They are located at the values of the physical masses of those particles with the same quantum numbers as the composite operator. A central result in QFT is the Lehman-Symanzik-Zimmermann [4] reduction formula. It relates the connected e~ Green functions with S-matrix elements, thus providing the link between Green functions and physical observables such as cross sections or decay rates. In its simplest version it can be written as

en

(2.46)

where qi and k i are the momenta of the incoming and outgoing particles, that should have the same quantum numbers as the field cf>. The R normalization factor is defined as the two point Green function (the exact propagator) residue of the pole located at M 2 . That is -2

Gc(k, -k) ~ k 2

R

_ M2

(2.47)

for k 2 ~ M 2 . Note that the (kf - M 2 ) and the (q~ - M 2 ) factors cancel the simple poles of the Green functions so that the on-shell limit is well defined. Starting from this simple reduction formula it is straightforward to make the appropriate generalization for the case of more complicated fields (spinors, vectors, etc.) or composite operators. It is important to emphasize here that, although most calculations in QFT are performed in terms of Green functions, the connection with observables is done through the reduction formula. Thus, only the residua of these functions at the corresponding mass poles have a physical meaning. In fact there are many examples in field theory where the Green functions can be defined differently, although they yield the same S-matrix elements. For instance, in standard renormalizable theories, the renormalized Green functions depend, for a given regularization and renormalization procedure, on an arbitrary scale J.L. However, due to the Callan-Symanzik renormalization group equations, the S-matrix elements are J.L independent (see Appendix C.l). Other interesting examples are gauge theories and non-linear sigma models (NLSM) where the Green functions depend on the gauge or the coset parametrization, but the S-matrix elements do not. Further on we will return in more detail to these important issues.

2.3 Quantum Symmetries and Ward Identities

31

2.3 Quantum Symmetries and Ward Identities As it was mentioned in the previous section all the physical information in a QFT can be obtained from the Green functions through the reduction formula. Thus, a natural question arises concerning the effect on the Green functions derived from a classical action when it is invariant under a continuous symmetry group G. The answer is that the symmetry of the action gives rise (provided that no anomalies are present) to some relations between different Green functions which are known as Ward identities [5]. To illustrate how to obtain these relations we will consider the case of a theory with the following classical action 8[4>]

= 8 0 [4>] + 8 1 [4>] ,

(2.48)

where 8 0 [4>] is invariant under the global G transformations but 8 1 [4>] explicitly breaks that symmetry. For the sake of simplicity we will assume both that 8 1 [4>] depends on the fields (but not on their derivatives) and that G is an internal symmetry not affecting the space-time coordinates. In other words, let us assume a Lagrangian L = LO + L1 transforming as

L = LO

+ L1 - - L + bL = L + i(Pb aL1 ,

(2.49)

where the fields transform linearly as

b4>i = i8 a(T a)iieJ>i .

(2.50)

Let us now consider the current J p.a =

8L

8( 8p.4»

Tad.

(2.51)

'F .

According to the discussion in the previous section, it would have been a Noether current was it not for the 8 1 [4>] symmetry breaking piece of the action. Therefore we can write 8P. J~

= baLl.

(2.52)

In order to obtain the Ward identities corresponding to the global symmetry G, one has to consider momentarily the effect of local transformations on the generating functional. In this case we have 4>'i(X)

= 4>i(X) + b4>i(x) = 4>i(X) + i8a(x)(T a)iieJ>i(x)

(2.53)

+ i8 a(x)b aL1

(2.54)

and

U(x) =

iJ~(x)8p.8a(x)

.

Then the generating functional can be written as

W[J] =

J

[d4>]e i (S[Pl+<JP» =

J

[d4>']e i (S[P'l+<JP'»

,

(2.55)

since 4> is just a dummy integration variable. Now, when the theory is not anomalous (see Chap. 4) the path integral measure can be defined so that

32

2. Global Symmetries in Quantum Field Theory

[dtf>] = [dtf>'], Le. there is a G invariant regularization procedure for all the divergent integrals. In such case one has W[J] = j [dtf>jei(S[q,'l+<Jje i (S[q,J+i+i<J~8"O" >+<Jq,>+<J6q,»

(2.56)

and by comparison with (2.55)

j dx j[dtf>jei(S[q,J+<Jq,» x {io a(x)8 a£I(X)

iJ: (x)8J.'oa (x) + J i (x)8tf>i(X)} = O.

+

(2.57)

We can now integrate by parts and use the Gauss theorem together with the field boundary conditions (the fields vanish at infinity). Thus we can make the derivative to act on instead of O(x), which then becomes a factor of the whole integrand. But O(x) is an arbitrary parameter so that the rest of the integrand satisfies

J:

j[dtf>jei(S[q,J+<Jq,» {i8 a£I(X) - i8J.'J:(x)

+ Ji(X)~~~~~~} = O.

Therefore, if we take a functional derivative with respect to

j [dtf>jei(S[q,I+<Jq,» {

_«sa £1 (X )tf>i

+ 8J.' JU(X)tf>i J.'

l

(xI)

l

(Xl)

+ Ji(X) ~~~~~~ itf>i

+ 8tf>i(x) 8ii1 8(x 80a(x)

Jil 1

(2.58)

(Xl) we find

(Xl)

Xl)} = 0 .

(2.59)

Let us now recall that the Green functions like (2.60) with the field operators in the Heisenberg representation, are written in the path integral formalism as

= N j [dtf>jei(S[q,]+<Jq,» A(x)tf>i 1 (XI)tf>i 2 (X2) ... tf>in (X n )

,

(2.61)

where N is some normalization constant. Notice that A(x) could be any composite operator as J~, 8a£1 or 8tf>i. Therefore, by setting J = 0 in (2.59), we arrive at the Ward identity a 8~ (0IT(J:(x)tf>i 1(Xl)) 10) - (OIT(8 £1 (X)tf>i 1 (xI) )10)

+ (OIT (~~~~~~ 8ii1 8(x or defining 8atf>i

= (Ta)ijtf>j,

so that 8tf>i = i8 atf> i oa

XI)) 10) = 0

(2.62)

2.3 Spontaneous Symmetry Breaking and the Goldstone Theorem

cW (OIT( J:(x )<J>i

l

33

(Xt}) 10) = (OIT(Oa £1 (X)<J>i l (Xt}) 10)

-io(x - xd(OIT(oa<J>i)jO) . (2.63) i Then, if we differentiate with respect to J 2(X2), ... , Jin(X n ) in (2.59) and we take J = 0 we find a~ (OIT(J:(x )<J>i l (Xl), <J>i2 (X2), ... ,<J>in(x n )) 10) = (0IT(oa£1(X)<J>i l (Xl),<J>i 2(X2), ... ,<J>in(xn))IO) n

L O(X -

Xj) (OIT( <J>i l (Xl), ... ,Oa<J>ij (Xj), ... ,<J>i n(X n )) 10),

(2.64) j=l which is the most general form of a Ward identity. It applies in any case, i.e. even when the symmetry is explicitly broken (by 51 [<J>] in our case) or when it is spontaneously broken, i.e. when the action is symmetric but the ground state or true vacuum is not. In particular, for the n = 0 case and taking £1 = 0 we find the simple Ward identity -i

(2.65) which can, in some sense, be considered as the quantum version of (2.11). The above derivation has been rather formal since our Green functions are not renormalized. In order to obtain the Ward identities for the renormalized Green functions it is necessary to implement a regularization method preserving the G symmetry. In case it is not possible to have such a method, the symmetry is said to be anomalous and then the classical symmetry is not present at the quantum level. The most obvious example are the scale transformations, since any regularization procedure will spoil this symmetry even if it was present in the classical action. Many other examples of anomalies will be studied in Chap. 4 but for the moment we will assume that none of our symmetries is anomalous. Let us consider again the case in which there is no explicit symmetry breaking or, in other words, £1 = o. Then it is possible to integrate the Ward identity in (2.64). In addition, if no spontaneous symmetry breaking term is present, it is possible to neglect the surface term (see next section). On such assumptions, we finally obtain (2.66) which is the simplest form of Ward identity associated with some global G symmetry. Using the reduction formula it is straightforward to obtain the constraints that this equation imposes on the 5-matrix elements.

2.4 Spontaneous Symmetry Breaking and the Goldstone Theorem In this section we will study in detail the case of spontaneous symmetry breaking, i.e. what happens when the symmetry group of the action G is

34

2. Global Symmetries in Quantum Field Theory

larger than the vacuum symmetry group H. In this case (assuming that there is no explicit symmetry breaking) the Ward identity in (2.64) reads 8~ (OIT( J:(x )q>i 1 (xd, q>i 2 (X2), ... ,q>i n (x )) 10) n

n

= -i

L o(x -

Xj )(OIT(q>i 1 (xd, ... ,oaq>ij (Xj), ... , q>i n (x n )) 10). (2.67)

j=1 For the sake of simplicity we are now going to consider a very simple Green function with just one field. Let us assume that only one of its components has a non-vanishing vacuum expectation value (2.68) which is constant because of translational invariance. Thus we have

Jdx8~(0IT(J:(X)q>i(xd)10)

= -i(0Ioaq>i(X1)10) = _i(OI(Ta)ijq>i (xdIO) = -i(T a)i1 v1

i=- 0 ,

(2.69)

i.e. in this case the surface term cannot be neglected as we did in the previous section when no spontaneous symmetry breaking was present. Let us now rewrite the above equation in momentum space. With that purpose, we multiply the integrand by exp[-iq(x - xd] which in the q -+ 0 limit is the same as multiplying by one. Integrating by parts, we then get

~~

J

dxe-iq(X-xtlqjL(OIT(J:(x)q>i(xd)IO) i=- 0,

(2.70)

which shows that the Green function

G~i(q) =

J

dxe-iq(x-x 1 ) (0IT(J:(x)q>i(X1))!0)

(2.71)

satisfies (2.72) This implies that for q '" 0 we have

G~i(q) '" q~cai . q

(2.73)

This result can be easily generalized to Green functions with more than one q> (see Fig. 2.1). As we have a pole in all these Green functions whenever q2 '" 0, then there should be a massless mode in the physical spectrum of the theory with the same quantum numbers as the spontaneously broken Noether current [6]. In the general case we will have an invariance symmetry group G for the Lagrangian, but the vacuum will have (Oloaq>IO) i=- 0 for some a values. In other words, this means that the corresponding Noether charge operators do not annihilate the vacuum, Le. QaIO) i=- O. If the group G is generated by

2.4 Spontaneous Symmetry Breaking and the Goldstone Theorem

35

2l

J~

q~

i2

rv -

q2

'In

Fig. 2.1. Diagrammatic representation of a Green function with a Noether current line corresponding to an spontaneously broken generator

T 1 , T 2 , ... ,T9 (g = dimG) we can assume without loss of generality that the last k generators are the broken ones, Le. (2.74)

a=g-k+l, ... ,g

and thus the remaining Ta generate the h dimensional subgroup H, with 9 = h+ k. In the following, and in order to simplify our notation, we will rename the G generators so that Ta = Ha for a = 1,2, ... , hand T a = x a for a = h + 1, h + 2, ... , h + k, so that the H a generate Hand X a are the broken generators. Therefore, in the general case, we will have a massless mode, which is called a Nambu-Goldstone boson (NGB) for every broken generator, i.e. we will have k NGB with k = 9 - h = dimG - dimH = dimG/ H. In fact, as we will see in the next chapter, the NGB fields can be identified in some sense with this quotient or coset space K = G / H.

Example 2.4.1. A simple and quite popular system with spontaneous symmetry breaking is the linear O"-model (LSM) introduced in the previous chapter and defined by the Lagrangian £ =

~a/LipT8!-'ip 2

(2.75)

V(ip2)

with ipT = (7[1,7[2, ... , 7[N ,0") and ip2 = ipTip. The above Lagrangian is O( N + 1) invariant but the potential is such that we find a minimum whenever ip2 = v 2 f= and therefore there are many possible vacua, all them related by O(N + 1) rotations. To define our quantum theory we have to choose a vacuum and, in principle, any direction will be equally valid. For simplicity we will take ip~ac = (0,0, ... ,0, v) which is obviously invariant under O(N) transformations on the first N components. Therefore in this case

°

G

= O(N + 1),

H

= O(N),

K = O(N

+ 1)/O(N) =

SN ,

(2.76)

so that 9 = N(N + 1)/2, h = N(N - 1)/2 and k = N. The dynamics of the NGB is described by the Lagrangian above. At very low energies one can neglect the massive mode H = O"-V and then imposing the constraint ip2 = v 2

36

2. Global Symmetries in Quantum Field Theory

one is left with the corresponding NLSM that we have already encountered in Chap. 1. Note that the NGB fields, i.e. the first N components of rJ>, after imposing the constraint n 2 + 0- 2 = v 2 are nothing but coordinates on the SN manifold. We will turn to this important point in the next chapter. Example 2.4.2. Another important system with spontaneous symmetry breaking is the case of two flavor massless quantum chromodynamics (QCD), which is supposed to provide an approximate description of the hadronic world at low energies (see Chap. 6). The corresponding Lagrangian is given by (2.77) where LYM is the standard Yang-Mills Lagrangian for the gluon fields, DJ.L = 8J.L +GJ.L' GJ.L = -i9sAaG~/2 with Aa the Gell-Mannmatrices, 9s is the strong coupling constant and QT = (u, d) are the u and d quark spinors. As is well known this Lagrangian is SU(3)c gauge invariant. In addition it is SU(2)L X SU(2)R x U(l)v X U(l)A globally symmetric. The first two subgroups are called chiral groups, because they act separately on the two chiral components of the quark fields, and the corresponding symmetry is called chiral symmetry. The chiral transformations are defined as

Q'

= ei8'tTaPLQ

Q' = ei8'flT

a

(2.78)

PRQ,

i.e., they correspond to independent rotations of the left and right components of the Q spinors. The U(l)v group acts as follows,

Q'

= ei8 Q

(2.79)

and its associated conserved Noether charge is nothing but the standard fermion number. Finally the U(l)A transformations are given by Q' =

e

i8n5

(2.80)

Q .

However these symmetries are not found explicitly in the hadronic physical QCD spectrum. This is why it is believed that chiral symmetry is spontaneously broken to SU(2)L+R (isospin symmetry). Therefore, in this case we have G = SU(2)L

H K = SU(2)L

X

X

SU(2)R

= SU(2h+R

(2.81 )

.u-

SU(2)R/SU(2)L+R

rv

SU(2)L-R .

Notice that, as K is three dimensional, we expect three NGB in the QCD spectrum. These NGB are naturally identified with the pions n±, no. In Nature they are not massless owing to the small quark masses appearing in the complete QCD Lagrangian which explicitly break the chiral symmetry, so that the above scheme is only approximate and pions are called pseudo-NGB.

2.4 Explicit Symmetry Breaking and the Dashen Conditions

37

However it explains why pions are so light when compared with other hadrons. Note also that pion fields are composite fields made of quarks. Further details about low energy QCD can be found in Chap. 6. The case of the UA (1) symmetry has to be dealt differently. As it happened with the chiral symmetry, it is not observed in the hadronic spectrum although its disappearance is not related with an spontaneous symmetry breaking but with the presence of an anomaly. In Chap. 4, we will study in more detail this other symmetry breaking mechanism. In conclusion, the only remaining symmetries are 5U(2)L+R' which is nothing but the isospin, and U (1) v, that is not spontaneously broken and gives rise to baryon number conservation by the strong interactions.

2.5 Explicit Symmetry Breaking and the Dashen Conditions We have already seen that whenever we find a system with spontaneous symmetry breaking of a given group G down to some subgroup H, there will be a set of degenerate vacua in correspondence with the different points of the coset space K. In order to build the quantum theory we have to choose one of these points as the quantum vacuum 10) and, in principle, this choice is completely arbitrary. In fact, the NGB modes can be understood as field oscillations around the chosen vacuum in the K manifold. However, in practice one is also interested in cases where, apart from the spontaneous symmetry breaking, one also has a small piece .c 1 (x) in the Lagrangian .c(x) which explicitly breaks the G symmetry (for the sake of simplicity we will assume that .c 1 (x) only depends on the fields but not on their derivatives). When such term is added to the symmetric Lagrangian .co (x) the degeneracy of the vacua is broken and only one of them remains as the true vacuum of the theory. If we want to make a consistent perturbative expansion of .c(x) we should be able to find this true vacuum. Otherwise even a soft explicit symmetry breaking interaction term can produce large changes in the system. To understand better this point we can think of a ferromagnet. As it is well known, at low enough temperature the microscopic interactions produce an alignment of the microscopic magnetic dipoles giving rise to a net macroscopic or spontaneous magnetization of the system even in the absence of an external magnetic field. The direction of the magnetization in space is completely arbitrary (but not its modulus) so that the coset space is in this case the sphere 52. Now imagine that we switch on an external magnetic field. This will produce a strong reaction in the system because the external field breaks the 52 degeneracy and the magnetization vector moves until it is parallel to the external magnetic field. Only when the external field is switched on adiabatically in the same direction of the previous spontaneous magnetization, the reaction will be weak and perturbatively computable.

38

2. Global Symmetries in Quantum Field Theory

Therefore, for a perturbative approach, it is important to have some method in order to know which is the appropriate vacuum for the system. The Dashen conditions [7] will help us to find this vacuum state. In the following we will briefly describe how they can be easily obtained and we will comment their intuitive interpretation. Let 10) be the true ground state of the system obtained after some small perturbation £l(X) has removed the K degeneracy. Any other K point can be obtained from this one through the formula (2.82)

When the perturbation is present, the variation of the energy of 1£1) is given at the lowest order of perturbation theory by (2.83)

Actually, we consider that 10) is the real vacuum if it is a minimum of the energy. Thus, it has to satisfy the extremum condition aL1E(B) aBa

I

= 0

(2.84)

0=0

or, expanding the exponentials in the last equation, (2.85)

which is the first Dashen condition. The second one is obtained by imposing that the vacuum is a true energy minimum and not only an extremum. 2

a L1E(B) aBaaBb

I 0=0

>0,

(2.86)

which leads to (2.87)

It is not difficult to check that this condition is related with the positivity of the pseudo-NGB mass matrix. The above conditions can be applied to the two examples discussed at the end of the previous section. Example 2.5.1. For the LSM, it is possible to introduce a new term in the Lagrangian that explicitly breaks the O(N + 1) symmetry to O(N) and gives masses to the NGB. According to the above discussion the new term has to be chosen so that it breaks the symmetry in the same direction on K = SN as our arbitrary vacuum choice. This leads to £1 = m 2 va where m becomes the pseudo-NGB mass at the tree level. Then the Dashen conditions read:

(0111"10) = 0 (OlaIO) > a

(2.88)

in agreement with the tree level results, one loop calculations and the large N limit (see Chap. 6).

2.6 References

39

Example 2.5.2. For the case of massless two flavor QCD, the perturbation is just a mass term for the quarks £1 = -mqQQ (with m q being the quark masses) which explicitly breaks the SU(2)L x SU(2)R symmetry down to SU(2)L+R' In this case we find

(OIQ'lQIO) = 0 (OIQQIO) < 0

(2.89)

in agreement with the phenomenological estimate (OIQQIO) ~ 2(-225 MeV)3 (see Chap. 6).

2.6 References [1] [2] [3] [4] [5] [6] [7]

E.L. Hill, Rev. Mod. Phys. 23 (1951) 253 F.J. Belinfante, Physica 6 (1939) 887; Physica 7 (1940) 305 S. Coleman, Aspects of Symmetry, Cambridge University Press, 1985 H. Lehman, K. Symanzik and W. Zimmermann, Nuovo Cimento 1 (1955) 1425; Nuovo Cimento 6 (1957) 319 J.C. Ward, Phys. Rev. 78 (1950) 1824 Y. Takahashi, Nuovo Cimento 6 (1957)370 Y. Nambu, Phys. Rev. Lett. 4 (1960) 380 J. Goldstone, Nuovo Cimento 19 (1961) 154 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965 R. Dashen, Phys. Rev. 183 (1969) 1245

3. The Non-linear

(J"

Model

According to the Goldstone theorem, systems with spontaneous symmetry breaking have one massless mode for every broken generator. As a consequence, these Nambu-Goldstone boson (NGB) modes are the natural degrees of freedom of the system at low energies. In this chapter we will illustrate how the non-linear u model (NL8M) provides a general and systematic description of the low energy NGB dynamics as well as how their effective Lagrangian can be obtained through a beautiful mathematical picture based on the geometry of homogeneous spaces.

3 .1 Introduction In Chaps. 1 and 2 we have already introduced the linear u model (L8M), which is based on the classical Lagrangian

.c = ~aJ.LipTaJ.Lip 2

V(ip2)

(3.1)

with ipT = (7r 1 , 7r 2 , ... , 7r N , (J) and ip2 = ipTip. The potential V (ip2) is chosen to produce a minimum whenever ip2 = v 2 = N F 2 =1= 0, so that the system exhibits an spontaneous symmetry breaking from the group O(N + 1) into O(N). The N NGB associated to this symmetry breaking can be understood as zero energy excitations of the system around the chosen vacuum state. If we choose, for example, ip~ac = (0,0, ... ,0, v) as the vacuum of our theory, then the other vacuum states are indeed zero energy modes satisfying the constraint ip2 = 7r 2 + u 2 = v 2 = N F 2 . When we introduce this condition in the Lagrangian we are left with a NL8M which just describes the dynamics of the NGB. As it is well known, any O(N + 1) transformation can be written as R = exp iBabMab with 1 ::; a < b ::; N + 1, where Mi/ are the group generators, which can be chosen as Mi/ = i(8j8f -8i8j), thus being antisymmetric in the ab and the ij indices. Following the notation for the generators introduced in Chap. 2, we have Ha "-' M ab for a < b < N + 1 and X a "-' MaN+! for a < N + 1. Note that under an infinitesimal O(N) transformation we have

7r /i =7r i

+ iBabMi/7rj

u' =u , A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

(3.2)

42

3. The Non-linear a Model

and under a broken generator transformation 7r'i

= 7r i + iBaMt:::/ u = 7r i + Bi VNF2

- 7r 2 .

(3.3)

That is, the 7r fields transform linearly with respect to O(N) but nonlinearly with respect to general O(N + 1) transformations. In fact the NGB fields can be chosen as the 7r coordinates of P. From this notation it is obvious that the NGB fields are nothing but coordinates on the sphere SN = O(N + l)/O(N), that is, the coset (homogeneous) space corresponding to our symmetry breaking pattern. Moreover, we will show later in this chapter that, given an analytical function fi(7r i ), with f rv O(7r 2 ), field reparametrizations in QFT such as 7r'i

= 7ri + fi(7r i )

(3.4)

do not change the S-matrix elements, although, in general, they give rise to different Green functions. Thus, we can choose as the NGB fields any set of analytical coordinates on the coset space (for instance, the SN sphere in the above example) and all them will provide the same S-matrix elements. For instance in the SN NLSM we can make any analytical change of variables as the one above and the resulting model is still appropriate to describe the NGB physics at the quantum level. In fact this result is quite general and applies whenever a spontaneous symmetry breaking occurs: Imagine a system with some global symmetry group G which is spontaneously broken to some subgroup H. According to the Goldstone theorem the spectrum of the theory will contain k = dim G dim H = dim G/ H NGB modes. If we assume that there are no more massless particles, only the NGB will be relevant in the low energy effective theory. As in the SN NLSM discussed above, these NGB fields can be chosen as any set of coordinates parametrizing the coset space K = G/H. The S-matrix elements are invariant under reparametrizations and thus the very low energy physics only depends on G and H. However, we still do not have a general action describing the low energy dynamics of the K NLSM, as we had for the SN case. In the following section we will discuss how this action can be found and to what extent it is uniquely determined by G and H.

3.2 The Geometry and the Dynamics of the Non-linear (T Model Let us start from the Lie symmetry group G and its subgroup H corresponding to the symmetry breaking pattern of some given model [1]. Let Hi be the H generators (i = 1,2, ... h = dim H), X a (a = 1,2, ... , k = dim G -dim H) the broken generators and T = (H, X) the complete set of G generators denoted by T a (a = 1,2,., h, h + 1, .. , h + k = 9 = dim G). For technical reasons to be discussed later we will assume G and H to be compact. In such case the commutation relations can be written as

3.2 The Geometry and the Dynamics of the Non-linear

17

Model

43

[Hi, Hj] =i/ijkHk [Hi,X a] =ifiab Xb [X a , X b] =ifabiHi

+ ifabeXe .

(3.5)

When the homogeneous space K is symmetric (see Appendix B.2), it is possible to arrange the latter definitions so that the fa be structure constant vanish, i.e. the commutator of any two X generators is a linear combination of H but not of X generators. We are interested in the low energy behavior of the system. Therefore, we will only use NGB fields or their derivatives to build the Lagrangian. Obviously we want this Lagrangian to be G invariant. Apart from that, we will also work with a covariant formalism in the K manifold. In this way, the classical description of the system will be independent of the K parametrization. That may not seem very important, since in Sect. 3.3 we will show that, anyway, the NLSM S-matrix elements are independent of the coordinate choice. However, most of our results will be stated in terms of Green functions, not S-matrix elements. The covariant formalism ensures that the very same expressions remain valid for any choice of coordinates. We will impose all these constraints by introducing a metric in the K space that will allow us to build invariant terms just by contracting indices. In the construction we will see that invariance under G implies that G itself is nothing but the metric isometry group. In order to build such a Lagrangian we have to introduce some geometrical objects on the coset space K. First of all we choose some coordinates 1r"(0: = 1,2, ... , k) on some patch of K, such that 1r" = 0 corresponds to the chosen vacuum state. A generic point 1r" will correspond to some G element l(1r) which is the canonical representative of a K element (a class). Notice, however, that we have many ways to assign a representative to a class in K, and each choice defines a map l(1r) : K --+ G. For example a simple choice would be l(1r) = exp(i1r"o"a X a ); but there are other possibilities and all them are related by some analytical change of coordinates on K, which preserves the vacuum point. The left action of some element 9 E G on l(1r) can be decomposed as

gl(1r) = l(1r')h(1r,g) ,

(3.6)

where h(1r,g) E H. Thus we can write h(1r,g) 1 + ioafl~Hi. The above equation defines the transformation properties of the 1r coordinates under the G group. In other words, let p E K be the point labeled with coordinates 1r and let p' (labeled by 1r') be the result of the transformation 9 ~ 1+ioaTa E G on p. Then the coordinates of both points are related by some non-linear transformation like

(3.7)

44

3. The Non-linear

(J

Model

In order to define a G invariant metric on K we introduce now l-ldl which can be written as l-ldl = (W~(Jr)Hi + e~(Jr)xa)dJr'" = wi Hi + e a X a = W + e . (3.8) From this equation it is easy to obtain the ~a and fl a in terms of ea and wi. The ea are k independent forms that can be understood as the vielbein (see Appendix B.l) defined on the K manifold and can be used in the standard way to build a G invariant metric; Wi is known as the canonical H connection and will be useful later to couple fermions to the NCB. Using (3.6) it is quite easy to obtain the transformation equations for w and e which turn out to be w' (Jr) = h( Jr, 9 )w( Jr )h- 1(Jr, g) + h( Jr, g)dh -1 ( Jr, g)

e'(Jr) =h(Jr,g)e(Jr)h-1(Jr, g) ,

(3.9)

i.e. we see that w indeed transforms as a connection, or, in other words, as a gauge field under the H group. Now we can define a metric as (3.10)

where K is some constant. It is straightforward to check that this metric is G invariant (but not the vielbein itself). Therefore we have been able to find a G invariant metric on K in a canonical way. By this we mean that the metric has G as its isometry group. In other words, with the G transformation defined in (3.9) we have g~/3(Jr)

= g"'/3(Jr) .

(3.11)

Notice that the ~::(Jr) are in fact Killing vectors on the K manifold corresponding to the isometry group G of the metric that will be defined later. In particular it means that we will have the so called closure relations (3.12)

where [x, yJ should be understood as the Lie brackets of the Killing vectors x and y (see Appendix B.l) Apart from the K constant, the above invariant metric is unique under some technical conditions to be discussed below. Once we have found a G invariant metric on the coset space K it is straightforward to write many invariant terms describing the low energy NCB dynamics. The simplest one has two derivatives and is given by LNLSM =

~g"'/3aJ.LJr"'aJ.LJr/3 ,

(3.13)

where the metric has been normalized so that it reproduces the kinetic term at zeroth order in the expansion in Jr, i.e. g",/3 = 8"'/3 + O(Jr 2 ). The above Lagrangian is that of the NLSM. It is covariant both in the space-time indices f-l, v, ... and in the K space ones a, (3, ... and it is G invariant as it can be easily shown using the isometry condition in (3.11)

3.2 The Geometry and the Dynamics of the Non-linear

(j

Model

Jdx~go{3(1l')8",1l'08"'1l'{3 Jdx~g~{3(1l")8",1l"08"'1l',{3 = Jdx~go{3(1l")8",1l"08"'1l',{3

SNLSM[1l'] =

45

=

= SNLSM[1l"] ,

(3.14)

where dx == d D x and D is the space-time dimension. For convenience, we will use in the following dimensional regularization and, hence, we will use an arbitrary D (see below). Also note that, in spite of its simple form, the NLSM Lagrangian contains interactions for any even number of NGB. In momentum space, all these couplings are proportional to p2/ v 2 where v is some dimensional parameter appearing in the metric go{3(1l') which provides an energy scale. For example, in the SN NLSM we have already found in Chap. 1 that 1l' 0 1l'{3

go{3(1l') =

8o{3

+ V 2 -1l' 2 '

(3.15)

Probably the most important property of the Lagrangian in (3.13) is its universality, since it only depends on G, H and the v parameter. Therefore, the very low energy NGB dynamics of two theories with the same spontaneous symmetry breaking pattern G ---+ H is essentially the same. At higher energies, when other p2/v2 powers become relevant, we have to consider new terms in the effective Lagrangian [2]. For example, up to order (p2/ V 2)2 one can consider terms like £- = £-NLSM + cl(go{3(1l')8",1l'°8"'1l'{3)2 + c2(go{3(1l')8",1l'°8 v 1l'{3)2 + ... +higher derivative terms (3.16) Thus the NGB low-energy dynamics at the next to leading order is not universal since it does not only depend on the groups G and H but also on the coupling constants Cl, C2, ... which encode the information about the underlying physics. These constants could be obtained from a more fundamental theory if it were available and manageable. Alternatively, the lowenergy constants could be fitted from the experiment (see Chap. 6 for the case of quantum chromodynamics). Similarly, other G invariant terms with more and more derivatives can be introduced provided they are both spacetime and K covariant. The general theory defined by £- above, with infinite higher derivative terms, is called the generalized NLSM. We remark that in (3.16), apart from the terms explicitly displayed, there appear in general four and higher derivative terms which can be built out of derivatives of the NGB field and both the metric and curvature tensors. The number of independent terms for a given number of derivatives depends on the coset space K. In addition, fermionic matter fields can also be coupled to the NGB in a G invariant way as follows: Let 'l/Ji be some fermionic fields belonging to some linear representation of the subgroup H whose generators 'Hi have the same dimension that the 'l/Ji multiplet. Then we can write the Lagrangian

46

3. The Non-linear

(T

Model

(3.17) where wp. = W~ap.1ro.'Hi. This Lagrangian is G invariant as it can be easily shown from (3.9). It can be used to couple chiral fermions too. However in such case some subtleties related to reparametrization anomalies should be carefully taken into account (see Chap. 4). To end this section we will discuss some technical details concerning the metric introduced in K. As it was mentioned before, this metric is G invariant, that is, it has G as isometry group. In general, however, it is not unique. This can be seen as follows. Given some NLSM based in two compact simple groups G and H, the broken generators x a span a representation of the subgroup H (see (3.5)). As a consequence, the NGB transform linearly under the H transformations, but non-linearly under the whole G transformations, as it was the case with the SN NLSM. However, this representation could be reducible. If this is the case, the x a generators decompose in r irreducible sectors X ai with i = 1,2, ... , r and the same happens with the vielbein that can be written as e i = e~i X ai where the vielbeins of each representation transform separately under the H transformation. Thus, the most general G invariant metric is not the one defined above but

(3.18) where "'i are r arbitrary constants. Hence we have an essentially unique metric on the coset K only when the broken generator X a representation of the subgroup H is irreducible. Another important issue concerns the possibility of suppressing the G and H compactness condition. As we mentioned before the NGB belong to a linear representation of the subgroup H. When G is not compact its generators may have either a positive or a negative metric, i.e. tr TaT b = f. a8ab /2 with f. a = ±. Therefore, if H is compact, it is possible to adjust the signs of the "'i parameters in order to have a consistent kinetic term. However such a procedure is not possible when H is not compact.

3.3 The Quantum Non-linear u Model In order to obtain a good physical description from the NLSM Lagrangian, it is necessary to quantize the theory. For that purpose we will use the path integral formalism, as in previous chapters. The only subtlety that we have to take into account when dealing with the NLSM is that any quantum theory is not uniquely defined by the classical Lagrangian but also by a measure in the field functional space. Therefore, in order to have a G invariant and K covariant quantum theory, both the Lagrangian and the measure should respect these symmetries. As a consequence, the proper definition of the generating functional is [3]

3.3 The Quantum Non-linear eiW[J] =

![d1r.jg]exP i(S[1r]+ <

S[1r] =SNLSM[1r]

J1r

(j

Model

47

»

+ higher derivative terms,

(3.19)

where S is the generalized NLSM action of the Lagrangian given in (3.16). Notice that 9 is the determinant of the K metric. Obviously, with the definition in (3.19), both the path integral measure and the action are K covariant and G invariant. It should be noted that the term (J1r) breaks the K covariance. This is not a problem, since it is possible to define a different functional where (J1r) is replaced by f dxra.Ja. with ra. defined as follows: Let us consider a point p of K with coordinates 1ra. and the origin 0 (7l"a. = 0), that is, the classical vacuum. We assume now that the 0 and p points are such that there is a unique geodesic curve joining them in their local neighborhood. Now let E be the distance between 0 and p along that geodesic and define ra. = 8E j81ra., which is then a vector. With those definitions, the external source J a. transforms like a vector and the whole generating functional becomes K covariant. However it does not generate the Green functions of the 1r fields but those of the composite operators ra.(1r). As far as, in general, 2 7l"a. = ra. + O(1r ), both generating functionals can be equally useful. Moreover, it is always possible to choose the coordinates as 7l"a. = ra. (geodesic coordinates). In that case, the above two definitions for the generating functional would be the same. Hence, using one definition of the generating functional or the other is equivalent to changing coordinates on the K manifold. In addition, the S-matrix elements are independent of the coordinate choice on the K manifold, as we will see in Sect. 3.4. Therefore, we can still keep on working with the functional in (3.19) and the physical predictions will only depend on the groups G and H. The ..j9 factor included in the functional measure in order to make it K covariant can be re-exponentiated, yielding a new term in the Lagrangian of the form f).£ = -

~OD (O)tr

log 9 .

(3.20)

Where the OD(O) is to be understood as f d Dkj(21r)D. For this reason, as it happens with gauge theories, it is extremely convenient to use dimensional regularization when dealing with this kind of models, since in this scheme the above integral vanishes. Then we can simply forget about the measure factor in the path integral [4]. In case we had decided to work in another regularization scheme, the OD (0) term in the Lagrangian would cancel other contributions which are also absent in dimensional regularization. Once we have chosen the regularization procedure we can use the standard methods to derive the Feynman rules and diagrams, in order to calculate the Green functions of the NLSM. The main properties of these Green functions are the following:

48

3. The Non-linear

17

Model

• The physical predictions they yield are independent of the coordinates chosen on K [5] (see Sect. 3.4 for details). • The Ward identities resulting from the G invariance of the (regularized) W[J] functional are known as the Weinberg low-energy theorems [6]. As we saw in the previous section, given G and H, the two derivative term in the action is unique up to a scale factor. Therefore, these theorems are universal and provide the very low-energy description of the NGB dynamics. However, the derivation of the Ward identities cannot be done in the way described in Chap. 2. The reason is that the NGB fields do not transform linearly and also that the action is a function of arbitrarily higher order field derivatives. We will briefly describe in Appendix C.3 an alternative method to obtain Ward identities in this more general case. • The counterterms needed to absorb the divergences are also G invariant. As far as all these terms are included in ,[ with their corresponding couplings Ci, the theory is renormalizable in a generalized sense, since here we have an infinite number of coupling constants. However, following the philosophy of the derivative expansion it suffices to calculate the Green functions only up to some given power of the external momenta. In such a case, only a finite number of terms and couplings contribute. In such sense, the low-energy theory is completely predictive [2].

3.4 Reparametrization Invariance of the S-Matrix Elements In this section we show how the S-matrix elements and hence the physical predictions of the NLSM are independent of the coordinate choice on the coset manifold K [5]. The only condition is that the change of variables should be of the general form (3.21 ) with f analytical and O(1r 2 ). For the sake of simplicity we will consider the case dim K = 1, i.e. when there is only one NGB field, but it will be straightforward to extend the proof to the general case. As a starting point we consider the power expansion of the new field 1r' in terms of the old field 1r, which can be written as (3.22) with al = 1. We will see below that a different normalization of the linear term is not relevant. Thus the Green functions calculated from the new fields (3.23)

3.5 Local Symmetries and the Higgs Mechanism

49

where the fields are in the Heisenberg representation, can be written as

GIn(X1,""X n )=

==

00

00

Pl=l

Pn=l

L ... L

apl···apn(O\T(nPl(X1), ... ,nPn(Xn))\O)

L ... L ap1 ' .. apnG;1, ... ,Pn(X1, ... ,Xn ). 00

00

Pl=l

Pn=l

(3.24)

Therefore the Green functions of the new field variables can be formally written as an expansion in terms of Green functions of composite operators which are powers of the old field variables. According to our discussion in the previous chapter, the Fourier transforms G~n (k 1, k 2, ... , k n ) of the connected Green functions have simple poles at k; = M 2 , with M the mass of any particle in the spectrum with the same quantum numbers as the composite operators. That also happens with the G~ Pl,P2' ..',Pn (k 1, k 2,···, kn ) Green functions, although in the general case the composite operator spectrum is different than that of the single fields. Therefore, when their Green functions are introduced in the reduction formula, they do not cancel the k; - M 2 factors in the on-shell limit, and they vanish. The only exception is the first Green function G~ 1,1, ... ,1(k 1 , k 2, ... , k n ) = G~(k1, k 2,.·., k n ) ,

since in this case the k; - M poles. Therefme we have G~n(k1,k2, ...

2

(3.25)

factors are exactly cancelled by the n field

,kn) rvG~(k1,k2, ... ,kn),

(3.26)

where the symbol rv means here that both Green functions yield the same S-matrix elements. Now it is straightforward to extend this argument to the case of k different NGB fields where the change of coordinates can be written as (3.27) n'C< = nC< + a~lP2"'Pk (n 1 )Pl (n 2)p2 ... (nk)Pk .

L

PI +P2+···+Pk2:2,

Pi2:0

Note that owing to the R; factors appearing in the reduction formula (the residues of the exact two-point functions poles), the above result also applies for transformations where the new and the old fields are normalized differently, i.e. when we have (3.28)

3.5 Local Symmetries and the Higgs Mechanism In previous sections we have discussed how the low energy dynamics of the NGB associated to a symmetry breaking pattern G ----+ H can be described by the Lagrangian

50

3. The Non-linear

I:- =

(j

Model

~gQ,8Crr)ol'1rQOI'1r,8 + higher derivative terms

,

(3.29)

which is globally G invariant and covariant both in the space-time and in the coset space K. In practice, it is also very interesting to consider the case when part of the G global symmetry is made local. This can be achieved in the standard way by introducing the appropriate gauge boson fields and covariant derivatives. In principle these gauge fields are massless since an explicit mass term for vector fields breaks gauge invariance. Nevertheless, when the gauge bosons are coupled to the NCB the former may acquire masses while preserving the gauge invariance of the theory. This phenomenon is known as the Higgs mechanism [7]. It plays a central role in our present description of the electroweak interactions, since it provides a natural way to give masses to the W± and Z bosons. For that reason we will now describe in some detail how the Higgs mechanism works. Specifically, in this chapter we will discuss this mechanism within the NLSM framework whereas the minimal standard model (MSM) version, in terms of the LSM, will be analyzed in Chap. 5. We also refer the reader to Chap. 7 for a more detailed description of the standard model (SM) symmetry breaking sector. Let us consider a system with spontaneous symmetry breaking G -> H (again we assume for simplicity that G and H are compact) so that the dynamics of the associated NCB is described by a globally G invariant Lagrangian. Now we consider some subgroup G C G with generators T a (a = 1,2, ... ,dim G = g) and commutation relations (3.30) where the fa be are the G structure constants. Note that in particular we can take the G to be G or H but it could be any other subgroup of G. If we want to make the system locally G invariant then, following the standard procedure, we have to introduce the gauge boson fields AI' = -igTa A~ where 9 is the gauge coupling. The G gauge transformations of the NCB and the gauge boson fields will be

1r /Q (x) =1r (x)

+ ~~(1r)Ba(x) A~(x)=A~(x) - oI'Ba(x) + gfabeBb(x)A~(x) Q

,

(3.31 )

where ~~(1r) are the Killing vectors associated to the G group and thus their Lie brackets (closure relations) are given by (3.32) The Lagrangian in (3.29) can be made G gauge invariant by replacing the NCB field derivatives by covariant derivatives defined as (3.33) Thus the G gauged NLSM can be described by the Lagrangian

3.5 Local Symmetries and the Higgs Mechanism

La = LYM

+ ~g"'.B(1r)DiL1r"'DiL1r.B +

51

higher covariant derivative terms. (3.34)

Notice that we have added the pure Yang-Mills term Ly M for the gauge bosons, which provides, respecting the gauge invariance, their kinetic term and interactions of three and four gauge bosons. We remark that the G gauge invariance of the complete Lagrangian La is a direct consequence of the closure relations. In the general case, gauging the G group breaks explicitly the global G invariance to G x R, where R is the maximal subgroup of G elements commuting with G. This fact gives rise to the existence of a correct vacuum according to the Dashen conditions (vacuum alignment) [8]. In addition, the NGB associated to the X a generators not appearing in G x R get masses. The remaining NGB either are absorbed by the gauge bosons through the Higgs mechanism or continue being massless. In order to make this more explicit it is convenient to expand the second term in the La Lagrangian 1

;;. g",.B(1r)DiL 1r'" DiL 1r.B 1

= ;;.g"'.B(1r)8iL 1r"'8iL 1r .B - g8iL 1r"'€", li AiL

li

g2_

+ 2g"'.B(1r)f~€1AiLIiAiLb.

(3.35)

It is important to remember that the metric defined on the coset K can always be written as g",.B(1r) = 8",.B + O(1r 2 ). In addition we have €~ = K~ + O(1r) where K~ are 1r-independent constants. Therefore we can write g2

2

-

1

-

b ,. . ., M -AIiAiLb + "'''( ) g",.B (1r )'>r::'"Ii'>r::.BAIiAiL ii iL -;;. lib iL v 1r ,

(3.36)

where

Mliii = g2 K~K~8",.B .

(3.37)

This 9 x 9 matrix can be diagonalized by a rotation of the gauge boson fields AiLli . In so doing, we can find a maximum of k positive eigenvalues and eventually other zero eigenvalues depending on the dimension of G and H and the relationship between their respective generators. This means that, at least at tree level, some of the gauge bosons have become massive through their coupling with the NGB. However things are not that simple since we can also expand the second term of the right hand side of (3.35) in powers of 1f to find -g8iL 1r"'€", li AiLli = -g8iL 1r'" K",liAiL li + O(1r 2) . (3.38) This term mixes the NGB with the gauge bosons and makes the physical interpretation of the fields appearing in the Lagrangian obscure. Surprisingly, the physics will become more transparent once we have quantized the theory. The simplest way to include the quantum corrections is by using perturbation theory in g. As is well known, the formulation of the gauge perturbation

52

3. The Non-linear a Model

theory is not trivial since the bilinear term in the Yang-Mills action has no inverse in the whole functional gauge field space and therefore the propagator cannot be defined in the standard way. Thus, in order to define a consistent propagator, one is forced to reduce the path integral integration to some smaller space where only one representative of every gauge class is considered. Such a representative is chosen by means of a gauge fixing function fa,(-rr, A), following the Faddeev-Popov procedure (see Appendix C.2). After applying this method we are left with a Lagrangian 1

L Q = La - 2f;,fa(-rr, A)fa(-rr, A)

+

J

-a 8fa(-rr, A)(x) b dye (x) 8e b (y) e (y)

(3.39)

that can be used to derive the Feynman rules. The e a and ea are anticommuting scalar variables known respectively as the ghost and antighost fields. In the case of gauge theories coupled to systems with spontaneous symmetry breaking, it is particularly convenient to work with the so called t'Hooft or renormalizable gauge-fixing conditions or RE-gauges (see Appendix C.2). They have the two following essential properties: first they yield an invertible bilinear term for the gauge fields so that their propagator is well defined; second, they cancel the unwanted mixing term appearing in the classical Lagrangian (3.38). In order to cancel that term, we can work in the following RE-gauge: Q fa(-rr, A) = 81J.A~ + gf;,-rr K Qa . (3.40) Therefore in the quantum Lagrangian L Q there is a contribution -g8J.LA~-rrQK Qa

(3.41)

,

which, after integrating by parts with the standard boundary conditions, cancels the unwanted mixing term. In addition we are left with an invertible bilinear term for the gauge boson fields and thus we can define a consistent propagator. However, throughout the previous sections we have been trying to obtain a K-covariant formalism and the gauge fixing function in (3.40) yields a noncovariant term, since the -rr coordinates do not transform properly. In case we want to keep the K-covariance we have to introduce the following gauge condition

fa(7f, A) = where

8J.LA~ + gf;, u-rr ~fQK~

,

(3.42)

f is some scalar function defined on K such that

~f = TQ + O(-rr 2) = -rr Q + O(-rr2) . (3.43) u-rrQ Hence, at the lowest order in the -rr expansion, (3.42) is nothing but the simple gauge fixing condition in (3.40). Again we obtain the same propagator and the mixing term is cancelled. That is why, in the following, we will consider the gauge-fixing condition in (3.42). Now we are ready to apply perturbation

3.5 Local Symmetries and the Higgs Mechanism

53

theory. However, it should be noted that a non-linear gauge-fixing condition as that in (3.42) generates radiatively a four ghost interaction [9] which is not present in the Faddeev-Popov formalism. That problem can be solved by using the more general quantization procedure discussed in Appendix C.2. In any case, the tree level results can be obtained with the standard formalism analyzed here. At this point, we would like to comment briefly on the structure of the counterterms needed for the renormalization of the gauged NLSM. Remember that we have just obtained it by replacing the ordinary derivatives by covariant derivatives. However, gauge invariance makes it possible to introduce other structures that cannot be obtained in this way. For instance F:vF/: vgo.{3(-rr)D p7fo. DP7f{3 , (3.44) where FJ.1.v is the standard strength tensor of the G gauge fields. As a matter of fact, some of these terms, which are gauge invariant, are needed to cancel some of the divergences coming from the original NLSM: Thus, they have to be taken into account. In fact, when dealing with a perturbative gauge theory, the relevant symmetry is not the gauge symmetry, since the quantized Lagrangian (i.e. the one obtained with the Faddeev-Popov method) is no longer gauge invariant. As it is well known the relevant symmetry is the so called BRS symmetry [10]. The invariance of the quantized Lagrangian under the BRS transformations gives rise to the corresponding Ward identities (now called the Slavnov-Taylor identities) for the Green functions. Finally, using the reduction formula, these identities imply the gauge invariance of the Smatrix elements. Therefore, the most general counterterms that one could expect are not necessarily gauge invariant but BRS invariant. We will turn back to this issue in Chap. 7. Summarizing, we find that, at the lowest order, the gauge boson field masses are given by the M ab matrix in (3.37). Indeed, whenever any of the broken generators x a of G is contained in the G subgroup, some of the gauge bosons will become massive through the Higgs mechanism. The remarkable fact is that we can thus describe massive gauge bosons without explicitly breaking the gauge invariance. On the one hand, gauge theories seem to be essential for our present understanding of the fundamental interactions. On the other hand, we know that the W± and Z are massive gauge vector bosons. That is why the Higgs mechanism plays a decisive role in the formulation of the electroweak theory contained in the SM (see Chaps. 5 and 7 for more details). In some heuristic sense one can see the Higgs mechanism as though the NGB degrees of freedom were transformed in the gauge boson longitudinal components. However the real connection between NGB and longitudinal components of the gauge bosons is not so simple. The precise relation is given by the so called equivalence theorem which will be studied in Chap. 7.

54

3. The Non-linear a Model

3.6 Topologically Non-trivial Configurations In this section we will study a very interesting (and in some sense unexpected) property of the NLSM, in relation to the topological structure of the coset space. As we have seen, the NLSM describes the low-energy dynamics of the NCB fields, that are nothing but maps from the space-time into the coset space. Up to now, we have only considered NCB fields, representing small oscillations around the' vacuum. Now we are interested in less trivial field configurations. The motivation of this study is to describe, not only NCB, but also other states of the QCD spectrum such as baryons. One may wonder whether that is possible within the NLSM without explicitly including more degrees of freedom. We will show that this is indeed the case. For that purpose, let us consider static field configurations with finite energy and therefore vanishing at spatial infinity. Accordingly, it is consistent to compactify the space to the sphere S3, so that our static configuration is a map from S3 into K. In general, the topology of K is not trivial and thus all these maps are not continuously deformable into each other. In other words, they are not necessarily in the same homotopy class. As it is discussed in Appendix B.3, the homotopy equivalence classes of maps from SN into some given space X define the so called Nth-homotopy group 7rN(X). Thus, the different static NCB configurations with finite energy are classified according to 7r3(K), the third homotopy group of the coset space. Hence, whenever this group is not trivial, there is at least one class of static configurations with finite energy which cannot be continuously deformed to the trivial one (the 7r(x) = 0 vacuum). In particular that means that starting from one of these non-trivial configurations, the system cannot evolve into the vacuum, since a continuous time evolution can be regarded as a homotopy transformation. As a consequence, these configurations do not disperse, no matter what the equations of motion may be. For this reason, they are known as topological solitons. From the point of view of the quantum theory, every field belonging to different homotopy classes corresponds to disconnected sectors of the Hilbert space of the theory. Thus we see that coset spaces with non-trivial 7r3(K) give rise to a new and very rich structure of states. In the NLSM, the topological solitons are called skyrmions after Skyrme, who in the early sixties suggested that they could be identified with baryons (nucleons) in a theory of mesons (pions) [11]. In order to illustrate in some detail the above ideas, we will concentrate on the coset space K = SU(2)L x SU(2)R/ SU(2)L+R = SU(2) = S3. This is the original system considered by Skyrme and it is also the NCB space associated with the spontaneously broken chiral symmetry of two flavor massless QCD (see Chap. 6). As we have already seen, in this model we have three NCB that are identified as the pions, representing small excitations of the vacuum. In addition we have 7r3(S3) = 71. and therefore infinite homotopy classes labeled by some integer number (see Appendix B.3).

3.6 Topologically Non-trivial Configurations

55

The key idea is to identify this integer number with the baryon number. In this picture, the pions belong to the trivial topological sector with zero baryon number and they are connected with the vacuum. One nucleon belongs to the sector with baryon number one, one antinucleon belongs to the -1 sector, 14C to the 14 sector and so on. Therefore, in this model, baryon number conservation by strong interactions is a consequence of the coset space topology. When dealing with skyrmions it is useful to parametrize the coset space 53 = 5U(2) in the exponential representation U(x) = exp(i7r£> (X)T£> Iv), which is an 5U(2) matrix-valued field (for the case of two flavor QCD, v = f7r' the pion decay constant, as it is shown in Chap. 6). The chiral coordinates 7r£>, should not be confused with the standard coordinates 7r'£> on the sphere (those with 7r,2 + (72 = v 2 ) introduced in Chap. 1. Then, the integer number labeling the homotopy class to which U(x) belongs, is given by (see Appendix B.3) t[U] =

24~2

f

dXfijktr(oiUU+OjUU+OkUU+) ,

(3.45)

where i,j and k run only through the spatial coordinates. In particular, the skyrmion configuration, belonging to the t = 1 sector, is U(X)

= exp(if(lxl)xaTa) ,

(3.46)

where x = x/lxl. The so called chiral angle f(lx\) satisfies the boundary conditions f(O) = 7r and f(oo) = O. For completeness, we will also give the expression for the above non-trivial field configuration in the standard coordinates 7r'£> (3.47) where U(x) = exp (

i7r a (X)T a ) v

=

VI~ - -;2 + i7r,aTa

(3.48)

defines the change from chiral to standard coordinates. Now we will turn to the important point of the mass and stability of the skyrmion. The energy of any static configuration can be obtained from the mass functional, defined in terms of the Lagrangian of the theory as M[U] = -

f

dXL .

(3.49)

For the case of the skyrmion, the classical solution is obtained by minimizing the above functional with respect to the chiral angle f(jxl). From the physical point of view, it is necessary for this solution to be stable. It is important to notice that a general Lagrangian describing the NCB dynamics, like that in (3.16) does not always yield a stable skyrmion. In practice, stability can be achieved, for instance, just by keeping only up to four derivative terms

56

3. The Non-linear u Model

in a certain region of the CI, C2 parameter space, which includes the model originally derived by Skyrme in [11]. Within this region, it is possible to reproduce the pion scattering data [12]' although not with the simple choice of Skyrme. Furthermore, even at one-loop, the data can be made compatible with an stable skyrmion [13]. Up to this moment we have only seen how the classical skyrmion arises and how it is possible to obtain its classical mass Mel. A more realistic approach is to quantize the possible excitations of that skyrmion configuration. At low energies, the relevant excitations are those corresponding to the zero modes, that is, rotations and isorotations. Therefore, neglecting other possible excitations, the skyrmion spectrum can be found by canonically quantizing those zero modes [14]. In this way we describe physical states with well defined spin and isospin quantum numbers. Those states would correspond to the low energy spectrum of QeD in the sector with baryon number equal to one (N, L\ ... ). Thus, as a first step, let us consider a space independent SU(2) matrix A(t) under which the skyrmion U(x) transforms as U ->A-I(t)UA(t) A(t) = b + ibaT a

(3.50) a with b5 + bab = 1. We can now replace the bk degrees of freedom and their conjugated momenta by operators satisfying the canonical commutation relations. Then, it is possible [14] to write the Skyrme model hamiltonian for the rotated U field in terms of the classical mass defined above and two operators 1 and} satisfying the SU(2) algebra. We identify these operators as the isospin and spin of the skyrmion, respectively. Moreover, we have 12 = }2, which is very natural, since in (3.46), isospin and spatial indices appear in a symmetric way. We get

o

b2 =

H = Mel

}2

+ 2.11 .

(3.51)

As it happened with Mel, the A parameter is a functional of f(lxl) and it depends on the constants of the model. It can be interpreted as the skyrmion momentum of inertia. Therefore, within this approximation, the t = 1 skyrmions are determined by the quantum numbers J = I and the energies

HI

J) = (Mel

+ J(~;

1)) I

J) .

(3.52)

In order to determine the physically relevant values of J, we have to know whether the skyrmions are bosons or fermions. It is very clear that bosonic states can be obtained starting from fermionic states by adding angular momenta in the standard way. What it is much more surprising is the possibility of having fermionic states in a theory like the NLSM where the fundamental fields are bosons. In fact, since nucleons are fermions, this should be the case if we want to describe, to some extent, the real world with this model. This

3.7 References

57

property, showed by Witten [15], is actually one of the most strange and important features that the skyrmion states can exhibit. We will briefly outline in what follows the main ideas leading to the quantization of the skyrmion. One possible way to determine whether a configuration is fermionic or bosonic is to perform a 271" rotation in the configuration space Q. The fermionic nature of skyrmions would then be related to the existence of a nontrivial first homotopy group 71"1 (Q) which would allow to define multivalued wave functions on Q. In our case,·the configuration space is Q = {U : S3 -> SU(2)} and 7I"1(Q) = 7l 2 . Hence, the possibilities are in principle opened: the skyrmions may be quantized as bosons or as fermions, corresponding to 1= J = 0,1,2, ... and 1= J = 1/2,3/2, ... respectively. In order to solve this problem, it is necessary to make use of the WessZumino-Witten (WZW) term, which we will study in detail in Chap. 4. This non-local term rwzw[U] has to be added to the NLSM to reproduce the effect of some anomalies present in the underlying theory (i.e. QCD). The key idea is to rotate the skyrmion field U(x) as in (3.50) with t E [0,271"] and A(t) a rotation matrix of angle t and to evaluate the amplitude for the process described by the rotated U(t, x) field. When such process is performed adiabatically, it turns out that the NLSM contribution to the amplitude is the same as that for the static skyrmion. However, the WZW action gives the extra contribution r wzw = iNc 7l", with N c the number of colors, which appears in the WZW term in order to reproduce properly the QCD anomalies. Therefore the amplitude for such a rotation gets a factor (-1 )Nc , that can be understood as an exp (i271" J) factor. In summary, skyrmions are fermions (bosons) if N c is odd (even). As in nature N c = 3 we arrive at the conclusion that skyrmions should be quantized as fermions and therefore the spectrum I = J = 1/2,3/2, ... corresponds to the well known nucleons and delta baryons. The properties of those skyrmion states have been studied in detail in the literature [16] and they provide an appropriate qualitative description (and in some cases even quantitative) of the lower states of the baryon spectrum.

3.7 References [1] [2] [3]

[4]

8. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239 C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247 L. Alvarez-Gaume and P. Ginsparg, Nucl. Phys. B262 (1985) 439 8. Weinberg, Physica 96A (1979) 327 J. Charap, Phys. Rev. D2 (1970) 1115 1.8. Gerstein, R. Jackiw, B.W. Lee and S. Weinberg, Phys. Rev. D3 (1971) 2486 J. Honerkamp, Nucl. Phys. B36 (1972) 130 L. Tararu, Phys. Rev. D12 (1975) 3351 T. Appelquist and C. Bernard, Phys. Rev. D22 (1981) 425 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, New York, 1989 D. Espriu and J. Matias, Nucl. Phys. B418 (1994) 494

58

[5] [6]

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

3. The Non-linear (]" Model S. Kamefuchi, L. O'Raifeartaigh and A. Salam, Nucl. Phys. 28 (1961) 529 S. Weinberg, Dynamic and algebraic symmetries, in Lectures on elementary particles and quantum field theory, eds. S. Deser, M. Grisaru and H. Pendleton, MIT Press, Cambridge, 1970 P.W. Higgs, Phys. Rev. Lett. 12 (1964) 132 S. Pokorski, Gauge Field Theory, Cambridge University Press, Cambridge, 1987 N.K. Nielsen, Nucl. Phys. B140 (1978) 499 R.E. Kallosh, Nucl. Phys. B141 (1978) 141 C. Becchi, A. Rouet and R. Stora, Comm. Math. Phys. 42 (1975) 127 T.H.R. Skyrme, Proc. Roy. Soc. London 260 (1961) 127; Nucl. Phys. 31 (1962) 556 J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Rev. Lett. 53 (1984) 747 A. Dobado and J. Terr6n, Phys. Lett. B247 (1990) 581 G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552 E. Witten, Nucl. Phys. B223 (1983) 422 and 433 I. Zahed and G.E. Brown, Phys. Rep. 142 (1986) 1

4. Anomalies

In this chapter, we will study different types of anomalies, both perturbative and nonperturbative, that appear in quantum field theory (QFT). We will consider in detail their physical implications as well as different regularization methods, consistency conditions and their topological interpretation. Our analysis will also include anomalies in the non-linear u model (NLSM) studied in Chap. 3. The connection with effective Lagrangians is established in detail through the Wess-Zumino-Witten anomalous term. To complete our study, we will also discuss the anomalies that arise due to scale transformations, known as trace anomalies.

4.1 Introduction Anomalies constitute an important feature of QFT with several crucial physical implications, both from the phenomenological and theoretical point of view. Since they were first discovered [1, 2, 3] using Feynman diagrams, they have been proved to playa fundamental role in many relevant contexts as, for instance, the decay of the neutral pion into two photons, the violation of the baryon and lepton number in the standard model (SM) or the understanding of the U (1) A problem. Furthermore, the cancellation of gauge anomalies is a fundamental requirement that any consistent gauge theory should satisfy. To be precise, we will say that we have an anomaly when a classical symmetry of the theory is absent at the quantum level. As we have seen in Chap. 2, any symmetry in a QFT leads to the corresponding Ward Identities (WI) for the Green functions. Anomalies will arise when extra (anomalous) terms appear in the WI due to the fact that the path integral measure is not invariant under that symmetry. The structure of this chapter will be the following: In Sects. 4.2 and 4.3 we will discuss the global axial anomaly both with diagrammatic and functional methods. Sects. 4.4,4.5,4.6 and 4.7 will be devoted to the analysis of gauge non-Abelian anomalies. We would like to remark that Sect. 4.7, in which we analyze the topological interpretation of gauge anomalies, is rather technical and can be omitted on a first reading. In Sects. 4.8, 4.9, 4.10 and 4.11 we treat the non-perturbative SU(2) anomaly, the one in the NLSM, the Wess-Zumino-Witten term [4, 5] and the trace anomaly respectively. A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

60

4. Anomalies

4.2 The Axial Anomaly, Triangle Diagrams and the 7r 0 Decay We will start with the Abelian axial anomaly in the simplest example in which most of the features of anomalies appear, that is, quantum electrodynamics (QED). The Lagrangian in Minkowski space is Leff

=

-~FJLvFJLV + ~(i 1lJ- m)7/J

(4.1)

with

FJ.LV =8J.LA v - 8vAJ.L DJ.L7/J = (8J.L - ieAJ.L)7/J .

(4.2)

The above Lagrangian is invariant under local U(l) transformations 7/J ---> eia (x)7/J so that the vector current, defined as jJ.L = ~'YJ.L7/J, is conserved. If the fermion was massless, there would be an additional invariance under global axial transformations 7/J ---> eia'Y57/J, whose associated Noether current is jJ.L5 = ~'YJ.L'Y57/J. For m -=J. one has, classically 8J.LjJ.L5 = 2imj5

°

(4.3)

8J.LjJ.L=0

with j5 = ~'Y57/J. As we mentioned in Chap. 2, when dealing with symmetries in QFT we are interested in the WI of Green functions containing current operators which, in the present case, are the vector and axial-vector ones. The problem is whether it is possible or not to maintain relations like those in (4.3) when quantum corrections are taken into account. For the consistency of the theory we expect the vector current conservation law in (4.3) to hold at the quantum level too. This is necessary since we could not gauge a symmetry (make it local) if it was not globally conserved. However, in principle, there is no reason for the axial current to be conserved in the quantum theory. In fact, we will see in a moment how the conservation of the vector current implies necessarily the appearance of an anomaly in the axial current. We will first show it by using diagrammatic techniques, as it was originally derived in [1, 2, 3]. To obtain the WI containing the axial current, we consider the following three-point Green functions:

GJ.Lv>,(Xl, X2, X3) = (0IT(jJ.L(xl)jv(x2)j~(x3) )10) GJ.Lv(Xl, X2, X3) = (0IT(jJ.L(xl)jv(x2)j5(x3))10) ,

(4.4)

and their corresponding Fourier transforms

GJ.Lv>.(-q,k 1 ,k2)= GJ.Lv(-q,k 1 ,k2)=

J J

d4xld4x2GJ.Lv>,(Xl,x2,0)eiklXl+ik2X28(q - k 1 d4xld4x2GJ.Lv(Xl' X2, 0)eiklxl+ik2X28(q - k 1

-

-

k 2) k2). (4.5)

4.2 The Axial Anomaly, Triangle Diagrams and the

71"0

Decay

61

Then, (4.3) implies the following identities in momentum space:

kiCJ-Lv),( -q, k 1, k 2) =0 = k~CJ-Lv),( -q, k 1, k 2) ),

-

-

(4.6)

q GJ-Lv),(-q,k1,k2)=2mGJ-Lv(-q,k1,k2),

which are obtained by differentiating the T-product in (4.4) and using equal time commutation relations between currents, like [j8(X1), jJ-L(x2)J8(x~ -xg) =

o.

'YV Fig. 4.1. Anomalous triangle diagrams

The relations in (4.6) would be the WI arising directly from the classical invariance. However, we also have to take into account that quantum effects (loops) can modify these identities. At one loop, the diagrams which correspond to the above three-point Green functions are those in Fig. 4.1. Their analytical expression is given, respectively, by

TJ-Lv),=

J

d4p (2n)4(-I)tr /1

+ [k1 TJ-Lv=

1/

k2

]

J

d4p (2n)4 (-I)tr /1

+ [ k1

1/

k2

[i p_

[i p_

m 'Y),'Y5

m 'Y5

p_

p_

i

i]

rj_m'Yv p_ /(l_ m 'YJ-L

i

i]

rj _ m 'Yv p_ /(1 _ m 'YJ-L

(4.8)

]

.

Now, by using rJ'Y5 = 'Y5(P- rj - m)

q),TJ-Lv>, = 2mTJ-Lv - i,1J-Lv

(4.7)

+ (p -

mh5

+ 2m'Y5

we get

(4.9)

62

4. Anomalies

with LlJ.lv

=

J i p- /C2 d4p

(21l")4 tr

[

i

i

p _ m l5'Yv p_ /Cl _ m II'

m IS IV

i ] + [/-L1 p- fIm II'

k

(4.10)

+--+ 1/ ] +--+ k 2

Naively, one could argue that the change of variables p ---+ p+ }t2 in the second piece in the r.h.s of (4.10) would give LlJ.lv = 0 and hence we would get (4.6) for the WI. However, the integrand of that piece in LlJ.lv does not vanish at p ---+ 00, giving rise to a linearly divergent integral. Hence we are not allowed to perform the above momentum shift. Furthermore, TJ.lv>, is also linearly divergent, so that we could have changed our choice of momenta in the diagram associated to TJ.lv>" simply by shifting p ---+ p + a thus obtaining a different result. In general, in d space-time dimensions, if f(p) is such that limp-+oopd-l f(p) f= 0 and limp-+oopd-l f, in (4.7) by p ---+ p + a in the first integral and by p ---+ p + b in the second one (that corresponding to the interchange of /-L, 1/ and k 1 , k2). Let us call the result TJ.lv>.(a, b). Then, by applying (4.11) and taking the Dirac traces in the leading term when p ---+ 00 we get -i TJ.lv>.(a, b) - TJ.lv>,(O, 0) = 81l"2 fpJ.lv>.(a - b)p .

(4.12)

Now, in order to calculate the WI, we multiply q>'TJ.lv>.(a, b) and take into account that LlJ.lv is again a shifted linearly divergent integral, so that (4.11) can be used once more to calculate q>'TJ.lv>,(O, 0). We obtain q>. TJ.lv>, (a, b) -_ 2mTJ.lv

i fpJ.lv>' + 41l"2

(1-"2 >'(

q a - b)P

+ k 1Pk2>.) ·

(4.13)

Notice that TJ.lv is finite and there is no shift ambiguity in the momentum integral. In addition, by following similar steps, we can also calculate kiTJ.lv>, and k 2TJ.l v>' . We arrive to

kiTJ.lv>.(a,b)=~2f(fPv>.ki(a 81l"

t

b - k2

k~TJ.lv>.(a,b)=~2f(fPJ.l>.k~(a-b+kl)(f. 81l"

(4.14)

We observe that there is no value for a - b that could make (4.13) and (4.14) satisfy the identities in (4.6). To select a value for a - b we have to impose, as commented before, the WI corresponding to the conservation of the vector current. Thus, we obtain a - b = k 2 - k1 and the anomalous WI becomes

4.2 The Axial Anomaly, Triangle Diagrams and the

0 1r

Decay

63

(4.15) which is the Adler-Bell-Jackiw (ABJ) anomaly for the axial current in momentum space [1, 2, 3]. It is important to observe here that higher order corrections to the triangle graphs do not give rise to any new contribution to the anomaly [6, 7]. This is basically due to the fact that those corrections are finite and then one can perform the previously discussed shifts in the integration variables. Remembering that jJ1. couples to the photon field AJ1.' the WI in (4.15) can be translated into an identity for current operators in position space: 2

!:l (.J1.5)-2· (.5) e J1.Vp U F F uJ1.J - zmJ - (471")2(; J1.V pu

0J1.(jJ1.) =0 .

(4.16)

Let us now turn to a physical application of this formula, in connection with the decay of the 71"0 meson. For this purpose we have to introduce fermion fields contained in a multiplet of some internal symmetry group. In this way, the theory under consideration has a bigger group of classical symmetries. Thus, we will need to generalize the Green functions considered in (4.4) in order to introduce the new group indices. That is G~~c,\ (Xl, X2, X3) = (0IT(j~(xI)j~(X2)j~C(x3))IO) G~~C(XI' X2, X3) = (0IT(j~(xt}j~(X2)j5C(X3)) 10) ,

(4.17)

where the non-Abelian vector, axial and pseudoscalar currents are defined, respectively, as: jJ1.a ="ifi-yJ1.Ta'lj; jJ1.5a = "ifi-yJ1.-Y5Ta'lj; j5a = "ifi-Y5Ta'lj;

(4.18)

with Ta the generators of the symmetry group. A good example is quantum chromodynamics (QCD), where we know that in the limit of massless quarks the Lagrangian is invariant under the SU(Nf)V x SU(Nf)A x U(l)A x U(l)v global group, N f being the number of light flavors. The SU(Nf)v x SU(Nj)A symmetry is spontaneously broken to the SU(Nf)v group of isospin. As we will see in detail in Chap. 6, we can identify the Nambu-Goldstone Bosons (NGB) of this symmetry breaking pattern with the pions when N f = 2, plus the kaons and the TJ meson when Nf = 3. Thus, the vacuum is not invariant under SU(Nf )A. Consequently, the axial charges Q5a constructed in the usual way from the axial currents j~a in (4.18), do not annihilate the vacuum, which implies qJ1.(0Ij~aI7l"a) =J 0, 1r a denoting the NGB (see Sect. 2.4). That is, the NGB have the same quantum numbers as qJ1.j~a and therefore we can use the Green functions involving

4. Anomalies

64

these currents to describe the pion dynamics. In particular, the a = 3 state corresponds physically to the neutral pion. Here we are interested in the 7f o decay into two photons, which is the dominant channel [8]. To describe this process, our only assumptions will be the spontaneous symmetry breaking mechanism and the minimal coupling between photons and the quark fields, through the vector electromagnetic current (4.19) That is, Q is the quark electric charge matrix. Notice that, for the moment, we have set N f = 3. We will then consider a Green function as the first in (4.17) where we take c = 3 and Ta = Tb = Q. We shall denote it by c~~t Here, T 3 is the third generator of SU(3) in the fundamental representation

T'

~.\' /2 ~ ~

C-1 0)

(4.20)

with Aa the Gell-Mann matrices. Following our previous derivation, we multiply the Fourier transform of the Green function by q>'. The traces of the group generators can then be factorized out and the remaining terms are precisely those calculated previously for the Abelian axial WI. Hence, the result is q

>.

QQ3 _

CJ.Lv>.

-2m

CQQ3 J.Lv

i

+27f 2tr

(Q2 2

A3 )

€J.LVPU

kP k U

(4.21)

1 2'

Now, let us show why the presence of the anomalous term in (4.21) is essential to correctly reproduce the 7fo ->" decay observed experimentally. This can be seen by inserting a complete set of states in the C~3.3 Green function and then taking into account that, due to the Goldstone theorem, j~3 has the same quantum numbers as the 7fo. In this way, the l.h.s of (4.21) can be related to the 7fo -> " decay amplitude as follows

b(k1,cl) ,(k2,c2)I7fo(q)) = i(27f)48(q - k 1 - k2)ci(kl)C2(k2)R~~7r(kl' k2,q)

R~~7r =

e2

J

d4xd4yeiktx+ik2Y(0IT

Jzm(x)J~m(Y)I7fO(q))

,

(4.22)

where Cl and C2 are the photon polarization vectors. Then, it turns out that the contribution from the non-anomalous term (that proportional to the quark mass) in (4.21) is not enough, by far, to reproduce the experimental value at low energies. On the other hand, the anomalous piece in (4.21) contributes to the amplitude at low energies (q -> 0) as

4.3 The Axial Anomaly and the Index Theorem

J~ R~~7r =

2.

65

2N

1;7[2 17r E~vpaki kg ,

(4.23)

which reproduces the experimental result. Notice that we have included an additional multiplicative factor of N c = 3 due to the color charge carried by the quarks. We remark that if we had taken N f = 2 then the result would have been the same, consistently with the fact that the pion is only composed of u and d quarks. Any other flavor does not play any role in this process. Thus, we see how if the WI was not anomalous the 7[0 would not decay at low energies (qA -> 0) in the massless limit. To conclude this section we would like to comment on the relation between the axial anomaly and the so called U(I)A problem, due to the absence of the U(I)A symmetry from the QCD spectrum. We cannot get rid of it by assuming an spontaneous symmetry breaking of U(I)A since, then, there should be a ninth Goldstone boson, with a mass of the same order of the pion mass, which is not observed in Nature. In fact, the lightest particle of the spectrum with the required quantum numbers is the 7]'(960)[8], which is too heavy compared to the NGB of SU(3) (that is, pions, kaons and the eta). However, as we have seen above, the U(I)A symmetry is not a real symmetry of the theory, due to the presence of the axial anomaly. Hence, in principle, we can think that the 7]' would acquire mass, somehow, as a consequence of the anomaly. That is, in the chiral limit it will not become massless as it happens with 7[, K and 7]8. Nevertheless, it is not clear which is the precise mechanism that quantitatively explains this property. In fact, different points of view on this subject can be found in the literature. For instance, we refer to [9, 10] and references inside those works.

4.3 The Axial Anomaly and the Index Theorem In the previous section we have seen how the axial anomaly appears in oneloop perturbative calculations. It can be shown that this result remains valid to all orders in perturbation theory [7]. On the other hand, in Chap. 2 it has been explained how to obtain generalized WI using functional integral methods. Here, we shall develop this technique in order to obtain anomalies and relate them to topological results. As an starting point, we will focus again on the Abelian axial anomaly, leaving for the next sections and Chaps. 5 and 8 the discussion of other cases of physical interest. From now on we will extensively use the functional integral method to calculate anomalies that was first developed by Fujikawa [11,12]. In these works, the connection between anomalies and the Atiyah-Singer index theorem was already noted. As it is shown in Appendix B.3, this theorem relates the index of the Dirac operator, which depends on the zero modes of its spectrum, with certain topological invariants, that depend on the gauge field. This link with topology has been extended [13], to other kind of anomalies, including the

66

4. Anomalies

gravitational ones (see Chap. 8) and to arbitrary dimensions of space-time. Furthermore, as we will see in detail in Sect. 4.7, gauge anomalies can be reinterpreted in terms of topological results. However, as we will discuss in Sects. 4.5 and 4.6, there are still some discrepancies between different methods, not affecting in any case the physical results. We will start by calculating the U (1) A anomaly for the case of a vector-like theory described by the following Lagrangian in Minkowski space (4.24) with f1J = 'YJ.L(8J.L + AJ.L) and AJ.L the vector gauge field which in general will be considered as non-Abelian (that is, we will deal with a QCD-like theory). For most of the applications in this book we will restrict ourselves to the SU(N) case. The functional integral methods that we are going to use, as well as the topological results, are only well defined in Euclidean space-time. For this reason we perform the Wick rotation as it is indicated in Appendix A, so that the gammlj. matrices and the if1J operator become hermitian. With the conventions in that appendix the Euclidean effective action for the gauge field is

e-r[A] = j[d'lj;][d1jjj exp ( - j d4 x1jj( f1J+ m)'Ij;)

== j [d'lj;][d1jj]e-SE[A,""'~J .

(4.25)

As explained in Chap. 2, to derive the WI associated to the U(l)A global symmetry of the classical Lagrangian, we have to consider local transformations on the fermion fields, that is

'Ij;-+eio:(xhs'lj; 1jj -+ 1jjei o:(x hs

.

(4.26)

When we consider infinitesimal transformations, the first order variation of the Euclidean Lagrangian under (4.26) is

bSE = i j d4 xa(x) [8J.L(1jj'YJ.L'Y5'1j;) - 2m1jj'Y5'1j;] .

(4.27)

If the theory was not anomalous we would get, by applying the techniques described in Chap. 2, the WI corresponding to the conservation of the axial current in the massless limit. However now we have to consider the Jacobian in the variation of the measure [d'lj;][d1jj] under the transformations in (4.26), which is going to give us an extra term in the WI. To calculate this Jacobian factor we expand the fermion fields in terms of the eigenfunctions of i1,1J, which form an orthonormal basis, since if1J is an hermitian operator. In order to have a well-defined eigenvalue problem we consider the Euclidean spacetime IR4 compactified to the sphere S4 with the usual boundary condition of vanishing fields at infinity. Then we can consider the set of eigenvalues and eigenfunctions of the Dirac operator, defined as

4.3 The Axial Anomaly and the Index Theorem

with

J

d4x4>;;,(x)4>n(x)=8nm

'lj;(x) = L

an4>n(x)

67

(4.29) (4.30)

""if = Lbn4>~(x) , n

n

where 4>n = (xln) and the coefficients an and bn of the expansion are elements of an anticommuting or Grassmann algebra. Then the measure is written as [d'lj;][d""if] = I1nm damdbn , which follows from the unitary change of basis defined in (4.30). The transformation in (4.26) implies that the coefficients an change as follows 'lj;

a~ =

->

eiet(xhs'lj;= L

LAmnan

a~4>n

A mn =

J

(4.31 )

d4x4>;;,(x)eiet(xhs4>n(x) .

n

Due to the anticommuting nature of the Grassmann variables an, the Jacobian of the transformation is given by the inverse of the determinant of A, i.e, [d'lj;] -> (detA)-l[d'lj;], which we can exponentiate again by making detA = exp(trlogA). The same arguments apply to [d""if] which gives the same contribution to the change in the measure. Infinitesimally we get

[d'lj;][d""if]->[d'lj;][d""if]exp (-2i A(x)

J

d4xa(x)A(x))

= L 4>~(xhs4>n(x) .

(4.32)

n

The anomaly factor A(x) is the one that, in the end, will modify the classical symmetry relations. However, that factor is not well defined as it is written in (4.32) and, therefore, we have to give it sense by means of some regularization method. We will comment on different possible regularization methods in Sect. 4.5. In our case, as i$>is hermitian, the most natural choice, compatible with the symmetries of the theory, seems to be that used originally by Fujikawa, in which a cutoff M is introduced as

A(x) =

A~

lim '"" 4>~(x)e-M'I,s4>n(x) =

M~=LJ n

(ij1»2

lim tr(xhse- Mr Ix),

M~=

(4.33)

where the trace runs over Dirac and internal indices. Comparing with (C.83) in Appendix C.4, we see that A(x) in (4.33) is written as the heat kernel G(x, x; M- 2). Furthermore, we note that (i $»2 = (aIL + A IL )2 - [TIL, ,v]FILv with FILv = aILA v - avAIL + [AIL' A v] is a second order elliptic differential operator. Then, following the techniques developed in Appendix CA, we can expand the anomaly factor in (4.33), in the 11M2 parameter. This is nothing but the Seeley-DeWitt expansion in the coincidence limit, displayed in

i

68

4. Anomalies

(C.92) in Appendix C.4 with coefficients an. From the expressions of the first coefficients in (C.99), we see that ao and a1 vanish after taking the Dirac trace. Thus, in the M - t 00 limit, only a2 survives, yielding (4.34) where f4123 = +1 and the trace runs now only over internal indices. Now, we include this term in the variation of the effective action, together with oSE in (4.27) and impose invariance under the change of integration variables. Then we get the WI 8 (J0IJ-s) = 2m(Jos) IJ-

+ _l_ lJ- vpC7 F F 16n 2trf IJ-V pC7'

(4.35)

which, in Euclidean space, is the same result that we had obtained analyzing triangle graphs (for the Abelian case). We now turn to the connection between the results obtained here and topology, through the index theorem. We refer to Appendix B.3 for the results and definitions concerning this important theorem. Let us consider the case of the Euclidean Dirac operator flJ = 'YIJ- (81J- + AIJ-) defined on the space-time manifold S4, for which the index theorem reads . . md tflJR

== n+

- n_ =

-1 -()2 2n

1 84

1 2 -trF, 2

(4.36)

=-i

v f lJ- PC7 FlJ- v FpC7 , iflJR = iflJPR and n± are the number of zero where trF 2 modes
sn

4.4 Gauge Anomalies So far we have considered anomalies in global (non-gauge) currents and it has been shown how they can be related to the index of the corresponding

4.4 Gauge Anomalies

69

Dirac operator. In the following we will concentrate in the case of non-Abelian gauge currents [13]. The absence of this kind of anomalies obviously possesses an enormous importance for the consistency of the theory, since they destroy gauge invariance, which is basic for unitarity and renormalizability. In this and the next sections we will obtain an explicit form for the non-Abelian anomaly by means of the so called Fujikawa method. We will also study the topological interpretation in terms of the index of certain Dirac operator and the Wess-Zumino consistency conditions. Let us begin by considering a chiral gauge theory in which the gauge field A couples only to right-handed fermions 'l/J (of course, the same is valid for left-handed fermions), and 'l/J transforms in a complex representation of the gauge group G. The effective action for this theory r[A] is written as e-r[AJ = J [d'l/J] [d;;J] exp (- J;;J IJR'l/J) ,

(4.37)

where (4.38) is the Dirac operator acting only on right handed fermions. The gauge current (4.39) is conserved at the classical level. Under an infinitesimal gauge transformation

9 = 1 - v with v = -ivaTa, the gauge field transforms as

+ 0iL)g ~ AiL - 0iLV - [AiL' v] = AiL - DiLv with the usual covariant derivative definition DiLv = 0iLV + [AiL' v]. AiL

->

g-l(A iL

same transformation the effective action changes as

r[A]

->

r[A - Dv]

=

r[A] - J dx (DiLv)aO;l;] .

(4.40)

Under the

(4.41)

Now it is straightforward to obtain the second term of this equation

or[A] oAa = -ig('l/J,·-tTaPR'l/J) = _ig(jiLa) , iL

(4.42)

and accordingly

ovr[A] == r[A - Dv] - r[A] = -i J dx va (DiL(jiL))a ,

(4.43)

where we have integrated by parts once. Therefore, the violation of gauge invariance is given by the non-conservation of the expectation value of the gauge current. Before presenting the explicit calculation of the effective action anomalous variation in (4.43), we will make some comments about how to define properly the left hand side of this equation, which will be useful when discussing the topological interpretation of this kind of anomalies.

70

4. Anomalies

Following the general scheme of gaussian integration, one could try to define e-r[A] as det( JbR) = It Ai, where Ai are the eigenvalues of i JbR' i JbR1/Ji = Ai1/Ji· However, the eigenvalue problem of i JbR is not well defined since i JbR maps positive chirality states, "(51/JR = 1/JR, to negative ones, "(51/JL = -1/JL and the negative ones to zero. Therefore i WR1/JR = >"1/JR does not make any sense. To avoid this difficulty we can formally introduce a Dirac spinor 1/J including both chiralities and define (4.44)

where

D = "(J.L(8J.L + AJ.LPR )

(4.45)

.

Now the eigenvalue problem is well posed since iD maps the space of Dirac spinors with both right and left chirality components into itself. Moreover, D has only gauge couplings to right fermions and, up to a global factor independent of the gauge fields, (4.44) defines the same gauge theory as (4.37). It will be useful later to note that iD only has right chirality zero modes which are precisely those of i WR' since i qL does not have any non-trivial zero modes. The iD operator is not hermitian and in general its eigenvalues will be complex and not gauge invariant, since g(D(A9))g-1 =1= D(A). However the absolute value of the eigenvalue product is indeed gauge invariant. This fact can be shown as follows, let us write Idet DI 2 = (det D)(det D+) = det(

= det( fJ L fJ R)(det W) ,

fJ L fJ R) det( WR JbL) (4.46)

where (4.47) is the ordinary Dirac operator. The Dirac determinant det Wcan be shown to be gauge invariant and therefore Idet DI 2 is also gauge invariant. This is equivalent to the gauge invariance of the real part of the effective action since

exp(-2Re(r[A])) = exp(-r[A] - r*[A]) = (detD)(detb+) <X (det W) .

(4.48)

Consequently only the imaginary part Imr[A], Le, the phase of det D may have an anomalous variation under a gauge transformation. In general, it can be shown [13] that gauge anomalies only appear when fermions transform in a complex representation of the gauge group, as in the case we are considering here. Moreover, the real part of the effective action is always gauge invariant whereas the imaginary part is responsible for the anomalous variation. Finally, let us compute explicitly the anomalous variation in (4.43) that, as we have just seen, only affects its imaginary part

4.4 Gauge Anomalies

8v (Imr[A]) = -i

J

dxva(D/J.(jlL))a .

71

(4.49)

The l.h.s. of this equation can be evaluated with different methods. We will first follow the Fujikawa procedure outlined in Sect. 4.3. Under an infinitesimal gauge transformation, we have

A-+A - Dv 'l/J -+ (1 - ivaT aPR)'l/J 1jj -+ 1jj(1 + ivaT aPL) .

(4.50)

Note that the gauge transformation only affects the right handed spinors, which are the only relevant modes in this model. The Jacobian factor for this transformation can be calculated in a similar fashion as we did in Sect. 4.3 for the global U(l)A case, so that 8v r[A]

=

-i

J

dx va(x) I:(nlx)T a,5(xln) .

(4.51)

n

From now on we will use the following notation: (xln) = ¢n are the eigenfunctions of D acting to the right, i.e., D¢n = An¢n and (nix) = x;t are the eigenfunctions of D acting to the left, that is, x;t D = A~X;t. It is important to stress that since iD is not an hermitian operator the right and left eigenfunctions are not in general the same. Moreover, it is possible that they could not constitute a complete set of eigenfunctions (see Sect. 4.5). In any case we will assume that the following analysis is justified. In order to give sense to the integral we may use again a formal gaussian regulator (see Appendix C.4)

8vr[A] = lim -i M --+(X)

= =

lim -i

M-+<x>

2~;2

J

J J

dx lim va(x) y-+x

dx lim

y-+x

I: x;t(Y)Ta,5e-~¢n(x) n

va(x)tr(YITa,5e-~ Ix)

dx vatr (Tat:I<,)./J.vOI<[A>'O/J.Av +

~A>.A/J.Av])

(4.52)

Therefore, using (4.43) the final expression for the anomaly in the gauge currents is

(D/J.(j/J.))a =

24~2tr (Tat:/J.vPlTO/J.[AvOpAlT + ~AvApAlT])

.

(4.53)

This result was first obtained perturbatively by Gross and Jackiw [14] and satisfies the Wess-Zumino (WZ) consistency conditions [4] that we will introduce shortly. As we have already commented, the origin of the anomaly in this approach lies in the impossibility to define a gauge invariant regulator for the Jacobian. We will show in Sect. 4.5 that in the literature there have been proposed different regulators which in general give different results. Notice that (4.53) does not transform covariantly. The use of a different

72

4. Anomalies

regulator can yield a covariant form of the anomaly although it violates the WZ consistency conditions.

4.4.1 The Wess-Zumino Consistency Conditions We are going to derive an alternative procedure to obtain the anomaly in nonAbelian gauge currents [4]. The basis of this method lies in the conditions that gauge symmetry imposes on the variation of any functional of the gauge fields. In order to show these constraints, let us define the generator of infinitesimal gauge transformations bv acting on an arbitrary functional p[A], as we did for the effective action. In a given manifold M, it is nothing but

bvp[A]

=

p[A - Dv]- p[AJ

=

fM tr ( VD J1. b~J1. P[AJ) ,

(4.54)

and we identify

bv =

fMtr(vDJ1.b~J1.)

(4.55)

as the infinitesimal generator of gauge transformations for such functionals. The generators Ov should satisfy the group commutation relations

[b Vt ,OV2] =

(4.56)

O[Vt,V2] .

The anomalous variation defined in (4.41) then should satisfy

bVt f tr (V2 DJ1. :~) - bV2 f tr (V 1 DJ1. :~)

=f

tr

([Vl,V2]DJ1.:~) ,

(4.57)

which are the WZ consistency conditions. It is possible to show [15] that the only solution, up to a normalization factor, of the above conditions is the following anomalous variation of the effective action:

f tr

(VDJ1.:~) ex f

(4.58)

Ql(v,A),

where Q~ is the first-order variation of the Chern-Simons form (see Appendix B.3)

Ql(v, A) = tr [Vd(AdA +

~A3)]

,

(4.59)

which gives precisely the result in (4.53) (up to the normalization factor) obtained through the gaussian regularization of the anomalous Jacobian.

4.5 Regularization Methods

73

4.5 Regularization Methods In this section we will concentrate on some different regularization methods proposed in the literature. As noticed in Sect. 4.4, the Dirac operator for a theory with chiral gauge couplings (that is, the gauge couplings of the right and left-handed components of the fermion field are different) is not hermitian. However in the Fujikawa functional methods, the hermiticity of the operator seems to be an essential feature in order to have a well defined regularization procedure. For instance, when the anomalous Jacobian is written in terms of the Dirac operator eigenfunctions, it is assumed that they form a complete orthonormal set, which in general is not true for a non-hermitian operator. Furthermore, if a Gaussian cutoff is introduced in order to regularize the measure (as in (4.52)) the change of basis to "plane waves" to calculate the coefficients of the expansion in inverse powers of the cutoff (see Appendix C.4) would not be justified. However, in Sect. 4.4 we have assumed that this procedure is well defined and then we have arrived to the so called consistent form of the anomaly (that satisfying the WZ consistency conditions). Therefore, for that case, this approach would be justified, which is confirmed with the topological analysis that we will present in Sect. 4.7. Moreover, this is the only functional method giving rise to the consistent anomaly. In contrast, in this section we will analyze two alternative methods to regularize the anomaly, proposed originally in [12], which avoid the difficulties appearing when working with non-hermitian operators and can be applied to more general situations. We will show that, for the same Dirac operator considered in Sect. 4.4, they yield a different form of the anomaly, known as covariant, since it transforms covariantly under gauge transformations. Remember that, as we previously remarked, the consistent anomaly is not gauge covariant. In the general case, when both right and left-handed couplings are present in the Lagrangian (for instance in the SM) the Dirac operator is of the form (4.60) Let us show how the above mentioned functional methods work for the calculation of the anomaly.

• Method 1 This method is based in the fact that, for the Dirac operator in (4.60) it is always possible to separate

~ WJ=~L JPB'l/JL + ~R JPA'l/JR JPB=·../'(fJJ1, + EJ1,) JPA ='YJ1,(fJJ1, + AJ1,)

(4.61)

and then consider 'I/J Land 'I/J R as independent integration variables in the functional integral. Notice that the mass terms which would mix the Land R sectors are omitted here. In any case, their presence would not change

4. Anomalies

74

our conclusions, since these terms do not contribute to anomalies for any physically relevant example. The operators i 1,1.>B and i f/JA are hermitian in Euclidean space. Then we can split the fermionic measure into the Land R parts as [d'l/J][d:;j}] = [d'l/JR][d:;j}R][d'l/JL][d:;j}L] and regularize each part separately with the Fujikawa method using f/JB and f/JA respectively. To compare with the results in Sect. 4.4, let us consider the case of a right-handed fermion multiplet, that is, we set BI-' = O. We remark that all our analysis in this section can be worked out similarly for the BI-' =I- 0 case, which is in fact the case of the SM Lagrangian (see Chap. 5), leading to the same conclusions. For BI-' = 0 we then have (4.62)

:;j} l,1.>t/J =:;j}R f/JA'l/JR +:;j}L ~'l/JL ,

and we can expand the field 'l/J R as 'l/JR= Lan¢~ n~O

(4.63) where if/JA¢n =An¢n ¢~,L=

.j2PR,L¢n, {

JdX¢~¢m=8nm

PR,L¢n,

An

=0 (4.64)

.

Notice that, with the normalization in the above equation, not only the ¢n eigenfunctions, but also the ¢;;,L are orthonormal, since {f/JA' "Ys} = O. Now we compute in this basis the Jacobian of the measure for the local transformation 'l/JR -+eiTaaa(X)'l/JR -;:r; -;:r; e-iTaaa(x) (4.65) 'f/R-+'f/R . Following the same steps as in Sects. 4.3 and 4.4 we get the infinitesimal anomalous change of the measure dJ-t-+ dJ-texp ( -i

with dJ-t =

J

(4.66)

dXaa(x)Aa(x))

TInm dandbm and the anomaly factor

Aa(x) given by

(4.67) n

n

Notice that, up to an overall 1/2 factor, this result is formally the same as that of the axial anomaly for a vector-like theory, analyzed in Sect. 4.3.

4.5 Regularization Methods

75

2

Now, we regularize the anomaly in (4.67) with the e-(ifJA)2/M factor in the same way as in previous cases. The final result for the divergence of the gauge vector current is then

(D!,(j!,))a =

~2€!'Vpcrtr(TaF!'vFpcr) .

(4.68)

321f

As we had anticipated, we observe that this result (the covariant anomaly) differs from that previously obtained for the consistent anomaly in (4.53). It is easy to check that the above expression for the anomaly indeed transforms covariantly under gauge transformations. • Method 2 We now turn to discuss another possible regularization procedure, which makes use of two hermitian operators, defined as Ht/J = I;} .wand H1f = .wI;}, .w being the Dirac operator. Their eigenfunctions Ht/J'Pn

=

/-Ln'Pn

(4.69)

H1f
J

J

then form an orthonormal basis (that is, dX'P~'Pm = dX
'l/J

eiaa(x)T aPR'l/J

""iii

""iiie-iaa(x)TaPL .

(4.70)

The regularization of the Jacobian is performed now with the operators

Ht/J and H1f for [d'l/J] and [d""iii] respectively, thus yielding the anomaly factor Aa(x) =

2) 'P~TaPR'Pn -

n

= Mlim

_00

=

2

2

2:('P~Ta P Re- H ,,/M 'Pn -
.J~oo 2:('P~Ta PRefJ~/M2 'Pn -
=

lim tr(xh'se-(ifJA)2/M2Ix) ,

M->oo

(4.71)

so that we arrive to the same result obtained applying method 1. This equivalence between these two basis has been shown in more general cases [12]. To summarize, we have shown that there are two possible forms for the non-Abelian gauge anomaly in a theory with chiral couplings, namely the

76

4. Anomalies

consistent and the covariant anomaly. In Sect. 4.6 we will show that it is possible to obtain one form of the anomaly from the other by redefining the gauge current. In any case, what is very important to remark is that both forms of the anomaly require the same conditions for their cancellation, as we will see in detail in Chap. 5 for the SM. Therefore, the difference between them cannot lead to any observable consequence. The reason is that the physical requirement that any theory should satisfy is precisely gauge invariance, through the cancellation of the gauge anomaly, as indeed happens in the SM. In contrast, we remark again that axial anomalies do have observable effects.

4.6 Ambiguities and Counterterms In Sect. 4.5 we have seen how gauge anomalies depend on the regularization procedure when the Dirac operator of the theory is not hermitian. In the following we will try to clarify this issue and show what is the relation between different forms of the anomaly [16]. In Sect. 4.4 we saw that gauge anomalies are defined as the gauge variation of the effective action or, in other words, that of the Dirac determinant in the presence of external gauge fields. Let us remark that the determinant of a differential operator is a singular object which requires regularization and, in general, renormalization in order to give it sense. The renormalized determinant is not unique, depending in general on the renormalization prescription. Since the divergences are always local polynomials in the external fields, one can take into account the different renormalization prescriptions by adding suitable local counterterms to the effective action. That is, anomalies, considered as the gauge variation of the effective action, are defined modulo the variation of a local functional in the gauge fields. This is a first source of ambiguity. On the other hand, as we saw in Sect. 4.7, the very definition of gauge anomalies as the variation of a functional imposes them certain consistency conditions. In the case of non-Abelian chiral anomalies, it can be shown that these consistency conditions imply that the anomaly cannot have a covariant expression. The anomaly, in turn, implies that the gauge current cannot transform covariantly. In spite of this fact, we have shown that some methods give for the anomaly expressions which are covariant. Let us clarify this issue. The consistent anomaly satisfies the WZ conditions given in (4.56) and this anomaly determines the transformation properties of the non-Abelian current, defined as = br[A] (J.J.L) a bAa· J.L Let us consider the gauge variation

bvr[A] =

f

dx

15;1;] bvA~ =

-

(4.72)

f dx(j~)(DJ.Lv)a f =

dxvaC a ,

(4.73)

4.6 Ambiguities and Counterterms

77

where G a = (DJ1- (jJ1-))a is the consistent anomaly. Now consider also the transformation defined as

8BA~ =B~ ,

(4.74)

where, by definition, B~ does not change under gauge transformations. Accordingly, the effective action changes as

8Br[A] =

J 8;1~] 8BA~ Jdx(j~)B~ =

dx

.

(4.75)

The commutator of these two transformations can be directly shown to satisfy

8B8v - 8v8B == 8[B,v] .

(4.76)

Now, acting with this operator on the effective action, we find

1 = - Jdx(j~)([BJ1-'

(8 B8v - 8v8B )r[A] =

(8v(j~))B~}

dx{8B(G a)V a -

v]t .

(4.77)

From this expression we can obtain the gauge transformation properties of the non-Abelian current (4.78) The first term in the right hand side gives the covariant transformation rule, while the second one is imposed by consistency. Accordingly, the gauge current (j~) will be covariant only if the anomaly vanishes. We can construct a covariant non-Abelian gauge current from the consistent one. We only need to find a local function X:; in the gauge fields such that its gauge transformation exactly cancels the non-covariant piece in (4.78), that is

Jdx(8vX~)B~ Jdx[v,XJ1-]aB~ J =

-

dx(8BGa)v a .

(4.79)

Therefore we can define the covariant current as G~) = (j~)

+ X~(A) ,

(4.80)

which satisfies 8v G:;) = ([v, GJ1-)]k This ambiguity in the current definition should not be confused with the previously mentioned ambiguity in the fermion determinant concerning the variation of a local functional of the gauge field. Notice that X:;(A) cannot be the gauge variation of any local functional, that is, there is no noc such that 8J'loc = dx(DJ1-v)a X:; since, in that case, adding fioc to the effective action, we would obtain a covariant anomaly satisfying the WZ consistency conditions, which is not possible, according to our previous discussion [16]. Let us consider the following example in order to clarify these ideas

J

78

4. Anomalies In (4.53) we have given the form of the consistent anomaly, namely

G a = 24171"2tr [Taf/LVPO"O/L (AvOpAO" Now from (4.79), we obtain

J

dx

(t5vX::)B~ =

-~2 4871"

([v, X/LlaB~) = -

+ ~AvApAO")]

J

dx(8 B Ga)v a

J

dXf/LVPO" 0/Lv aBttr{ (Tan

+ TbTa)FpO"

-TanApAO" - nTaApAO" - TaApnAO"} . It is possible to obtain a polynomial X:: =

XI:

(4.82)

satisfying this equation [16]

~2f/LVPO"tr{Ta(AvFpO" + FpO"A v -

AvApAO")} .

4871" Finally by taking the covariant divergence in (4.80), we obtain

(D/L(J/L))a =

(4.81)

(4.83)

~2f/LVPO"tr{TaF/LvFpO"},

(4.84) 3271" which is the covariant anomaly. Notice that the cancellation condition for covariant and consistent anomalies is nevertheless the same, tr(T a{T b,T C }) =

O.

4.7 Topological Interpretation of N on-Abelian Anomalies In previous sections it has been shown how the Abelian anomaly can be related to the index of the Dirac operator. Now we will show that non-Abelian gauge anomalies can also be related to the index of a certain operator defined in two more dimensions [13, 17]. Let us consider a four dimensional Euclidean space compactified to S4 and a semisimple, simply connected gauge group G. Now consider a oneparameter family of gauge transformations g( B, x) with 0 ::; B ::; 271" which satisfies the boundary conditions g(O,x) = g(271",x) = 1 (the identity element in the group). For simply connected gauge groups, the maps g : S1 x S4 ........ G are classified by the fifth homotopy group 7I"5(G) since S1 x S4 looks topologically like S5. Let us recall that 71"5 (SU (N)) = 7l if N ::::: 3 (see Appendix B.3). We take a topologically trivial reference gauge field A for which the Dirac operator does not have zero modes. Consider now the one-parameter family of gauge fields A9(11) obtained from the reference gauge field A by means of the gauge transformation g(B)

A9(11)(x) = g-1(B,x)(A(x)

+ d)g(B,x) .

(4.85)

4.7 Topological Interpretation of Non-Abelian Anomalies

79

Let us introduce now the D and 1) operators defined in Sect. 4.4. There we showed that \ det DI is gauge invariant and, accordingly, under a gauge transformation only the phase of det D might have an anomalous variation, i.e.

e- r [A"(8)j

= detD(A9(1J)) = I detDleiw(A,B) = (det

(4.86)

.fX:A))1/2 eiw(A,B) ,

where w(A, B) is the anomalous phase variation induced by the gauge transformation g(B). Let us now define a two-parameter family of gauge fields which is obtained from the above defined one-parameter family by interpolation between A = 0 and A9(B), that is

At,B = tA9(B), (0::; t ::; 1) .

(4.87)

As it can be seen in Fig. 4.2, the parameter space (t, B) defines a disc D 2 . In the boundary of the disc (t = 1), denoted 8D 2 rv 51, we have

IdetD(A 1,B)1

=

I detD(A9(B»)1

= Constant

i- O.

(4.88)

It does not vanish since by assumption we have taken the Dirac operator for the reference gauge field without zero modes, and it does not depend on B because I det D(A)I is gauge invariant. However, the anomalous phase eiw(A,B), which is not constant, defines in the boundary a mapping

5 1(rv 8D 2)---+5 1 B---+w(A, B) ,

(4.89)

whose winding number is given by

n=

~ 21l"

27r r 8w(A,B) dB. io 8B

(4.90)

We will show below that n is related to the index of a six-dimensional Dirac operator. As we mentioned above, Idet D(A9(B»)1 = I det D(A)\ i- 0 and the operator D(A9(B») does not posses zero modes. However, the operator D(At,B), with (t, B) in the interior of the disc, may have them since it is not obtained from D(A) by means of a gauge transformation. Suppose then that D(Ati,Bi) has one zero mode. Since det D( Ati ,Bi ) is a regularized product of eigenvalues, it vanishes. We will assume that operators with zero eigenvalues are isolated points (ti' Bi ) in the parameter space. Therefore the phase of det D(At,B) can only be homotopically non-trivial around these (t i , Bi ). Moreover, the winding number of the phase around each (ti' Bi ) will be determined by the vanishing eigenvalue at (t i , Bi ). That is, if An(t, B) vanishes in (ti, Bi ) then it can be written as

An(t, B) = j(t, B)eiwi(t,B) ,

(4.91)

80

4. Anomalies g(9)

1

S=dD

A

2

'------'----I

A

Fig. 4.2. The parameter space (t,B)

where J(ti, ei ) = 0. Then the winding number of the eigenvalue phase around (ti' ei ) is mi

= -1

27r

1 Ci

(4.92)

d deWi(t, e)de ,

where Gi , parametrized bye, is some contour around (t i , ei ), as depicted in Fig. 4.2. Deforming continuously and combining the contours Gi around each (t i , ei ), the global winding number will be a signed sum of the corresponding local winding numbers

1

8 w (A,e)de = Lmi. n = -1 8e 27r s' ,.

(4.93)

These interior points (ti, ei ) are in a one-to-one correspondence with the zero eigenvalues of a six dimensional Dirac operator that we are going to build now. The operator .JZ>6 is built in the following manner. Let us define a gauge theory in D 2 x S4 with coordinates (t, e, x), where S4 is the compactified space-time manifold. The boundary term in this manifold can be avoided by adding another D 2 x S4 piece with coordinates (s, e, x). With this two patches we have a S2 x S4 manifold without a boundary. As it is shown in Fig. 4.3, we call the upper piece UN and the lower one Us. The equator is t=s=l. We now define a six-dimensional gauge field or connection on the whole S2 x S4 as AN(t,e,x)=(At,Ao,AJl) =

(0,g-1 :eg,A~O)

As(s,e,x)=(As,Ao,AJl ) = (O,O,A Jl ),

(4.94)

where A is the reference gauge field introduced at the beginning. For this global field to be well defined as a connection, in the equator (t = s = 1) we should obtain AN from As by means of a gauge transformation like

4.7 Topological Interpretation of Non-Abelian Anomalies

81

t= 1

8=1

Us

Fig. 4.3. The upper and lower patches of the 8 2 x 8 4 manifold

AN = g-l(e,x)(As

+ L1)g(e,x)

(4.95)

,

where L1a + de ae a + dt at a . - d + de + dt -- dx J1. axJ1.

( 4.96 )

This is indeed the case since in components (in the equator) (4.95) reads (O,g-l :e 9 ' A

1

,e) = (g-l :t g, g-l :e g, g-l Ag + g-l a~J1.g)

(4.97)

because agjat = 0 and A 1 ,e = g-l(A + d)g. Let us consider now the Dirac operator ./2>6(A) with A the connection defined in (4.94). Then the index theorem states that (see Appendix B.3) ind ./2>6 (A) =

nR -

nL

=

-\

r

487l" } S2 xS4

trF3

(4.98)

with F = L1A+A2 and nR(L) the number of zero modes of right (left) chirality (chirality is now defined in a six dimensional space). It is possible to show [17] that to each zero mode of the Dirac operator ./2>6 we can associate a winding number +1( -1) around the points (ti' ei ). This number is +1 or -1 depending on whether the vanishing eigenvalue has a right or left chirality. In summary, the total winding number of the phase of det iJ (the anomalous variation of the phase) defined in (4.93) is nothing but the index of the ./2>6 operator . 1 r L mi = md./2>6 = 27l" }

21r

n =

0

de

aw(A, e) ae .

(4.99)

"

Once we have this expression we can compute the non-Abelian anomaly. Since trF3 is a closed form, we can write it locally as (4.100)

82

4. Anomalies

where Qs(A, F) is the corresponding Chern-Simons form (see Appendix B.3). For the gauge field in (4.94) and using Stokes theorem, it is possible to obtain

r

J S2 XS4

trF

=j

3

D2

=

XS4

r

JS1XS 4

trF~ +

j

D2

XS4

Qs(AN,FN)1

trF~

t=l

-

r

JS1 XS4

Qs(As,Fs )!

5=1

' (4.101)

where the minus sign appears due to the different orientation of the upper and lower patch boundary. Recalling the definitions of AN and As we see that

r

Qs(As,Fs)1 = 0 (4.102) JS1XS 4 5=1 since Qs(As,Fs) does not have a dB component and therefore it cannot be a volume form for Sl x S4, hence ind W6 =

--=!.r Qs (A 1'O + g-l~gdB, F9(O») 487r 3 } Sl x S4 oB

,

(4.103)

where F9(O) = dA9(O) + (A9(O»)2. As Ow/ oB measures the anomalous local variation of the phase under an infinitesimal gauge transformation given by v(B) = g-l (B)o/ oBg(B) therefore, in terms of the effective action we can write

h4tr(vD8~r[A]),

i:Bw(A,B) =

(4.104)

and finally (4.99) and (4.103) allow us to write

dB8 u r[A] =dB

h4

tr (VDI-' 8Jl:]) = idow(A, B)

=-1-1 Q1(vdB 2 4 , A9(O) , F9(O») 247r

S4

,

(4.105)

where Ql(vdB, A9(O), F9(O») is the linear term in dB in the QS(A9(O) + vdB, F9(O») expansion in powers of v. That is the only term contributing in (4.103). Therefore (4.105) can be rewritten, using (4.59), as

Bu T[A] = _1_2

r tr [Vd (A9(O) dA9(O) + ~(A9(O»)3)] 2

247r JS4

(4.106)

Finally, we can conclude that the anomalous divergence given by (4.43) and (4.106) is (taking B = 0, i.e. 9 = e)

(DI-'(j!-'})a =

24~2 tr [TaEK,),I-'V OK (A,),OI-'Av + ~A,),AI-'Av)] ,

(4.107)

which is in agreement with the result obtained using the Fujikawa method with the gaussian regulator given by the operator b 2 . As we saw in Sect. 4.4 it satisfies the WZ consistency conditions.

4.8 Non-perturbative Anomalies

83

4.8 N on-perturbative Anomalies In this section we will study another kind of gauge anomalies that can appear in some theories. The main difference of such anomalies when compared to those analyzed in the previous sections is their relation with the so called large gauge transformations, i.e., gauge transformations that are not connected with the identity. The most important example of non-perturbative anomalies was discovered by Witten [18] for the case of an SU(2) theory coupled to an odd number of chiral fermion doublets. In order to see clearly the origin of this anomaly, let us consider the model described by the Lagrangian

.c=.cyM+i"7fiL'YJ1.DJ1.7/JL,

(4.108)

where .cYM is the standard Yang-Mills action for the SU(2) gauge field AJ1. and 7/J is and SU(2) doublet. The Euclidean effective action for the gauge field is thus defined as

e-r[Aj = ![d7/JL][d"7fiL]e- S E[A,'PL,;;J;d

(4.109)

and S E is the Euclidean action corresponding to the above Lagrangian. In order to compute this functional one uses the standard formula (4.110) where we take the square root of the fermionic determinant since we are integrating only the left-handed component. As we saw in Sect. 4.4, the determinant is defined as the product of the eigenvalues of some Dirac operator, with the space-time compactified to an S4 sphere. The problems of such definition come from the fact that, in this model, gauge transformations are maps g(x) from S4 into SU(2). Hence, the gauge field transforms as usual as (4.111) but, as far as 7l"4(SU(2)) = Z2, we have two different equivalence classes of gauge transformations, depending on whether they are homotopically connected with the identity or not (see Appendix B.3). The non-perturbative anomaly reflects the impossibility to define the fermionic determinant in (4.110) in such a way that it is invariant under non-trivial or large gauge transformations. When considering the Dirac equation iD(A)7/J = >"7/J, the eigenvalues are real numbers and for every eigenvalue>.. there is an eigenvalue ->.., since iD(A)7/J = >"7/J implies iD(Ah57/J = ->"'Y57/J. Therefore to define the square root of the fermionic determinant we can, for a given AJ1., take the product of the positive eigenvalues only. Notice that, in principle, we have two possibilities for the sign of the square root in (4.110). The above choice fixes the sign of (detD)1/2, which for infinitesimal transformations should change

84

4. Anomalies

o

1 t

Fig. 4.4. Possible flux of the eigenvalues from t=O (AJL(x)) to t=1 (At (x))

smoothly with AI-" so that there is no further freedom if we restrict ourselves to gauge transformations belonging to the same homotopy class. Let us see what happens if we change from one class to another. In particular consider the variation AI-'(x, t)

= A~(x)t + AI-'(x)(1 -

t)

(4.112)

t E [0,1] ,

which connects AI-' with A~ in (4.111), g being a non-trivial gauge transformation. The spectrum of the Dirac operator is the same at t = and at t = 1 but the individual eigenvalues can (and in fact they do) rearrange themselves as t is varied from t = 0 to t = 1. For example, the simplest possibility is shown in Fig. 4.4 in which a positive eigenvalue An changes its place with -An. Thus if (det D(A))1/2 was defined at t = 0 as the product of the positive eigenvalues then at t = 1 (after following the eigenvalues continuously) (detD(A9))1/2 is the product of many positive and one negative eigenvalue so that

°

(detD(A))1/2 = -(detD(A9))1/2.

(4.113)

It can be shown [18], by using some version of the Atiyah-Singer index theorem, that for any possible rearrangement of the eigenvalues, there is always an odd number of them that change their sign and therefore (4.113) always holds. Hence, integrals such as (4.114) vanish identically because the contribution of any gauge field A is exactly canceled by the A9 contribution. Thus, the expectation value of any gauge invariant operator O(x) is zero and the theory is not well defined. This result can also be extended to the case of an odd number of fermionic doublets or other gauge groups G with 7r4(G) = 7L. 2 , like Sp(N) for any N. The important point here is that if we had an even number of fermion doublets, there would not be an anomaly, since every doublet would give a minus

4.9 Non-linear

(7

Model Anomalies

85

sign in (4.113). In particular, in the 8M the gauge group is 5U(2)L x Uy(l) and therefore this kind of non-perturbative anomaly could occur. Nevertheless it is not present due to the fact that there are four doublets per family; three (Nc = 3) made out of quarks, and another one of leptons.

4.9 Non-linear

(j

Model Anomalies

In Chap. 3 we showed how fermionic matter fields, as well as chiral fermions, can be coupled to NGB in an G invariant and K = G / H covariant way. In such case the NGB-fermion interactions are described by the Lagrangian 12 m = 7jji'y/l.(0/l.

(4.115)

+ w/l.)PL 1/; ,

where the 1/;i are fermionic fields belonging to a linear representation of the subgroup H with generators 11. i , wJ-I. = w~OJ-l.1ro.11.i and w~(1r) is the canonical H connection introduced in Chap. 3. In spite of the fact that this Lagrangian is G invariant and K covariant, these properties could be spoiled at the quantum level, due to the presence of the so called non-linear (7 model (NLSM) anomalies [13]. When fermions are integrated out we can find the contribution T f to the NGB effective action coming from the Lagrangian above, which in Euclidean space time (compactified to 54) reads e- r /[w(7l")]

=

J

[d1/;][d7jj]e -

JdxL.=(w(7l"),,,';;;J;)

.

(4.116)

Note that the dependence of T f on the NGB fields is only given in terms of the canonical connection w(1r). Moreover, with such a coupling of the fermions to the H connection we can expect that a mechanism similar to that producing the gauge anomalies studied in previous sections, could lead here to a new anomaly. The main difference now is that the presence of this anomaly would not make the theory gauge non-invariant (since we are not dealing with a gauge invariant theory) but only not globally G invariant. As we have already discussed in this chapter, gauge theories with gauge anomalies are inconsistent. On the contrary, a NL8M coupled to fermions with anomalies, like that described above, is not inconsistent although its physical predictions are not G invariant. Thus their initial appeal as theories depending only on the symmetry breaking pattern, i.e. on G and H, is lost and they are not good candidates to describe the low-energy behaviour of a system with that symmetry breaking. In order to calculate this anomaly, we can just translate the results found for gauge anomalies in Sects. 4.4 and 4.7. For that purpose, we recall (see Chap. 3) that, under an isometry transformation 9 E G given infinitesimally by 9 ~ 1 + eaTa, the H canonical connection W transforms as W -t

w' =

W -

D(eana)

(4.117)

with na defined in Sect. 3.2, so that, we can write for the corresponding change on Tf

86

4. Anomalies

r

t5rf[W]=~ Q~(ea[2a,w) 241r J S4

- - - -i2l e a tr ( n Hal" p.vpUBp. [ WvWpW u 241r

S4

1 + -WvWpW u] )

2

.

(4.118)

Therefore this anomaly breaks the invariance of the quantum theory under the isometry group G and therefore the Green functions do not satisfy the Ward identities corresponding to that invariance. Nothing ensures us then that the NGB could not become massive or that their dynamics could not depend on the particular parametrization of the K manifold. An important observation about the above equation for the NLSM anomaly is the following: 1r( x) is an application 1r : S4 -+ K. Thus, (4.118) can be understood as the integral of a four form in K. But any four form necessarily vanishes on a manifold of dimension less than four. Thus, regardless of the fermionic sector, the NLSM anomaly vanishes whenever the dimension of the coset space K, i.e. the number of NGB fields, is less than four.

4.10 The Wess-Zumino-Witten Term 4.10.1 Anomalous Processes in QeD

In Chap. 3 we have seen, for SU(2), how the low-energy dynamics of the NGB is well described in the framework of the NLSM. We have built the effective Lagrangian with the lowest possible number of NGB field derivatives, that has all the symmetries of the theory. However, we observe that, with that model, we cannot reproduce anomalous processes involving NGB. For instance, if we think about QCD with the flavor symmetry SU(3)L x SU(3)R (see Chap. 6), a typical example is the process K+ K- -+ 1r+1r-1r0 . Another example, when the electromagnetic gauge field is included, is the 1r0 -+ II decay that we have studied in Sect. 4.2. Then, we need an effective action, to be added to the NLSM, from which we can obtain the anomalies of the underlying fermionic theory, like those analyzed in this chapter. As it is usually done, let us parametrize the NGB fields as U(x) = exp(i1r a Aa /F), with U(x) E SU(3) ~ (SU(3)L x SU(3)R)/SU(3)L+R. The anomalous processes mentioned above clearly violate the L ~ R symmetry, which means U ~ ut, or, in other words, (_l)NB with NB the total number of NGB in a given vertex (remember that the 1ra are pseudoscalar fields, so that 1ra -+ _1r a under L ~ R). Thus, the existence of such anomalous reactions implies that (_l)NB is not a symmetry of QCD. However, the NLSM has this symmetry naturally incorporated and that is the reason why it cannot give rise to such anomalous processes. Actually, what it is a good symmetry of QCD is the parity transformation P that changes the spatial vector components and interchanges L ~ R simultaneously. Therefore, we can write P = poe _l)NB, acting on a vertex

4.10 The Wess-Zumino-Witten Term

87

of NGB fields, where Po denotes the transformation x ~ -x, t ~ t, with (x, t) any four-vector in the theory. Then, if we want to build a term that violates (_I)NB, we have to be sure that it also violates Po. This forces us to include the Levi-Civita symbol EJ.Lvpu. With these ingredients we can try to write down a term with the lowest possible number of derivatives, preserving also chiral symmetry, Lorentz invariance and charge conjugation. Thus, that term would be (4.119) but it vanishes identically due to the cyclic property of the trace. Hence, we see that it is not possible to build our anomalous action with the symmetries required and to lowest order in derivatives, at least if we want to have a local Lagrangian. 4.10.2 The Non-local Anomalous Effective Action A possible alternative is to abandon locality and try to write a non-local integral functional with the same requirements. With that purpose, one of the simplest possibilities is to introduce a continuous dependence on a fifth integration variable t. Without loss of generality we can choose t E [0, 1]. Then, we will have fields Ut(x), which, for a given t, are mappings from the compactified Euclidean space-time 54 into 5U(3). Thus Ut(x) is nothing but an homotopy (see Appendix B.3). As far as the fourth group of homotopy 7r4(5U(3)) = 0, we can continuously connect any two fields U(x) and U'(x) with an homotopy Ut(x) such that Ut=o = U and Ut=l = U'. Specifically, let us choose an Ut(x) field such that Uo = 1 and U1 = U(x), with U(x) the NGB field in terms of which we want to construct the action. Of course, by consistency, the physics should not depend on the choice of the homotopy Ut , that is, on the path that we follow in the fifth dimension t to connect the NGB field U(x) with the identity map. In other words, our action, though non-local, should depend only on the value of the field Ut on the four-dimensional spacetime, which is the boundary of the five-dimensional manifold I x 54, with 1= [0,1]. In fact, this condition is going to suggest us which is the functional that we are looking for. With that purpose, let us write the anomalous action as (4.120) with Ws a 5-form whose explicit expression we want to determine. As we have previously said, a different choice of the interpolating field, say U£(x), should drive to the same quantum theory. This implies that the difference r[Un - r[Utl has to be 27rni with n E 7l., since we have e- r in the Euclidean path integral. In particular, this also has to be true if we choose an homotopy U[ such that U~ = U and U{ = 1. It is possible to think of U(x) as defining a four dimensionat"sphere in the 5U(3) manifold. Then, Ut is homeomorphic

88

4. Anomalies

to a five-dimensional disk whose boundary is U(x) and U{ is then the disk with opposite orientation. Thus, the difference between the two functionals is nothing but an integral over the SS manifold. Therefore, with this choice, the above consistency requirement reads ( Ws = iss

27rni

nEll. .

(4.121)

Now we recall that 7rs(SU(3)) = 71. so that (4.121) can be satisfied by choosing Ws precisely as the winding number that labels the maps from SS into SU(3), with a suitable normalization factor. To find the precise form of that winding number we will make use of the index theorem formulated on the six-dimensional manifold S6 (we refer to Appendix B.3 for details about notation, the index theorem and the properties of the Chern-Simons forms) ind

iW6

=

(-i)3 { trP 3 6 27r

(4.122)

iS6

with .(2)6 the six-dimensional Dirac operator for a given gauge field A. On S6 we have two patches homeomorphic to IR 6 , namely S6 = Dt U D(;, being Dt two six-dimensional disks. In each one of these disks we can define a gauge field A ±. The intersection between the two patches is homeomorphic to SS and, on it, A+ and A - are related by a gauge transformation 9

A+ = g-l(A-

g: SS

+ d)g

----+

SU(3) .

(4.123)

On each patch we have, by definition, trP3 dQs(A, P) with Qs the Chern-Simons 5-form. Now, we write (4.122) as the sum of the two integrals over Dt and apply Stokes theorem to both integrals. Then we choose the particular gauge configuration A- = 0, A+ = g-ldg and we get ind

iW6

=

6(~;)3 ~s Qs(A+) = 60;~)3 ~s tr(g-ldg)s ,

(4.124)

where we have used Qs(O) = 0 and Qs(g-ldg,O) = l~tr(g-ldg)S, as showed in Appendix B.3. The result in (4.124) gives us the form of the winding number for mappings of SS into SU(3), since the index of the Dirac operator is an integer and the r.h.s of (4.122) is a topological invariant. Consequently, we have a functional satisfying the condition in (4.121)

r

N rwzw[U] = 24O 2 7r =

~ 2407r

J

Ix S4

tr(Ut-1(d + ddUt)S

{I dt ( d4xfijklmtrMiMjMkM1Mm

io

i S4

Uo(x) = 1 ; U1(x) = U(x) (4.125) 1 with M i = Ut- ai Ut and N an integer to be determined below. Here we have d = ap,dxP" J.1 denoting S4 indices, dt = atdt and i, ... m = 1, ... ,5. The functional in (4.125) is the Wess-Zumino-Witten (WZW) anomalous action

4.10 The Wess-Zumino-Witten Term

89

[4, 5]. As we have commented before, a crucial property of this functional is that it only depends on U(x) and not on the homotopy Ut which interpolates between 1 and U(x). It is not difficult to show that (4.121) automatically ensures that (4.125) satisfies this property.

4.10.3 The WZW Term with Gauge Fields Up to now, we have proved that the WZW five-dimensional Lagrangian has the required symmetries and that it is independent of the choice of the homotopy. We now turn to see how it indeed reproduces the anomalies of the theory and hence the anomalous processes. First, we will show it in detail for the case of the non-Abelian gauge anomaly studied in Sects. 4.4 and 4.7. As we did there, we will consider the case of a right-handed fermion multiplet coupled to a non-Abelian gauge field A R . Thus we will need the extension of the WZW term in (4.125) when a gauge field is included. For that purpose, we will consider the Chern-Simons form for A R "transformed" by Ut (instead of the "pure gauge" configuration Ut-1(d+ddUt in (4.125), without any gauge field) [13]. We will show below that this extension of the WZW action will enable us to reproduce the gauge anomaly, which in this case affects the 5U(3)R current. Thus, we have N rwzw[U,ARl = 24O 1r

2

f

IxS4

Q5(A~' +Vt,Fj{')

(4.126)

with A~' =Ut-1(A R + d)Ut

Vt =Ut-1dtUt Fj{' = (d

+ dt)(A~' + Vt) + (A~' + Vt)2

.

(4.127)

Let us show how the non-Abelian anomaly is indeed recovered with this functional. Under a right gauge transformation gR the NGB and gauge fields change as

U -+gi/U AR-+AfJt = gR1(AR +d)gR.

(4.128)

Note that Q5(A~') = 0, since A~' and Fj{' are, respectively one and twoforms only in 54 and Q5 is a 5-form polynomial in A and F. Furthermore, we have = 0 and if we expand Q5 in powers of Vt, only the first term in the expansion survives. This term is, by definition, Q~ (Vb A~'). Hence the integrand in (4.126) can be written as

v;

Q5(A~'

+ Vt, Fj{t)

= Q~(Vt, A~') .

(4.129)

Let us recall the result in (4.105), where now the one-parameter gauge transformation appearing in that formula, is given by Vt. Then we get

90

4. Anomalies

(4.130) where we have identified the variation of r[A ~t] under Vt with d t , since dtA~t = dVt + [A~t, Vt], which is the covariant derivative of Vt. Remember that here r is the effective action for the gauge field, defined in (4.37) and that A~ is invariant under the gauge transformation given in (4.128), since

(A!Jf)9F/U = A~R9F/U = A~ .

(4.131)

In addition, r[ARl has the correct anomalous variation, by definition. Then, (4.130) implies that rwzw correctly reproduces the non-Abelian gauge anomaly. The role of the normalization constant N will become clear in a moment. The above analysis can also be extended to include both left and righthanded gauge fields, yielding the complete left-right (LR)-handed WZW term. In that case, the analysis is a bit more involved than that presented here, as it is necessary to add a suitable counterterm in order to reproduce LR anomalies [13]. An important point is that the rwzw gauge field dependent part is always local and can be separated from the non-local gauge field independent part. Let us show this property for the case of rwZW[U,A R ] in (4.126). For that purpose, recall the formula for the variation of the Q5 form under a gauge transformation, given in (B.76) in Appendix B.3. A direct application of that formula to the integrand in (4.126) gives rwzw[U,ARl

=~ 2401r

r

JIX84

tr(Ut-1(d + ddUt )5

+N

r (Y4(A R , U-1dU) ,

J 84

(4.132)

where (Y4 is a 4-form, whose explicit expression is known (see [13]). Another approach when trying to incorporate gauge fields in the WZW term is the trial and error method [5]. Notice that the usual procedure of replacing derivatives by covariant derivatives, in order to find a gauge invariant effective action, cannot be applied here because rwzw is non-local, and, then, is not given as a manifestly covariant SU(3)L x SU(3)R action. Instead, one starts by finding the variation of the WZW term under a gauge transformation, and then, by trial and error, finally arrives to a gauge invariant expression. Notice, however, that this method can only be applied to the so called anomaly-free subgroups of G = SU(3)L X SU(3)R' since, as we have just seen, the WZW effective action is anomalous in the non-Abelian gauge currents corresponding to the G symmetry, as it should be if we want to reproduce anomalous processes. In that sense, the method discussed above, starting directly from the Q5 form with gauge fields, is more general.

4.10 The Wess-Zumino-Witten Term

91

4.10.4 Anomalous Processes and the WZW Term A particularly interesting example of an anomaly-free subgroup is that of electromagnetism. Thus, the trial and error method previously commented can be applied to this case. As a matter of fact, we will show how the low energy limit of the 11'0 --+ II decay amplitude is obtained with the gauged WZW term. This is consistent with the fact that the WZW reproduces the anomaly in the J~3 axial current, which is responsible for the leading order amplitude, as we have discussed in Sect. 4.2. Let us then consider infinitesimal transformations for the fields under the electromagnetic group U(X) --+U(x)

+ ia(x)[Q, U(x)] 1

Aj.t(x) --+Aj.t(x) - -8j.ta(x) , (4.133) e where Aj.t is the (real) electromagnetic field, e is the electric charge and Q is the quark electric charge matrix. Now, either the trial and error or the topological method can be followed in order to obtain the gauged WZW action, which is given in Euclidean space by Twzw[U,Aj.t]

= Twzw[U] + xtr

r Aj.tJj.t JS4

2 e 2N 2411'

r fj.tvP"'8j.tAvAp

JS4

(4.134)

[~Q8",UQU-l - ~QUQ8",U-l + Q 2U- 1(8",U) + Q2(8",U)U- 1]

with Jj.t =

~~:~ fj.tvP"'tr[Q(8vUU-l)(8pUU-l)(8",UU-l) +Q(U-18vU)(U-18pU)(U-18",U)]

(4.135)

and Twzw[U] given in (4.125). We see again how the Aj.t dependent part only appears in the local term. There we can find a contribution proportional to 1I'°AA when we parametrize U(x) = exp(i1l'a>"a/ F1r) , with >..a the GellMann matrices and F1r ~ 93 MeV the pion decay constant. This term, in the Euclidean Lagrangian corresponding to the action in (4.135) turns out to be, after integrating by parts . 2N r _ ze j.tVp'" F j.tv F p",1I' , L.- 1r 0 .... -y-y - 9611'2 F f 4.136

°

(

)

1r

which correctly reproduces the 11'0 --+ II amplitude, displayed in (4.23) in momentum space, when N = N c, the number of colors. This value for N is also consistent with the result in (4.130), since the non-Abelian gauge anomaly also carries a factor N c from the color charge of the fermion fields (the analysis in Sects. 4.4 and 4.7 was carried out for only one fermion multiplet).

92

4. Anomalies

Another important point is that the WZW term, although non-local, can be made local term by term in the pion field expansion. For instance, without any gauge field the leading term is given by

rWZW [U] = 240-iNc 2p5 7r

1r

= 24~i~~5 7r

=

2

1r

-iN 40 2~5 7r 1r

j

j . Ix 8 4

1

tr

( €

ijklm!'l A!'l A!'l A!'l A!'l A) u(rrUj7rUk7rurrrUm7r

+ O( 7r 6)

tr(€ijklmfJi(frfJjfrfJkfrfJliTBmfr))

+ O(7r6 )

Ix8 4

tr(€J1.Vo.{3iTfJJ1.frfJvfrfJo.frfJ{3fr)

84

+ O(7r 6 )

,

(4.137)

where the latin indices run from 1 to 5 and we denote fr = 7r U Au. The above expansion enables us to identify, for instance, a piece in the Lagrangian which correctly describes the anomalous process K+ K- -+ 7r+ 7r-7r 0 . Thus, having fixed the normalization integer of the WZW term with one given process, we are able to obtain predictions for the amplitude of any anomalous process involving NGB, to leading order at low energies. 4.10.5 The SU(2) WZW Term We would like to comment now on the WZW term for the two flavor case. Due to the properties of the Pauli matrices, the WZW term in (4.125) vanishes identically for SU(2). However, this is not true for the gauge field dependent local pieces, as it should be, because the 7r 0 -+ 'Y'Y process remains the same when only two flavors are considered. Now let us take the SU(2) group as embedded in SU(3). For instance, a left-handed gauge transformation would be given by some matrix []L

=

(9t

~)

(4.138)

with 9L E SU(2). The fields U(x) change under SU(3h as U -+ U[]L. When considered in this way, the WZW term rwzw[U = g£1 (that is, for the gauge transformation of the identity field U = 1) does not vanish in general. In fact, it reproduces the non-perturbative SU(2) anomaly discussed in Sect. 4.8. This can be seen by considering the difference rwzw[U

= 1, A L = 0] - rwzw[U = [/L, A L = gL1dg£1 .

(4.139)

Let us remember that 7r4(SU(2)) = ~2 and there are two different classes of gauge transformations 9L : S4 -+ SU(2). From the expression for rwzw[U, ALl it can be shown that the above difference vanishes when 9L belongs to the trivial sector (that connected with the identity) and it is equal to i7r N c for the non-trivial one. Then the Euclidean path integral acquires an anomalous factor (_l)Nc . We have one fermion SU(2)L doublet per color, so we arrive to the same conclusion as in Sect. 4.8 for an SU(2) theory coupled to an odd number of chiral fermions (which in our case are only the quarks)

4.11 The Trace Anomaly

93

using the anomalous effective Lagrangian for QCD with only NGB degrees of freedom. Summarizing, the WZW effective action reproduces the anomalies studied in this chapter, in terms of only NGB degrees of freedom. Although non local, it only depends on the fields defined in the four-dimensional Euclidean space-time and it can be written as a local action when expanded in terms of pion fields to a given order. The topological analysis of non-Abelian gauge anomalies presented in Sect. 4.7 has allowed us to establish the connection between them and the WZW term. We conclude by remarking that the presence of the WZW action is also essential to explain why the skyrmion should be quantized as a fermion [5], as it was already discussed in Sect. 3.6. For that purpose we have to rotate adiabatically the skyrmion field U E SU(2) embedded in SU(3), a 2n angle. It can be shown, similarly to the case of the SU(2) nonperturbative anomaly commented above, that r wzw changes by inN c under such rotation, which is parametrized by a field belonging to the non-trivial homotopy class of SU(2). This implies that the skyrmion has to be quantized as a fermion, since N c is odd, as we stated in Chap. 3.

4.11 The '!race Anomaly In the previous sections we have studied different kinds of anomalies which are related to chiral fermions. In this section we will consider a completely different anomaly known as the trace anomaly. It occurs in many QFT in four dimensions and produces the breaking of the scale or dilatation symmetry. This is due to the fact that these theories lead to divergent integrals which are only meaningful through renormalization. Such procedure necessarily introduces some dimensional parameter, thus breaking the possible scale invariance of the action [19]. To illustrate this effect, let us consider a system described by the action S[4>] =

j dx t

.ck(4), fJ4» ,

(4.140)

k=l

so that the generating functional for the Green functions is defined by (4.141)

However, as it is written, such a definition yields meaningless results for the Green functions, since the measure, which is is not well defined, leads to ultraviolet divergences. To avoid those divergences we have to consider the regularized generating functional eiWI\[JJ = j[d4>]AeiCS[PJ+CJP}) ,

(4.142)

94

4. Anomalies

where [d!P1A means that we are only integrating over fields whose w frequencies satisfy w < A, where A is some ultraviolet cutoff. Now the generating functional is well defined, although the limit A --+ 00 is not. Therefore to give sense to the theory we have to consider a renormalized Lagrangian by introducing the appropriate A-dependent counterterms (we assume that the model is renormalizable in the standard sense). This leads to the following definition of the renormalizedgenerating functional eiWR[J]

=

lim j[d!PR1Aei(J dx Z=:;1 .c.dtPR,atPR)- Z=:;1 ~k( ~ ).c.k(tPR,atPR)+(JtPR») A-DO

=

lim j[d!PR]Aei(J dx Z=:;1 Zk( ~ ).c.k(tPR,a
=

lim j[d!PR]Aei(J dX.c.Cff(tPR,atPR;~)+(JtPR»), A-DO

(4.143)

where Zk are the renormalization constants (see Appendix C.1). Therefore, even though S[!pl is scale invariant, the action that we really use to define the renormalized generating functional is scale dependent. In other words, the renormalization procedure gives rise to an anomaly in the scale symmetry. A more detailed description of this fact can be given as follows: First one considers the scale transformations X' =eax ~ (1 + o)x !p' =e-ad!p ~ (1 - od)!p A' =e- a A ~ (1 - o)A .

(4.144)

Then, assuming that S[!p] is scale invariant, it is easy to find

eiWR[Jj

=

lim j[dPR]Aei(J dx Z=:;1 Zk( A-DO

e-;A ).c.k(tPR,atPR)+(JtPR»)

. (4.145)

But

e-aA) (A) dZn 2 Zn ( -J..L- = Zn ;, - dlogAo+O(o).

(4.146)

Therefore, the scale transformation does not leave the effective Lagrangian invariant but it introduces an extra term r:r _ U-'-eff -

..[!--. -0

dZk r L.J dlo A -'-k

k=l

g

_

-

-0

TJ.1. J.1. '

(4.147)

where, by definition, TJ.1.1/ is the energy-momentum tensor, chosen so that it is traceless for scale invariant Lagrangians at the classical level (see Chap. 2). Maybe, the most interesting example of scale invariant four dimensional theories are the pure Yang-Mills theories. They are scale invariant at the classical level, since there is no dimensional parameter in the Yang-Mills action. Due to gauge invariance, and using the background field method,

4.12 References

95

they can be renormalized with only one Z constant (see Appendix C.3), since ZI = Z3 and then the bare coupling gB = Z;I/2 gR . Thus, the renormalized Lagrangian can be written in terms of the bare Lagrangian as

L~M = LYM(A o ) = Z3LYM(AR) .

(4.148)

Therefore, in this case, the anomaly in (4.147) introduces an extra term f: r _ dZ3 r _ . IJ. ULYM - -ndlogALYM - -nTIJ. .

With the usual /3(gR)

=

(

4.149

)

/3 function definition

dgR -dl og/-L

(4.150)

and using go = Z; 1/2 gR, it is easy to obtain dZ3 _ 2Z3 /3 dlog A - - gR .

(4.151)

Now remembering the relation olJ.DIJ. = T/: where DIJ. is the Noether current corresponding to classical scale invariance, we finally find (4.152) Thus, starting with a scale invariant theory at the classical level, the renormalization gives rise to the breaking of the scale invariance, which is reflected in the above exact equation. This important effect is called the trace anomaly. Hence, the quantization of Yang-Mills theories introduces a dimensional scale which was not present at the classical level. For example, One could consider QCD without quarks, or even massless QCD with quarks, which are both scale invariant theories at the classical level. In spite of that, the trace anomaly will introduce a dimensional constant, denoted by AQCD, (not to be confused with the cutoff A introduced above) which sets the scale of the physical processes described by the theory. The importance of the trace anomaly can be perhaps stressed by remembering that a completely scale invariant theory at the quantum level (i.e. without scale anomaly) can only have in its spectrum zero energy states. Somehow, the extremely rich hadronic spectrum of massless QCD is supposed to come entirely from the trace anomaly.

4.12 References [1] [2] [3] [4]

s. Adler,

J. S. S. J.

Phys. Rev. 177 (1969) 2426 Bell and R. Jackiw, Nuovo Cimento 60A (1969) 47 Adler in Lectures on Elementary Particle and Quantum Field Theory, Eds: Deser et al., MIT, 1969 Wess and B. Zumina, Phys. Lett B37 (1971) 95

96 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18] [19]

4. Anomalies E. Witten, Nucl. Phys. B223 (1983) 422; B223 (1983) 433 S. Adler and W.A. Bardeen, Phys. Rev. 182 (1969) 1517 W.A. Bardeen, Phys. Rev. 184 (1969) 1848 Particle Data Group (R.M. Barnett et al.), Phys. Rev. D54 (1996) G.A. Christos, Phys. Rep. 116 (1984) 251 G.'t Hooft, Phys. Rep. 142 (1986) 357 K. Fujikawa, Phys. Rev. D21 (1980) 2848, D22 (1980) 1499 (Erratum) K. Fujikawa, Phys. Rev. D25 (1982) 2584 L. Alvarez-Gaume and P. Ginsparg, Ann. Phys. 161 (1985) 423 D.J. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477 R. Stora, Progress in Gauge Fields, Eds. G. t'Hooft et al. (N.Y. Plenum Press), 1984 B. Zumino Relativity, Groups and Topology II, vol 3, Eds. B.S. DeWitt and R. Stora (North Holland), 1985 W. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421 L. Alvarez-Gaume and P. Ginsparg, Nucl. Phys. B243 (1984) 449 E. Witten, Phys. Lett B117 (1982) 324 R. Crewther, Phys. Rev. Lett. 28 (1972) 1421 M.S. Chanowitz and J. Ellis, Phys. Lett. B40 (1972) 397 J. Collins, A. Duncan and S. Joglekar, Phys. Lett. D16 (1977) 438

5. The Symmetries of the Standard Model

The standard model (SM) provides a detailed description of the electroweak and strong interactions, which is compatible with all the presently known experimental observations and data. Our aim in this chapter is to review its main properties, paying an special attention to its symmetries and the way they are realized. The motivation of this review is to establish the framework for the analysis in the following chapters, in which we will apply the effective Lagrangian formalism to some of the less-known sectors of the SM.

5.1 The Elements of the Standard Model The SM is a renormalizable quantum field theory (QFT) that describes the electroweak and strong interactions of quarks and leptons, which are the most elementary components of matter known at present. One of the essential features of this model is that both the electroweak and the strong forces are introduced as gauge interactions [1]. In this description it is possible to define three separated parts: First there is a matter sector which is made of fermionic fields; second, there are vector boson gauge fields and, finally, the so called symmetry breaking sector (SBS). This sector is needed in order to provide masses for the fermions and the electroweak bosons and it is not as well established as the other two. In this chapter we will study the minimal Standard model (MSM) where the SBS is given by a doublet of self interacting scalar fields. However, in Chap. 7, we will consider more general versions of the SM with the same matter and gauge boson content, but without any specific choice of the SBS. Let us now briefly review the main components of the MSM. 5.1.1 Matter

In the SM, matter is described by fermionic fields which are organized in three generations, as it is shown in Table 5.1. Each generation is made of two quark flavors (u and d like) [2] and two leptons (neutrino and electron-like). All these particles are accompanied by their corresponding antiparticle with the same mass and opposite charges. A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

98

5. The Symmetries of the Standard Model

Table 5.1. The SM fermionic content organized in families, following the notation used in the text muon family

t:

electron family e

J.I.

T

N

Vo

vI'

VT

Notation

1:Q

u 'D

u Cli Q d

cCli

sQ

tau family t Cli

bQ

electric charge

-1 0

273 -1/3

For further convenience it will be useful to introduce the following notation for the quark fields:

U = (u(\ ca., ta.) V=(da., sa., ba.) ,

(5.1)

where the ex index labels the color degree of freedom, as well as

N = (ve , v/-L' v T ) (5.2)

£=(e-,jL-,T-)

for the leptons. In all cases each component belongs to a different generation. As far as in the MSM there are no right-handed neutrinos, we have N = PLN, besides, they are massless. 5.1.2 Gauge Fields Within the SM there are two types of gauge interactions: a) The strong interactions are introduced in the SM by a vector SU(Nc ) gauge theory [3] which is called Quantum Chromodynamics (QCD). All quarks transform in the fundamental representation of this group and thus ex runs from 1 to N c . Here, N c stands for the number of colors, which in the real world is N c = 3. This number can be measured from the ratio between hadronic and leptonic decays of e+ e- and from the nO -+ I I decay (see Chap. 4). In addition we will see that some information about N c can be obtained theoretically from anomaly cancellation conditions. An infinitesimal color gauge transformation for an arbitrary quark q = u, d, s ... is given by:

qa.

-+

qa. _ i()~(x)

;(3 q(3 ,

).a

(5.3)

where ).a /2 are the SU(Nc ) generators in the fundamental representation. These transformations only affect the quarks since leptons are colorless; that is, Nand £ are SU(Nc ) singlets. The corresponding gauge fields G~ are called gluons and transform in the adjoint representation and thus a runs from 1 to N; - 1. The coupling constant will be denoted by 9s. b) The electroweak interactions are based in the gauge group SU(2)L x U(l)y [4]. Quarks and leptons are organized in pairs like

5.1 The Elements of the Standard Model

Q= (

~)

£ = (

-i ).

99

(5.4)

This notation is motivated by the fact that the left components QL and £L are indeed SU(2)L (or weak isospin) doublets. However, this is not the case for the right components, which are singlets and do not transform under SU(2)L. We can thus write the infinitesimal SU(2)L transformations as £ ~ £ - iBi)x)T a PL £

(5.5)

Q ~ Q - iB'L(x)T a PLQ .

Here T a = T a /2 where T a are the Pauli matrices; thus the weak isospin component T3 is 1/2 and -1/2 for the upper and lower member of the doublet, respectively. Finally, the U(l)y or hypercharge transformation for a given fermion 7/.J = e, v,u,d ... is (5.6)

where YL, YR are the fermion hypercharges listed in Table 5.2. The numbers of this table may seem quite arbitrary but, as we will see later, they are connected in a deeper and subtle way with the absence of gauge and gravitational anomalies in the SM. The SU(2)L vector bosons are denoted by W~, where now a = 1,2,3. The coupling constant will be called g. The U(l)y or hypercharge interactions are mediated by the BI-' gauge boson and the coupling will be denoted by g'. Table 5.2. Hypercharge assignments in the SM

UR

2/3 0 2/3

y

T3 Q

UL

1/6 1/2 2/3

'DR

-1/3 0 -1/3

'DL

1/6 -1/2 -1/3

NL

-1/2 1/2 0

CR

-1 0 -1

CL

-1/2 -1/2 -1

Therefore, the Lagrangian describing the interactions of matter and gauge bosons can be written as £y M + 12m where _ 1

£YM-tr 2g s2

G

I-'V

12m =iQ f/JQQ

GI-'V

+ -2g12 trWI-'VW I-'V

1B I-'V - - I-'vB 4

+ i l llh£ .

(5.7)

(5.8)

Notice that we are using the conventions for gauge fields listed in Appendix A. The Dirac operators in (5.8) are f/JQ=/I-'D; = /1-' [&1-' f/J£=/I-' D{; =

+ GI-' + WI-'PL + ig'(yLQPL + YJ2PR)BI-'] /1-' [&1-' + WI-'PL + ig'(Yf PL + Yk PR)BI-'] ,

where the hypercharge matrices are given by

(5.9)

100

5. The Symmetries of the Standard Model yQ,.c _ L,R -

(Y~:-X 0

0

yf:~

)

(5.10)

.

The above covariant derivatives are defined so that the .cy M + .c m is invariant under local (gauge) SU(Nc ) x SU(2)L x U(l)y transformations. Therefore the gauge fields transform as follows:

e a --+ e a _ I-'

W:

I-'

--+ W:

BI-'--+BI-'

-

c ~ g8 aI-' ()ac (x) + fabc()bc (x) e 1-"

a= 1

2 1 Nc -

~al-'()L(x) + fabc()t(x)W~,

a= 1

3

+ ~al-'()Y(x). g'

(5.11)

r

Where bc and fabc are the structure constants of the SU(Nc) and SU(2) groups, respectively. In view of the Lagr;angian, it is important to remark that the hypercharge coupling not only depends on the fermion type, but also on its chirality, according to Table 5.2. Therefore, the left-handed and right-handed couplings are different, so that the U(l)y interactions are not vectorial and break the discrete P and C symmetries, although they still respect CP invariance. Finally, the gauge fields that appear in (5.7) and (5.8) do not correspond to those observed in nature. From those equations, the electric charge, defined as Q = y + T 3 , couples to the following linear combination: " __ Ar

glW~ + gBI-' . )g2

+ g'2

(5.12)

This field is nothing but the photon, which is the gauge boson corresponding to the electromagnetic gauge group U(l)EM. Its coupling is vectorial for all fermions (the neutrinos are chargeless) and proportional to Qe, where e is the electron charge, which is given by e=

gg' )g2

(5.13)

+ g'2

The orthogonal combination to (5.12) is the Z electroweak boson ZI-' =

gW3 - g'B I-'

)g2

+ gl2

I-'

(5.14)

so that we can write: AI-'=sin()wW~ + cos ()wBI-' ZI-'=cos()wW~ - sin ()wBI-' ,

(5.15)

where tan ()w = g' / 9 and ()w is called the Weinberg angle. The following commutation relations, [Q, T 3 ] = [Q, Yj = 0, imply that the photon and the Z boson are electrically neutral. The other WI-' combinations with well

5.1 The Elements of the Standard Model

101

defined electric charge, can be obtained from [Q, T 1 ± iT2 ] = =F(T1 ± iT2 ) and they are given by: W iL± =

1 (W 1 =F zW . 2) . V2 iL iL

(5.16)

Then it is easy to check in the Lagrangian that the only gauge fields that couple to the photon are W±. Notice that these definitions are consistent with the fact that the photon does not couple to neutral fields and, in particular, there is no photon self-interaction. 5.1.3 The Symmetry Breaking Sector As it has already been commented, the SBS has to be included in the SM in order to provide masses for the Z and W± electroweak bosons [5], as well as to all the fermions but the neutrinos. The need for such a sector comes from the fact that mass terms like

Mar W:WiL a

or

- mlftlJl = -m(lftLlJIR

+ lftRlJIL )

(5.17)

would spoil the SU(2)L and U(l)y gauge invariance. To avoid this problem the only known solution is to couple the SM to some other system, the SBS, which drives the spontaneous symmetry breaking of SU(2h x U(l)y down to U(l)EM. The mechanism responsible for such a breaking is not known, although many different procedures have been proposed in the literature. In the following we will concentrate on the most popular version, which is the MSM, whose SBS is chosen as simple as possible. Namely, it is built from a self-interacting complex scalar SU(2)L doublet
= (DiL¢)t(DiL¢) - V(¢)

+ .c YK

,

(5.18)

where, following the above hypercharge assignment: D iL ¢ = ( aiL

.g' + z2 BiL -

a a) ¢.

. T W z9 2 iL

(5.19)

In the last equation, V(¢) is the scalar field potential and .cYK are the Yukawa terms that will be described at the end of the section and which are also SU(Nc ) x SU(2)L x U(l)y gauge invariant. The potential V(¢) is chosen ad hoc to break spontaneously SU(2)L x U(l)y down to U(l)EM' The simplest choice is given by: (5.20)

which corresponds to the MSM, where J.L2 > O. At tree level, we find a minimum of this potential for space-time independent field configurations whenever (5.21 )

102

5. The Symmetries of the Standard Model

Notice that we do not have a unique classical vacuum state satisfying the last equation, however all minima are related by global SU(2) rotations. In order to build a QFT we have to choose one specific vacuum. As a consequence there is an spontaneous symmetry breaking which provides the Z and W± masses, although preserving the gauge invariance of the model. This effect is known as the Higgs mechanism, already explained in Chap. 3. In order to illustrate such phenomenon, let us adopt the usual choice of the scalar vacuum, 4>~ac = (0, v / J'2,), which is a minimum of the classical potential with zero electric charge (so that the photon will remain massless) but with the other three electroweak charges different from zero. It is now convenient to define the shifted field 4>' = 4> - 4>vac. Replacing this shifted field in the SBS Lagrangian of (5.18), it is easy to see that the term with covariant derivatives yields a mass for the Z and W± bosons, which at tree level are given by:

Mw=gv 2

Mz = Mw

cosBw

.

(5.22)

Experimentally it has been found that v ::= 250 GeV, M w ::= 80GeV and M z ::= 91 GeV. Let us change the notation slightly, in order to write the Lagrangian in a more compact form, by defining ¢ = ir 2 4>* and the 2 x 2 matrix M = J'2,(¢, 4», so that the previously chosen vacuum state is now written

(~~

Mvac

)

In general the M matrix can be written as M = (7 + iraw a , where (7 and w a are real scalar fields. With this parametrization, and once 9 and g' are set to zero, it is now easier to see that the scalar Lagrangian

.eM =

1

"4tr(oILMoILMt)

J.L2

+ ""4trMtM -

>.

16 (trMtM)

2

(5.23)

is invariant under the following SU(2)L x SU(2)R global transformations:

M'

= URMUl

UL,R

= exp ( -iaL,R

r;) ,

(5.24)

whereas the chosen quantum vacuum is only invariant under SU(2)L+R transformations. Therefore, the 7fa are the Nambu-Goldstone Bosons (NGB) that would have appeared from the global spontaneous symmetry breaking SU(2)L x SU(2)R ~ SU(2)L+R if we had set the gauge couplings to zero. However, when considering the real values of 9 and g', the Higgs mechanism comes into play making the NGB disappear from the physical spectrum and the gauge bosons become massive. That is the reason why these 7fa fields are usually called "would-be NGB". This symmetry breaking pattern will be an

5.1 The Cabibbo-Kobayashi-Maskawa Matrix and Weak CP Violation

103

essential feature when building the effective Lagrangian for a general SBS of the SM in Chap. 7. The a field is equal to v in the vacuum, so that small excitations around this state can be described by the field H = a-v. Again from the Lagrangian, we can read that the H field, which is called the Higgs boson, has a tree level mass given by

MiI

= 2.\v 2

.

(5.25)

Notice that, after the Higgs mechanism, this field does not disappear from the spectrum and indeed it corresponds to the only particle in the MSM which has not been discovered yet. Finally, the Yukawa terms which describe the interactions between the scalar fields and matter, can be written as (5.26) where Hu

o

(5.27)

are the matrices containing the Yukawa couplings. Thus, by replacing a = H - v in (5.26) we can read from those terms with just one v, the following fermionic mass matrices

MQ=vHQ + h.c. Mt:.=vHt:.

+ h.c..

(5.28)

There are also other terms with one h field which give rise to interactions between matter and the Higgs field. Notice that this interactions are proportional to the mass of the corresponding fermion. This completes the formulation of the MSM classical Lagrangian, which is given by: (5.29)

In Sects. 5.3 and 5.4 we will study some of the effects and constraints that appear when the SM is considered as a QFT. But before let us describe some consequences of considering massive fermions.

5.2 The Cabibbo-Kobayashi-Maskawa Matrix and Weak CP Violation In the preceeding section we have introduced the SM Lagrangian. In particular we have described the Yukawa terms, i.e. those responsible for the interaction between fermions and the SBS particles. These Yukawa couplings give rise, through the spontaneous symmetry breaking, to a mass matrix for fermions and electron-like leptons. However, this mass matrix is not diagonal

104

5. The Symmetries of the Standard Model

in the family space, namely, its eigenstates are not states of a definite family. In order to find the physical fermionic fields, which diagonalize the mass matrix, we have to perform some unitary transformations between different chiral fermion generations. The physical fields will be then defined as

1J'L,R

UE,R 1J L,R

U'L,R

Ut,RUL,R

[' L,R

Uf,R[L,R,

(5.30)

so that the Yukawa terms become r

'--YK

= -v

Q-' (UI£ HuU~t

0 )' L O U r HvU~t Q R

-, ( 00 UfHt:U~t 0 )LR ' + h.c.. -VLL Thus, the diagonal mass matrices will be given by i it M diag = V UiH L i uR ,

(5.31 )

where i = U, 1J, [. It can be easily checked that the redefinition of the fields does not change the form of vector forces as the electromagnetic or strong interactions, where for example we have: (5.32)

However, that is not the case for weak interactions. Indeed, in the L m Lagrangian, the term that describes the WI' interactions with fermions is given by

iQ,pWpPLQ = ~(UL,pW;UL -

+ .../2UL,pW;;1JL + .../2Th,PW:UL

3

-1J L,PWp1Jd ,

(5.33)

which shows that the interactions that change the quark flavor are only mediated by W±, but not by the Z nor the photon. This fact is usually known as the absence of flavor changing neutral currents. It is a very important property of the 8M, since those currents have not been observed in Nature. In terms of the rotated fields the charged flavor changing terms are

.,J2 - PWg-UL, I' 1J£ 2

uUVt + h .c. = gV2 - U-'£, PWI' U£ £ 1J'£ + h .c., 2

'-v--'

( ) 5.34

V CKM

so that the physical states couple with the charged currents through the matrix

VCKM =

ul£uft ,

(5.35)

which is called the Cabibbo-Kowayashi-Maskawa (CKM) matrix [6]. It is a n g x n g unitary matrix, Le.

5.2 The Cancellation of Gauge Anomalies in the Standard Model VCKMVdKM

= 1.

105

(5.36)

n g being the number of generations so that, at first sight, it depends on n~ real entries. However, by rotating the phases of the U and 1) fields it is possible to reduce the number of independent parameters

n;

---->

n; - (2n g

-

1) = (n g

-

1)2 .

(5.37)

Thus, for n g = 2 we are left with just one parameter, which is known as the Cabibbo angle, 8c, and thus in this case we can write v:ng=2 = ( CKM -

cos 8c sin8c

sin Bc ) cos8c

For n g = 3, which is the physically relevant case, we have four independent parameters, which are customarily parametrized as three angles (8 1 , 82 and 83) and one phaSe 8. The CKM matrix can then be written as: VCKM=

(81~~C1 8182C1

C2C3 82C3

~~~~3ei6 C1C28~~::C3e:66) + C283ei6

,

C18283 - C2 C3 e "

where 8i = sin 8i and Ci = cos 8i . Therefore the flavor changing charged current terms in (5.34) have an effective complex coupling due to the 8 phase appearing in VC K M. This fact leads to CP violating interactions or, according to the CPT theorem, to time reversal non-invariant interactions. We have thus shown how the SM includes an arrow of time. This feature is very welcomed since CP violating interactions were observed a long time ago in the KO kO system [7]. The CKM matrix could provide a natural explanation for this phenomenon, although the precise details of the possible connection to hadronic physics still remain unclear.

5.3 The Cancellation of Gauge Anomalies in the Standard Model In Chap. 4 we have seen that chiral gauge theories, like the SM, are potentially anomalous. Such anomalies in the gauge currents could lead to inconsistencies in the quantization of gauge theories. Therefore, the SM is potentially inconsistent and thus, in this section we will study whether the SU(Nc ) x SU(2}L x U(l)y invariance of the classical Lagrangian can be maintained at the quantum level. With that purpose, we are going to select the quantum numbers of the different chiral fermions such that all gauge anomalies cancel. As we will see, this can be done by an appropriate hypercharge assignment. Moreover, we will show in Chap. 8 that when the gravitational interaction is included (at the classical level) the cancellation of gauge and gravitational anomalies completely determines the values of the hypercharges for the different chiral fermions (see Table 5.2) and gives rise to the quantization of the electric charge. In the following we will describe how to calculate

106

5. The Symmetries of the Standard Model

the possible gauge anomalies that could arise in the SM. From now on we will be working in Euclidean space-time (see Appendix A). Due to the chiral couplings of the U(l)y and SU(2)L gauge bosons, the Dirac operators for quarks and leptons defined in (5.9) are not hermitian, in fact

(i -It'Q)t =i'y{L D; = i'y{L(8{L+ G{L + W{LPR + ig'(Yj PL + yLQ PR)B{L) (i -It'c)t =i,{L D~ = i,{L(8{L + W{LPR + ig'(Yi, PL + Yf PR)B{L) .

(5.38)

As we saw in Chap. 4, when dealing with non-hermitian operators there is no natural operator to be used as regulator in the anomaly calculation. Instead several methods have been proposed that lead to the same cancellation conditions, although they yield different values for the anomalies. In the following we will use the second method presented in Sect. 4.5. As we have seen in the preceeding section any infinitesimal gauge transformation of an arbitrary fermion can always be written as

'lj;(x)-+'lj;(x) - iaa(x)Ta'lj;(x) 7jj(x)-+7jj(x) + ia a(x)7jj(x)T a ,

(5.39)

where now we write generically Ta as any of the SM group generators. Let us recall that the existence of an anomaly means that the integration measure in the generating functional changes as follows:

[d'lj;d7jj] A"

-+

[d'lj;'d7jj'] = [d'lj;d7jj] exp

(i Jd x aa(x)Aa(x)) 4

~ J~= [~¢~T"e-HZ;' ¢n - ~{~T"e- H~;' {nJ '

where cPn(x) and

~n(x)

HE,c

(541)

are, respectively, the eigenfunctions of the operators

Hi'c = (i -It'Q,c)t (i -It'Q,c) = D;'c D~,c 1/J

(5.40)

~b{L, ,vj[D;'c, Df'c]

= ('T1IQ,C)('T1IQ,C)t = DQ,CD{L _ ~[{L V][DQ,c DQ,C j t-'f' t-'f' {L Q,C 4" , {L' v

. (5.42)

Therefore (5.41) is already appropriate to perform a Seeley-de Witt expansion in powers of M- 2 (see Appendix C.4). The first coefficient ao is the same for both pieces in (5.41) and it cancels. The al term also vanishes, when taking the group and Dirac traces. Thus the only contribution to the anomaly is that of a2, as we saw in Chap. 4, whose explicit expression is given in (C.99). The results for the SM groups are the following

SU(Nc ):

A~(x) =

(4:)2 tr

C';

(a2 (Hj,x) -a2

a SU(2)L: A't(x) = (4:)2tr (T (a2 (Hj, x) PL

(H~,x)))

5.3 The Cancellation of Gauge Anomalies in the Standard Model

-az (H~, x) P

R) )

107

+ (Q -+ £)

= (4~)Z tr ((yLQ PL + Yj PR) az (Hi, x)

U(l)y : Ay(x)

- (yLQPR + YjPL) az (H~,x)) + (Q -+ £).

(5.43)

The precise form of these anomalies are given below, when discussing their cancellation conditions. First, let us recall the relationship between the anomaly and the divergence of the gauge current vacuum expectation value. We will consider in detail the SU(Nc ) case, but the procedure is the same for the other two groups. From (5.3) the induced anomalous change in the effective action reads e- r [G-D9 c ,W,BJ =

=

where 12m now (see Appendix ant derivative jJ.L = _ij~)..a /2 (DJ.Lu~))a

ei J d4x9~(D,.(j:;W

J

4 [dQdQJeJ d x.c=e i J

d4x(9~(x)A~(x))

(5.44)

,

denotes the Euclidean version of the 8M matter Lagrangian A). As usual, DJ.L = oJ.L + [GJ.L,·J is the gluon field covariand j:;a = Q,J.L )..aQ/2. Therefore, for the gauge current we can write

= A~(x)

(5.45)

.

Hence, as we have already commented, the consistency of the gauge theory requires the cancellation of the Aa for every gauge group. Let us now see how this can be achieved in the 8M. With that purpose, we replace the az coefficients in (5.43) and take the trace over Dirac and internal indices. Thus, after a lengthy but straightforward calculation, we get the following gauge current divergences - SU(Nc )

_ ( D J.L (.J.L))a Jc

-

1 I J.LvpaG a B 321rz9s9 E J.LV pa

~

(5.46)

~

all quarks

The only way to cancel this expression is that the summatory above vanishes. Conventionally, however, the cancellation is imposed on each family separately, which is more restrictive. In other words, we demand

L(YL - YR)

(5.47)

= 0

u,d

for each generation. The expressions for the other two families are obtained by replacing u and d by their corresponding u and d-like quarks. - SU(2)L

(DJ.Lut))a

=-

32~z99'EJ.LVpaW:vBpa (Nc

L all quarks

YL +

L

YL).

all leptons

(5.48)

108

5. The Symmetries of the Standard Model

Again the cancellation condition is imposed on each family separately and, for the first generation, it reads:

Nc

I>L + LYL = O. u,d

(5.49)

v,e

The conditions for the mUOn and tau families are the same.

- U(l)y 2

D J.L]Y ( -J.L)

-

1 95 167r 2 [ 4

X

(Nc

a

a

10 J.Lvpae J.LV e pa

,2

~

L....

(

all quarks

L

all quarks

(yI -

9 YL - YR ) + 2

y~) + L

all leptons

2

+ 98

fJ.LvpaW:v W;a

(Nc L all quarks

(yI -

10 J.LVpaB J.LV B pa

YM)

YL + L all leptons

YL)] .

(5.50)

The cancellation of these terms, again imposed on each family separately, gives the conditions

0= L(YL - YR) u,d

O=NcLYL + LYL u,d

v,e

(5.51 ) u,d

v,e

for the first family, as well as similar expressions for the other two generations. Notice that the conditions in (5.47) and (5.49) are the same as the first two in (5.51). From Table '5.2 it is easy to check that these equations are satisfied in the 8M generation by generation, provided N c = 3. This is a highly nontrivial fact since historically the hypercharge assignment was based on pure phenomenological grounds and only later it was realized that the 8M is free of gauge anomalies with that choice. We should remark that the above COnditions do not completely determine the hypercharges. Nevertheless, within the M8M, the gauge invariance of L YK imposes two additional conditions. Namely, for the first generation, we have 1

-YR - -YR=-Y'" 2 2 dIu

3 d 3 u "2YR + "2YR =Y", ,

(

5.52

)

where Y", is the Higgs doublet hypercharge. With this new constraint, all the hypercharges are determined, up to a normalization constant, which can

5.3 Baryon and Lepton Number Anomalies in the Standard Model

109

G{3 Ba G.., W{3 Ba W.., B{3 Ba B.., Fig. 5.1. Feynman diagrams contributing to the standard model gauge anomalies

be fixed by demanding Qe = -1. Alternatively, in Chap. 8 we will see that, without assuming any specific 8B8, the hypercharges can be fixed from the cancellation of gravitational anomalies. Finally, (5.51) can be obtained in a straightforward way by considering the triangle diagrams in Fig. 5.1. Apart from the gauge anomalies we have also studied in the previous chapter the so called non-perturbative anomalies. These anomalies appear in any chiral SU(2) theory with an odd number of fermion doublets, so that, in principle it could also affect the SU(2)L group. However, we have one lepton doublet and N c quark doublets per generation. Hence the total number of SU(2)L doublets is N c + 1, which is even for N c = 3. Therefore we see that, not only the gauge anomalies, but also the non-perturbative ones do cancel generation by generation in the 8M.

5.4 Baryon and Lepton Number Anomalies in the Standard Model From the 8M classical Lagrangian, (5.29), it can be checked that the following vector currents 1-

j~=-Q'Y~Q

Nc

jt =Z'Y~.c

(5.53)

110

5. The Symmetries of the Standard Model

are conserved classically. They are associated, respectively, to the following global transformations Q' =eiO
£'=eiO
(5.54)

The associated charges are called the baryon and fermion numbers, respectively. However such conservation laws are spoiled for the quantized theory [8]. Throughout this section we will make use of the methods presented in Chap. 4, in order to obtain the anomalies which are responsible for the quantum violation of the baryon and lepton numbers. With that purpose, we will obtain the anomalous Ward identities associated to these symmetries. We will then consider the local and infinitesimal transformations in (5.54)

+ iaB(x)Q £~£ + iaL(x)£ .

Q~ Q

(5.55)

Let us study the baryonic current. (the results will be equally applicable for the leptonic case). We will follow similar steps as we did in Chap. 4 for the calculation of the U(l)A anomaly. Thus, the first transformation in (5.55) yields for the effective action: e-r[G,W,Bj

= J [dQdQ

4 ]eI d x.cMSM(Q,Q... )

= J[dQdQ

]e- I

(5.56)

4 4 d xiO
where the dots stand for the rest of the 8M fields which are irrelevant in this calculation. Using the same regularization method as in the previous section, we obtain for the regularized anomaly: 1

A(x) = (41T)2tr(a2(HQ,x) -a2(HQ ,x)) .

(5.57)

Therefore, identifying the integrands in (5.56) we arrive to the anomaly in the baryonic current, which is

8ttj~ ~;:: (g; W:vW;cr + g,2 BttvBpcr L =

all quarks

(YL - Yk)) .

(5.58)

Following the same steps with the leptonic current, we obtain: ttVPcr .tt _ E (g2 a a , 2 8 tt h - 321T2 2 W pcr W ttV + 9 BpcrBttv

~

LJ all leptons

(2

2 ))

YL - YR

.

(5.59)

Therefore the baryon and lepton numbers are not conserved. However, we see that B - L will be a conserved current, provided the following relation is satisfied:

5.5 The Evolution of the Coupling Constants

111

(5.60) all quarks

all leptons

which is the case for the usual SM hypercharge assignment (see Table 5.2). Indeed the above equality holds for each family separately. The phenomenon of Band L violation but B - L conservation could be of physical relevance for baryogenesis. As we can all observe, our world seems to be made of particles rather than antiparticles; or in other words, the universe constituents are mostly baryons and leptons. Notice that B - L conservation means that if the global number of baryonic matter is decreased by one, then the number of total antileptons is increased, again by one. As we have already seen, the SM slightly breaks CP invariance, which means that certain particle reactions have not the same rate for the corresponding antiparticles. Both B and CP violation are two of the three conditions [9] required to generate an excess of matter with respect to antimatter in the universe. The third condition is that the system is out of thermal equilibrium. In principle, the SM contains all the ingredients to explain baryogenesis qualitatively. However, at present, no physical mechanism within the SM, has been able to yield the observed magnitude of the Universe baryon asymmetry. Furthermore, it is even possible that the baryon anomaly could erase asymmetries produced beyond the SM.

5.5 The Evolution of the Coupling Constants It has been shown by 't Hooft that gauge theories, either spontaneously bro-

ken or not, are renormalizable [10]. Therefore, the formulation of the SM in terms of gauge interactions ensures its renormalizability. In addition to the gauge, non-perturbative and global anomalies, we have already mentioned in the previous chapter the trace anomaly. This anomaly occurs in four dimensional QFT since a scale or cutoff has to be introduced in order to regularize the divergent integrals that appear in the calculations, thus breaking the explicit scale invariance at the quantum level. Notice that in the SM the scale invariance is already broken in the classical Lagrangian due to the presence of masses; nevertheless, the trace anomaly will give another contribution to this breaking. Indeed, for the case of pure Yang-Mills theory we obtained in Sect. 4.11

aDJ.L--(3(g)trF J.L g3 J.LV FJ.LV ,

(5.61)

where DJ.L is the Noether current associated to the scale symmetry of the pure Yang-Mills action and (3 is the standard beta function, which describes the evolution with the renormalization scale f.1 of the gauge coupling 9 (see Appendix C.1). If we want to consider this equation for the SM we should also include the above mentioned explicit scale symmetry breaking contributions.

112

5. The Symmetries of the Standard Model

However, at energies much bigger than any mass scale, we can neglect the effect of these terms. Then we can have an approximate idea of the scale evolution of the 8M gauge couplings at very high energies (when compared with the electroweak symmetry breaking scale, v ~ 250GeV) [11]. To one loop it is not difficult to find how the couplings run with the scale

d9i ~ b 9; (Ji(9i) = d(logJ.L) - i (411")2 ,

(5.62)

where, for further convenience, we have defined 91 = 9',92 = 9 and 93 = 9s· The bi coefficients are given, at one loop, by 20 b1 =gn9

b2 = -

(2; - ~ng )

b3 = - (11 -

~ng)

,

(5.63)

where n g is the number of generations and the Higgs sector contribution has been neglected. Now it is immediate to integrate the above differential equation to find:

O:i (J.L ) =

O:i(J.LO) b

2

1 - O:i(J.LO)~ log ~

(5.64)

with

O:i = -9t . 411"

(5.65)

The qualitative behavior of these couplings as a function of the energy can be found in Fig. 5.2. Looking at this figure one could reasonably assume that the three couplings reach the same value at some large scale M, i.e. O:i(M) .= 0:0. From (5.13) the electromagnetic coupling satisfies O:e1- = o:~ + o:i- Then we can use the experimental values 0:3(Mz) ~ 0.11 and O:e1-(Mz ) ~ 128 to find M ~ 10 15 GeV. As well, we can define sin 2 Bw = O:EM/0:2 and obtain sin 2 Bw(Mz ) = 3/8. This value is not very far from the experimental one and in fact this was one of the strongest hints that led to the idea of grand unified theories (GUT) . In these models, the 8M gauge group is a subgroup of a larger group like, for instance, SU(5) or SO(10) which are spontaneously broken to SU(Nc) x SU(2)L x U(I)y at the grand unified scale M. However, after the precision measurements performed at LEP (large electron-positron collider), of the 8M couplings at the M z scale it has been established, with a large statistical significance, that the three curves do not meet in a single point. Nevertheless, the amazingly close evolution of the three curves at high energies has given rise to many elaborated extensions of GUT, including supersymmetry, different group choices and intermediate symmetry breakings. In general, these models introduce

5.6 The Strong CP Problem

QEM

-----

113

______

M

Fig. 5.2. The running of the couplings up to a possible unification scale

many heavy particles that could be used to explain some cosmological problems like dark matter, baryogenesis, neutrino fluxes, etc., although they are constrained by other observations, mainly the proton lifetime, since it will decay via these new particles and interactions. The effective Lagrangian formalism could be particularly suitable to deal with the low energy effects of these models but, as far as we are only concerned with the SM, they are beyond the scope of this book.

5.6 The Strong CP Problem We will now turn to study one of the most strange aspects of the SM which is still not very well understood, namely the so called strong CP problem. It relates in an unexpected way the quark mass matrix coming from the Yukawa couplings, the U(l)A anomaly and the so called 8-vacuum term. This term appears in the effective action due to the highly non-trivial vacuum structure of any non-Abelian gauge theory such as QCD. In Sect. 5.2 we have seen how the quark mass matrix can be diagonalized to obtain the physical states. This diagonalization gives rise to observable effects through the CKM matrix which is considered responsible for the CP violation that occurs in the K - KO system. In this section we are going to introduce another source of CP violation coming from the strong interactions. In the following we will concentrate in the SM light quark sector, namely, the u, d and s. As it has already been discussed, in the massless quark limit this system has a global U(3)L x U(3)R = SU(3)L X SU(3)R x U(l)v X U(l)A symmetry at the level of the action. However the physical hadronic spectrum does not correspond to an

114

5. The Symmetries of the Standard Model

SU(3)L X SU(3)R symmetry but to an SU(3)L+R. It is then natural to assume an spontaneous symmetry breaking from SU(3)LXSU(3)R to SU(3)L+R. The U(l)v is preserved even at the quantum level as an exact symmetry an gives rise to the baryon number conservation by strong interactions. However, the U(l)A symmetry is anomalous and this anomaly has very important physical implications like the possible solution of the r/ problem. More precisely, let us define the axial current (5.66)

which is conserved at the classical level (in the massless limit). As we saw in Sect. 4.2, we obtain the following anomaly equation:

o ).J.L5 = _ 3g; tJ.LvpuC a C a J.L 321T 2 J.LV pu

(5.67) 1

where we have only written the gluon contribution which is the only relevant for the present analysis. On the other hand, as it will be discussed in detail below, the axial anomaly is closely related to the so called B-vacuum coming from the very rich vacuum structure of non-Abelian gauge theories. In order to understand such relation, let us first give a brief overview of the main properties of the ground state of quantized Yang-Mills theories an its main observable effects. As we will see, the complexity of the non-Abelian gauge theory vacuum is due to the very rich topological structure of these models, which plays a very important dynamical role. For the sake of definiteness we will start by considering a pure (without matter) SU(2) gauge theory. Thus the action is given by

SYM[Aj =

~ jdxtrFJ.LvFJ.LV 2g

.

(5.68)

This Lagrangian is invariant under the following SU(2) gauge transformations A~(x) = g-l(x)(AJ.L(x)

+ oJ.L)g(x)

,

(5.69)

where

g(x) = e- ira ",a(x)/2

.

(5.70)

In order to illustrate the physical meaning of the Yang-Mills vacuum, we are going to work in the gauge

Ao(x) = 0 .

(5.71)

In spite of this gauge-fixing condition we still have the residual gauge invariance

(5.72) where we have used the notation x = (t, x) and the i index runs through the spatial components. In other words, we still have the freedom to make

5.6 The Strong CP Problem

115

time-independent SU(2) gauge transformations on the spatial components of the gauge field. From now on we will impose the additional restriction lim g(x) = 1 ,

Ixl-oo

(5.73)

which will define the acceptable spatial gauge transformations. The reason for such definition is that in quantum gauge theories, Gauss law can only be imposed as a weak condition, i.e., as a constraint to obtain physical states but not as an operator equation. As a consequence, it can be shown [12, 13] that gauge equivalence only holds under gauge transformations which satisfy the above boundary condition. As far as the admissible gauge transformations satisfy the above constraint, we can identify the sphere at spatial infinity with a single point. That is, we can compactify the space as an S3 sphere so that the gauge transformations can be considered as maps g(x) : S3 ---. SU(2). However, not all these maps can be deformed to the trivial one, but they are classified in different equivalence (homotopy) classes according to the homotopy group 7r3(SU(2)) = 71. (see Appendix B.3). Hence, any spatial admissible gauge transformation can be labeled by an integer winding number which is given by (5.74) 5.6.1 The

(J- Vacuum

Now we will consider the possible classical vacuum states of the SU(2) gauge theory. The minimum of the energy is reached whenever FJ.l.v = O. This condition immediately leads to the perturbative vacuum Ai = o. However this is not at all the only possibility, since any pure gauge field Ai(x) = g-l (x)Big(x) will also have a vanishing strength tensor and thus zero energy. Therefore, at the classical level, we have many vacuum configurations corresponding to different homotopy sectors, classified by their winding number. In addition, it can be shown that the energy barrier between two of those topologically different vacua is finite [13]. In Fig. 5.3 we have represented a schematic view of the energy barrier between different classical vacua. At the quantum level we will have a set of degenerated states In) corresponding to the local minima of the topological sector labeled by the integer n. Those states are invariant under small (topologically trivial) gauge transformations but they are not under non-trivial gauge transformations. More precisely, if gm is a gauge transformation with winding number m (W[gml = m) and U(gm) is the operator that implements this transformation in the Hilbert space of the quantum theory, we find

U(gm)ln) = In + m) .

(5.75)

However, the physical vacuum Iphys) should be invariant (modulo one phase) under both, small and large, gauge transformations. Hence, we can write

116

5. The Symmetries of the Standard Model

Iphys) as a linear combination of the In) states. Then, using (5.75), the above requirement of invariance for the true vacuum leads to 00

(5.76) n=-oo

where we have introduced the notation IB) for the physical vacuum to make explicit its dependence on the undetermined phase B [14]. Thus we have (5.77) Therefore, none of the states In) is the real vacuum. What really happens is that, as far as the energy barrier between the different In) states is finite, tunneling gives rise to the true vacuum state IB), which is a quantum superposition of the In) states. In Fig. 5.3 we have represented such tunneling effects by a discontinuous line. Let us now consider a gauge invariant operator acting on the Hilbert space of the theory, so that under a gauge transformation with winding number one gl we have

°

[0, U(gdl = 0,

(5.78)

and hence (5.79) Thus we have

(BIOIB') = 0

(5.80)

whenever e is different from B'. In other words, theta vacuum states with different phases cannot be connected through any gauge invariant operator and, in particular, they are orthogonal. As a consequence the whole relevant part of Hilbert space corresponds to just one value of the phase B which can be considered as a new parameter of the theory since the states corresponding to other B values are completely decoupled.

In+l> Fig. 5.3. Pictorial view of the energy barrier between different classical vacua and the effect of a gauge transformation of winding number one

5.6 The Strong CP Problem

117

5.6.2 The Role of Instantons As we have mentioned above, quantum tunneling between different In} states is responsible for the appearance of the B-vacuum structure of the ground state. In order to understand the nature of these transitions from the path integral point of view, one has to consider Euclidean gauge field configurations AJ.L(t, x) interpolating between some given pure gauge field Ai(x) = g;;1(x)8i gn (x) with winding number n for the initial time t = -00 and some other pure gauge field Ai"(x) = g;;.1(x)8i gm (x) with winding number m for the final time t = 00. In order to find those interpolating configurations, let us consider an Euclidean gauge field AJ.L(t, x) which is pure gauge AJ.L = g-18J.Lg on the space-time boundary 53. We will denote by q = W[g] E 7L. the winding number of g, as given in (5.74). This number can also be written as (see Appendix B.3)

q --

- -12

3211"

1 84

(5.81 )

d X t r€ J.LVP(jFJ.LV F p(j.

Now, to make contact with our previous discussion, let us consider the A o = 0 gauge and deform the space-time boundary from 53 into a cylinder as in Fig. 5.4 (note that this deformation will not modify the topology of the problem nor the values of the winding numbers). Thus, for any x, the top and the bottom surfaces of the cylinder correspond to t = 00 and t = -00 respectively, while the curved surface stands for the spatial infinity (Ixl = 00) for any time t. At this point, we can use the freedom to make time1:= 00 ,

any

IxI = 00

~--+--'--'----

1:=-00

,

x

,

any

1:

any X

Fig. 5.4. Representation of the cylinder obtained by topological deformation of the S3 space-time boundary

118

5. The Symmetries of the Standard Model

independent gauge transformations to set g = 1 at t = -00 by means of a large gauge transformation, so that n = 0 and q = m. Then, as far as we are in the Ao(t, x) = 0 gauge, we can write symbolically

Ao(t, oo) = g

-1

a

(t, 00) atg(t, 00) = 0,

(5.82)

which means that g(t, 00) is t independent and as we have set g( -00,00) = 1 then g(t, 00) = 1 for any timet. Notice that this is nothing but the constraint in (5.73). Hence, the contribution of the curved surface to the q integral vanishes and we get

q = -1-2 247r

1 1

- - -12

247r

t=oo

··k f.'J

t=-oo

trAmAmA m 'J

··k trAf f.'J

k

AjAk = m - n.

(5.83)

Therefore, the Euclidean gauge field AJ..l(t,x) (which is a pure gauge in space-time infinity) interpolates from t = -00 to t = 00 between pure gauge fields with gauge functions gn(x) and gm(x), which satisfy gn(oo) = gm(oo) = 1. Thus, in order to obtain the tunneling amplitude from the state In) to 1m), we have to consider the Euclidean gauge fields with winding number q = m - n, which we will denote by A~. Among such fields, there is an special subset which minimizes the Euclidean Yang-Mills action. As it is discussed in Appendix B.3 they are (anti)self-dual fields which are solutions of the equations of motion and are known as q-instantons (as a technical remark, note that the solution given in Appendix B.3 for the 1-instanton should be transformed to the Ao = 0 gauge). Instantons are specially relevant in the context of semiclassical calculations, where the path integral is restricted to solutions of the equations of motion and hence the instantons are the main contribution to the tunneling processes. Moreover, as the instanton action has a well defined value, the Boltzman factor (see Appendix B.3) (5.84)

where W[A] is given in (5.74), can be evaluated exactly thus yielding complete analytical results [13].

The (}-Term The complicated and rich topological structure of the vacuum in non-Abelian gauge theories can be effectively taken into account by adding a term to the Yang-Mills action. For that purpose, let us consider again the transition amplitude for two different e-vacuum states Ie) and Ie'). In the path integral formalism it is given by

5.6 The Strong CP Problem

(8'le- iHt I8) =

00

00

L L

e- im9' ein9 (mle-iHtln)

m=-oon=-oo

=

L L 00

119

00

ein9-iq9'-in9'

j

[dA~]e-SYM[Aj

m=-oon=-oo

=

21f8(8 - 8')

f f

eiq9 j[dA~]e-SYlvdAl

q=-oo

=

21f8(8 - 8')

j[dA~] e-SYAodAI+sf[Aj ,

(5.85)

q=-oo

where H is the Hamiltonian, q = m - n and [dA~] means integrating to any gauge field that interpolates between two states nand m. Notice that, in the second step above, we have performed the Wick rotation to Euclidean spacetime. Hence, apart from reobtaining that different 8-vacua are decoupled, we conclude that the effect of the 8-vacuum amounts to adding to the Euclidean Yang-Mills action the term i8 j dX € !J.Vpu tr!J.v F Fpu, S 9E[A] -- 321f 2

(5.86)

where we have used (5.81). Note that this term is a total divergence and therefore does not contribute to the equations of motion, although it leads to different quantum theories depending on the 8 parameter. In addition, it violates the P and T symmetries, i.e., it violates CP according to the CPT theorem. Finally we would like to comment that the above analysis can be extended to any other SU(N) gauge theory with N ~ 2, since then 1f3(SU(N)) = 7l and all the topology arguments used here are the same.

5.6.3 The Strong CP Problem Now it is very simple to apply the whole previous discussion to QCD. In particular, as the sf term in QCD violates CP, it could produce an electric dipole moment for the neutron, whose measurement would provide an estimate of the 8 parameter. Let us show in detail how this mechanism works. Notice that the U(l)A anomaly in (5.67) has the same form as the 8-term. Thus, one could think of absorbing the latter with the anomalous Jacobian of an appropriate axial redefinition of the quarks. However, as we will see below, the existence of the mass matrix makes impossible such a redefinition in QCD. We are going to start from the Euclidean QCD partition function for the lightest three quarks which, including the 8-term, is given by

Z(8, M) = j[dq][dij][dG!J.] x exp [- j

(5.87)

dxGG~vGa!J.v + q f1Jq + (qLMqR + h.c) + ~~~€!J.vPuG~vG~u)] ,

120

5. The Symmetries of the Standard Model

where q is a vector in flavor space, with components qr = U, d, s for T = 1,2,3 and M is the 3 x 3 mass matrix. According to our discussion in Sect. 5.2, we can always diagonalize M, although in general its eigenvalues will be complex, i.e. m eiOu

M

u0

=

o

(

0 0)

md eiOd

0

0. msetlis

with m r real and Dr some arbitrary phases. Now we can perform the following change on the quark dummy integration variables: (5.88)

Note that this is not a U(I)A rotation, since the Or phase is different for each quark. The effects produced by this change are the following: on the one hand, it is not difficult to see that the phases in the mass matrix change as Dr ---. Dr + Or' On the other hand, due to the U(I)A anomaly mentioned above we generate a term g2 expio dX_s_ElJ.vpaea e a (5.89)

J =

6471"2

IJ.v

pa

with 0 Lr Or' Then, the 0 angle changes as 0 ---. 0 - o. Now, let us take Or = -Dr, so that we get a real mass matrix (physical basis). Notice that, with this choice, Lror = - LrDr = -arg(detM). Hence, we obtain for the partition function

Z(O,Dr)=Z(e,O) e=o

+ arg(detM)

.

(5.90)

Therefore, our first conclusion is that we can rotate to the physical basis by shifting the 0 parameter to the effective given in (5.90), with M the original mass matrix. Our next step is to analyze the physical effects of For that purpose, we consider the physical basis and perform again a chiral rotation

e

e.

(5.91)

qr ---. q~ = e-il3r'Ys/2qr

chosen so that we can remove the term, that is, we take Lr /3r = Of course, that yields a complex mass matrix. However, let us write the quark mass term in a slightly different form, with the additional assumption « 1 (which is the physically interesting case, as we will see below) and hence /3r « 1. Thus,

e

7jM physq

=

L mr(cos /3r7j~q~ -

e. e

i sin/3r7j~'Y5q~)

r

(5.92) r

where M phys = diag(m u , md, m s )' The quark operator linear in /3r in the last equation above breaks CP and yields an electric dipole moment for

5.7 The Symmetries of the Standard Model

121

the neutron. Now we recall the QCD symmetry breaking pattern SU(3)L x SU(3)R ----> SU(3)L+R in the chirallimit. Then, in order to consider the terms in (5.92) as a true perturbation when m r =I- 0, we have to impose them the Dashen conditions (see Sect. 2.5). It can be shown [15] that, using the freedom to perform SU(3)L-R rotations (which is equivalent to change the origin in the coset space), these conditions imply (5.93)

e

which, together with the equation L r f3r can be solved for the f3r in terms of the m r and O. Finally, the Lagrangian responsible for the neutron dipole moment can be written as rdip

L

3m u m d m s

'0-

=1

mumd

+ mums + mdms

(5.94)

,,",-

~~~~.

r

The estimated theoretical value for the neutron dipole moment de depends on the models for the nucleon structure. It is on the range de rv 10- 15 e·em [16]. The present experimental bound on de is given by de < 1.1 X 10- 25 e· em [17] which means:

ex

e< 10- 10 .

(5.95)

e

Note that in order to have such a small value we need an extremely fine tuning to cancel the 0 parameter with arg(detM). However, the 0 parameter is related only with the QCD vacuum, whereas the mass matrix M comes from the Yukawa couplings and finally from the SBS of the SM, which is very poorly understood from the experimental point of view. What is called the strong CP problem is why these two quantities, which apparently are not related, cancel each other with such a high precision. The only possible solution in the framework of the MSM is to have at least one massless quark, since then the CP violating term £dip above is not observable. However, the present data strongly suggest the opposite, i.e. there are no massless quarks. If this is really the case, the solution of the strong CP problem probably lies outside the SM. In principle, for SU(2)L' there should be another 0 Lagrangian, in terms of v fields. Nevertheless, an appropriate global vector transformation like that in (5.54) can generate the very same SU(2)L 0 term, through the anomaly in (5.58). In contrast to the QCD case, such a redefinition does not change the mass matrix and we can cancel the 0 parameter. Hence, the SU(2)L 0 term does not lead to any observable effect.

W:

5.7 The Symmetries of the Standard Model We end this chapter by reviewing the main symmetries of the SM, paying an special attention to the way in which they are realized. This analysis will be

122

5. The Symmetries of the Standard Model

very useful in the next chapters, where we will study different phenomenological effective Lagrangians in order to describe the low-energy behavior of some of the less known sectors of the SM. In view of the preceeding sections, the status of the SM symmetries can be summarized as follows:

• Poincare, Scale and Conformal Symmetries: Poincare is a global symmetry. However, when gravitation is included (see Chap. 8) the Lorentz subgroup becomes local and there is no translational invariance. Scale invariance is broken explicitly in the Lagrangian as well as by the trace anomaly through the renormalization process. Therefore, the SM is not invariant under conformal transformations. • Gauge Symmetries: The SM Lagrangian is SU(Nc ) x SU(2)L x U(l)y gauge invariant. The hypercharge assignment is done so that the gauge and non-perturbative anomalies cancel generation by generation (gravitational anomalies do cancel too, but that will be studied in Chap. 8). Thus the SM is a consistent SU(Nc ) x SU(2)L x U(l)y gauge invariant QFT. However, the vacuum is not SU(2)L x U(l)y invariant and therefore this symmetry is spontaneously broken down to the electromagnetic group U(l)EM. Such a breaking gives masses, through the Higgs mechanism and the Yukawa couplings, to the W± , Z electroweak bosons and to the fermions, respectively. • Discrete Symmetries: As far as the electroweak interactions are chiral, they break the P and C symmetries although they conserve CPo However, the complex phase appearing in the CKM matrix breaks this CP symmetry. In addition, there is another possible, but not very well understood, source of CP violation in the strong interactions (the strong CP problem). For local QFT there is a strong result that ensures CPT invariance. As a consequence, CP violation implies T violation, Le., the SM provides a microscopic time arrow. • Baryonic and Leptonic symmetries: The baryon and lepton numbers Band L are conserved classically although there is an anomaly that spoils this conservation at the quantum level. However, even taking into account the anomaly, B - L is still conserved in the SM. As we will see in Chap. 8 that is not the case when gravitational fields are present. • Approximate symmetries: a) When 9 and g' and the Yukawa couplings are switched off, the SBS of the SM presents a SU(2)L x SU(2)R global symmetry which is spontaneously broken to SU(2)L+R. When turned on again, this global symmetry breaking induces the gauge SU(2)L x U(l)y symmetry breaking down to U(l)EM thus providing the masses for the electroweak gauge bosons.

5.8 References

123

b)By neglecting the masses of the lightest N f quark flavors the SU(Nc ) QCD Lagrangian has a global SU(Nf)L x SU(Nf)R x U(l)v X U(l)A symmetry. In view of the physical hadronic spectrum, the chiral SU (Nf) L xSU(Nf)R seems to be spontaneously broken to SU(Nf h+R thus giving rise to the appearance of 1 (pseudo)NGB (see Chap. 6). The U(l)v symmetry is exact and implies baryon number conservation by strong interactions. However the U(l)A symmetry is anomalous. This anomaly is welcomed since it could explain the 1]' problem.

NJ -

5.8 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

[12] [13] [14] [15] [16] [17]

C.N. Yang and R.L. Mills, Phys. Rev. 96 (1954) 191 M. Gell-Mann, Phys. Lett. 8 (1964) 214 G. Zweig, CERN preprint 8182/Th. (1964)214 O.W. Greenberg, Phys. Rev. Lett. 13 (1964) 598 M.Y. Han and Y. Nambu, Phys. Rev. 139 (1965) BlO06 H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 478 (1973) 365 S.L. Glashow, Nucl. Phys. 22 (1961) 579 A. Salam and J.C. Ward, Phys. Lett. 13 (1964) 168 S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264 P.W. Higgs, Phys. Rev. Lett. 12 (1964) 132 N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531 M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652 J.H. Christenson, J.W. Cronin, V.L. Fitch and R Thrlay, Phys. Rev. Lett. 13 (1964) 138 G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8 A.D. Sakharov, Prisma ZhETF 5 (1967) 32 G. 't Hooft, Nucl. Phys. B33 (1971) 173; Nucl. Phys. B35 (1971) 167 H. Georgi, H. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451 R. Jackiw, Rev. Mod. Phys. 52 (1980) 661 R Raj araman, Solitons and Instantons, North-Holland, 1982 C.G. Callan, RF. Dashen and D.J. Gross, Phys. Lett. 63B (1976) 334 V. Baluni, Phys. Rev. D19 (1979) 2227 J.F. Donoghue, E. Golowich y B.R. Holstein, Dynamics of the Standard Model, Cambridge University Press, 1992 Particle Data Group (RM. Barnett et al), Phys. Rev. D54 (1996)

6. The Effective Lagrangian for QeD

In previous chapters, we have studied the formal development of the effective Lagrangian formalism. In Chap. 3 we have seen how to obtain the non-linear sigma model (NLSM) Lagrangian. This is the lowest order term in the derivative expansion that describes the dynamics of the Nambu-Goldstone Bosons (NGB), which are the low-energy excitations ofthe vacuum in any theory with an spontaneously broken global symmetry. In addition, in Chap. 4 we have studied the WZW effective action, which correctly reproduces the anomalies whenever they are present in the underlying fermionic theory. In the next three chapters, we will apply the effective Lagrangian formalism to describe the low energy dynamics of some physically relevant cases; namely, Quantum Chromodynamics (QCD), the spontaneous symmetry breaking sector of the standard model, and gravity. In this chapter we will deal with QCD and its low-energy dynamics using the so called chiral effective Lagrangian. The main idea of this formalism is to perform an expansion in the energy parameters, such as quark masses and NGB external momenta. Generically, these parameters are denoted by p. These techniques were originally developed by Weinberg [1, 2] and other authors [3] to O(p2). Later on, Gasser and Leutwyler [4, 5] extended the analysis up to O(p4). In the following we will present the leading and next to leading order chiral Lagrangian and analyze the one-loop results in chiral perturbation theory (ChPT), as well as the renormalization of the O(p4) Lagrangian. In addition, we will discuss in detail the physical application of the one-loop ChPT to pion and kaon scattering, studying the amplitudes and the problem of unitarity.

6.1 The QeD Lagrangian Our starting point will be then the QCD Lagrangian. As it is well known, QCD describes the strong interaction between quarks and gluons through a non-Abelian SU(Nc ) gauge theory, N c being the number of colors, which in the real world is N c = 3. As a further complication, we will have N f different types of quarks (flavors). The Lagrangian of QCD in terms of quark and gluon degrees of freedom is given by A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

126

6. The Effective Lagrangian for QeD .cQCD =q(i

f/J - M)q -

1

4G~vG~v

+ .cFP

Aa D J.L=J.La z.9s G aJ.L2 G~v=aJ.LG~ - avG~

+ 9srbcGtG~

,

(6.1)

where a = 1, ... 8, G~ are the gluon fields and 9s is the strong interaction coupling constant. The quark field q represents a column vector in both color and flavor spaces, M is the quark mass matrix in flavor space given by M = diag (mi,'" ,mNf) with mi the masses of the different quarks. The Aa are the Gell-Mann matrices, so that Aa /2 are the SU(3) generators in the fundamental representation, and bc are the SU(3) structure constants.

r

Finally, the Faddeev-Popov term .cFP includes the Lagrangian for the ghost fields and the gauge-fixing term (see Appendix C.2). In principle, we could think of using standard perturbation theory with the Lagrangian in (6.1). In fact, we can derive the usual Feynman rules for a Yang-Mills theory and calculate the corresponding Feynman diagrams up to a given order in the coupling constant 9s. However, things are not that simple, since QCD is asymptotically free, which means that, in contrast to the case of quantum electrodynamics (QED), its running coupling constant decreases at high energies. On the contrary, at low energies it is expected to increase, which hints that the asymptotic states of the theory cannot be the free quarks. Intuitively that leads to the confinement of quarks and gluons inside hadrons. In fact, the success of the parton model in describing Bjorken scaling [6] in deep inelastic lepton-nucleon scattering, confirmed the existence of asymptotic freedom. Later on, it was shown [7] that the only renormalizable field theories that can exhibit this property are non-Abelian gauge theories, leading to the formulation of the QCD Lagrangian in (6.1) by different authors [8, 9]. For the case of QED, we can interpret the coupling constant decrease at long distances as the result of the charge screening due to the presence of electron-positron virtual pairs. Extending this physical picture to the color charge, we could think of an anti-screening effect in QCD, which is due to the non-Abelian nature of the gluonic interactions. To be more precise, we will now turn to see how asymptotic freedom indeed appears in QCD [10, 11]. The one-loop diagrams calculated with the Lagrangian in (6.1) give rise to infinities which have to be absorbed in the renormalization of the fields and the coupling constant. This ensures the finiteness of the perturbative calculations, since QCD is a renormalizable theory. Let us concentrate on the case of 9s. The relation between the bare coupling constant 9? and the renormalized one 9s can be written as 0 (62) . 9s = Z 1-lZ2 Zl/2 3 9s ' 2 where Z~/2 and are the renormalization constants for the q and G~ fields respectively. On the one hand, Z2 can be obtained from the diagrams contributing to the quark propagator, whereas Z3 comes from those corre-

zi/

6.1 The QeD Lagrangian

127

sponding to the gluon propagator. On the other hand, Zl is obtained from the diagrams contributing to the quark-gluon vertex. We assume now that we are in a region in external momentum space such that the renormalized coupling constant remains small, and therefore it is meaningful to restrict ourselves to one loop. In addition, in the renormalization procedure an arbitrary energy scale f.L necessarily appears, which allows us to define the beta function from (6.2). To one loop (see Sect. 5.5) it reads {JQCD =

a9s = - ( 11 - -2Nf ) -9~f.Laf.L 3 l67r 2

+ 0 (9s5)

(6.3)

so that, at least at leading order, (JQCD is negative for N f :::; 16. Let us turn now to the Callan-Symanzik renormalization group equations (RGE). As it is explained in Appendix C.l, the RGE imply that the renormalized coupling constant varies with the scale, which is usually called the "running" of the coupling constant. That is, if we rescale the momentum p by p --+ >..p the RGE imply that the Green functions will behave as if the coupling constant 9s is replaced by an effective one 9s(9., t) with t = log >.., satisfying the following differential equation:

Ogs(9., t)

at

=

{J

QCD

(- ) 9s

(6.4)

9s(9s, 0) = 9s ,

which gives the evolution of the effective coupling constant with the scale, starting from a given point. The fact that {JQCD is negative then implies that 9s decreases with energy and then the asymptotic freedom follows. Notice however that we have arrived to the negative {J function value in (6.3) precisely by assuming that we are in the momentum region in which the one-loop calculation has sense, which turns to be that of large momentum. Then, strictly speaking, (6.4) would be valid only in that region in momentum space. What is commonly believed is that this behavior of the QCD coupling constant can be extrapolated to the low-energy region, thus leading to a non-perturbative regime. As we have already commented, the fact that the asymptotic states of strong interactions are not free quarks, but hadrons, in which quarks are confined (which is a hint of a strong interaction) gives support to this idea. Another technical remark concerning the above RGE study is that it is well defined only in Euclidean momentum space. Thus, finally we can integrate (6.4) in the region of large Euclidean momenta, using the one-loop (J function in (6.3) to obtain 2

l27r

as(q ) = (33 _ 2Nf)log(q2/A~CD) ,

(6.5)

where as = 9s/47r and we have redefined >..2 = q2 / f.L2, with f.L the renormalization scale. Written in this form, the evolution of the coupling constant with the scale only depends on a single parameter AQCD , which is known as the

128

6. The Effective Lagrangian for QeD

QCD scale and is defined in terms of J1. and cx s (Ji.2) through 2 2 127l" logA QCD =logJ1. - cx (J1.2)(33-2N ) (6.6) s f We would like to remark that N f in (6.5) should be taken as the number of flavors with mass smaller than the scale q2, since we are in the deep Euclidean momentum region in which q2 should be much bigger than any mass scale considered. Notice that this is an application of the decoupling theorem studied in Chap. 1, which in this case states that other flavors with mass bigger than the q2 scale decouple. In fact, it was the proof of this statement what motivated the formulation of the decoupling theorem. Thus, the value of A QCD depends on both the renormalization scheme and the number of light flavors considered. Furthermore, since A QCD appears in (6.5) through a logarithmic dependence, it is particularly difficult to determine its precise numeric value from as measurements (for instance those from the Large electron Positron Collider (LEP) at CERN), although in most schemes it is in a range from AQCD = 100 MeV to 400 MeV.

6.2 QeD at Low Energies As we have seen, perturbative methods in QCD only make sense in the deep momentum region. Therefore, the description of the low-energy QCD in terms of the Lagrangian in (6.1), with quark and gluon degrees of freedom, is not appropriate. Moreover, at low energy the relevant degrees of freedom are those of the lightest particles in the theory, which are the pions when only the strong interaction is considered. In order to describe efficiently the dynamics of such particles, it is crucial to take into account how the different symmetries of QCD are realized. Specifically, as we have seen in previous chapters, for instance in the study of the 7l"0 decay in Chap. 4, it is the chiral symmetry that will playa fundamental role. Apart from the U(1)v (baryon number) and the (anomalous) U(1)A' in the massless quark limit the Lagrangian in (6.1) is invariant under SU(Nf h x SU(Nf )R. The latter corresponds to the following transformations: qL

LqL

qR

RqR,

(6.7)

where L, R E SU(Nf) and qL,R = PL,M. Accordingly, the following vector (L + R) and axial (L - R) currents are conserved classically a

J~ =7j'Y/-LT q T a q, J /-L5a =q'Y/-L'Y5

(68) .

where T a are the SU(Nf ) generators in the fundamental representation.

6.2 QeD at Low Energies

129

However, this symmetry does not appear in the particle spectrum of the theory, in which we do not observe any chiral degeneracy, that is, particles with the same quantum numbers but with opposite parity. Furthermore, it is not difficult to see that, if this was a true physical symmetry of the theory, the vacuum expectation values of products of axial currents should be equal to those of vector currents. However, this is in contradiction with the experiments on semileptonic weak decays, in which those vacuum expectation values have been shown to behave differently [12]. Consequently, SU(Nj)L x SU(Nj)R cannot be a symmetry of the vacuum. In addition, we know that the vector isospin symmetry SU(Nj )L+R indeed shows up in the particle spectrum, since we observe particles of approximately equal masses within the same isospin multiplet. For instance, the pions (11"+,11"-,11"0) form an isospin triplet with m 1r ± ~ m 1r o, as a result of the fact that m u ~ md. Then we are faced with an spontaneous symmetry breaking (SSB) pattern in which SU(Nj)L x SU(Nj)R breaks down to SU(Nj )L+R. The Goldstone theorem that we have studied in Chap. 2 tells us that there will appear as many NGB as the number of broken generators. Thus, in our case we are breaking SUL-R(Nj) and there are NJ-1 NGB fields. Furthermore, since SUL-R(Nj) is the group of axial transformations, the corresponding NGB have to be pseudoscalar particles, which is again a consequence of the Goldstone theorem. This picture is again consistent with experimental observation; for N f =2 the NGB are precisely the pions, which are the lightest particles in the spectrum. They are not massless and, in a strict sense, the pions are just pseudo-NGB. Nevertheless their masses are very small compared to the scale Ax, below which the low-energy limit has sense. As it is discussed in Sect. 6.3, it is reasonable to take Ax ~ 1GeV. Thus, to ease the notation, from now on we will refer to the light mesons simply as NGB. We recall that our symmetry breaking pattern is exact only in the massless quark limit. Since the u and d quarks have a small mass, this description is only approximate, which is reflected in the fact that the NGB are not really massless. However, we expect it to be a good approximation for the case N j =2 (m u ~ md ~ 1 MeV). For Nj=3, the strange quark has to be considered and the eight NGB are now the three pions together with the four kaons and the eta. Nevertheless, the mass of the strange hadrons is of the order of three or four times the pion mass and consequently the description of their dynamics in terms of the SSB pattern is not expected to work as well as for pions. The identification of the lightest hadrons with the NGB of the chiral symmetry allows, via the Goldstone theorem, to derive certain interesting relations between vacuum expectation values involving NGB degrees of freedom and quark currents like those displayed in (6.8). Such relations are obtained using the current algebra of the axial and vector currents, as well as the fact that, according to our SSB pattern, the axial charges do not annihilate the vacuum. As a consequence, (see Sect 2.4) we can define

130

6. The Effective Lagrangian for QeD

(0181' J~a(O)I¢}(p)) = 8ab FbM; ,

(6.9)

which is known as the partial conservation of the axial current (PCAC) , where ¢} are the NGB fields and M b their corresponding masses. For instance, for Nf=2 and assuming isospin symmetry, Fb = F", ~ 93 MeV with F", the pion decay constant, that has been measured in semileptonic processes. Relations like that in (6.9) were extensively used in the late sixties and early seventies to derive low-energy theorems [13, 14] involving NGB and relate these theorems with physically measurable quantities. In particular, (6.9) is very useful in order to derive a very important relation known as the GellMann-Oakes-Renner [15] formula when Nf = 3 in the low-energy limit

M; Mk 3M; (OlqqIO) (6.10) 2m m +ms 2(m + 2m s ) F:; where (OlqqIO) is the quark condensate and we have taken m u = md = m. In addition, we have used (OlqqIO) = (OluuIO) = (OlddIO) = (018810) =1= 0, which

follows again from the Goldstone theorem applied to our present SSB pattern. Notice that the quark condensate can be understood as the order parameter for the chiral symmetry of QCD, since it vanishes when the chiral symmetry of the vacuum is restored. Moreover, it can be shown that Fa = F", to the leading order, also for N f = 3. The relation in (6.10) states that for a nonvanishing quark condensate, the squared masses of the NGB are proportional to the quark masses and thus vanish in the chirallimit, consistently with the Goldstone theorem. It also leads to the Gell-Mann-Okubo [16] mass relation

4Mk - M; - 3M; = O.

(6.11)

The modern effective Lagrangian approach that we are going to describe here will allow us to derive in a very simple and elegant way all the lowenergy theorems like those displayed in (6.9) and (6.10). Moreover, we will be able to calculate their next to leading order corrections in the low-energy expansion.

6.3 The Chiral Lagrangian at Leading Order Following our previous discussion, our task will be to find an effective Lagrangian containing only NGB degrees of freedom and respecting all the symmetries of the underlying theory, which in our case is QCD. As we are interested in the low-energy dynamics, this program will be developed up to a given order in derivatives or, in other words, in the NGB external momenta and the quark masses. Thus, we will have a low-energy expansion with respect to some energy scale Ax. It is reasonable to expect that the presence of resonances in the particle spectrum would spoil this approach. Hence, we can take Ax of the order M p ~ 770 MeV, since p is the lightest resonance. That choice seems consistent [17, 18] with Ax ~ 41rF", ~ 1 GeV, that will appear in the loop expansion.

6.3 The Chiral Lagrangian at Leading Order

131

Let us concentrate on the N f =3 case. The peculiarities of N f = 2 will be commented in Sect. 6.5.3. From now on we shall use the exponential parametrization of the NCB fields

U(x) = expi

Aa 7r a (X) F '

(6.12)

where F is some energy parameter that will be related with the NCB decay constants Fa, as we will see below in detail. Notice that we use the notation a 7r for the NCB fields, including not only the pions but also the kaons and the eta. The field U transforms under the group SU(3)L x SU(3)R as

U

--+

RULt .

(6.13)

To lowest order in derivatives, the Lagrangian describing the NCB dynamics is that of the NLSM, as we have discussed in Chap. 3. We shall also include a mass term proportional to the quark mass matrix, which explicitly breaks the chiral symmetry, although it is still invariant under SU(3)L+R in the limit in which all the quark masses are the same. With the parametrization in (6.12) the Lagrangian of the NLSM can be written as

LNLSM =

~F2tr {a/LUa/Lut + 2BoM(U + ut)}

,

(6.14)

where B o is some constant to be determined later. The normalization factor has been chosen so that the usual kinetic term for the NCB is recovered when U is expanded in powers of the 7r fields. Let us recall that we had already met the first term of the above Lagrangian when we built the NLSM in Chap. 3, (3.14). With the present parametrization, the coset metric is 90i{3

=

2F

2

aut) tr (au a7r Oi a7r{3

(6.15)

.

Next, we will perform a change of basis to that with the quantum numbers of the physical mesons (in the following, we will call it the physical basis). Then, U(x) is given in (6.12) with ~7r0 Aa 7r a = v~2 (

+

~178

7r-

7r+ 1 0 1 - ;/27r v'6178

K-

-gO

+

where the 7r~ are the physical NCB fields. In the physical basis and in the limit m u = md = m, the NCB mass term in (6.14) is diagonal, and thus we can read from it the NCB masses

M;=2mB o Mk=(m

+ ms)Bo

M; = ~(m + 2m )Bo , s

(6.17)

which are consistent with (6.10) and thus we see that the form of the mass term in (6.14) is the correct one at leading order with the identification

132

6. The Effective Lagrangian for QCD

B oF 2 = -(OlqqIO). We remark that the above relations are valid only at leading order, where we can identify the masses and decay constants appearing in the NLSM Lagrangian with the physical ones. The corrections to these values will be obtained in Sect. 6.4. In addition, we notice that isospin breaking effects (mu -I- md) imply a 7r°T/S mixing in the mass term in (6.14)

[5].

With the Lagrangian in (6.14), apart from the mass relations obtained above, we can also calculate amplitudes to lowest order for processes involving NGB fields, such as 7r7r scattering. As it is shown in Appendix D.3, these amplitudes can be obtained from a single one, A(s, t, u), by using isospin and crossing symmetry (D.17). As usual, s, t, u are the Mandelstam variables also defined in Appendix D.3. Then, if we expand U in powers of NGB fields in the NLSM Lagrangian and identify the relevant terms coming from the fourpion vertices we get the very low-energy prediction obtained by Weinberg [1]

A(s, t, u) =

s-M 2 F2 1r

,

(6.18)

1r

where we have set the pion decay constant and mass appearing in the Lagrangian to their physical values, which is consistent to leading order as it was explained above. The formula in (6.18) gives a good approximation near threshold. However, it is necessary to consider higher order corrections to (6.18) when we require the amplitude to satisfy certain relations which follow from unitarity (see Appendix D.1), at least perturbatively in the chiral expansion. We will discuss this point in detail in Sect. 6.6 Before dealing with the effect of loops and next to leading terms, we would like to have a systematic method to derive from the effective Lagrangian Green functions involving currents, as for instance that appearing in (6.9). They will allow us to obtain NGB decay constants, masses and amplitudes. In order to incorporate such features in this framework, we will follow the technique of external fields, according to Gasser and Leutwyler [5]. Let us then consider local SU(3)L x SU(3)R transformations L(x), R(x). The idea is to make the QCD Lagrangian in (6.1) invariant under this local group. Then we will have to couple external axial ajL(x) and vector vjL(x) fields to the quarks, in order to gauge the chiral symmetry. Moreover, we will also couple an external scalar field s(x) to take into account mass effects and a pseudoscalar p( x). The QeD Lagrangian gauged with these external fields then reads 1 LQCD [q, q, a, v, s, p] = q[rjL(iDjL+vjL+'Y5ajL)-(s-iP'Y5)]q-4G~vG~v, (6.19) where the external fields are matrix-valued functions in flavor space. As long as we want to obtain SU(3) vector and axial-vector currents (see below), we will take vjL and ajL as traceless. The above Lagrangian is invariant under the following set of local transformations

6.3 The Chiral Lagrangian at Leading Order

133

qL-+L(x)qL qR-+R(x)qR ljJ.(x) -+L(x)ljJ.(x)Lt

+ i (ojJ.L(x)) Lt(x) TjJ.(X) -+R(X)TjJ.(x)Rt(x) + i (ojJ.R(x)) Rt(x) s(x) + ip(x)-+R(x)(s(x) + ip(x))Lt(x) , where we have defined TjJ. = vjJ. + ajJ. and ljJ. = vjJ. - aw

(6.20)

Now we turn to the connection with the effective Lagrangian. We assume that, after integrating out in some way the quarks, the only relevant degrees of freedom at low energies are the NGB fields collected in U. Then, the resulting effective action, which will now depend on the external fields, admits a representation in terms of a path integral with only NGB. That is, at low energies eir[v,a,s,p] =

=

J J

[dq] [dq]e i J d 4 x.cQCD [il,q,a,v,s,p] [dU]e i

Jd

4

x.c e ff[U,a,v,s,p]

,

(6.21)

where we have omitted the gluon fields, whose effects will be discussed in Sect. 6.5.2. We remark that, in general, £eff will be a series with operators of increasing energy dimension, that is (6.22) Therefore, we can derive Green functions with vector, axial vector, scalar and pseudoscalar currents simply by taking functional derivatives with respect to their corresponding external fields. For instance, for Green functions involving the axial currents defined in (6.8) we have

J5a(x) = O£eff jJ.

oa~(x)

(6.23)

and so on for the other currents. Now we will make use of an important result, namely, the invariance theorem [19]. This theorem states, for the present case of QCD with N f flavors, that if £QCD[q,q,a,v,s,p] is invariant under the local SU(Nf)L x SU(Nf)R transformations in (6.20) and the theory is anomaly free under that group, then £eff[U,v,a,s,p] is also invariant under the same group. The condition for the absence of anomalies holds for Nf = 2 but it does not for N f = 3. As a matter of fact, for Nf = 3 the effective action r[v, a, s,p] is not invariant under the group transformations but exhibits a gauge anomaly of the type analyzed in Chap. 4. However, we have seen in that chapter that the WessZumino-Witten (WZW) effective action correctly reproduces the anomalies of the theory. As a matter of fact, for N f = 3 the invariance theorem states [19] that £eff is the sum of £wzw and the invariant, non-anomalous, piece. Notice that the WZW term is O(p4) in the effective Lagrangian.

134

6. The Effective Lagrangian for QeD

The invariance theorem then implies that we have to build our effective Lagrangian in an invariant way, up to a given order in the derivative expansion. This can be easily achieved by replacing derivatives by covariant derivatives acting on the U fields as

D/loU =0/loU - ir/loU + iUl/lo D/loUt =O/lout + iUtr/lo - ilJJ:Ut .

(6.24)

This replacement is enough to ensure the invariance of the terms in the Lagrangian that only depend on the U field. However, we also have to take into account all possible gauge invariant combinations of U and external fields, up to a given order, that are not obtained with this simple procedure. In Sect. 6.4.1 we will see how such combinations appear in the ,[(4) Lagrangian. Notice that the vector and axial vector external fields v, a are considered as O(p), whereas s + ip is O(p2), which is consistent with the low-energy expansion since v/lo and a/lo appear linearly in the covariant derivatives in (6.24). In addition, replacing s = M gives rise to the quark mass term, which we have seen above that is proportional to the pion mass squared, at least to leading order. We will see below that this power counting scheme is indeed consistent with the low-energy theorems that we have discussed in the previous section. To the lowest order in the chiral expansion, that is O(p2), the most general term with all the required symmetries is ,[(2)

=

41 F2tr

{D/loUD/loUt +X ut +XtU}

(6.25)

with X = 2Bo(s+ip). Notice that the above Lagrangian reduces to the NLSM in (6.14) when we take v = a = p = 0 and s = M. With the Lagrangian in (6.25) we can derive all the low-energy theorems simply by taking functional derivatives with respect to the appropriate external fields. For instance, taking into account (6.19), we can obtain the quark condensate as

(OlqqIO) = -

8r[V~:~ s,p] !s=M,v=a=p=o

'

(6.26)

where s(x) = sO(x) + sa (x)>.a /2. With the above equation applied to the lowest order Lagrangian in (6.25) we get (6.27) and then, using (6.17) we can eliminate B o and recover the Gell-MannOakes-Renner relation in (6.10). In addition, we have a definition of the physical decay constants from the PCAC relation in (6.9), by replacing the corresponding expression for the axial current in terms of the pion fields. In this way, we obtain again the result F1f = F K = Fry = F + O(M 2 ), as predicted by the low-energy theorems.

6.4 The Chiral Lagrangian to Next to Leading Order

135

Next, we will turn to calculate the effects of chiralloops and higher order terms in the effective Lagrangian, which will allow us to obtain corrections to the low-energy theorems, as well as to physical amplitudes such as that of 1r1r scattering given in (6.18).

6.4 The Chiral Lagrangian to Next to Leading Order In the previous section we have derived the effective Lagrangian to leading O(p2) order. Our task will be now to find the next to leading order O(p4) corrections, which can arise from two different sources: one-loop contributions from the £)2) Lagrangian and those coming from £(4) at tree level. Moreover, we should consider the anomalous WZW term, which is O(p4) too. This procedure can also be extended to higher orders, and it is usually known as the modern chiral perturbation theory (ChPT). This technique has been successfully developed from the first works of Gasser and Leutwyler [4, 5]. The divergences appearing in loops coming from £(2), have to be regularized and renormalized. For that purpose, we will use the dimensional regularization method. In Chap. 3 we have seen that this is a suitable method to get rid of the 8D (O) factors that arise when one tries to formulate the NLSM invariantly under reparametrizations. Concerning the renormalization problem, we will show that it is possible to absorb the divergences appearing up to O(p4) in the £(4) parameters. In this sense, we say that the theory is renormalizable only up to a given momentum order, although in general it is not strictly renormalizable. Indeed there will be an infinite number of terms in (6.22) which could give rise to new divergences to be absorbed with an infinite number of counterterms. In addition, the one-loop diagrams can also yield finite contributions to the physical observables, as we will see in detail below.

6.4.1 The

.c(4)

Lagrangian

According to the above discussion, our first step will be to build the most general £(4) Lagrangian invariant under Lorentz, chiral, parity and charge conjugation transformations. However, as we are dealing with this Lagrangian only at tree level, we can make use of the equations of motion at O(p2) in order to eliminate some of the O(p4) terms. These equations for the NGB fields are

(D 2 U)Ut - UD 2 Ut

1

+ UX t - Xut - "3tr(Uxt - xut)

= O.

(6.28)

The above relations are obtained from the Lagrangian £(2) in (6.25) by imposing the usual variational constraint 8£(2) /8.1 = 0 with .1 parametrizing a small variation of the U field. For instance, we can choose U' = UeiLl with .1 t = Ll and trLl = O. Furthermore, we have to impose the constraint

136

6. The Effective Lagrangian for QCD

det U = 1 by adding a Lagrange multiplier, which can be determined by setting U(x) = Un, a constant field. It yields the last term in the l.h.s. of the above equation [5]. The use of (6.28) will become clear in the following sections when we expand the NGB fields near a classical configuration Thus, the O(p4) Lagrangian can be written as a linear combination of the minimal set of terms satisfying the symmetry requirements. That is .c(4)

=

L1(tr( DJ.LUt DJ.LU))2 + L 2tr(DJ.LUt DvU)tr(DJ.LUt DVU) +L 3 tr(DJ.LUt DJ.LUDvUt DvU) + L 4 tr(DJ.LUt DJ.LU)tr(xtU +Lstr(DJ.LUt DJ.LU(XtU + UX t )) +L6(tr(xtU + XUt))2 + L7(tr(xtU - xut)f +Lstr(xtUxtu + Xutx ut ) R DJ.LUDvUt + F L 8J.LUt DVU) -iLgtr(FJ.LV J.LV R UFLJ.LV L trUt F + 10 J.LV +H1tr(F!!vFJ.LvR + F(;vFJ.LvL) + H 2trx t x,

+ Xut )

(6.29)

where, as usual, we have introduced the strength tensors for the external fields

F/!v=8J.LTV - 8VTJ.L - i[TJ.L,T v ] F(;v =8J.L lv - 8v lJ.L - i[lJ.L' Lv] .

(6.30)

As we have commented before, the relations in (6.28) have been used to write some terms, as tr(DJ.LU DJ.LUt)tr(Ut DJ.LDJ.LU) , as a linear combination of the operators displayed in (6.29). Hence, apart from the Hi, there are 10 low-energy constants (LEC), denoted L i , in the O(p4) Lagrangian that, once renormalized, have to be determined in order to make physical predictions. We remark that it is the underlying theory, which in our case is QCD, what determines the actual values of the LEC, since the operator basis has been obtained only by symmetry requirements. The terms with the Hi constants in (6.29) only depend on external sources and, therefore, they do not have any physical relevance. However, they are needed to absorb some ofthe divergences appearing in the chiral loops. They can also generate contact terms when calculating Green functions with external fields. 6.4.2 One-Loop Renormalization We now turn to analyze the effect of one-loop diagrams in ChPT to O(p4). With that purpose, we will make use of the Heat-Kernel (HK) method developed in Appendix CA. Let us then consider small variations of the U field near a classical configuration U that satisfies the equations of motion in (6.28). We choose to parametrize those variations as U = uei~u with ~ hermitian and traceless,

6.4 The Chiral Lagrangian to Next to Leading Order

137

uu t = 1 and uu = U. This parametrization is related as ~ = uL1u t to the one we used to derive (6.28). Replacing this field configuration in the O(p2) Lagrangian, retaining terms up to O(e) in the expansion in powers of ~ and integrating by parts, we get for the action

J

d4 x.c(2) [U] =

J

d4 x

{ .c(2)

[U] -

~2 ~aDab~b + O(e) }

D=dl-'dl-' + (J ab dab I-' = b aI-' + rab I-' r: b= -~tr([,,\a, ,,\b]([u t , 0l-'U]- iutrl-'u - iull-'ut)) (Jab = str([,,\a, L11-'][,,\b, L11-'] 1

L11-'=utDI-'Uut

+ {,,\a, ,,\b}(uX tu + U t xut ))

a=1 ... 8.

(6.31)

The linear terms in ~ in the above expansion vanish, since U is a solution of the equations of motion in (6.28). The effective action for the classical field U, defined in (6.21), can now be calculated to one loop by integrating the ~ fields in the path integral measure. Thus, up to an irrelevant constant, we get -

i

r[U,v,a,s,p] = "2logdetD.

(6.32)

First of all, we want to extract the ultraviolet divergent part of the effective action in order to see if we are able to renormalize it with the LEC. With such purpose, we regularize the determinant in the above formula by using the HK method with the dimensional regularization scheme. As it is shown in Appendix CA, the ultraviolet infinities in D = 4 - f, only appear up to a2 in the Seeley de Witt expansion ((C.92) in Appendix CA). Let us now write, as usual, log det D = Trlog D and notice that the D defined in (6.31) is a second order elliptic differential operator. Hence it is written in a suitable form to apply the HK techniques. Moreover, we would have to add a mass term to regularize the infrared behavior of the integral in (C.88). However, such a mass term is already present in D, since, if we set all the external fields to zero except s = M then D can be recast, in the physical basis given in (6.16), as Dg b = (0 + M~)bab . (6.33) Hence, the infrared limit is well-behaved if the quark masses are different from zero and, therefore, we will only concentrate on ultraviolet divergences. For the technical purposes explained in Appendix C.4 we need to add a mass scale J.L2 to D, since we are interested in the logarithmic dependence with the scale appearing in dimensional regularization. Then, using (C.93) for the present case, we can write (up to irrelevant constants)

6. The Effective Lagrangian for QeD

138

r[U, v, a, s,p] = 2(4:)D/2

+ p,D- 4 r

J [r (1 - ~) 4

d x

(2 - ~)

p,D-2 tr (]"

tr C12rJ.LvTI.LV

+ ~(]"2) + ... J,

(6.34)

where rJ.LV = [dJ.L' d v ] and D is the space-time dimension. According to our previous discussion, the terms that we are neglecting in the expansion in (6.34) are not relevant for our purposes when D ---+ 4, since they remain finite. Notice that the poles in the above expression appear in the Euler r functions and they do so for D = 2 and D = 4. In dimensional regularization, these divergent terms only depend logarithmically on p,. Hence, for D = 2 and D = 4 we are interested in the first and second contributions in (6.34), respectively. Now we recall the definitions in (6.31), from which we have, after some algebra, for Nf flavors

Nf -t NJ -1 t--t tr (]" = -tr(DJ.LUDJ.LU ) + - N tr(x U + U X) , 2 2 f tr(TJ.LV rJ.LV) =

Nftr{~DJ.LutDvUD 4 J.L UtD v U - ~DJ.LUtD 4 J.L UDvUtD v U R DJ.LUDvU t _ iF L DJ.LU t DVU - iFJ.LV J.LV R UFLJ.LV _ ~tr(FR FJ.LvR + F L FJ.LVL)} _ U t FJ.LV 2 J.LV J.LV ,

tr

(]"2

(6.35)

(6.36)

= ~(tr DJ.LU t DJ.LU)2 + ~tr(DJ.LUt DvU)tr(DJ.LU t DVU) +

~NfDJ.LUt DJ.LUDvU t DvU + ~tr(DJ.Lut DJ.LU)tr(xtU + xut)

f + N tr(DJ.LU t DJ.LU(XtU + UX t )) 4

+ NJ; 2 (tr(x tU + XUt ))2 8 f

NJ - 4 t- t-t -t t + - N tr(x Ux U+X U xU +X X)· 8 f

(6.37)

The contribution given in (6.35) has the same form as the NLSM, thus leading to the well-known conclusion that the NLSM is renormalizable in two space-time dimensions, since the D = 2 pole in (6.34) can be absorbed in the definitions of F and BoM (that is, the NGB decay constants and masses) in (6.14) and, for D = 2, there are no further divergences. On the other hand, from (6.36) and (6.37), we see that the terms in the D = 4 pole in (6.34) are all of the same form of those displayed in (6.29), except for the case of the first contribution to the r.h.s of (6.36). However, for N f = 3 there is an SU(3) identity that allows us to write that term as

~ tr(DJ.LU t DVUDJ.LUt DvU) = -2tr(DJ.LU t DJ.LUDvU t DvU) + ~(tr( +tr(DJ.LUt DvU)tr(DJ.LU t DVU) ,

DJ.LU t DJ.LU))2 (6.38)

6.4 The Chiral Lagrangian to Next to Leading Order

139

and then we arrive to the desired result, namely, that all the divergences coming from the NLSM at one-loop can be absorbed in the LEC appearing in the Lagrangian .[(4) at tree level. From (6.34), (6.36) and (6.37), their renormalization, for N f = 3, is explicitly given by

Lr =L

'Yi>"

i -

H[ =Hi

L1 i >"

-

D4{ 2- -10g47l' -1 +'Y } D- 4

1 >"=--j.L 327l'2

(6.39)

with 'Y the Euler constant. The 'Yi renormalization coefficients are listed in Table 6.1. Table 6.1. Values of the renormalization coefficients for Nf /1

/2

/3

/4

/5

3

3

o

1

3

32

16

8

8

/9 11

144

o

5

4s

/10

=3

Ll 1 5

1

"4

24

6.4.3 The Effective Action to One Loop Up to now, we have proved the renormalizability of the effective Lagrangian to O(p4). It remains to show how to obtain physical observables, that will depend on the values of the renormalized LEC. In other words, we need the finite parts of the one-loop diagrams whose divergences we have just analyzed. As we will see, this finite part of the effective action cannot be written as a four-dimensional integral of some effective Lagrangian, although a systematic approach will be provided in order to extract physical information for any given observable. Such information can be used, for instance, to fit the renormalized LEC from some set of processes. We would then be able to make physical predictions for other processes. In order to carry out this program, let us write the D operator in the physical basis given in (6.16) as

D = Do

+V

,

(6.40)

where Do is given in (6.33) and

V = {tJ.L, oJ.L} + tJ.L tJ.L + 0'.:1 + O'~ t::b =

-~tr([>..~, >"~]TJ.L)

O'~ = ~tr([>..~, L1J.L] [>../l, L1J.L])

O'~b = ~tr( {>..~, >"'b}(u~t u + u t ~ut)) -

DabM; ,

where the TJ.L and L1J.L operators are defined in (6.31). The defined in (6.16).

(6.41) >..~

matrices are

140

6. The Effective Lagrangian for QeD

The next step is to expand the determinant in (6.32) in terms of the external fields, that is r[U,v,a,s,p] i i i 1 = 2"logdetDo + 2"Tr(Do V) - 4Tr(DolVDolV)

+ O(V3)

.

(6.42)

According to the arguments developed in the previous section, the NGB have the same quantum numbers as the axial currents Jza, which are obtained from (6.23). Hence, neglecting O(a~) corrections, as in (6.42), we can derive up to four-point NGB Green functions at one-loop. Moreover, through (6.9), we can also obtain the value for the different Pa. decay constants. The same arguments apply to the other sources vf.L, sand p. It is important to note that r depends on vf.L, af.L' sand p, not only explicitly but also implicitly through U, which is a solution of the equations of motion in (6.28) in the presence of those external sources. The second term in the expansion in (6.42) gives rise to one-loop diagrams with only one vertex, that is, tadpole-type contributions. The third term yields both tadpoles and graphs with one loop and two vertices. Notice that the tadpole contributions contain the poles in D ~ 4 renormalizing the constants in £(4). We remark that in the oneloop diagrams calculated in this way the renormalization scale JL appears in logarithms. In the physical observables this scale dependence cancels exactly with that of the coupling constants in (6.39). We will check this property in some particular cases. As a first example of the applicability of the above methods, we can calculate the one loop O(M) contributions to the formula in (6.27) for the quark condensate. For that purpose, we only have to keep s =f:. 0, setting all the remaining fields equal to zero. In addition, since the equations of motion do not have s dependence, in this case we do not have to care about the implicit dependence of U commented before. The functional derivative of the effective action with respect to s can easily be calculated, with the result (m u = md = m)

(OluuIO) = (OlddIO) = - p Bo{ 1 - 2 2

(018sI0) = _p B o { 1 - 2 2

K1= K

I/.

2=

-

,-a -

(~JL1l" + JLK + ~ JL7)) + mK1 + K 2 } (2JLK + ~JL1l" + ~JL7)) + msKl + K 2 }

o( r 8B r) F,2 2L 8 + H 2 o

(A 32BoLr 2m + m s ) --pr 6 o M2 a 3211"2

P6

100" I::.>

M

2 a

Jl2 .

(6.43)

6.4 The Chiral Lagrangian to Next to Leading Order

141

Notice that we can now use (6.17) in (6.43), which is consistent at O(p4). Then it is easy to check explicitly the cancellation of the scale logarithms. Another important remark is that the dependence on the constant HJi. cannot be extracted from experiment and thus there is an ambiguity in the definition of the quark condensates. However, one still can make predictions by using some relations among the different condensates. For instance, if m u f=. md then, by eliminating the K 1 constant in (6.43), one can relate (OluuIO)j(OlddIO) with (OluuIO) j (018810) [5] in terms of measurable quantities. The relations in (6.17) for the NGB masses as well as the definitions of the physical decay constants also present next-to leading order corrections. Both cases can be analyzed if we consider two-point Green functions containing two axial currents. Then, what we have to keep now in the effective action is the dependence on the axial field a w In this case there is the additional difficulty of solving the equations of motion in order to find the dependence 7r = 7r[aJ.L][5]. Once this is done, the two point Green function (OIT J~aJ~bIO) follows by functional derivation, using (6.23). It has a dominant contribution with poles in momentum space that are identified as the physical masses of the NGB. Moreover, using (6.9) as a definition of the physical decay constants and inserting a basis of one-particle states in the previous Green function one can also read the values of those decay constants. We will give here the result for the masses in the limit m u = md = m

M; = 2mBo {I

+ J-L7f

-

~J-L1) + 2mK3 + K 4}

Mk = (m + ms)Bo {I + ~J-L7J + (m + m s)K3 + K 4}

M; = ~(m + 2ms)Bo {I + 2J-LK - ~J-L1) + ~(m + 2ms)K

3

+2mBo { -J-L7f

+ ~J-LK + ~J-L1)} + K s ,

+ K 4} (6.44)

whereas the decay constants are given by F7f =Fo{1- 2J-L7f - J-LK

FK=Fo {1F1)=Fo

+ 2mK6 + K 7 }

~J-L7f - ~J-LK - ~J-L7J + (m+m s)K6 +K7 }

{I - 3J-LK + ~(m + 2ms)K6 + K

where

K3 =

8~o (2£'8 Fo _

£5) 16Bo

r

K4=(2m+ms)~(2£6

o

-£4)

7}

,

(6.45)

142

6. The Effective Lagrangian for QeD

128

K s = m s - m )2 A

(

9

B6 F.2 (3L7 + Lar) 0

- 4B K 6 - 2OLrS FO

oLr (6.46) ( A ) 8B K 7=2m+ms F.2 4· o Therefore, we conclude that the three decay constants Frr , F K and FTJ that were the same to leading order, become different to next to leading order. In addition, the masses given in (6.44) yield the following violation of the Gell-Mann-Okubo mass relation in (6.11) _ (4M'k - M; - 3M;) 6 2 2 r r r M2_M2 = F.2(MTJ-Mrr)(Ls-6La-12L7) TJ rr 0 4M'kJlK - M; Jlrr - 3M;JlTJ (6.47) -2 M2 -M2 TJ rr This value for LlGMo will be used in the next section to obtain information on the L 5, L'7 and L 8 constants. LlGMO =

6.5 The Low-Energy Constants 6.5.1 Phenomenological Estimates In previous examples, we have seen how some of the renormalized LEC of the O(p4) effective Lagrangian indeed appear in the chiral Lagrangian calculations of the physical observables. Therefore, if we knew the numerical values of those constants we would be able to make physical predictions for all the observables. Unfortunately, this is not the case. As we have previously mentioned, the information of the underlying theory is encoded in the values of the LEC. In other words, they discriminate between different theories and, hence, could be obtained from QCD. However, as we will comment in detail in Sect. 6.5.2, it is not possible to extract the LEC directly from the QCD Lagrangian without making additional assumptions. Thus, different approaches have been used in order to estimate the values of the LEC. The most commonly used is that followed originally by Gasser and Leutwyler [5]. It consists simply in fitting the LEC with some set of physical processes and then make predictions for the rest of the observables. To illustrate how this method works, let us consider the ratio FK / Frr . From (6.45) we get, up to O(M)

FK 1 3 5 ( A) B o L r ( ) p=1-2JlK-"4JlTJ+"4Jlrr+4ms-m F2 sJl rr

rr

=1- 0 01 .

+ 4(M'kF2rr

M;) F(M ) s TJ'

(6.48)

6.5 The Low-Energy Constants

143

where in the last line we have used (6.44) to the relevant order and set the scale fJ. = MTf (remember that the LEC are scale dependent, whereas the observable FK / Frr; is not). The usual conventions are to take fJ. = M 71 or fJ. = M p , the mass of the rho resonance. Then, with the experimental value FK / F7f = 1.23 ± 0.02, which is obtained from semileptonic decays [20], we get L5CM71) = (2.3 ± 0.2) x 10- 3 . Another measurable quantity that can be used to obtain information on the LEC is the violation of the Gell-Mann-Okubo mass formula LlCMO defined in (6.47). With the value of L 5 obtained before and the experimental value LlCMO ~ 0.21 [20] we get 2L 7 + L 8 = 0.4 x 10- 3 , at the scale fJ. = M 71 . Notice that L 7 is scale independent since it is not renormalized at one loop. Concerning its value, it is worth mentioning that it can be determined assuming TI - TI' mixing in the chiral Lagrangian. Then, it can be shown [5] that the TI - TI' mixing angle e shows up in L 7 . Now, with the value e ~ -20 0 , which can be obtained for instance from the analysis of 7f0, TI, TI' ---. 'Y'Y decays [21], it is obtained that L 7 = -0.4 X 10- 3 . The remaining LEC could in principle be determined just by looking at the appropriate process. However, in some cases this is not an easy task and the LEC values have to be given using additional information. This happens, for instance, with L 4 and L 6, since, in the large-Ne limit these two constants vanish [5, 22]. Indeed, in such regime it is found that £7 is O(N;), 2L 1 - L 2 , L 4 and L 6 are 0(1) and the remaining LEC are O(Ne ). The deviations of this rule for L 4 and L 6 have been estimated in [5] and they appear in Table 6.2. The behavior of 2L1- L'2 has been checked in [23], by analyzing semileptonic reactions K ---. 7f7flv. The estimates for Ll, L 2 and L 3 given in Table 6.2 were also obtained in that work, from experimental data on such reactions. In addition, L 2 can be measured in D-waves in 7f7f scattering [24,4, 5]. Another example in which additional information can be used to set the values of the LEC is in the determination of L 9. This constant is related to the pion electromagnetic form factor that, by Lorentz and gauge invariance, can be written as (6.49)

where J';;M is the electromagnetic current and q = PI - P2. The above matrix element can be calculated using again the techniques that we have developed before. Thus, we set llJ. = rlJ. = eQAIJ. with e the electric charge, Q the quark charge matrix in (4.19) and AIJ. the photon field. In this way, we get the effective action gauged with the electromagnetic field. This allows us to calculate all kind of low-energy processes involving photons and NGB. In addition, we can obtain the electromagnetic current by functional derivation of the effective action to one loop with respect to the photon field, as we have previously done for the case of the axial current. The result for the pion electromagnetic form factor is

144

6. The Effective Lagrangian for QCD

Table 6.2. Values of the Nf = 3 LEC at the J.L = M1) = 548.8 MeV scale

Constant

LI

L~

L3 L~

L5 L6 L7 L'8 L9 LIo 2L 7

+ L'8

G 7r (q2) = 1 + q2 X

2L g

Value x 10 3

Source

0.65 ± 0.28 1.895 ± 0.26 -3.06 ± 0.92

1r1r scattering

0±0.5 2.3 ± 0.2 0±0.3 -0.4 ± 0.15 1.1 ± 0.3 7.1 ± 0.3 -5.6 ± 0.3 0.4 ± 0.1

Large N c FK/FTr Large N c I TJ-TJ mixing

1 (

[ F27r - 961r 2 F27r

and

K

->

1r1rlv

Mk/ M;,L5

rare pion decays

Ll GMO , L5

1

3)

M; + "2 log -Mk 4 ] 2 +"2 + O(q) J..L J..L

log - 2

(6.50)

.

Let us parametrize the pion form factor in the usual way as G 7r (q2) 1+(r;)q2+0(q4) where (r;) is, by definition, the pion electromagnetic charge radius. Experimentally it is found to be (r;) = (0.439 ± 0.008) fm 2 [25], from which we obtain Lg(M,,) = (7.1 ± 0.3) x 10- 3 . However, the constant L g

can also be determined from the so called vector meson dominance (VMD) description, in which a low-energy theory with couplings between NGB and vector resonances is considered [26, 27, 28J. Then, the resonance fields are integrated out and one gets predictions for the LEC. In particular, in the VMD approach one finds for the electromagnetic form factor 2 1 G 7r (q ) = 1 - q2/MJ

(6.51)

that gives us, when compared to (6.51) the value for Lg = F;/2M; ~ 7.3 x 10- 3 which is in good agreement with the one previously obtained. Alternatively, one can measure the value of L g from rare pion processes such as 1r+ ----+ e+ve / or /1r+ ----+ /1r+ [5][29J. Analyzing the corresponding amplitudes to O(p4) in the present framework, one is led to expressions in which L g and L 10 are involved. The values for these constants displayed in Table 6.2 are obtained when comparing to experimental data. Then, the listed value for L g can be taken as an input in order to give a prediction for the electromagnetic charge radius, as explained before. To conclude, in Table 6.2 we have compiled the numerical values proposed in [5, 23], for the LEC, for the case of N f = 3 at the scale M". In the next section, we will analyze the different methods proposed to extract information for the LEC from the QCD Lagrangian.

6.5 The Low-Energy Constants

145

6.5.2 Theoretical Estimates In the previous section, we have discussed different examples in which the LEC of the effective Lagrangian can be determined from experimental data. As we have already commented, the LEC encode the physical information of the underlying theory, QCD, in the low-energy regime. Thus, in principle, if we could solve the theory at low energies, with the QCD Lagrangian in terms of quark and gluon degrees of freedom, we should be able to predict the values of the LEC. Unfortunately, this is not possible without additional assumptions. We will discuss in this section some of the different approaches that can be follow in order to determine theoretically the LEC from the QCD Lagrangian.

The LEe in the Large-Ne Limit A common feature of the various ways of extracting information for the LEC from QCD, is the use of the large-Ne limit. It is well known that many diagrams turn out to be subdominant in the liNe expansion [22]. Although the theory is still non solvable in this limit, there are several properties that can be qualitatively explained. For instance, an interesting result is that the NGB loops are subdominant and thus the U field can be treated classically, that is, satisfying the equations of motion in (6.28). The dependence with the U field can be included in QCD in the different ways that will be commented below. Hence, after integrating out the quark and gluon degrees of freedom, we would get the low-energy effective action for U and the information for the LEC. Notice that, within this approach, we obtain a classical effective action. Another assumption is that, in a later stage, the effects of NGB loops can be taken into account, in the way explained in previous sections. There are several ways of including the dependence of the action on the U field when we start from the QCD Lagrangian. One possible approach is to rotate the quarks under SU(Nf), which gives rise to an anomalous Jacobian that can be regularized using the methods studied in Chap. 4. The variation of the effective action under an arbitrary rotation can be integrated up to a given momentum order. Then, when such a rotation is parametrized by U, it is possible to obtain the low-energy effective Lagrangian [30]. As a further refinement, the effect of the gluon fields can be included in the large-Ne limit [31]. Another alternative, also in this line, is based on the FaddeevPopov method [32]. The approach that we will consider in more detail here, introduces the dependence on U through an explicit coupling with the quarks in the QCD Lagrangian [31]. For that purpose, we will consider, as in previous chapters, the typical Yukawa coupling between the quarks and the U field. We will also include external axial and vector sources, as explained in previous sections and, for simplicity, we shall work in the chiral limit M = O. Then, our starting point will be the following QCD effective action in Euclidean space

146

6. The Effective Lagrangian for QeD e-r[v,a,Uj =

BE =

J J

[dGj[aqj[dq]e-SE[q,q,G,v,a,Uj

d4 X[q,JL (a JL -

+M(qRU(x)qL

==

J

[dG]e-rq[G,v,a,Uj

i;s >..aG~ + vI' (x) +'5aJL(X)) q

+ qLUt(X)qR)] ,

(6.52)

where [dG] includes the exponential of the Yang-Mills action and the ghost and gauge-fixing terms. We are using the conventions for the Euclidean Dirac matrices in Appendix A and M is some energy parameter. It will be useful to write the M coupling as a mass term, which can be done by means of the following change of variable for the quark fields

QL = ~qL

(6.53)

QR = ~tqR

with e(x) = U(x). In this basis, the Euclidean action in (6.52) reads (see Appendix A for notation) BE =

J

4

-

d X QDQ

D=,JL (aJL _1 [~ t (aJL rJL -"2

i;s >..aG~ + rJL + '5(,1') + M

+ r JL)~ + ~(aJL + lJL)~ t]

_ 1 t t] . ~JL-"2[~ (aJL+rJL)~-~(aJL+lJL)~

(6.54)

The above action can be interpreted as that of a constituent quark model, M being the constituent mass [17]. A typical value of M ~ 320 MeV gives a good numerical agreement for the various LEe that are obtained with this method [31]. Let us point out that the change of basis given in (6.53) gives rise to a Jacobian in the fermion measure, which can also be regularized in the same way as we did in Chap. 4. For instance, if the method based on the operators DtD and DDt is used, with D given in (6.54), it can be checked that this Jacobian only depends on the external sources and, therefore, it will not be of interest here. In fact, we will be only concerned about the real part of the effective action q defined in (6.52), which is given by

r

_

_

1

rR = Re rq[v,a,U,G]- -"2logdetD D. t

(6.55)

It can be shown that the imaginary part of the effective action gives rise to the WZW term [33]. Indeed, in Sect. 1.6 we have shown, within this approach, how to obtain the local WZW term at O(7f5). In this section, we will focus on the real part, which is the one that provides information on the LEC. In addition, we will neglect the effect of gluon fields as a first approximation

6.5 The Low-Energy Constants

147

developed in Appendix C.5. In particular, we recall the formulae for the (function regularization method, which in our case read d log detVtV= - d/DtD(s)ls=o

(DtD(S) ==

;~;)

1

00

dTT s- 1

J

d4x tr(xl exp( -TVtV)lx) ,

(6.56)

where the symbol tr denotes the trace over Dirac and flavor indices and, from (6.54), the VtV operator is given, without gluon fields, by

V t V=\1J.L\1J.L + M 2 + E \1J.L=8J.L + rJ.L + '5~J.L E= _aJ.LV RJ.Lv - 2M,J.L'5~J.L ,

(6.57)

where aJ.LV = [,J.L, ,V]/4 and RJ.Lv = [\1 J.L' \1 v). The vtv operator in (6.57) is suitably written to apply the Seeley-de Witt expansion in (C.92). We have 00

(xl exp( -T(E + \1 J.L \1J.L))lx) =

n

~ an(x) (4:T)2 .

(6.58)

It is important to note that the above expression is not a low-energy expansion, due to the presence of the scale M. Remember that, in the chiral power counting, the external vector and axial fields are O(p), the same as the spatial derivatives acting over U or ~. Thus, if the gluon fields are not included, on the one hand we have that the first piece in E in (6.57) is O(p2), while the second one is O(p) and RJ.Lv is O(p2). On the other hand, let us perform a dimensional analysis in terms of T- 1 . From (6.58) it is clear that E has dimensions of T- 1 , whereas \1 J.L has those of T- 1 / 2 . Now, again from (6.58) it follows that, to a given an coefficient contribute all possible invariant combinations with dimension T- n , of the operators E, RJ.Lv and \1 J.L' Then, with this argument, it is not difficult to see that, neglecting the gluon fields, only the coefficients with n :::; 4 in (6.58) contribute up to O(p4). The relevant an coefficients can be found, for instance, in [34). The first ones (n :::; 2) are given in Appendix C.5. For n = 3,4, the terms contributing up to O(p4) are 1

1

a3(x) = -6E3 - 12 ({E, [\1 J.L' [\1J.L, EJ)} + [\1 J.L' E)[\1J.L, ED + O(p6) 1

a4(x)= 24E4 + O(p6).

(6.59)

The next step is to replace the expansion (6.58) in (6.56). The M 2 piece in VtV allows us to solve the Gaussian T integral. Then, we take the traces of the Euclidean Dirac matrices and we get the effective action for rJ.L and ~J.L' Finally, it can be written in terms of the U and external sources, with the definitions given in (6.54). It is important to remark that, in order to arrive to the expression in the operator basis given in (6.29), it is necessary to use the equations of motion for the U field, given in (6.28) up to O(p2). The final

148

6. The Effective Lagrangian for QeD

result for the effective Lagrangian up to O(p4) is therefore that in (6.29), but now we get values for the LEC in terms of the M parameter and the scale p,. In Minkowski space those values are the following (we will not consider the Hi constants):

Nc M 2 1 p,2 F11"2 = 47r2 og M2

L 1 -~L-~ - 2 2 - 3847r2 Nc

L

Nc 3 = - 967r2

Nc L lO = - 967r 2 .

L 9 = 487r 2

(

6.60)

Setting N c = 3, it is immediate to check that the L i above are reasonably close to the values in Table 6.2. If the quark masses are taken different from zero, the same procedure can be followed for the operators depending on X in the effective Lagrangian. To O(p2), these terms are displayed in (6.25) and, therefore, we get a prediction for the quark condensate within this approach. To O(p4), from the Lagrangian in (6.29) we obtain information for L i with i = 4, ... , 8. The results are _1 _ Nc 3 ( p,2 ) v=2"I(Qq)1 = -87r 2M log M2 +1 L 4 =L6 = 0 2

L 5 = _Nc MF1I" 167r 2 4v

(

p,2 logM2

+ 1)

N c (MF; p,2 1) L 7 =- 967r2 ~ log M2 - 12

L- 167r2 N [(MF; _M F;) 10g L -~] 8v 16v 2 M2 24 2

c

8-

(6.61)

Notice that we obtain a vanishing value for L 4 and L 6 in the large-Nc limit, as we discussed in Sect.6.5.1 (see Table 6.2). Using now as an input M '2' 890 MeV, F1I" '2' 93 MeV and 2v '2' (-225 MeV)3, we get values for L 5 , L 7 and L 8 also in a reasonable agreement with those in Table 6.2, taking into account that we have neglected both chiral loops and gluonic effects.

Gluonic Corrections We will briefly discuss how to incorporate the effect of the gluon fields, within this approach. Of course, a perturbative treatment in terms of the QCD coupling constant would be meaningless at low energies. Thus, we will make use again of the large-Nc limit, where it is possible to use the following approximation (exp -X)

'2'

exp -(X)

(6.62)

with (X) = J[dG]X. In other words, we replace the different gluonic operators appearing in the effective action by their vacuum expectation values. The

6.5 The Low-Energy Constants

149

lowest order operator in gluon fields is then the gluon condensate (G~vG~V) and the one at next order would be the triple gluon condensate (GGG). In order to include the gluonic effects, we will follow the same steps as before, starting again from the Dirac operator in (6.54), but with the following replacements

\l

J-L

----> \l

J-L

_ ig s >..uG u 2

J-L

E---->E - (jJ-LvGJ-LV RJ-Lv---->RJ-Lv

+ GJ-LV

(6.63)

.

When the gluon fields are considered, the calculation in terms of the Seeley-de Witt expansion gets more involved. The reason is that all the gluonic operators displayed in (6.63) are 0(1) in the chiral power counting and, hence, it is necessary to go beyond n = 4 in the an coefficients. In addition, it is not clear, in principle, to which order in gluon condensates we should calculate. However, the corrections proportional to the triple gluon condensate have been analyzed in some examples, and they always turn out to be subdominant, using different estimates for the numerical value of (GGG) [31]. For instance, the gluonic corrections to the value of in (6.60) are given by

P;

p2 1r

= Nc M 2

41l"2

~ x

J1.2

g;

~og M2 + 24Nc

(G~vG~V)

M4

1

+ 360Nc

g~(GGG)

M6

)

+ ....

(6.64)

We remark that, by retaining only the gluon condensate corrections, it has been found [31] that the numerical values of the LEe are improved with respect to those in (6.60) and (6.61). In these analysis, the gluon condensate is an input parameter whose precise value is very difficult to set, due to its nonperturbative behavior. For instance, a value based on e+ e- ----> hadrons data is as(G~vG~V)/1l" c:= [410 ± 80] MeV 4 [35]. In the next section we will analyze the peculiarities of the two flavor case. 6.5.3 The Nt = 2 Case As a consequence of the decoupling theorem, the two light flavor case can be obtained from N f = 3 when m s is much bigger than m. Then, the very same derivation of the effective Lagrangian and the effective action applies for Nt = 2 [4]. In fact, since SU(2) is a subgroup of SU(3) all the expressions obtained above would be directly applicable. However, it turns out that not all the operators in the £(4) Lagrangian in (6.29) are linearly independent, since there are identities in SU(2) that do not hold in SU(3), as for instance (6.65)

150

6. The Effective Lagrangian for QeD

that allows us to express some of the operators as linear combinations of a minimal set, that now consists only of seven terms plus contact terms. In addition, in the SU(2) case we will find more useful to work with a parametrization for the U fields as 0(4) vectors. Namely, U == (U0, Uj) with j = 1,2,3, and Uo, uj real fields, with the constraint UTU = 1. Then, the covariant derivatives acting on U are defined as

\lJ.LUo=8J.LUo + a~Ui \l J.L U i = 8J.L Ui + €ijkvjJ.L Uk - a iJ.L' UO (6.66) where the sources vJ.L' aJ.L are parametrized in the same way as U. Let us define now the fields X=2B(so,pi) (6.67)

X=2B(po, -i) , where the value of B is related to B o. In the limit is given by [5]

B=

B o {I -

~JLTJ -

m = 0, m s i- 0 this relation

16;;1 (L 4(JL) - 2L 6(JL))} ,

(6.68)

which is obtained simply by comparing the expressions in (6.43) with the corresponding ones obtained in SU(2). With the above notation, the SU(2) chiral effective Lagrangian up to O(p4) reads Leff

=

~2 \lJ.LUT\lJ.LU + ll(\lJ.LUT\lJ.LU)2 +l2(\lJ.LUT\lvU)(\lJ.LUT\lvU) + l3(XTU)2 +l4(\lJ.L XT \lJ.LU) + l5(UT FJ.LvFJ.LVU) +l6(\lJ.LUT FJ.Lv\lvU) + l7(X T U)2 +h1XT X + h 2FJ.L v FJ.LV + h 3XT X ,

(6.69)

where FJ.Lv = [\lJ.L' \lv]. The renormalization procedure for the li constants is the same as for the L i , but the constants "Yi and L1 i change. The corresponding values are given in Table 6.3. We will also find useful to define scale-independent LEC Ii by

Ii

T

"Yi

= 321r 2

(-l i + Iog M; ) JL2

i

= 1, ... ,6.

(6.70)

Notice that, up to the different normalization, the li constants are nothing but the l[ at the scale of the pion mass. Consequently, they have to be used very carefully when dealing with the chirallimit M", = 0, although, of course all the apparent divergences in that limit cancel. The values of these constants together with the processes from which they can be obtained are listed in Table 6.4. We remark that the LEC l[ or

6.6 The Problem of Dnitarity in ChPT

Lr

151

can be related with by comparing the results obtained for the different observables in which they are involved and taking the limit m s >>> m. With the effective action derived before, now written for the SU(2) case, we can obtain the next to leading order corrections to the pion scattering amplitude, which at the lowest order is given in (6.18). The full expression for the amplitude is detailed in Appendix D.3.

Ii

Table 6.3. Values of the renormalization coefficients for N f /1

/2

1

2

"3

/3

"3

/4

/5

/6

2

/7

,,11

o

2

<1 3 1

12

Table 6.4. Values of the LEC for the N f Constant

h

12 13

14

Is

16 h

Value (-0.62 ± 0.94) (6.28 ± 0.48) (2.9 ± 2.4) (4.3 ± 0.9) (13.9 ± 1.3) (16.5 ± 1.1) ~ 0(10- 3 )

=2

o

= 2 case

Source D-wave -rr-rr scattering and K -> 1r1rlv N f = 3 mass formulae Nf = 3, FK / F-rr -rr

->

w/

EM pion charge radius -rr 0 - TJ mixing

6.6 The Problem of Unitarity in ChPT Throughout this chapter we have seen how to apply the effective Lagrangian formalism to describe the low-energy hadronic·phenomenology. By means of the chiral expansion, we are able to obtain amplitudes for different processes as series in the external momenta and masses. In the following sections we are going to study how well this series satisfy some general properties which follow from quantum field theory (QFT). Let us first concentrate in unitarity. In Appendix D.l we have given a brief review of some general and well known unitarity constraints that any two body elastic amplitude should satisfy, as well as the definitions of the partial waves for several cases of interest. In order to illustrate the unitarity behavior of the amplitudes obtained from ChPT, we are going to study first the elastic NGB scattering. In such case, the relevant quantum numbers for the process are the isospin I and the angular momentum J. It is customary to project the amplitudes in partial waves tIJ of definite I and J. Then, above the two body elastic threshold and below any other inelastic threshold, unitarity imposes the constraint (see Appendix D.l)

152

6. The Effective Lagrangian for QeD

ImtIJ =

(T I

tIJ

2

1

(6.71)

,

where (T is the integrated phase space of the particles involved in the process, whose precise form is given in (D.S). For all means and purposes, we only need to know that (T ---. 1 when s ---. 00. Apart from logarithmic factors, we have already seen that any amplitude in ChPT is obtained as a series in positive powers of the energy and mass. From now on we will use the Mandelstam variable s = 0(p2). In particular we will have

tIJ(s) = t}~(s)

+ t}~(s) + 0(S3)

,

(6.72)

where t}~ is real and O(s), t}~ is 0(s2), etc. Thus, our amplitudes will basically behave as polynomials in s. It is then clear that at some energy any polynomial will violate the unitarity bound in (6.71). As a rough estimate we expect that tIJ ~ 1 when the terms in the expansion are of order one. Remembering the discussion in Sect. 6.3 we expect that to happen when VS ~ Ax ~ 1 CeV, although in practice it happens much before. However, the ChPT amplitudes satisfy the unitarity condition perturbatively, i.e. order by order, so that at 0(s2) we find

Imt}~(s) = (T I t}~(s)

2

1

(6.73)

and therefore the violation of unitarity is an 0(s3) effect that should be negligible at low energies, precisely where we want to apply ChPT. According to the previous arguments, we expect the interactions between the NCB to become strong at energies close to 1 CeV. Nevertheless, even below that energy one of the most relevant features of strong interactions comes into play; namely, the resonances, which saturate the unitarity bound in (6.71). However, we have already seen that the amplitudes obtained from ChPT do not respect unitarity and thus they cannot reproduce any resonant behavior. This fact can be intuitively understood as follows: if we want to mimic a resonant shape with a polynomial, we will need at some point that the contribution from the highest energy power dominates, but at that moment the chiral expansion is meaningless. It is then of the most relevance to implement a formalism able to reproduce such resonant states. In the literature several techniques have been suggested to accommodate resonances in the framework of ChPT. Among others: • The explicit introduction of resonance fields with their mass and width fitted from experiment [26, 27, 28, 36]. Although this is not a predictive approach, the amplitudes thus obtained provide a good description of the experimental data and they can be used for further analysis of other processes. • To impose strict unitarity by means of dispersion relations, which reflect the general analytic properties of the amplitudes [37, 38, 39]. • The large-N limit, N being the number of NCB, where we obtain the leading terms in an liN expansion which again satisfies unitarity only

6.6 The Problem of Unitarity in ChPT

153

perturbatively. The advantage is that now the unitarity breakdown does not behave as an energy power [40]' but as 0(I/N 2 ). There are other unitarization procedures, but we will mainly concentrate in the last two, since they are based on a more fundamental approach. Let us now recall that light-meson interactions playa key role in hadron physics at energies below 1 GeV.. Indeed, the most abundant products of hadronic processes are pions and kaons. That is one of the main motivations to study processes involving these particles. In what follows, we will analyze both 7r7r and 7r K elastic scattering, since they have the lowest thresholds and thus they are the best candidates to apply ChPT. In both processes there is a strong resonant behavior that cannot be ignored. In particular, for two pion scattering there is a p meson which appears at 770 MeV in the I = J = 1 channel, whereas for 7r K --+ 7r K the lowest lying resonance, called K*, has I = 1/2, J = 1 and a mass of 890 MeV. Once we had explained how to obtain a good parametrization of these reactions by means of unitarized ChPT amplitudes, we will illustrate how it can also be used to describe other hadronic processes. 6.6.1 Unitarity and Dispersion Relations

Next, we will study the unitarity properties of chiral amplitudes from the point of view of dispersion theory. This approach makes use of the analytic structure of any amplitude obtained from a relativistic QFT, when the real Mandelstam variable s is extended to the complex plane. The existence of thresholds in the different s, t and u channels in a given amplitude is responsible for the appearance of a right and a left cut on the real s axis of the partial waves. As it can be seen in Appendix D.2, a dispersion relation is a direct consequence of the Cauchy theorem, which relates the value of a function in a given point with an integral over a surrounding contour. In the case of scattering amplitudes, when this contour is taken to infinity, we are only left with integrals over the right and left cuts. In this section we want to study the properties of chiral amplitudes at 0(S2), which therefore grow as S2 at large s. Thus, in order to ensure the convergence at lsi --+ 00 of the integrals, it is convenient to apply the Cauchy theorem not to the amplitude itself, but to the amplitude divided by s3, which is usually known as making three subtractions (see Appendix D.2). That is why, for elastic scattering of two particles a and b, we are interested in the following dispersion relation: tJj(s)

s31°°

= Co + CIS + C 2 S 2 +-7r

(M a +Mb)2 S

ImtJj(s')ds'

'3 ( ,

S -

S -

. )

tf

s31-

+ -7r

0

(6.74) 00

ImtJj(s')ds'

S

'3 ( ,

S -

S -

. )

tf

154

6. The Effective Lagrangian for QeD

Let us remark first that the C i constants depend on the masses M a , M b . Second, the integrals do not yield polynomial contributions in 8, but complicated nonanalytic functions containing cuts, like logarithms, square roots, etc. The use of this general property is a common tool to modify the chiral amplitudes so that they respect unitarity. It has been used in many contexts for different processes and a complete account of the works on the subject would require a chapter on its own. Nevertheless we will try to illustrate the general use of dispersion relations with a method which is very easy to implement and yields extremely good results.

The Inverse Amplitude Method The aim of this section is to explain and justify a simple procedure to obtain unitarized amplitudes [39]. We will concentrate on the elastic scattering of two NGB, that is, we will unitarize the chiral amplitudes, at order 0(8 2 ), of the following processes: 1r1r 1r K

----+

1r1r

----+

1r K

.

As we have already commented, our amplitudes grow as 82 when 8 -. 00, and thus the best suited dispersion relation for our problem has three subtractions, precisely as that in (6.74). Let us remember once more that in ChPT the amplitudes are obtained as truncated series in the mass and external momenta, as in (6.72), and that the first order tJ~ reproduces the low-energy theorems. Introducing this series in (6.71), we obtain the following perturbative unitarity conditions

ImtJ~(8)=O ImtJ~(8)=(JabtJ~2

(6.75)

on the real 8 axis above threshold. Notice that we have written (Jab, with a, b = 1r1r, 1r K instead of (J since the phase factor is different depending on the particles involved in the process, but its high-energy properties are still the same. Now, at each order of perturbation, we can write again the three-times subtracted dispersion relation of (6.74). We find

tJ~(8) =ao + al8 (I)

2

tIJ(8)=bo +b I 8+b 2 8

31

8

+1r

(6.76) 00

(M a +Mb)2

(Jab t(O)2( IJ 8')d 8' 13( I .)

8

8 - 8- u

( (I) +LCt IJ ),

where LC is the left cut contribution and we have replaced the imaginary parts in the integrand with the help of (6.75). Notice, by counting energy powers, that the polynomial part has been expanded in terms of the masses M a , M b of the particles and then: Co ::: ao + bo + ... , C I ::: al + bl + ... and

C2

:::

b2

+ ...

6.6 The Problem of Unitarity in ChPT

155

Up to the moment we have just written, in a more complicated way, that

tIJ(s) ~ t~~(s) + t~~(s). The main idea of the inverse amplitude method is

that we can also apply the three times subtracted dispersion relation to the inverse amplitude l/tIJ(s), since it has exactly the same cut structure and the same or even better s -+ 00 limit. For further convenience, instead of the inverse amplitude, we will use another one which is normalized as follows: (0)2

(6.77)

G(s) = tIJ /tIJ.

Notice that t~~2 is real and thus we are not changing the analytic structure. The dispersion relation is then

G(s) = Go + GIS + G2S2 s31

+ 1r

00

(M a +Mb)2

ImG(s')ds' S'3( S, - S - u. )

+

LC(G)

PC

+,

(6.78)

where PC stands for the pole contribution in G(s) coming from the zeros of t(s). But now in the right cut we have t(0)2 1 ~ I J m t IJ

1m G --

-

_t(0)2 IJ

ImtIJ - _t(0)2 t IJ 12 I J a ab

1

,

(6.79)

which can be exactly calculated from ChPT. Thus we are able to calculate explicitly the whole right hand cut contribution. This is not the case in the left cut, but we can still use the S2 approximation as we did for t(s), that is

1m G --

-I m t(I) IJ 1ImtIJ tIJ 12 '" IJ

_t(0)2

-+

(1) LC(G) ~ -LC(tIJ) .

(6.80)

Only the subtraction coefficients GO,G l and G2 are still unknown in (6.78). But if we expand in powers of the particle masses Ma,Mb , we find that: Go = ao - bo + ..., G I = al - bI + ... and G 2 = -b2 + ..., so that we can finally write

t(0)2 G(s) =.J...:L ~ ao + als - bo - bIs - b2s 2 tIJ 31 00 ')d s' _ LC( (1») _~ a t(0)2( IJ s '3( , .) t IJ , 1r (M",+MIJ)2 S S - S - u

(6.81)

where only LC is calculated in the S2 approximation. Notice that the pole contribution of G(s) has been neglected. We will comment on that below. Therefore we have just found (0)2

t I J '" tIJ -

t(O) _ t(I) IJ IJ

(6.82)

and therefore t(O)

t

'"

IJ -

IJ (1) (0)' 1-tIJ/tIJ

(6.83)

156

6. The Effective Lagrangian for QeD

which, apart from logarithmic factors, is the formal [1, 1] Pade approximant of (6.72). The most relevant fact about this result is that only the left cut integrand has been approximated to 0(8 2). In contrast, from the point of view of dispersion theory, the pure chiral expansion is an approximation in both the left and right cut integrals. As a consequence the amplitude above satisfies exactly the elastic unitarity condition, (6.71), below the inelastic threshold. At the same time, we have not spoilt the good low-energy properties, since, if we expand again (6.83), we recover tIJ = t}~ + t}~ + 0(8 3 ). However, we have made several approximations. First, we have neglected eventual pole contributions that correspond to zeros of the amplitude. Such zeros indeed exist but they are located below threshold [41]. Second, we have made a very crude estimation on the left cut in (6.80). Nevertheless the method is still valid. On the one hand, these two approximations seem justified at high energies due to the dumping 8,3(8-8') factor in the integrals. Thus, apparently, the relative error will be bigger at small 8. But, on the other hand, we have just seen that the inverse amplitude method at low energies, only differs from the ChPT result by 0(8 3 ) and hence the contribution of the left cut and the poles are subdominant. As we will see, the results do confirm these expectations and it is possible to fit the data with values of the LEC which are very close to those in Table 6.2. Before looking at the results, we have to introduce the most extended parametrization of elastic scattering. The experimental measurements are usually analyzed in terms of the phase shifts 8IJ(8), which are defined from the corresponding partial wave amplitude, as follows:

tIJ(8) =

eiOIJ(s)

sin8IJ(8)/aab(8)

(6.84)

for

8 > (Ma + M b )2 and below any inelastic threshold. With this definition, we can now present the comparison with experimental data of the results obtained with ChPT amplitudes unitarized with the inverse amplitude method. The curves in Figs.6.1 and 6.2 represent the phase shifts 8IJ (8) of the partial amplitudes for 7r7r and 7r K elastic scattering in different (I, J) channels. They have been obtained using the SU(3) formalism and the L i parameters given in Table 6.3. It is important to notice that the fit has been performed up to rv 1 GeV and tuning the exact values of the resonance masses M p = 770 MeV (which implies 2£1 + £"3 - L 2 = (-3.11 ± 0.01)10- 3 and MK* :::::: 894 MeV. The best values at Jl = M." are

£~ ,

= (0.41 ± 0.20) 10- 3 ,

3

L 3 = (-2.44 ± 0.21)10- ,

£~

= (1.48 ± 0.33) 10- 3 (6.85)

which lie within the errors of the LEC given in Table 6.2. Remarkably, once the masses are properly obtained, it is also possible to calculate the widths, which then become a pure prediction of the approach. The results

6.6 The Problem of Unitarity in ChPT

.,.,'"

157

~180 . - - - - - - - - - - - - - - - - - - - - - ,

g.

160

111T

~ 1T1T

~ <0140 ~

"£ ~ 120

'"o

.c

0..00

eo 60

40 20 0

'".,., 100

~

.,a.

~

~

J f"£

.,'" '"o

&.

80

60

40

400

300

700

800

900

vs

1000

(Mev)

,,. , ,,.,,,,." ,_ ,,,. #-=# 1111

{i? ? ~.+.

~ 1T1I

0

~-

20

o

-20 400

500

600

700

eoo

900

vs

1000

(Mev)

Fig. 6.1. Pion elastic scattering phase shifts {jIJ obtained from the lAM fit to the correct M p • The dotted straight lines stand at {j = 90 0 . The shaded areas cover the error bars of the fitted parameters with the constraint 2£1 + £3 - £2 = (-3.11 ± 0.01)10- 3. The data come from: [42] (6), [43] (0, D), [44] (x),[45] (0), [46] (
158

6. The Effective Lagrangian for QeD ....... '80 r - - - - - - - - - - - - - - - - - - - - ,

., ~ "'160 ., II)

nK ~nK

~

C:: ,40

.0 ~

., l20 II)

o ~'OO

--_.....

80

__._-----....,.......---------

60 40

20

o

~6-!-:5O:-"':'700~~75O=!£~800~~8:-:5O:---'''-:9~00~--::9-!-:5O~7:IOOO:'=:-'-'-:":101:-:5O:-'"'''":1~,00 ";s (Mev)

....... 8 0 r - - - - - - - - - - - - - - - - - - - - ,

., .,'" II)

.,

't)

~

~ .0

nK ~ nK 60

S

'0 ~

:c

40

., II) II)

o ~ 20

o

-20 650

700

750

800

850

900

950

1000

1050

I 100

";s (Mev)

Fig. 6.2. 1rK elastic scattering phase shifts 8[J obtained from the lAM fit to the correct M p and MK-. The shaded areas cover the error bars of the fitted parameters with the constraint 2£'i + £3 - £2 = (-3.11 ± 0.01) 10- 3. The dotted straight line stands at 8 = 90°. The experimental data come from: [49] (e), [50] (*), [51] (0), [52] (0), [53] (0) and [54] (6)

6.6 The Problem of Unitarity in ChPT

T p = 149.9 ± 1.2 MeV

T K * = 41.2 ± 1.9 MeV

159 (6.86)

are quite close to the experimental values of T p = 150.3 ± 1.0 MeV and T K * :::::: 50 MeV. (There are two K* resonances, a charged and a neutral one, with slightly different masses and widths. That splitting is due to other effects. The K*(892) mass and width given above are rounded averages.) Despite the success of the approach, it is not convenient to identify the constants in (6.85) with the LEG of ChPT, even though we had kept the same names. That is the reason why we have written Li instead of L i . As a matter of fact, the above values have been obtained with a non-perturbative formalism, which somehow takes into account more terms that just those coming from pure ChPT as well as high-energy data in the fit. It is then natural that the precise values are not exactly those obtained from the ChPT series and low-energy measurements. In any case, the remarkable feature of the method is that it is able to reproduce two resonances, which appear whenever 8(8) = 7r/2, at the (1,1) channel for 7r7r scattering and in the (1/2, 1) for 7r K, in perfect correspondence with the p and K* resonances. Even using the low-energy parameters in Table 6.3, two resonances are found in the right channels, although within 10% to 15% of their actual values. The figures speak by themselves showing a considerable improvement in the unitarity behavior. This should not be considered as a criticism to 0(8 2 ) ChPT since it is only intended to work in the low-energy region.

The Inelastic Case Up to the moment we have illustrated the dispersive methods with elastic processes, but they also provide unitarization procedures for the inelastic case. The most widely studied are K --+ 7r7rlv [55, 56] and 'Y'Y --+ 7r7r. Here we will concentrate on the latter since the analysis is much simpler and the improvement of the unitarized results is more apparent. The first one-loop ChPT 'Y'Y --+ 7r7r calculations [57, 58], which are 0(8), showed that the threshold behavior of the neutral channel was poorly reproduced, whereas for the charged channel it yield a reasonably good description. In contrast, and even though this process is not dominated by any resonance below 1GeV (so that unitarization may seem unnecessary at first glance), several dispersive studies [59]' based on the low-energy theorems [13, 14]' were able to fit very well the experimental data at threshold. The problem is that the lowest order amplitude vanishes for the neutral channel, which intuitively comes from the fact that the tree level production is forbidden because the 7r 0 cannot couple directly to photons. Nevertheless, the reaction can take place at one-loop, via 'Y'Y --+ 7r+7r- --+ 7r 0 7r 0 . At present, there have been performed two loop ChPT calculations which display a much better agreement with data [60]. Let us first review some general features of the process. As we did for 7r7r elastic scattering, it is also convenient to work

160

6. The Effective Lagrangian for QeD

with partial waves FfJ which now are also projected on the helicity basis. We define the total helicity as A = A1 - A2, where A1 and A2 are the helicities of the incoming photons. Therefore we find that always I A I~ J. We refer to Appendix D.l. for further details on the partial wave definition, as well as on possible initial and final states. Physically, the most relevant channel is the one with lowest angular momentum J = o. Consequently A = 0 and below the kaon threshold we can consider a Hilbert space with three states: one with two photons and parity p = + and two with two pions and isospin I = 0 or 1= 2. To save notation we will call them: 1+ >,10> and 12 >. Thus we are interested in the following S-matrix: S =

(

S++ S+o S+2) S+o Soo S02 . S+2 S02 S22

(6.87)

Assuming T invariance, S is a symmetric matrix and the unitarity condition sst = 1 reads: L:n SinS~f = 8if . Keeping just the dominant terms in O'.~M' we are left with

I Soo1 2 =1 I S 221 2 =1

S~o

+ S~oS+o=O

S~2

+ S;2S+2 =0.

(6.88)

From these equations it is easy to see that: Soo = exp (i280 ) and S22 = exp (i28 2 ) where 8/ is the 8/ 0 phase shift for 1r1r elastic scattering defined in (6.84). From the relation between amplitudes and S-matrix elements (see Appendix D.1), it can be shown that the phase of S is twice the phase of the amplitude so that we can write

Flo =1 Flo I eifh

4m;

< 8 < 4mJ<

.

(6.89)

What we have just seen is that the phase of :PJJ is nothing but the 8/ J phase shift of 1r1r elastic scattering. We could then think of repeating the inverse amplitude method on the 'Y'Y --; 1r1r amplitudes. However, it is not possible to apply the same procedure since FfJ does not satisfy the elastic unitarity condition in (6.71). Instead, using that the phase of FfJ is the same as that of t / J, we find that

ImFfJ

=

CJFfjtIJ .

As it happened in the 1r1r --; obtained as series in the energy '1""11 _ '1""11(0) J/J - JIJ

(6.90) 1r1r

case, in ChPT the amplitudes FfJ are

+ J/J '1""11(1) + '1""11(2) J/J ... '

(6.91)

where F:JO) is 0(8) and reproduces the low-energy theorems, F:)1) is 0(8 2 ) and provides an analytic structure with a left and a right cut. As in the

6.6 The Problem of Unitarity in ChPT

161

elastic case, any ChPT amplitude will not satisfy (6.90) exactly, but only perturbatively Im.1':p) = a.1':;(O)t}~ .

(6.92)

Thus we have found a similar problem to the elastic case. As a matter of fact we could unitarize the I I ----+ 7m scattering with a Pade approximant: .1' rv .1'(0) /(1 - .1'(1) /.1'(0»), but then we would have to define t rv t(O) /(1 .1'(1) /.1'(0»), since otherwise they will not satisfy (6.90) exactly. This approach is consistent, since the last definition of t also respects the elastic unitarity condition. However, as we are not using the information in t(l), it is not the best we can do. Indeed, it has been shown that these Pade approximants improve the threshold description but they are not so good at higher energies [61]. That should not be very surprising since now they are not justified from dispersion theory. In any case, what we are interested in is the amplitude for physical values of s, which are located along a cut on the complex plane. It can be shown (see the Muskhelishvili-Omnes problem on Appendix D.2) that the cut structure is obtained properly once the phase shift is correctly reproduced. As a consequence, we can make profit of the 7m elastic scattering phase obtained with the inverse amplitude method [62]. Remembering that both t and .1' in a given channel I, J should have the same phase, we can simply obtain a unitarized F by calculating I .1' 1 with ChPT and then imposing the 7m scattering phase shift. That is -

.1' =1 .1'

1itT =1 .1' Ie, t

i6

(6.93)

where 8 is calculated with the inverse amplitude method. In this way .1' satisfies the unitarity condition (6.90). The cross sections for I I ----+ 7r7r are shown in Fig. 6.3, both for the neutral and charged channels. The solid lines have been obtained using (6.93) with the Li parameters in (6.85). In the charged process the dotted line represents the 0(1) ChPT prediction, which is dominant at all energies, so that the unitarization corrections are almost negligible. On the contrary, unitarity seems to play an important role in the neutral channel even at low energies, providing a good description near threshold. 6.6.2 The Large-N Limit In this section we are going to review another method that has been proposed to obtain amplitudes from the effective Lagrangian with a better unitarity behavior [40]. The central idea comes from the fact that in two flavor massless QCD, the SU(2)L x SU(2)R symmetry is broken to SU(2)L+R and the three NCB can be understood as coordinates on the S3 manifold since (6.94)

Fig. 6.3a,b. Unitarized cross sections for 'Y'Y --> 7r0 7r0 and 'Y'Y --> 7r+ 7r-. The continuous line has been calculated using £9 + Lo = 0.4 10- 3 and the phase shift obtained from the lAM. The dark shaded area covers the uncertainty in £2, the light area does also include the uncertainty from £9 + Lo = (0.4 ± 0.6) 10- 3. The experimental data come from: (a) Crystal Ball [63], (b) MARK II [64]

6.6 The Problem of Unitarity in ChPT

163

...J;V 1

N

~ N

N

Fig. 6.4. Generic bubble diagram. Note that every loop carries an N power, whereas each vertex contributes with an 1/N factor

We can now extend this symmetry pattern to O(N

+ l)/O(N)

rv

(6.95)

SN

and we will then have N NCB fields. As we have already seen in Chaps.l and 3, their dynamics will be governed at low energies by a NLSM with N degrees of freedom. The approach that we will follow in the following is to consider a large-N expansion. At first glance, this construction may seem quite unrealistic, but it is interesting since it is rather simple to extract closed formulas for the leading liN contribution. Indeed, as we are now dealing with a number N of NCB, whenever we form a loop with them we will find an N factor. Furthermore we will see that each vertex will carry at least a liN factor. In this way we expect that the leading contributions will be obtained from the so called bubble graphs, like that in Fig. 6.4. The sum of all the diagrams with one, two, three, etc. bubbles can be understood, formally, as the geometric series] + ]2 + ]3 + ... where] is the momentum integral of one bubble, which can be easily obtained in dimensional regularization. As we have already mentioned, in the chiral limit it is fairly simple to obtain at leading order closed analytical expressions for the amplitudes. That is the reason why this approach has been frequently used as the starting point to other more realistic approximations. In this section we will illustrate how this method works, obtaining the leading liN order prediction for NCB elastic scattering.

NGB Scattering Amplitude From now on we will use the standard coordinates in the SN sphere to describe our NCB. Therefore, we can write the Lagrangian as follows: L-NLSM =

1 "29ab(1r)8IJ.1ra8IJ.1rb

- ~8 8IJ. ~ (1r a8IJ.1ra )2 - 2 IJ.1ra 1ra + 2 N F2 _ 1r2

'

a, b = 1 ... N .

(6.96)

We have already seen in Chap. 3 that when quantizing this model, the derivative couplings give rise to more terms coming from the determinant of the functional measure. However, using dimensional regularization such contributions vanish and therefore we will use that scheme throughout our calculations.

164

6. The Effective Lagrangian for QeD

In order to calculate these NGB loops, which dominate in the leading term of the liN expansion, we have to obtain the large-N leading order of the NGB propagator. Nevertheless, it is very easy to realize that, as far as the NGB are massless and we are working in the dimensional regularization scheme, we get the following simple result for the NGB propagator Dab(p 2 ) =

i8ab

-2--'

P

+ 1E

+0

(N 1) .

(6.97)

.

Once again, let us recall that the NGB elastic scattering in any channel, can be obtained from a unique amplitude A(s, t, u), as it is shown in (D.17). Indeed at O(lIN) we find A(s, t, u) = A(s), that can be obtained from the graphs in Fig. 6.4 and the Feynman rules coming from the Lagrangian in (6.96). Namely, A(s) is given by

A(s)

N~2 + N~2 (N~(S)) N~2

=

+ N ~2 (N ~(s)) N ~2 (N ~(S)) N ~2 + ...

(6.98)

As commented before, the formula above can be seen as a geometric series in l(s)s/2F 2 , where l(s) is an integral that appears very frequently in one loop calculations and is given by .

2

11(k ) =

=

J

D

d qJ1!

1

(27r)D q2(k _ q)2

16~2 [N< + 2 + log ( :::2 )] = 16~2 log ( e~~:)

,

(6.99)

where J..L is an arbitrary energy scale. The far RHS is the result obtained when a cutoff A is used to regularize the divergent integral. Formally summing the series, we find

A( )

s s = NF 2 1_

8

1

1 (€2 A 2)

321l"2F2 og

(

6.100

)

-8

This amplitude can be understood as though it was coming from a model valid for s « A2. It presents a very interesting feature: at low energies it reproduces the slNF 2 behavior in (6.18), expected from the Weinberg lowenergy theorems. This allows us to give a physical meaning to F, identifying F; = NF 2 . Renormalization However, up to this moment all our calculations have been purely formal, since l(s) is indeed divergent. Once more, it is possible to introduce counterterms with more derivatives, together with their corresponding parameters that will absorb the divergences and will run with the scale as it happens in

6.6 The Problem of Unitarity in ChPT

165

the usual renormalization procedure. In so doing, an infinite family of counterterms have to be added to the NLSM Lagrangian, so that we can absorb the infinite number of different divergences that will appear. One possible choice of these counterterms is

.cet = ~ ~7f2 (_O)k+1 7f2 + LJ 8N

k=1

F2

.

0(_1) . N2

(6.101)

Then the total tree level contribution to the A(s) function is

Atree(s) =

N~2 [90 + 91 ;2 + 92 (;2) 2+ ... J (6.102)

where we have defined

90 == 1.

···cx Fig. 6.5. Generic n-loop diagram contributing to the NGB scattering amplitude

Now we have the tools for computing the complete renormalized A(s) function. The generic n-Ioop diagram contributing to this amplitude is shown in Fig. 6.5. In each of its vertices we have to consider all possible combinations of the infinite couplings introduced above and then we have to sum over the number of loops. The final result is

Hence we can write the last formula as follows:

(6.104) Or, formally summing the geometrical series

A( S )

=

Atree(s) I(s)

1 - -2-N Atree(s)

(6.105)

166

6. The Effective Lagrangian for QeD

The last equation is quite compact but we still have to show that it can be written explicitly in terms of only finite quantities such as the renormalized coupling constants. In order to do that it is useful to write (6.105) as follows: 1

A- (s) =

A~~e(s) -

=

where

(~:)

1

A- (s) - 2(::)2 log

A-1 (s)

--1

A

2(::)2 [Ne + 2 + log

(~:)] (6.106)

,

has been defined as

(s) = Atree(s) -1

N

- - 2 (Ne

3271"

+ 2)

(6.107)

and therefore

A(s) =

A(s) -

N

(6.108)

1-'2·

1 - A(s)2(411")210gCs)

Now we can expand A(s) in powers of s/F 2, as we did in (6.102) with A tree s ~ R (s)n A(s) = NF2 Lgn (/-L) F2 . n=O

(6.109)

As the notation suggests, the coefficients of this expansion are the renormalized coupling constants (with g{;(/-L) = 1). They depend on the arbitrary scale /-L in such a way that A(s) is /-L independent. At this point, it is useful to introduce the following function: (6.110) It can be considered as a generating function for the renormalized coupling constants g;; (/-L) since

R 1 oncR(s; /-L) I gn (/-L) = n! o(s/F2)n S/F2=O .

(6.111)

Note that with this definition, the f3n functions can be easily generated from

oCR(s; /-L) o(log/-L)

00

dg;;(/-L) ( s)n

= ~ d(log/-L) F2

00

(

S

= ~f3n F2

)n

(6.112)

So that we can now write the amplitude A( s) as

A(s)=_s_ CR(s;/-L). N F2 1 - 2(411")2F2 s CR(.) I (£) S, /-L og -s

(6.113)

The dependence of the generating function on the scale /-L can be obtained just by requiring the complete amplitude A( s) to be /-L independent. The result is

6.6 The Problem of Unitarity in ChPT

GR(S. ) = ,Jl 1+

GR(Sj S

2(41f)2F2

Jlo).

G R( Sj Jlo ) log (1-£2) ~

167

(6.114)

Therefore, once the S dependence of the generating function is known at some scale Jlo, the above equation gives us its corresponding S dependence at any other arbitrary scale Jl. Notice that the knowledge of this S dependence is equivalent to knowing the values ofthe infinite renormalized couplings at the given scale. Nevertheless, the above equation contains the information on the scale evolution ofthe g;;(Jl). Indeed, this infinite set of evolution equations can be obtained in a very straightforward way by writing (6.114) as

g;;

~ R( ) (~)n =

L

n=O

gn Jl

L::=og;;(Jlo)(~)n

p2

1+

R( )( S )n 1 (1-£2)· Lm=O gn Jlo F'I og ~

S""OO

2(41f)2F2

(6.115)

Identifying the coefficients of the sl p2 powers in both sides of this equation it is not difficult to reproduce the evolution equations for the different couplings, which means that we have completed the renormalization of the NGB scattering amplitude. The divergences appearing in (6.103) have been reabsorbed in the renormalized constants g;;(Jl), collected in the generating function GR(Sj Jl). Note that the introduction of an infinite number of couplings is not surprising since we know that the NLSM is not renormalizable in the usual sense; even more, it reflects the fact that there is an infinite number of different theories compatible with the low-energy theorems. In addition it is important to notice the good unitarity properties of the amplitude in (6.113). First of all, it has acquired a right cut coming from the log(Jl2 1- s) appearing in the denominator. Second, carrying out the standard partial wave decomposition, it is not difficult to find that

Imt~bcd = N t~bcd 1

2

1

+0

(~2)

,

(6.116)

where t~bcd is the partial wave amplitude obtained from (6.113) with a total angular momentum J. That is, once more the unitarity constraint is satisfied perturbatively, although this time, and that is the relevant point, the size of the breaking terms is not related to any momentum expansion. Note that the above equation holds for any generating function GR(Sj Jl). The interesting point is that many of these amplitudes have poles in their second Riemann sheet. That is indeed the case of the linear sigma model (LSM), that we will next discuss as an example. These poles can be interpreted as real physical states or resonances of the corresponding model providing a very rich phenomenological potential to the large-N expansion of ChPT. But even more, the rejection of models with poles in the first Riemann sheet (ghosts), which are physically unacceptable, provides a way to constraint the available region in the coupling constant space.

168

6. The Effective Lagrangian for QeD

The Linear Sigma Model as a Particular Case From all the models with an O(N + l)/O(N) symmetry breaking pattern, the simplest one is the L8M. Its NGB scattering amplitude, at leading order in liN, is given by: 1

S

ALsM(S) = N F 2 ·

2

1- M£(J-t) - 2(411")2F2log(~s)

(6.117)

.

This formula will be calculated in 8ect.7.8 (see (7.124)) in the context of the minimal standard model. It can be checked that the above equation can be reproduced from the NL8M amplitude in (6.113), with the following replacements: F2

gf(f.l) = Mk(f.l) g;;(f.l) = (gf(f.l))n = R

G (s;f.l)=

L n=O 00

(

F

(M~~f.l)) n

2

M2() R f.l

) n (

S ) n

F2

=

1-

1

~

,

(6.118)

Mn(J-t)

where MR(f.l) is the renormalized mass of the additional scalar field that appears in the L8M. Moreover, the above equations and (6.114) can be combined to give the evolution equation for the renormalized mass Mk(f.l) (that will also be obtained in Chap. 7)

M 2( ) R f.l =

Mk(f.lo)

1-

M~(J-to)

2(411")2F2

1-'2'

log(~)

(6.119)

Considering the relation Mk(f.l) = 2AR(f.l)NF 2 ofthe renormalized L8M it is also possible to find the evolution equation for the standard running coupling constant AR(f.l) in the large-N limit

A = R(f.l) 1_

AR(f.lO) I (1-'2)· (411")2 og ~

An(l-'o)N

(6.120)

Therefore, at least in the large-N limit, the L8M appears as a particular case of the renormalized (in the sense described above) NL8M. In addition, the amplitude in (6.117) has a pole in the second Riemann sheet, which is nothing but a physical scalar resonance. This shows that, in contrast with the standard loop expansion, the leading order of the 1 IN expansion of ChPT can naturally accommodate a resonant behavior.

NGB Phenomenology in the Large-N Limit The O(N + 1) L8M of the previous example is not the theory that reproduces the hadron interactions at low energies, so that we have to go back to

6.6 The Problem of Unitarity in ChPT

169

our general discussion. In principle, as it happened for the case of the 0(p4) ChPT Lagrangian, there are two possible approaches in order to determine the renormalized coupling constants in (6.113). From the theoretical side one could try to compute these parameters directly from the underlying theory, that is, QeD. From the phenomenological point of view, one can try to fit these constants from experimental data, for example, in low-energy pion scattering. As it can be guessed, the theoretical approach is extremely involved and, at present, there are no theoretical estimates available. However, there is a phenomenological determination, that we will briefly review, which is well suited to illustrate the quality of the large-N approach at leading order in the chiral limit. When using (6.113) to fit the experimental pion scattering data we are faced with an infinite number of parameters, i.e. the scale J-L and the values of the renormalized coupling constants g~ at J-L, which apparently makes the problem unmanageable and meaningless. There is, however, one way around, which is to consider only those particular cases where all the coupling constants but a finite set gf-, g!i ... g~, do vanish at some scale J-L. Such models are just defined by a finite number of parameters (J-L and the k coupling constants renormalized at this scale) and therefore can be used to fit the experimental data. In particular, one can consider the extreme case when all the renormalized couplings are zero at some scale J-L, i.e. gk(J-L) = 0 for all k > o. The model thus obtained from the generalized NLSM has only one parameter (the J-L scale or the cutoff 11). Hence, it can be considered as the NLSM renormalized at leading order of the 1jN expansion, (6.100). As we have already seen, the usual way to present results in 7r7r scattering is in terms of phase shifts. These are obtained from the corresponding partial waves amplitudes with definite isospin and angular momentum. Notice, however, that the proper generalization ofthe standard SU(2) isospin projections to the O(N) case is the following [65]: To=NA(s) + A(t) T 1 =A(t) - A(u)

+ A(u)

(6.121)

T 2 =A(t) +A(u) , whereas the partial waves are now given by

J1

aIJ = - 1 T1(s,cosO)PJ(cosO)d(cosO). (6.122) 647r -1 From the above formulae, and keeping in mind that A(s) is O(ljN), it can be seen that the aoo amplitude is 0(1) but, for instance, all as well as a20 are O(ljN) and therefore they are suppressed. Thus we have NA(s) aoo(s)=~ all(s)=O

+0

+0

(~)

(1) N

170

6. The Effective Lagrangian for QCD

.,'"

.-..

(a)

~ 60

.,

0'

~ 8

~

:c .,'" '"o .r; a..

50

40

30

20

10

o '-'::J~"""""'~-'-'-.L.40~0'-'-L..LJ45'-:'0""""'-':5~00-:!-'-""""55~0L..W~6'::-'0'::!0""""'-':6~50-:!-'-.L..LJ ..Is (Mev)

00

E

(b)

20

o

.D

...•.•.

-.SH.5

ro ci

....

15

\III

~

",12.5

o

u

"6

10

7.5

-}

5

2.5

o

'-'::30'-:'0..........-':3:-=50::-'-"'-':c40!::0~ .....4~570-':-':-'::5~00~~55'-:'0 ..........-':6~OO-:!-'-......,.65!::0~~7~00

..Is (Mev)

Fig. 6.6. (a) 800 in -rr-rr scattering. The continuous line represents the O(m;/ F 2 ) l/N fit. The dashed line corresponds to the m". = 0 limit and the dotted one is the standard one-loop ChPT prediction. The data are the same as in Fig. 6.1. (b) 'Y'Y ~ -rr0-rr 0 cross section. The continuous line represents the leading large-N result up to O(m;/F 2 ). The experimental data are the same as in Fig. 6.3

6.6 The Problem of Unitarity in ChPT

a20(S)=0 + 0

(~)

.

171

(6.123)

From previous considerations, it is then easy to realize that in this approximation the partial waves have the proper cut structure and that they satisfy perturbatively the elastic unitarity condition, i.e. Imaoo =1 aOO 12 +O(l/N), above the unitarity cut. The results of applying this approach to the I = J = 0 1m scattering channel are shown in Fig. 6.6. Indeed, the curves correspond to a model that only has one nonvanishing parameter at a given scale J.L. As discussed above, the only parameter in this specially simple case is nothing but the very scale J.L, which in Fig. 6.6 has been fitted to J.L ~ 775 MeV. Intuitively one is then tempted to interpret the J.L, or the cutoff in (6.100), as a cutoff signaling the range of applicability of the approach. Amazingly, the fitted J.L value is very close to the p mass, which can be understood as though the cutoff was setting the scale where new physics can appear. Thus we cannot expect the model to work beyond that point. However, we would like to stress that the renormalization method is completely consistent and the results are formally valid at any energy independently of the goodness of the fit. Before concluding this section, let us very briefly comment on two last issues: • First, notice that, for the sake of simplicity, we have restricted ourselves to introduce the large-N approximation in the massless or chiral limit. However, the more realistic approach which include the pion mass effects has also been worked out in [66], although a detailed presentation of these works is beyond our scope. Nevertheless, the results for pion scattering are shown in Fig. 6.6. There it can be seen that the introduction of pion mass effects considerably improves the large-N results near threshold, were obviously the chiral limit is too crude. • Second, the large-N limit can also be extended to inelastic processes as, for instance, 'Y'Y --> 1l'o1l'o. The predictions for this reaction are given in terms of the previous pion scattering fit. Although we will not detail the calculations, in Fig. 6.6 we have included a plot of the corresponding crosssection. Again there is an excellent agreement with the experimental data. To summarize, the large-N expansion provides an elegant and very useful complementary approach to the standard one-loop ChPT calculations, by improving their unitarity behavior and it offers the possibility for a non adhoc description of some resonances. This can be specially useful for the topic covered in the next chapter, where we will introduce the effective Lagrangian approach to the symmetry breaking sector of the standard model.

172

6. The Effective Lagrangian for QCD

6.7 References [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

[33]

S. Weinberg, Phys. Rev. Lett. 17 (1966) 616; Phys. Rev. 166 (1968) 1568; Physica A96 (1979) 327 S. Weinberg, Phys. Rev. D9 (1974) 3357 R, Dashen and M. Weinstein, Phys. Rev. 183 (1969) 1261; S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C.G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247 J. Gasser and H. Leutwyler, Ann. Phys. (N. Y) 158 (1984) 142 J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465 J.D. Bjorken, Phys. Rev. 179 (1969) 1547 S. Coleman and D.J. Gross, Phys. Rev. Lett. 31 (1973) 851; A. Zee, Phys. Rev. D8 (1973) 4038 S. Weinberg, Phys. Rev. Lett. 31 (1973) 494 H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. B47 (1973) 365 D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; Phys. Rev. D8 (1973) 3633 H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346; Phys. Rev. 14C (1974) 274 RD. Peccei and J. Sola, Nucl. Phys. B281 (1987) 1; C.A. Dominguez and J. Sola, Z. Phys. C40 (1988) 63 F.E. Low, Phys. Rev. 96 (1954) 1428 M. Gell-Mann and M.L. Goldberger, Phys. Rev. 96 (1954) 1433 M. Gell-Mann, R.J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195 M. Gell-Mann, Caltech Report CTSL-20 (1961); S. Okubo, Prog. Theor. Phys. 27 (1962) 949 A. Manohar and H. Georgi, Nucl. Phys. B234 (1984) 189 H. Georgi, Weak interactions and Modern Particle Physics, Benjamin/Cummings, Reading, MA, 1984 H. Leutwyler, Ann. Phys 235 (1994) 165-203 Particle Data Group (RM. Barnett et al), Phys. Rev. D54 (1996) J.F. Donoghue, B.R. Holstein and Y.-C. R. Lin, Phys. Rev. Lett. 55 (1985)2766; Erratum 61 (1988) 1527 G. 't Hooft, Nucl. Phys. B72 (1974) 461; B75 (1974) 461 E. Witten, Nucl. Phys. B160 (1979) 57 S. Coleman, in Aspects of symmetry, Cambridge University Press, 1985 C. Riggenbach, J. Gasser, J.F. Donoghue and B.R. Holstein, Phys. Rev. D43 (1991) 127 J. Gasser and H. Leutwyler, Phys. Lett. B125 (1983) 321; Phys. Lett. B125 (1983) 325 S.R. Amendolia et ai, Nucl. Phys. B277 (1986) 168 G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B321 (1989) 311 G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B223 (1989) 425 J.F. Donoghue, C. Ramirez and G. Valencia, Phys. Rev. D39 (1989) 1947 J.F. Donoghue and B.R Holstein, Phys. Rev. D40 (1989) 2378 A. Andrianov and L. Bonora, Nucl. Phys. B233 (1984) 232 A.A. Andrianov, Phys. Lett. B157 (1985) 425 D. Espriu, E. de Rafael and J. Taron, Nucl. Phys. B345 (1990) 22-56; Erratum B355 (1991) 278 N.!. Karchev and A.A. Slavnov, Theor. Mat. Phys. 65 (1985) 192 J. Bijnens, Nucl. Phys. B367 (1991) 709

6.7 References [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]

173

A.E.M. Van de Ven, Nucl. Phys. B250 (1985) 593 R.D. Ball, Phys. Rept 182 (1989) 1 R.A. Bertlmann, C.A. Dominguez, M. Loewe, M. Perrottet and E. de Rafael, Z. Phys. C39 (1988) 231 V. Bernard, N. Kaiser and D.G. Meissner, Nucl. Phys. B364 (1991) 283 Tran N. Truong, Phys. Rev. Lett. 61 (1988) 2526; 67 (1991) 2260 A. Dobado, M.J. Herrero and T.N. Truong, Phys. Lett. B235 (1990) 134 A. Dobado and J.R. Pelaez, Phys. Rev. D47 (1992) 4883 A. Dobado and J.R. Pelaez, Phys. Lett. B286 (1992) 136 S. Adler, Phys. Rev. B4 (1965) 1022 S.D. Protopopescu et al., Phys. Rev. D7 (1973) 1279 G. Grayer et al., Nucl. Phys. B75 (1974) 189 M.J. Losty et al., Nucl. Phys. B69 (1974) 185 P. Estabrooks and A.D. Martin, Nucl. Phys. B79 (1974) 301 V. Srinivasan et al., Phys. Rev D12 (1975)681 L. Rosselet et al., Phys. Rev. D15 (1977) 574 W. Hoogland et al., Nucl. Phys B126 (1977) 109 R. Mercer et al., Nucl. Phys. B32 (1971) 381 H.H. Bingham et al., Nucl. Phys. B41 (1972) 1 D. Linglin et al., Nucl. Phys. B57 (1973) 64 M.J. Matison et al., Phys. Rev. D9 (1974) 1872 S.L. Baker et al., Nucl. Phys. B99 (1975) 211 P. Estabrooks et al., Nucl. Phys. B133 (1978) 490 J. Bijnens, G. Colangelo and J. Gasser, Nucl. Phys. B427 (1994) 427 T. Hannah, Phys. Rev. D51 (1995) 103 J. Bijnens and F. Cornet, Nucl. Phys. B296 (1988) 557 J.F. Donoghue, B.R. Holstein and Y.C. Lin, Phys. Rev. D37 (1988) 2423 D. Morgan and M.R. Pennington, Phys. Lett. B192 (1987) 207, Z. Phys. C37 (1988) 431, C48 (1990) 623, Phys. Lett. B272 (1991) 134 S. Belluci, J. Gasser and M.E. Sainio, Nucl. Phys. 423 (1994) 80 J.F. Donoghue and B.R. Holstein, Phys. Rev. D48 (1993) 137 A. Dobado and J.R. Pelaez, Z. Phys. C57 (1993) 501 H. Marsiske et al. The Crystall Ball Coll., Phys. Rev. D41 (1990) 3324 J. Boyer et al., Phys. Rev. D42 (1990) 1350 M.J. Dugan and M. Golden, Phys. Rev. D48 (1993) 4375 A. Dobado and J. Morales, Phys. Rev. D52 (1995) 2878; Phys. Lett. B365 (1996) 264

7. The Standard Model Symmetry Breaking Sector

In this chapter we are going to review the mass problem in the standard model (8M) and how it is solved by means of an spontaneous symmetry breaking. This mechanism also admits a phenomenological description in terms of an electroweak effective Lagrangian, that is built based on just a few firmly established experimental facts. We will see that its applicability is mainly restricted to an strongly interacting symmetry breaking sector (8B8). In that context, it is particularly useful the so called equivalence theorem (ET). We will review in detail its derivation and applicability. Finally, we also present several applications to the Large Electron Positron collider (LEP) and the Large Hadron Collider (LHC).

7.1 The Mass Problem We have already shown, in Chap. 5, how gauge invariance plays a decisive role in the 8M formulation. The existence of a local symmetry implies the appearance of massless gauge fields, which are nothing but the vector bosons mediating the fundamental interactions. This is clearly the case of quantum electrodynamics (QED), where the U(l)EM gauge invariance produces a massless photon, thus yielding a long range electromagnetic force. On the contrary, the weak interaction has a short range and its mediating particles are massive. Indeed, we have already seen that with the Fermi-FeynmanGell-Mann model and the measured value of the Fermi constant, it was possible to estimate that M w ~ 100 GeV. Even more, the intermediate vector bosons have been found at CERN, and their masses have been obtained at LEP and the Tevatron to a much better precision Mz~91GeV

Mw

~80GeV.

(7.1)

Hence, it would seem that we are forced to introduce a gauge boson mass term in the Lagrangian, thus spoiling our gauge invariance and the possibility to describe all the electroweak interactions under the same formalism. However, as we saw in Chap. 3, there is a way, known as the Higgs mechanism [1], to reconcile massive bosons and gauge invariance. This mechanism was first suggested in the context of condensed matter physics to explain some collective density fluctuations in plasma (plasmons) that, apparently, were A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

176

7. The Standard Model Symmetry Breaking Sector

produced by a finite range electromagnetic field, or a massive photon [2]. Later it was generalized as a relativistic field theory and it was proved its renormalizability [3]. In order for the Higgs mechanism to work, we need a SBS that, on its own, presents an spontaneous symmetry breaking from a group G to another one H. The Goldstone theorem [4] then implies that there would be as many Nambu-Goldstone Bosons (NGB) in the spectrum as the number of broken group generators. However, when part of the G group is made local, we have to introduce a set of gauge fields whose quantization spoils the Goldstone mechanism. While SOme of the NGB (all in the SM) disappear from the spectrum, the gauge fields acquire a longitudinal component, thus becoming massive (see Sect. 3.5). The simplest known way to implement the Higgs mechanism in the SM is to introduce an SU(2)L x U(l)y gauge invariant term for a scalar doublet with a potential whose minimum is degenerated in SU(2}L+R. Therefore there will be three NGB that, due to the Higgs mechanism, will become the gauge bosons longitudinal components. Since they disappear from the spectrum such fields are usually called "would be" NGB. The fact that the scalar doublet transforms linearly under the group is responsible for the appearance of a new scalar particle in the physical spectrum which is called the Higgs boson and presents a vacuum expectation value v ~ 250 GeV. The SM with this simple SBS is known as the minimal standard model (MSM) [5]. But that is not the only possibility. Indeed, we have seen in Chap. 3 how to implement a Higgs mechanism within the effective Lagrangian formalism, irrespective of the linear or non-linear character of the gauge transformations, that is, without explicitly introducing more degrees of freedom than those of the would be NGB. Although up to now we have only been discussing the gauge bosons mass terms, massive fermions are also in conflict with invariance under chiral groups. The usual fermionic mass term is given by (7.2) and thus it is clear that a gauge transformation only in the left-handed component will not leave it invariant. In the SM, not only SU(2)L, but also U(l)y are chiral gauge groups, since they transform the right and left fermionic Components differently. Therefore it is not possible just to introduce a term like that in (7.2) without spoiling again the gauge invariance. However, a gauge invariant fermionic mass term can be obtained by introducing Yukawa couplings in the SBS. We have already seen this possibility in Chap. 5, when dealing with the MSM. In that case, the fermions interact with the scalar doublet via a Yukawa term whose lowest order is precisely the mass term in (7.2). The rest of the expansion of the Yukawa interactions compensate for a chiral gauge transformation, so that the complete Lagrangian remains invariant.

7.1 The Mass Problem

177

Although the MSM is the simplest and most widely used implementation of the Higgs mechanism in the SM, it presents several problems and opens questions: • First of all, the Higgs boson itself, which still remains to be discovered. • The Naturalness problem. The Higgs mass is affected by quadratic radiative corrections that would make it of the order of the scale of new physical phenomena, as for instance the Planck scale. If we want it to be in the experimentally allowed range, we have to ask for an extremely fine and unnatural cancellation of divergences. • Triviality problem. There are serious hints that models with self-interacting scalar fields, which are also coupled to gauge bosons and to fermions as in the MSM, can only be consistently defined as free theories and therefore we are left without a SBS. In such case the MSM should be considered as an effective theory, valid only below some given energy scale. • It has also been shown that a vacuum expectation value for an elementary field of the order of v ~ 250 GeV, would yield a constant term in the cosmological evolution equations which is not compatible, by many orders of magnitude, with present cosmological models. • The predictive power of the MSM is limited by the existence of 17 free parameters. In particular, the scalar selfinteraction constant, and therefore the Higgs mass, is not fixed, thus leaving undetermined the strong or weak nature of the SBS. For all these questions, the MSM, instead of being considered as a definitive fundamental theory, is usually understood as an effective theory. It is believed that, although it is valid at the present reachable energies, it will have to be modified when new physical phenomena appear at higher scales. Indeed there are many other proposals or even extensions of the MSM trying to solve the mass problems. In most cases they introduce new particles and interactions. Among others, there are several implementations of supersymmetry [6]' which for each field introduce a supersymmetric partner and whose minimal version possesses a second doublet in the SBS. Other formulations make use of a dynamical symmetry breaking, as for instance: Technicolor models [7], that mimic at a higher scale the quantum chromodynamics (QeD) symmetry breaking, composite models [8], etc. In view of the situation, it is then very interesting to implement a general description of the SBS, taking into account only the well established information. With that purpose we will now give the most general effective Lagrangian describing the Higgs mechanism that provides masses for the ZO and W± gauge bosons. First of all, we review the present knowledge of the SBS, which is basically the following: -

a) There has to be a system coupled to the SM displaying a symmetry breaking pattern from a global G group to a subgroup H. Such a breaking yields three NGB that through the Higgs mechanism will give masses to

178

-

-

7. The Standard Model Symmetry Breaking Sector

the intermediate vector bosons. Thus if K = G / H, we have k == dimK = 3. b) In order to give masses to the gauge bosons, they have to be coupled to the NGB and therefore the gauge group SU(2)L x U(I)y has to be contained in G. At the same time, we want to keep the usual electromagnetic gauge invariance, so that U(I)EM C H. Thus we can write 9 == dimG 2 4, h == dimH 2 l. c) We do know the gauge boson masses. But even more, from the value of the Fermi constant, obtained from the muon decay (see Chap. 1), we find the symmetry breaking scale v M w = gv 2

GF ~ l.I71O- 5 GeV- 2

v

-

~

(7.3)

250GeV

Incidentally, in the MSM, this scale is the Higgs boson vacuum expectation value, but we have seen in Chap. 3 that it can also be defined without explicitly introducing such a field. d) Finally, with the present determination of the vector boson masses and the Weinberg mixing angle, it has been observed that the p parameter, which measures the relative strength of the charged and neutral weak currents, is very close to one Charged currents

PExp

= Neutral currents ~ 1 .

(7.4)

It has been shown [9] that whenever the so called custodial symmetry SU(2)L+R is included in H, then the P = 1 value is only affected by small radiative corrections (7.5)

due to the U(I)y coupling, which, as we will see in short, explicitly breaks the custodial symmetry. This condition imposes that h 2 3. We will see in Sect. 7 that in the MSM indeed satisfies such a constraint. In the following we will only consider models whose SBS before gauging respects this custodial symmetry. If we now want to build an effective Lagrangian, these constraints may, at first glance, not seem very restrictive; however, there is just one possible choice of the G and H groups compatible with them all. Indeed, we have seen in Chap. 3 that our NGB take values on the K manifold, where the condition of G invariance is nothing but saying that G is the isometry group of the K metrics. But the maximum number of isometries in a manifold of dimension k is k(k + 1)/2, and thus we find that

k(k: 1) 2 9

= k+h .

(7.6)

7.1 The Effective Lagrangian for the SM Symmetry Breaking Sector

179

Now, using conditions (a) and (d) we can write

k(k;1)=62:9=3+h2:6

=>

g=6,h=3,

(7.7)

that is,

G = SU(2)L H

K

X

SU(2)R

= SU(2)L+R = SU(2)L-R .

Therefore, the K manifold where the NCB live is isomorphic to the S3 sphere (remember that S3 '" SU(2)). Notice that SU(2h-R is an axial group and, as a consequence, the NCB are pseudoscalar particles. Once we know the G and H groups we can apply the techniques described in the preceeding chapters to obtain the effective Lagrangian, to order O(p4), for the 8M 8B8.

7.2 The Effective Lagrangian for the SM Symmetry Breaking Sector We have just seen that in the 8M the NCB are fields taking values in a manifold K = SU(2) ~ S3. In principle, we could choose any coordinates in this manifold, since in Chap. 3 we have shown how to build a Lagrangian in a K covariant formalism. However, as it was the case of the QCD low-energy effective Lagrangian discussed in the previous chapter, the coset manifold is itself a group, which allows us to parametrize this space in terms of group elements U(x) E SU(2). The precise relation between these U(x) fields and the NCB can be defined in many ways and there are different parametrizations in the literature. For instance, some authors collect the NCB in a 2 x 2 SU(2) matrix, as follows

U(x)

= exp CW:T<»

Q

= 1,2,3.

(7.8)

T<> being the Pauli matrices. Another parametrization, which has also appeared very frequently when dealing with SU(2) groups is

U(x) =

VI~ - ;2 + iT<>W<> -v

Q

= 1,2,3.

(7.9)

In this way U(x) is again a 2 x 2 SU(2) matrix and, in fact, this parametrization corresponds to the standard coordinates defined on the S3 sphere (see Chap. 3). Our aim in this section is to define the effective Lagrangian for the 8M 8B8 or, for short, the electroweak chiral Lagrangian (EChL). With that purpose we will follow the steps described in Chap. 3. Thus we start from the two derivative term of the non-linear sigma model (NL8M) based on the coset

180

7. The Standard Model Symmetry Breaking Sector

SU(2)L X SU(2)R/SU(2h+R. Next we have to gauge the SU(2)L x U(l)y subgroup by considering the appropriate covariant derivatives and including the Yang-Mills terms for the gauge fields. Finally, in order to have well defined gauge field propagators we have to include the corresponding gauge fixing and ghost terms. Thus the EChL can be written as = v: tr [DJ.LUtDJ.LU]

£eff

+£R~

+ £FP

+~

tr [WwWJ.LV

+ BJ.LvBJ.LV]

,

(7.10)

where the vector boson fields W: and BJ.L have been parametrized in the following SU(2) matrices. _ -i W a

a

_ -i

3

WJ.L=2

J.LT,

BJ.L=2 BJ.L T

(7.11)

Notice that this notation is slightly different from the conventions in Appendix A. In this context it is usual to make explicit the coupling constants g and g'. (There is also a difference in a sign for BJ.L)' Thus, the covariant derivative and the field strength tensors are defined as

DJ.LU=,oJ.LU - gWJ.LU + g'UBJ.L , WJ.Lv=.oJ.LWV - ovWJ.L - g[WJ.L' Wvl ,

3

_ -i

BJ.LV=2BJ.LVT .

(7.12)

Expanding the first term of this Lagrangian we find the well known 8M mass terms for the gauge bosons

~ v2g2(W~WJ.LI + W;WJ.L2) I I I2 2g'2B + _v BJ.L - _v gg'W 3BJ.L 8 J.L 8 J.L 4 J.L'

+_v 2g2W 3WJ.L3

(7.13)

so that the physical fields are given, as usual, by ±

WJ.L

=

W J.LI +~'w2J.L

J2

AJ.L =sw W;

+ cw BJ.L ,

(7.14)

where cw = cos Ow, Sw = sin Ow and the Weinberg angle is defined by tanO = g' /g. Let us now discuss briefly the most important transformation properties of the Lagrangian in (7.10) at different levels. Our starting point has been, once

7.2 The Effective Lagrangian for the SM Symmetry Breaking Sector

181

more, the Lagrangian for the SU(2)L x SU(2)R/SU(2h+R NLSM, which is invariant under global SU(2)L x SU(2)R transformations like

U'(x) = 9LU(X)9~

(7.15)

with (gL,gR) E SU(2)L x SU(2)R. But later we have included in the NLSM Lagrangian the gauge fields through the covariant derivatives defined in (7.12) and the Yang-Mills terms. The Lagrangian thus obtained is SU(2)L x U(I)y gauge invariant, or in other words, it is invariant under the local transformations

U'(x) = gL(X)U(X)gt(x) 1

W~(x)=gL(x)W/l(x)gl(x) - -gL(x)8/lg1(x) 9

B~(x) =B/l(x) - ~gy(x)8/lgt(x) 9

(7.16)

,

where

a gL(X) = eiOa (X)7 /2 E SU(2)L,

3

gy = ei04 (x)7 /2

E U(I)y. (7.17)

Notice however, that due to the 73 in the exponent, the global SU(2)L x SU(2)R and the custodial SU(2)L+R invariances mentioned above are lost by the introduction of the U(I)y gauge field. Such a breaking will yield corrections to the Ward identities that we would have had in case the global symmetries had been exact. Nevertheless, we can expect that these corrections are small so that they can be calculated perturbatively in terms of g'. That is the reason why the custodial symmetry ensures that p::::: 1 + 0(g'2) But, up to this moment, we have been dealing with the classical Lagrangian. As it had already been noticed in Chap. 3, the quantization of gauge theories involves new terms that have to be added to the classical Lagrangian (see Appendix C.2). In this case, the gauge fixing terms are 3

£GF

= __1_ 2)r)2 _ _1_(14)2 . 2~w

2~B

a=l

(7.18)

As we have seen in Sect. 3.5, the most common choice of fa is a class of gauges known as t'Hooft, Rf, or renormalizable gauges (see Appendix C.2), which have the desirable property of cancelling the w - W mixing term that appears in (7.10). Usually, they have the form

a f a =8/lw/la - gv~w 2 w , i = 1, 2 , 3.

f4=8/lB/l+g'V~B w3 2

.

(7.19)

When dealing with the SM we will then have four gauge fixing functions, one for each electroweak vector boson. The corresponding Faddeev-Popov ghost term is given by

182

7. The Standard Model Symmetry Breaking Sector

(7.20) where the variation of the gauge fixing function is obtained using the gauge transformations in (7.16). Once we have added the gauge fixing and the Faddeev-Popov terms, the gauge 8U(2)L x U(l)y invariance is lost but it is replaced by BRS invariance (see Appendix C.2). As it is well known, this symmetry relates the Green functions by means of the Slavnov-Taylor identities, which somehow encode the original gauge symmetry of the classical action and ensures the gauge invariance of the physical observables. We will detail this discussion later on. In order to obtain physical amplitudes, we need a set of Feynman rules that can be derived from the Lagrangian in (7.10). Those amplitudes will describe the low-energy dynamics of the would-be NGB associated to the global spontaneous symmetry breaking 8U(2)L x 8U(2)R into 8U(2)L+R. Through the Higgs mechanism, that symmetry breaking will provide masses for the electroweak gauge bosons without spoiling the 8U(2)L x U(l)y gauge symmetry. However, it is very important to stress here that the Lagrangian in (7.10) has been defined using only the coset structure and the gauge symmetry. As a consequence, in any symmetry breaking mechanism respecting that precise group structure, the low energy dynamics of the would-be NGB and gauge fields will be described by the Lagrangian in (7.10). Somehow, the predictions of this Lagrangian are universal, and that is why they are known as low-energy theorems [10].

7.3 The O(p4) Lagrangian and One-Loop Renormalization As it was mentioned in Chap. 3 the gauged NLSM is not renormalizable in the usual sense. In particular, the one-loop Green functions derived from (7.10) are divergent and these divergences cannot be absorbed only with the renormalization of the wave functions and the couplings already present in this Lagrangian. For example, the four NGB Green function gets one-loop contributions proportional to the external momenta to the fourth, but there is no similar four NGB vertex in the original Lagrangian. Thus one is forced to introduce new counterterms with four NGB (covariant) derivatives or gauge fields. In this section we will present the full O(p4) Lagrangian and we will see how it is able to absorb all the divergences produced at one-loop from the EChL in (7.10).

7.3.1 The O(p4) Lagrangian In principle, in order to look for possible counterterms one should consider any four dimensional BRS invariant operator, since this is the only symmetry

7.3 The O(p4) Lagrangian and One-Loop Renormalization

183

present in the EChL. However, it can be shown, using dimensional regularization and in the Landau gauge, i.e. in the ~ ----+ 0 limit, that all the new counterterms are SU(2)L x U(l)y gauge invariant and thus they do not depend on ghost fields [11]. Hence, the Landau gauge is very useful since the NGB decouple from the ghosts. However, beyond one-loop there are no well established results and one should rely only on the BRS invariance, as we will do later. Therefore, in order to build the effective Lagrangian, we are just interested in all the independent terms, made out of U(x) matrices, with dimension two or four, which are invariant under the gauge SU(2)L x U(l)y symmetry. We also have to impose that the SBS before gauging has to be G = SU(2)L X SU(2)R invariant. This is due to the fact that the only source of the global SU(2)L x SU(2)R symmetry breaking in the Lagrangian in (7.10) is the introduction of the U(l)y field. A complete set of possible one-loop counterterms for the EChL containing the whole set of SU(2)L x U(l)y gauge invariant and CP-invariant operators up to dimension four is the following

[12J

2

£0 = aog'2 V [Tr (TVJL)J2 4

£2 = a 2

ig'

2

B JLv Tr(T[VJL, VV])

£3 = a3gTr(WJLv[VJL, VV]) £4 = a4 [Tr(VJLVv)J2

£s = as [Tr (VJLvJL)f £6 = a6Tr (VJLVv ) Tr (TVI') Tr (TV V)

£7 = a7Tr (VI' VI') [Tr (TV V)]2

g2 2 £s = aS4" [Tr (TWJLv )J £g = ag ~ Tr (TWJLv ) Tr (T[VJL, VV]) £10 = alO [Tr (TVI') Tr (TVv )J2

£11 = anTr ((D JL VJL)2) £12 = a12Tr (TDJLD vVV) Tr (TVI')

184

7. The Standard Model Symmetry Breaking Sector 1

£13 = a132 [Tr (TDJL Vv)] £14

2

= ga14EJLVpaTr (TVJL ) Tr (TVvW pa ) ,

(7.21)

where we have used the definitions T=UT 3 Ut

(7.22) It is immediate to see that £0 has dimension two, whereas the others are dimension four operators. All the undetermined electroweak chiral parameters ai, which are dimensionless, playa double role: First, they have to absorb the divergences appearing in the one-loop Green functions obtained from the Lagrangian in (7.10). Second, when properly renormalized, these couplings will take different values for different electroweak symmetry breaking models, thus providing a parametrization of the unknown SBS dynamics. Therefore, in contrast to the Lagrangian in (7.10), which is universal, the a;'s are sensible to the precise physical mechanism producing the symmetry breaking. That is, their values are model dependent. However, not all the above operators have to be taken into account. As it was previously mentioned, all the counterterms have to be SU(2)L x SU(2)R invariant in the limit when g' vanishes. This is because we are assuming from the beginning that the pure NGB interactions are described by an SU(2)L x SU(2)R/SU(2)L NLSM, which is by definition SU(2)L x SU(2)R invariant. When we make this model SU(2)L x U(l)y gauge invariant, we saw that the U(l)y interactions explicitly break the SU(2)L x SU(2)R global symmetry. Thus all the counterterms breaking this symmetry should vanish when the U(l)y gauge fields are removed from the model. It can be easily checked that all operators containing the T factor are not SU(2)L x SU(2)R invariant. Therefore, those operators with T factors, which survive when switching off the U(l)y gauge field, should be discarded. Following this argument, we can get rid of £6, £7, £8, £9, £10, £12, £13 and £14. Incidentally, note that this last term violates C and P although it is still CP invariant. In practice, we are only interested in the computation of the on-shell Smatrix elements obtained from the Lagrangian in (7.10) to one-loop, plus the counterterms considered at tree level. Hence, in order to restrict even more the number of terms in the O(p4) Lagrangian, one can also consider the gauge fields equations of motion obtained from (7.10), which are 2

0JLBJLV = -ig' V tr(TV V) + ... 2

4

_ V v +... DJLW JLV -g4V

(7.23 )

With the help of the above equations, it can be shown that £n vanishes on-shell. In conclusion, the only operators to be considered in the O(p4) Lagrangian according to our initial hypothesis are £0, £ 1, £2, £3, £4 and £5'

7.3 The O(p4) Lagrangian and One-Loop Renormalization

185

7.3.2 The Covariant Formalism Let us remember that the Lagrangian in (7.10) and (7.21), which describes the SBS, is nothing but the O(p4) gauged NLSM that we introduced in Sect. 3.2, although it is written for a particular coordinate choice. As we have already commented, it is quite common to use other coordinates in the NGB coset. We have seen that the S-matrix elements do not depend on the parametrization and thus the coordinate choice may seem irrelevant. However, as far as the derivation of the ET proof deals with Green functions, it will be very convenient to work in a covariant formalism. Therefore, following Chap. 3, we rewrite covariantly the Lagrangian in (7.10) and (7.21) in the usual way _ 2 1 g",{3 () W DjJ.w"'DjJ. W{3

.ceff -

+ higher covariant derivative terms +.c~M + .c~F + .c~p +.c~M + .c~F + .c~p ,

(7.24)

where, as we have a different gauge fixing for U(l)y and SU(2)L, we have separated their corresponding terms in the above Lagrangian. For the SM, the covariant derivatives are given by

DjJ.w'" = 8jJ.w'" where

l~(w)

gl~(w)W:

- g'y"'(w)BJ1. ,

(7.25)

and y"'(w) are given by the gauge transformations of the fields

c5w'" = l~(W)fL(X) + Y"'(W)fY(X) c5WJ1.a = ~9 8J1. faL (X) + fabcWJ1.bfCL (X) 1

c5BJ1. = ..,8J1.fY(X) .

(7.26)

9

For instance, for the standard coordinates of (7.9) it is easy to see that

l~(W)=-~(f",a"'(W"'(- c5",aJv 2 - w2 )

Y"'(W)=-~(f"'3"'(w"'( - c5"'3Jv 2 - w2 ) 2

(7.27)

.

At this point we would like to add a technical remark on the gauge fixing function in the covariant formalism. Notice that if we use an with a functional dependence on the NGB fields as in (7.19), then the gauge fixing term in the Lagrangian is no longer invariant under reparametrizations, since the coordinates w a do not transform covariantly. Therefore, in order to keep the covariant formalism, we have to impose that fa depends non-linearly on W as follows:

r

r=8J1.W:

+~w :!",l",a(w)

f4 =8J1. BjJ.

+ ~B 88 y"'(w) , f w'"

(7.28)

186

7. The Standard Model Symmetry Breaking Sector

where, if ro: are the geodesic coordinates (which transform covariantly) in the NGB coset, then f is some arbitrary scalar function of the NGB fields satisfying af = ro:

OwO:

+ O(w 2 ) c::: Wo: + O(w 2 )

(7.29)

.

Notice that now the a = 1 ... 4 index is related to the a 1 ... 3 index through io: a and yO: which, as we saw in Chap. 3, are Killing vectors and thus transform covariantly. This is just a technical point needed to work in the covariant formalism but, for all means and purposes in what follows, we will only be interested in the lowest order in the w expansion of which is the same either for (7.19) or (7.28).

r,

7.3.3 One-Loop Renormalization

As the NLSM in (7.10) is not a renormalizable theory, increasing the number of loops in a calculation implies the appearance of new divergent structures of higher dimension. Nevertheless, it is an effective theory that can be renormalized order by order in the p2 expansion. As a matter of fact, all new divergences generated in a one-loop calculation can be absorbed into redefinitions of the effective operators given in (7.21) thus obtaining finite renormalized Green functions. Such redefinitions not only affect the fields, but also the parameters in (7.21). Indeed, the renormalized quantities can be written in terms of the bare ones as 1/2

Boj./. = ZB lV,a

OJ./.

o: Wo

=

Z1/2 W

Z-1/2 (9 '_ 89, ') B

Bj./.'

wa

j./.'

wO: , -- Z1/2 w

90

Z-1/2 (9 _ 89, ) w

Vo

Zw1/2 ( v - 8v ) ,

~ow = ~w

(1 + 8~w), (7.30)

where the subscript 0 refers to bare couplings an fields. It is important to stress here how we perform any calculation. Note that the renormalized one-loop 1PI Green's functions R of the effective theory can be splitted in three parts

r

r R = rT

+r e +r

L ,

(7.31)

where the superscript R denotes the renormalized function and the superscripts T, C and L denote the tree level, counterterm and loop contributions, respectively. Following the general philosophy of the effective Lagrangians described in Chap. 3, L is calculated only from the Lagrangian in (7.10) and therefore it does not include any ai coupling. Let us illustrate the previous statement by turning off for a moment the Wand B gauge fields. Inside

r

7.3 The O(p4) Lagrangian and One-Loop Renormalization

187

loops, the operators in (7.21) yield O(p6) contributions, where p stands for the external NGB momenta. But in the energy expansion we are keeping terms up to O(p4), so that it is enough to consider loop diagrams from the Lagrangian in (7.10) as well as the tree level from both (7.10) and (7.21). When the gauge fields are turned on, the ordinary derivatives are replaced by covariant derivatives and then, for power counting purposes, the gauge fields count as a momentum and thus we can repeat the above argument. That is, we only make loops with the lowest order Lagrangian. Once we have clarified this issue, let us proceed with the renormalization. Following [13] we will consider an on-shell renormalization scheme which will be appropriate to obtain the relevant observables. This means that throughout the calculation the renormalized Wand Z masses are fixed to their physical values M w and Mz. Within this scheme, the Weinberg angle is defined in terms of these masses as

Mw

Cw == cos Ow = M

. (7.32) z The whole set of on-shell renormalization conditions that define the onshell scheme can be written in terms of the renormalized gauge boson selfenergies as follows: (7.33) The renormalized self-energies can be obtained as usual, just by adding the tree level and one-loop diagrams coming from (7.10) and the tree level contributions from (7.21) (according to our previous discussion we only consider the operators i = 0,1,2,3,4 and 5). The result is E~(q2)=E~(q2)

+ (s~8Zw + C~8ZB) q2 + s~g2(-2aOl)q2,

Ea,(q2) =E{v(q2)

+ 8Zw

(q2 - M~) - 8M~,

E§(q2)=E~(q2)

+ (c~8Zw + S~8ZB) (q2 +2g'2aooM~ + (2s~g2aol) q2,

E~Z(q2)=E~z(q2) + swcw (8Zw

- M~) - 8M~

- 8ZB )q2 - swcw

-(c~ - s~)gg'aOlq2 ,

M~

(8;' _8:) (7.34)

where

8M~=M~ (8Z

w -

8Mz2 =Mz2

(

8Zw

-

8 8V 2: - 2 v

-

8Zw ) ,

2 8g 8v - cw8Zw 2 2) 2cw- 2s 2w -8g' - 2- Sw8ZB , 9 g' v

188

7. The Standard Model Symmetry Breaking Sector

M~ = (9 2

+ 9'2)V 2/4,

(7.35)

and we have written the renormalization constants of the effective theory as Zi == 1 + 8Zi . The Rand L superscripts denote renormalized and one-loop contributions, respectively. From the above equations we have

8M~ 8Mar 2 89 2 89' 2 2s w - + 2cw- + Sw (8Z w - 8ZB). (7.36) M~ Mar 9· 9' Finally, by requiring these renormalized self energies to satisfy the on-shell renormalization conditions in (7.33) and taking into account that the U(I)y Slavnov-Taylor identities on the renormalized self-energies imply 89' = 0 [13]' one gets the following results for the values of the counterterms in terms of the unrenormalized self-energies and bare aOi's -- - -- =

8Mar = 17t\r(Mar ), 8M~=17~(M~)

89 9

+ M~ (29,2 aoo + 2s~92aol)'

-1 17;z(O) - M2Z ' SwCw

89'

-=0 9' , 2

(E~(M~)

_ 17t\r(Mar)) Mar 2a +2laoo + 29 Ol

8Z w= c S2

M~

+ 2cw 17;z(O) _ 17'£(0) Sw

M~

"I

8Z = 17hr(Mar) _ 17~(M~) _ 2 sW 17;z(O) _ 17'£(0) B M2W M2Z cw M2Z "I -29,2 aoo .

(7.37)

Once we have renormalized the gauge boson masses, couplings and wave functions, the only quantities that still need renormalization are the electroweak chiral parameters ai. Their renormalization conditions can be chosen as those of the MS scheme (see Appendix C.1). The divergent parts of the aOi parameters, or equivalently the divergent part of 8ai, which are fixed by the symmetries of the effective theory, are 1 3 1 1 8aoldiv = 161r2 SNf> 8alldiv = 161r 2 12 Nf '

-1

1

8a 4!div = 161r 2 12 Nf '

-1

1

8asldiv = 161r 2 24 N f

•

(7.38)

The other parameters appearing in (7.21), which were discarded in our previous discussion, do not need renormalization. In addition to the above

7.4 The Heavy Higgs and QeD-Like Models

189

divergent parts, other finite contributions could be included in the ai definition, thus yielding other renormalization schemes. However, the above choice seems to be particularly simple. As usual, the renormalized parameters ai become explicitly dependent on the renormalization scale /1. This dependence can be obtained in a straightforward way from the above equations and is given by

ao(/1) = aO(/1o) a2(/1)

1

3

/12

1

1

/12

1

1

/12

+ 161r2 slog /1~'

= a2(/10) + 161r2 24 log /1~' a3(/1) = a3(/10) -

a4(/1) = a4(/10) - 161r2 12 log /1~'

1 1 /12 161r2 24 log /1~' 1 1 161r 24

/12

a5(/1) = a5(/10) - ---log 2

/1~

.

(7.39)

The numerical values of the renormalized couplings at the scale /10 depend on the model we are describing with our EChL. For /10 ~ 1TeV, ai(/10) is typically 0(10- 3 ) (see next section).

7.4 The Heavy Higgs and QeD-Like Models In the previous section we have developed a formalism which is able to reproduce the low-energy dynamics of the electroweak SBS using only very general symmetry arguments. Obviously many different physical dynamics can be accommodated in such a phenomenological description. The information about the precise underlying dynamics is then encoded in the actual value of the chiral couplings ai, so that different symmetry breaking models will lead to different chiral parameters. In order to illustrate this idea we will consider in this section two well-known models which produce the required global symmetry breaking pattern SU(2)L x SU(2)R ~ SU(2)L+R. 7.4.1 The Heavy Higgs Model The first of our examples will be the MSM that we have already introduced in Chap. 5. In this case the SBS is described by a self-interacting complex scalar SU(2)L doublet. The potential is chosen ad hoc in order to reproduce the required symmetry breaking and the corresponding Lagrangian is given by

£/fif = (DIJ-¢»tDIJ-¢>+/1 2q;t¢>->..(q;t¢»2,

(7.40)

where (7.41 )

190

7. The Standard Model Symmetry Breaking Sector

which under the SU(2)L x U(l)y group transforms linearly as

¢(x)

--+

¢'(x) = ei«y(x)+
(7.42)

That is the common notation for the MSM. Nevertheless we are going to introduce a different one which is very useful to understand the MSM in terms of the effective Lagrangian used in the previous section. Indeed, it is very convenient to collect the four fields ¢b ¢2, ¢3 and (7 as follows: first we define J> = iT 2 ¢* and then we form a 2 x 2 matrix with J> and as columns. That is

M = Hence

Y2 (J>,¢) == + iTaWa (7

(Wl,W2,W3)

(7.43)

.

= (-¢2,¢1, -¢3)' Then it is fairly simple to obtain the

M(x) transformation law M(x)

--+

.
M'(x) = e'

.
(7.44)

In summary, we can recast the Lagrangian in (7.40) as follows:

£'tif = ~tr [(D~M)tD~M]

- :2 trMtM - lA6(trMtM)2.

(7.45)

The physically relevant situation is when p,2 > 0, since then the potential has a minimum whenever M satisfies 1

-trMt M

2

p,2

= (72 + w 2 = -A -= v 2

(7.46)

which means that the lowest energy state is not unique and thus there is an spontaneous symmetry breaking. Notice that the above Lagrangian is invariant under --+

a

M'(x) = e'~ M(x)e' .far

M(x)

.fBT 2

a

,

(7.47)

which is nothing but the usual SU(2)L x SU(2)R symmetry that we have been assuming for the general case, throughout the previous section. By construction, we can always write M(x) = p(x)U(x), where p(x) is a real scalar field and U(x) is an SU(2) field. Therefore, the minimum of the potential corresponds to field configurations where p(x) = v, whereas U(x) remains arbitrary, which means that the vacuum states are indeed an SU(2) manifold. Indeed, the U(x) matrix parametrizes the coset space SU(2)L x SU(2)R/SU(2)L+R. We have thus identified in the MSM the general group structure that we used in the formalism of the previous section. For any other state which is not a minimum, we can always write p(x) = H(x)+v. In that way, H(x) represents the perturbations of the system around a minimum. From the Lagrangian in (7.45), it is very easy to show that this field, the so called Higgs boson, has a tree level mass given by M'JI = 2AV 2 . When the self-coupling A goes to infinity the Higgs mass also goes to infinity and we are only left in the SBS with the U(x) field. In this limit we recover the O(p2) EChL in (7.10), where only NGB and gauge bosons appear.

7.4 The Heavy Higgs and QeD-Like Models

191

In order to obtain the terms in the O(p4) EChL [13], we just have to realize that, for finite values of the self-coupling A, the Higgs boson will mediate interactions between the NGB and gauge bosons. By making a derivative expansion of these interactions we can obtain four derivative terms like those in (7.21) and thus it is possible to find the ai values corresponding to the MSM. However, we know that the one-loop corrections coming from the lowest order term are also O(p4) and should also be taken into account. Thus we have to follow a renormalization procedure similar to that used in the previous section (but now also Il, v and A have to be renormalized). Once more, we can use an on-shell scheme with the same renormalization conditions for the self-energies of the gauge bosons, although in this case, we have an extra on-shell renormalization condition for the Higgs boson

Eji(Mk) =

o.

(7.48)

Moreover, we will also require the renormalized Higgs tadpole, i.e. the term in the renormalized Lagrangian linear in the Higgs field, to vanish, and, in addition, that the tree level relations Mk = 2AV 2 and A = (g2 Mk )/(8M'fv) hold both for bare and renormalized quantities. This set of conditions defines completely the renormalization scheme. As far as the MSM is a renormalizable theory, it is possible to calculate any renormalized Green function in terms of the renormalized parameters and one energy scale. Indeed, the relevant two, three and four point Green functions have been calculated to one-loop in the limit of large renormalized Higgs mass, which will also be denoted by MH in the following. By comparing with the one-loop results obtained with the effective Lagrangian in the previous section, one is left with a set of equations where the unknowns are just the bare chiral parameters. This system has a unique solution which can be used together with the definition of the renormalized ai to find [13J aO

(Il) =

16~2 i (~ - log ~ )

(7.49)

Thus, using these definitions we can describe with our effective Lagrangian formalism the dynamics of the MSM, for energies well below the renormalized Higgs mass. That is why this whole approach is most interesting when the Higgs mass is very large and it is mostly used in the context of heavy Higgs physics.

192

7. The Standard Model Symmetry Breaking Sector

As a consistency check of the result, note that the chiral couplings dependence on the renormalization scale J.L (not to be confused with the tachyonic mass parameter appearing in the MSM Lagrangian) is precisely the one obtained in the previous section, which is model independent. In addition, and according to previous discussions, there are no contributions from those operators which break the SU(2)L x SU(2)R symmetry when the U(l)y gauge field is turned off.

7.4.2 QeD-Like Models The other example that will be considered is the simplest technicolor model [7]. We will assume that the SM SBS is nothing but a rescaled version of a two flavor massless QCD-like model with Nrc colors. It is usual then to talk about technicolors instead of colors, techniflavors instead of flavors, technipions, technirhos, etc. As it has been already discussed in Chap. 6, there are many reasons to assume that the dynamics of this system gives rise to an spontaneous symmetry breaking of the chiral symmetry SU(2)L x SU(2)R into the isospin symmetry SU(2)L+R. But that is exactly the very same pattern we are looking for in the electroweak SBS. This analogy makes it possible to identify the would-be NGB with the corresponding technipions and use again the effective Lagrangian formalism described in previous sections. Once more we are interested in the value of the ai chiral parameters that reproduces this model. However, as the dynamics of this system is much more complicated than that of the MSM and perturbation theory breaks down at low energies, it is not possible to perform a complete parameter calculation from the techni-QCD Lagrangian. Therefore, the only thing we can do is to use the large-N limit to obtain estimates of these parameters, as we did in Sect. 6.5, although now N is the number of technicolors Nrc. In this way, and neglecting the contribution of the technicolor condensates, one finds Nrc

al

= - 9611"2 '

Nrc

a3

= 9611"2'

Nrc a --4 - 19211"2'

(7.50)

Nrc as = - 38411"2 ' ai

= 0,

i

= 0,6,7,8,9,10,11,12,13,14.

This result deserves some comments. First we observe that there is no dependence on the renormalization scale, which is due to the very nature of the large N calculation. As we saw in the previous chapter, in this approximation one is using the whole effective action for pions up to four derivatives, but only at tree level. Since there are neither NGB nor gauge boson loops, there

7.4 Phenomenological Determination of the Chiral Parameters

193

is no reason to introduce a J-L dependence in the chiral couplings. However, we know from QCD that this problem is not very relevant phenomenologically, since the dependence on the J-L scale, being only logarithmic, is numerically rather small. Note, for instance, that in the MSM case, the tree level approximation would lead to a5 = v 2 /(8M'k) and ai = 0 for i =1= 5. As it was expected, we have found again that the only nonvanishing chiral parameters are those corresponding to operators which remain SU(2)L x SU(2)R invariant when the U(l)y gauge field is turned off. Finally, we would like to stress that up to now, no technicolor model has received general acceptance. The main problem of these theories is the explanation of the fermionic masses since it is difficult to avoid the appearance of unwanted flavor changing neutral currents. However, the above discussed SU(2)L x SU(2)R technicolor theory is a well defined toy model for the electroweak symmetry breaking, leaving aside the fermion mass problem. It is useful as an academic or reference model to compare with the physical predictions of the MSM.

7.5 Phenomenological Determination of the Chiral Parameters In previous sections we have seen how it is possible to parametrize the most general electroweak symmetry breaking low-energy dynamics. Indeed, given some theory for the symmetry breaking, we can in principle obtain the corresponding values of the chiral parameters, much as we did with the MSM or the QCD-like model. However, we could also try to obtain these parameters directly from experiment. In fact, we can follow here a similar strategy to that used in the Chap. 6 for the description of low-energy hadron dynamics. There, the effective Lagrangian was considered as a link between the fundamental theory (QCD in that case) and experimental data. The main differences (apart from the factor v / F", rv 2500 in the scale) are that the electroweak SBS does not have a well established fundamental theory and, in addition, the available experimental information is very scarce. However the situation is changing very quickly with the new data coming mainly from LEP I at CERN and the Fermilab Tevatron and will improve considerably with the future data both from LEP II and LHC, at CERN too. In this section we will briefly describe how the different parameters appearing in the EChL could be measured or at least bounded experimentally. This information will bring some light on the SBS in a model independent way and will be useful to test which is the most suitable theoretical model when compared with reality.

194

7. The Standard Model Symmetry Breaking Sector

7.5.1 Precision Tests of the Standard Model (Oblique Corrections) Let us now consider a set of observables which are useful to study the SBS effects at relatively low energies. Apart from the standard QED effects, which in principle can be isolated, the relevant ai-dependent contribution to these observables will come through the gauge boson self-energies (the so-called oblique corrections). At the level of accuracy that nowadays we can hope to reach, it is enough to work at one loop. In such case the oblique corrections are mostly dominated by top quark contributions. Once these contributions have been separated, there is still some dependence on the SBS which, as far as we work at one-loop, can be described with the effective Lagrangian formalism discussed in the previous sections [14J. The definition of the observables that we will consider here is the following: • First we define the relation between G F (the Fermi constant obtained from muon decay) and the W mass e2

GF

v'2

1

(7.51 )

8M?vs~ 1 - Llr '

where we are still using the same notation as in previous sections as well as the on-shell renormalization scheme. Thus, in order to measure Llr one has to compare the GF value fitted from the muon lifetime, TJ1., with the treelevel formula (see Chap. 1) written in terms of the on-shell renormalized masses and couplings as

G-2 =

TJ1.m~

1927l"3

(1 _

8m~) m2

[1

+ ~ (25 _ 7l"2)] ,

(7.52)

27l" 4 J1. where we have explicitly extracted the 0(0') QED corrections. Hence Llr contains the radiative corrections to the naive tree level formula in Chap. 1. By that we mean, not only the oblique corrections, but also the vertex and box contributions, which are detailed below and that, at this order of the approximation, do not depend on the ai's. • The second observable was already discussed at the beginning of this chapter and it is the ratio between neutral and charged currents at q2 = 0 F

JNc(O) P = 1 + Llp = Jcc(O) .

(7.53)

In the effective theory both observables are defined in terms of the renormalized self-energies as follows (incidentally, is the same definition as for the MSM [16]) Ll - E~(O)

Eft, (0)

P=M2-~'

z

w

7.5 Phenomenological Determination of the Chiral Parameters

E R (0)

2 ) 2srv w log crv .

(

g2

7 - 4s

..dr == ~fv + 161r2 6 +

..

,

195

(7.54)

J

(vertex+box)

Then, using the on-shell prescriptions given in Sect. 7.3, we can write the observables in terms of the unrenormalized self-energies and bare chiral couplings (aOi) as follows:

..d _ E~(O) _ Etv(O) p- M2

M2 w

z

2sw E;z(O) M2 Cw z

+

+

2'2

9 aoo,

-2g 2(aoo + aOl) + (vertex + box) .

(7.55)

After the explicit loop calculation, the result is

..dp=

1~:2 [~:t

(

-NE

[11 (N

g2 ..dr=--2 161r 12

E

+ log :~ ) + h(Mfv, Mi)] + 2g'2 aoo ,

2 - I oMfv g2- ) +f(Mw,M z2 )] -2g 2 (aOO+aOl) ,(7.56)

J.L

with 2 2 1 2 h(Mw,Mz)=-logcw

(

crv

17 2) --7+2s w 4srv

+17- -5-srv 4

8 crv

5 -1 + -3crv 17) 2- - 4 2 2 Cw sw swcw -srv(3 + 4crv )F(Mi, M w , M w )

2 2 2 f(Mw,M z ) = logcw

(

-2-

2 +I2(cw)

(

sw

2

1

crv 2 + -2-h(CW) + - 12 (43s 2w - 38) +-1

crv ) 1- -2-

sw

2

8c w

+-(154s w - 166cw) + - 2 - +..dO! 18 4cw 2 ) + ( 6 + 7 - 4srv 2 logcw , 2s w where F(Mi, Mw,Mw )

= 1 - 2J(crv)

2 (4

-J(cw) 8cw

[15,16] and

5( 1) 1) + - - - -8- -

2 2 2 Il(CW)=4cw(3+2cw)--

9

5 +24c w

34crv 3

6

1

3

6crv

(7.57)

196

7. The Standard Model Symmetry Breaking Sector

2

83c~

12 (cw) = - 9

7c~ - -7 - -1 + -91 - - 13 - -1- + (4 3cw + -) 9

6c~

_ J(c~) (4C c~

tv

2

4__1_)

12ctv

+ 17c~

3

-

12

24c~

(7.58)

12c~

with 2 ) - (4 2 . 1 )1/2 J( Cw - Cw - 1) 1/2 arctan 2 . (4c W -l 40a Mz Lla=0.0602 + --log G V . (7.59) 9 rr 90 e From these formulae it is possible to write the observables in terms of the renormalized coupling as

Llp

[3

g2 s~ = 16rr 2 "4 c~

log

7Mtv + h(Mw2 , M z2 )]

[11 7Mtv + f(Mw , M z )

g2

Llr = 16rr 2 -12 log

2

2 ]

+ 2g /2 aO (f-L) , 2

- 2g (ao(f-L)

+ a1 (f-L))

. (7.60)

As it should be, after renormalization these two observables are finite and f-L independent, which can be easily checked by using the ai evolution equations. Therefore, a precise measurement of these quantities could (after removing the top quark contribution which has not been included here) provide some information on several chiral parameters, namely ao and a1. In the literature there are other conventions but we will not discuss them here. 7.5.2 The Trilinear Gauge Boson Vertex At LEP II, it will be possible to probe in some detail the nature of the ZW+W- and ,W+W- vertices in the reaction e+ e- - 7 W+W- [17J. Thus, besides checking the expected gauge structure of such processes, it will be possible to get some information about the chiral couplings contributing to those vertices. In the literature it is customary to parametrize an arbitrary ZW+W- or ,W+W- vertex as follows: VNWW = igNWWgi' (WJv WJ.L N V - WJ.LV WJ.Lt N V)

. WtW +zgNWW!\;N J.L v NJ.LV

+ gNWW MAN2

(wtPJ.L WJ.LNVP) v ,

(761) .

W

where NJ.L represents either a photon or a Z boson field, NJ.LV is the corresponding strength field tensor and

gl'ww =-e gZWw=-gCW·

(7.62)

7.5 Phenomenological Determination of the Chiral Parameters

197

It is possible then to relate the above parameters with the chiral coupling just by performing the explicit calculation with the effective chiral Lagrangian and comparing terms. The relations thus obtained are the following:

gJ -

1=0

Z

g2

gl -1=--y- a 3 Ii"'l

Cw -1=g2(a2 - a3 - ad

liZ -1=-g2 a3

+ g/2(a1

- a2)

.\"'1=0

.\z=O.

(7.63)

Hence it seems possible, at least in principle, to get some information on the aI, a2 and a3 chiral parameters at LEP II. In practice, however, the analysis is quite complicated since other constants like ao will also playa role through the two point functions entering in the e+e- ---4 W+W- calculation. In any case, the total number of produced W+W- pairs will obviously determine how strictly the values of the chiral parameters will be bounded at LEP II. 7.5.3 Elastic Gauge Boson Scattering Probably the most direct way to find phenomenological information about the electroweak chiral parameters is through the study of the elastic scattering of the electroweak gauge bosons. In particular, the longitudinal components of these gauge bosons are closely related, through the Higgs mechanism, with the NCB associated to the spontaneous global symmetry breaking of SU(2)L x SU(2)R down to SU(2)L+R. This relation is more evident at large energies and is rigorously stated by the ET, which will be studied in detail in the next section. Therefore, by studying the interactions of longitudinal gauge bosons, we expect to probe the almost unknown NCB dynamics. As we will see next, it seems quite likely that we could determine the values of the most relevant chiral parameters accurately enough to rule out some of the currently proposed symmetry breaking models. The best place to probe those gauge boson interactions will be the LHC, a high luminosity proton-proton collider that will operate at a center of mass energy of about 14 TeV. Among an enormous amount of low transverse momentum (FT ) and high multiplicity events, it will be possible to identify a small fraction of them where a pair of high-FT gauge bosons is produced. Some of these pairs are created directly from quarks inside the colliding protons, but others are produced through the elastic scattering (fusion) of other gauge bosons. As we have just argued, such processes are the ones we are interested in, since they can be used to probe the SBS. A schematic representation of VV ---4 VV fusion processes, where V = W, Z, " and direct gauge

198

7. The Standard Model Symmetry Breaking Sector

a)

b)

Fig. 7.1. (a) Gauge boson fusion process; (b) A typical direct gauge boson production from the proton quark content. The black dots represent the unknown four and three gauge boson vertices

boson production from quarks is shown in Fig. 7.1. The whole description of these reactions involves the calculation of the following cross sections:

+X PP~(Vl V2 ~ V 3 V4 ) + X pp~ (qij ~

V3 V4 )

taking into account all the different polarization states. The reactions between brackets contain the relevant physical phenomena and they are usually referred to as "subprocesses", since they will always be produced inside a bigger process. Customarily, we have denoted by X any other particle that does not take part in these subprocesses. In a first approximation, the above cross section can be written as a(pp ~ (qql ~ V3 V4 )

111 = ~ 11 ' 11

=

~

+ X)

dXldx2dcos(}h(xl, Q2)fj(X2' Q2)

d:~(} (qql ~ V3 V4)

',J

a(pp ~ (V1 V 2 ~ V 3 V4 ) + X)) dXldx2dcos(}h(Xl,Q2)!i(X2,Q2)

',J

fPL dfT dTdTJ M8ry d cos () (Vl V 2 ~ V 3 V4) A

(7.64)

where h(x, Q2) is the i quark (or antiquark) distribution function. As usual, x represents the proton momentum fraction carried by that (valence or see) parton and Q2 is its invariant energy. The luminosity function 8 2 L /8f8ry represents the probability for a gauge boson pair V1 V2 , with some given helicities,

7.5 Phenomenological Determination of the Chiral Parameters

199

to be radiated from a quark pair qiqj. The variables f and f} are defined as Xl,2 = Vfe±iI, where (Xl, X2) are the fractions of quark momenta carried by the two radiated gauge bosons. In a first approach, this luminosity function can be obtained in the so called effective W approximation [18], which is a generalization of the well-known Weizsacker-Williams approximation [19] and where the initial gauge bosons are taken as real. Using this method it is possible to write

2 8 L 8'8' = T

'"

II

1- Xi

(7.65)

!Vi-,-

i=1,2

Xi

for the longitudinal gauge bosons and

2 8 L 8f8'

'"

=

II ! i=1,2

. (Xi)2 + 2(1

V.

2x.

- x)

•

(1og ~) 4M2

2

(7.66)

Vi

for the transverse gauge bosons. Notice that we have used S to denote the subprocess center of squared mass energy. The values of !Vi depend on the particular gauge boson V = W, Z as well as on the type of quark i it comes from Q

!W=--241fs w

Q

[

Q

[

!ZUii = 1611"s&(1 _ s&) !ZdJ= 161fs&(I- s&)

for processes like pp -+ (W'Y Williams approximation

1

2 2]

8 1 + (1 - "3sw)

2 2]

4 1 + (1- "3sw) -+

W Z)

+X

(E

we have the standard Weizsiicker-

2 Q 2 +(I-x)2 ) !qj-y(x) = -2 eq log 2 ' 1f X mq

(7.67)

where eq , m q and E are the quark charge, mass and energy respectively. Thus, with the above formulae it is possible to compute the total number of (longitudinal) gauge boson pairs that will be produced at the LHC in terms of the qij -+ Va V4 and VI V2 -+ Va V4 cross sections. From the practical point of view this will not be so easy, since there are many uncertainties in this calculation, among others: the proton distribution functions, the validity of the effective W approximation, experimental cuts, backgrounds like those coming from the quark loop subprocess 99 -+ qij -+ ZZ [20], or possible failures on the reconstruction of the final bosons. All these problems should be kept in mind, but they are beyond the scope of this book. Our aim is to present the effective Lagrangian approach and with that purpose, we will just concentrate on the calculation of subprocess amplitudes. Indeed, one can try to compute the VI V2 -+ Va V4 amplitudes (and also the qij -+ Va V4 when the quarks are included) by using the effective Lagrangian given to O(p4).

200

7. The Standard Model Symmetry Breaking Sector

For example, the diagrams contributing to the W+W- -+ ZZ scattering can be found in Fig. 7.2, where, in particular, the four leg vertex is given by i9

2

(

-c~ + la3 + :; as) [29JLv9AP -

4 +i9 (a4

+ as)+[9JLA9vp + 9JLP9vA] .

Cw

9JLA9vp - 9JLP9vA]

(7.68)

.

In addition, one should include the one-loop diagrams coming from the two (covariant) derivative term, but those contributions have not been computed yet. A preliminary but detailed study using the tree level result has shown that the chiral couplings a4 and as could be probed at the LHC, but that does not seem to be the case of a3 [21]. Such calculations are extremely involved, since one has to project the amplitudes of the processes into the different polarization states because, as we have seen above, the initial luminosity depends on the gauge boson helicities.

z

w+

z

w+

z

w-x

z z

z

z

/ w-

z

Fig. 7.2. Diagrams contributing to the W+W- -+ ZZ reaction at tree level in the effective Lagrangian approach. The continuous line represents the would-be NGB

Nevertheless, in the next section we will explain in detail the ET, which provides the precise relation between NCB and longitudinal gauge bosons at high energies. Moreover, this theorem greatly simplifies any calculation involving longitudinal gauge bosons, by relating it to a similar one with NCB, which are scalar particles and therefore much easier to handle. Finally, we want to remark that the approach followed in this section has an important drawback. Namely, that the effective Lagrangian formalism is just considering terms up to four (covariant) derivatives. Thus, the results can only be applied in the low-energy regime, i.e., where p2/(47rv)2 can be considered small. Such a constraint will always be present, even if we were taking into account higher terms in the chiral expansion. In practice that

7.6 The Equivalence Theorem

201

means that the amplitudes computed with the electroweak chiral effective Lagrangian should not be trusted beyond 1 or 1.5 TeV. Note that the effective W approximation is a high energy one. Moreover, the particular form of the amplitudes does not make it possible to reproduce any resonant behavior, which is one of the most characteristic features of an strongly interacting scenario. We have already met a similar limitation when dealing with chiral perturbation theory (ChPT) in Chap. 6, where we saw that the natural solution is unitarization. In Sect. 7.8 we will also address such techniques within the electroweak effective Lagrangian approach.

7.6 The Equivalence Theorem 7.6.1 Introduction

In this section we will give a proof of the ET which is valid in the framework of the effective Lagrangian formalism. At first, the derivation of the ET was performed within the MSM, which presents two differences with respect to the most general effective Lagrangian approach: first, all the fields appear in linear representations of the symmetry groups and thus they transform linearly under gauge transformations; second, the MSM is a renormalizable theory and the amplitudes of the different processes respect the unitarity constraints, in particular, they do not grow with the energy. Intuitively, NGB and gauge bosons are related via the Higgs mechanism, since the NGB become the gauge bosons longitudinal components by means of a gauge transformation. Indeed, the gauge fixing functions on (7.19), in momentum space, are somehow telling us that pl-'W: rv w a . But a longitudinal polarization vector is cl{ rv pI-' at high energies, so that we find w a rv cl{ And, naively, that is the ET [22]. But that was a purely classical reasoning, only valid at the Lagrangian level. In order to formulate rigorously the ET, within quantum field theory, we have to use Green functions and S-matrix elements. Indeed, we have already seen how a gauge symmetry in the Lagrangian yields a set of relations between Green functions, which are known as Ward identities. Therefore we are looking for a general Ward identity that relates NGB Green functions with those containing longitudinal gauge bosons. Later we will have to translate these identities into relations between physical amplitudes and, in so doing, we will have to keep in mind that the amplitudes, in the effective Lagrangian formalism, are obtained as an expansion in powers of momenta. However, there are some technical issues that complicate the derivation of the theorem.

W:.

• First, the quantization of gauge fields requires the introduction of a gauge fixing and a Faddeev-Popov ghost term in the Lagrangian (see Appendix C.2 for details). Therefore, the quantum Lagrangian of a gauge theory

202

7. The Standard Model Symmetry Breaking Sector

is no longer gauge invariant, although it can be shown that the physical observables are still gauge independent. This fact is due to the existence of a new quantum symmetry, known as the BRS symmetry [23], which provides us with the necessary Ward identities that ensure gauge invariance and that in this context are called Slavnov-Taylor identities. Hence we have to use these Slavnov-Taylor identities to find the relations between the Green functions of gauge bosons and those containing NGB, that will lead us to the ET [24] . • Second, one has to take into account renormalization effects which yield small corrections to the theorem [25] . • Finally, within the effective Lagrangian approach, the high energy limit is not well defined, since the amplitudes are obtained as expansions in momenta. Only very recently it has appeared a rigorous formulation of the ET taking into account the chiral expansion [26] The plan of the ET proof can be followed in Fig. 7.3. Starting from the gauge invariant classical Lagrangian, we obtain the quantum version which is invariant under a set of BRS transformations. This Lagrangian has to be renormalized in order to derive Slavnov-Taylor identities between renormalized Green functions. Notice that we have already accomplished part of this program in previous sections. The next step is to use the Lehman-SymanzikZimmermann reduction Formula to obtain S-matrix elements, which are the physical observables. The result is known as generalized equivalence theorem (GET) from which we will derive the ET itself, that relates the S-matrix elements of longitudinal gauge bosons only with those S-matrix elements where every longitudinal vector has been replaced by its associated NGB. In the rest of the section, we are going to derive the ET following this plan, but it has been written so that it is not necessary to go through the following two subsections, which are rather technical. Indeed, if the reader is just interested in the final statement of the ET and why the effective Lagrangian formalism is a special case, it is possible to skip from here to Sect. 7.6.4 and the exposition is still self-contained.

7.6.2 The Slavnov-Taylor Identities Unified SU(2)L and U(I)y Notation Throughout the following subsections we will be using the covariant formalism, but before going any further, we will define a more condensed notation in order to deal at the same time with both gauge groups, SU(2)L and U(I)y. Thus we introduce the Killing vector L~(w) with a = 1,2,3,4 as L~(w) = gl~(w) for a = 1,2,3 and L'4(w) = g'yO:(w), where l~ and yO:(w) were given in (7.27). We also redefine the completely antisymmetric symbols fabe with a = 1,2,3,4, as fa be = g€abe for a = 1,2,3 and fab4 = O. Finally the gauge field with a = 1,2,3,4 will be defined as = for a = 1,2,3

V:

V: W:

7.6 The Equivalence Theorem

203

Gauge Invariance

Quantum Lagrangian BRS Invariance

Renormalization Slavnov-Taylor Identities Reduction Formula

Generalized Equivalence Theorem

THE EQUIVALENCE THEOREM

Fig. 7.3. The proof of the equivalence theorem

and V: = B w Using this notation, the two covariant derivative term in (7.24) can be written as

'12 9o.{3 (w) Dp.wo. DP.w{3 =

1

'2 9o.{3(w)8p.wo.8P.w{3 - 9o.{3(w)8p.wo. L~(w)vp.a 1 +'29o.{3(w)L~(w)L~(w)vp.av: .

(7.69)

Hence, the gauge boson mass matrix is just the zeroth order term of 9o.{3L~L~ when expanded in powers of the NCB fields. With this notation our quantum Lagrangian is also simplified, namely

Leff= ~9o.{3(W)Dp.Wo. DP.w{3 + LYM + LCp + Lpp +higher covariant derivative terms .

(7.70)

As we know, it is completely determined Once we specify the gauge-fixing functions. We will use those in (7.28), which now can be written in a much more compact way

(7.71)

204

7. The Standard Model Symmetry Breaking Sector

(Notice that we are using one single gauge parameter, but that will not be true after renormalization.) Renormalized BRS Transformations We have just obtained a quantum Lagrangian for our model, which is the second step in Fig. 7.3. As it was mentioned before, the Lagrangian in (7.70) is no longer gauge invariant. Nevertheless it is invariant under a set of BRS transformations, which are the quantum analog of gauge symmetry and ensure the gauge invariance of the physical observables. In Appendix C.2 we have listed the complete set of tree level BRS and anti-BRS transformations. In the functional formalism, it is usual to introduce an auxiliary Fa field, which, up to some normalization, can be identified with the gauge fixing function (see Appendix C.2). In that way, we can easily obtain Slavnov-Taylor identities containing the gauge-fixing function by taking functional derivatives with respect to Fa. But the BRS transformations in Appendix C.2 are still not enough for our purposes, since we have already explained that in order to make physical predictions, we have to renormalize the quantities in our theory, as we did in Sect. 7.3. Let us write the renormalization relations of (7.30) using the unified SU(2)£ and U(I)y notation, together with those needed for the ghost fields Vo~(x) = zja)1/2 V:(x) w(j(x) = Z~"')1/2w"'(x)

Fo(x) = Z~a) Fa(x) cg(x) = Z~a)1/2ca(x)

g~a) = Z~a)g(a)

Vo = Z~/2V ~~a) = zja\(a) cg(x) = Z~a)1/2ca(x) ,

(7.72)

where g(a) = 9 for a = 1,2,3 and g4 = g' and a parenthesis in the indices means that they are not summed. The first three Z3 are the same due to the gauge invariance of the classical Lagrangian. As a remark, notice that again we have more than a single gauge-fixing parameter. Once we have our Lagrangian written in terms of renormalized quantities, we can also define a renormalized set of BRS and anti-BRS transformations as follows: SR[W"'] = x(a)LRa(w)ca

SR[vp.aj = x(a) D~~cc (a)

X SR[Caj = --2-!'R bcc

SR[caj

b

CC

= x(a)~

sR[Fa] = 0

~

(7.73)

so that the Lagrangian in (7.70) is invariant under 'TJSR and ijsR, 'TJ being an arbitrary anticommuting parameter. In the above transformations we have also used the following renormalized objects

L'"Ra (w) = Z(e»-1/2 Z(a)1/2 L'"a(w) w 3

7.6 The Equivalence Theorem

205

- zeal Z(a)I/2 fa f Ra be9 3 be x(a)=Z~a)I/2/Z~a).

(7.74)

With these definitions, we are now ready to use the functional formalism to obtain the desired relations between renormalized Green functions.

The Slavnov-Taylor Identities In the next two subsections we will make an intensive use of the methods developed in Chap. 2. Let us first remember that the generating functional for connected Green functions is W R[J]

=

J

=

L

[dA]e i J d

n=l

4

x{LQR(A(x»+A i (x)J;{x)}

J

d4Xld4X2 ... d4x n GR il,···,i"(Xl, ... , xn)Jil (Xl) ... J i" (X n )

= (21r)4 L

n=l

Xb(4)

JIT ~~i4 i=l (

)

(L Pi)Ji, (-pd· .. Ji" (-Pn)G R

il, ... ,i"

(PI, ... Pn).

(7.75)

Notice that from now on we will be using the Green functions in momentum space. The above equation can be understood as a functional series expansion in the J i currents. To simplify our notation we have denoted the complete renormalized quantum Lagrangian as LQR and A stands for any field; that is, Ai = wO'., V;:, ca, ca, Fa. If we now perform the change of variables A' (x) = A(x)+1JsR[A(x)] the generating functional should not be affected; then, using the BRS invariance of the Lagrangian and the measure (we are supposing that there are no anomalies and thus the gauge theory is consistent) we are left with

L Jd4x < sR[Ai] >J Ji(x) == I[J] = O.

(7.76)

t

In view of the structure of BRS transformations, they can be expanded in sums of products of fields as follows

< sR[Ai ] > J

L S:J;··i" < Ail'" A in > = L S:J:··i b(nlWR[J] . , bJ ... bJ

=

J

n

n

n

il

(7.77)

i"

Looking at the BRS transformations, (7.73), we see that the above expansion is just a linear contribution in most cases. However, when Ai = VIL, ca there are bilinear terms, and when Ai = wO'. has an infinite number of terms in the effective Lagrangian formalism. That is due to the non-linear character of the

206

7. The Standard Model Symmetry Breaking Sector

symmetry realization. On the contrary, in the MSM the BRS transformation of w is linear in the fields and therefore there is one single term. Thus (7.76) can be written as

I[J] = " " Sil ... i ~ ~ Ai •

X

n

J

4

4

4

d qd k 1 '" d k n _ 1

n

(27r)4n

b(n)WR[J] J i ( -q) bJil (q - kd··· bJin (kn-d

= O.

(7.78)

Just by taking functional derivatives with respect to the currents Ji and then setting J = 0, we can generate relations between Green functions. We have already seen that we are interested in those identities involving F fields, since they are nothing but the gauge-fixing functions that relate the NGB and gauge bosons. As an illustration we will now derive the Slavnov-Taylor identity for one F and any other field A j . Later we will use this result when applying the reduction formula. Example: Two-Point Green Functions with an F Field. Let us then start with: (7.79)

b b I[J]I =0. bJCb (-k) bJj(p) J=O

Notice that we have taken one J c derivative since the F field only appears in one BRS transformation, which is precisely s[c]. By looking at the BRS transformations in (7.73), and using (7.78), it can be seen that the only possible contributions come either from A j = VI' or A j = wOo, due to the fact that a c field should always be accompanied by a c field. Therefore we get X (a) (0.(0) L Ra

+ ..:1 0.2a ( p2)) Gcacb () p +

x(a)ipl' (1

+ ..:1 3 (p2)) Gcacb(p) +

X(b)

I77h)G p bw ( P) =0

V ~(b)

o<

X(b)

I77h)G p bv a(p) = 0,

V~(b)

(7.80)

"

where we have made the following expansion of L'Ra(w) in powers of w

L'Ra(w) = L~~)

+ L';t.(l)W(3 + L';t.'Y(2)w(3w'Y +... .

(7.81)

Remember that L ~~O) is the same either in the effective Lagrangian approach or in the MSM. It will be the only contributing part. In (7.80) the ..:1 3 term is defined as

ipl'..:13(p2)

= rRdcG;a~b(P)

J(~:~4

GV,fCCcb(p - q, q,p) .

(7.82)

7.6 The Equivalence Theorem

207

In the effective Lagrangian formalism, due to the non-linear character of the s[w]' the ..::1 2 contribution has a complicated expression that we can expand as

Q( 2) (1)Q13 -1 ( )1 (27r)4 d q G 13 c b( ) ..::1 2a P = L Ra Gcceb P w c e P - q,q,p 4

+ ... ,

(7.83)

where the dots stand for similar terms with more w fields. In case we were dealing with the MSM, the BRS transformation of w would be linear and thus the terms that we have represented by the dots would not be present; instead there would be a contribution from the Higgs field, so that we would have to replace

..::12'a(p2) ---. ..::12'a(p2)

+ ..::1 1ab (p2)

= G~C~b(P)

X

(7.84)

(1)Q131 d q ( ) H 1 d q ( )\ (L Ra (27r)4Gwl3cCebp-q,q,p +L(a) (27r)4GHcaecp-q,q,P'), 4

4

where Lfa) = g/2 for a =1,2,3 and L~) = g' /2. It is important to notice that all these ..::1 terms are 0(g2), since the ccV, cCw and ccH vertices carry at least one power of 9 or g'. Therefore they will not contribute at lowest order when expanding in the electroweak coupling. At this point, and in order to simplify the notation, we are going to introduce the following operator:

D'lu(p) =iPIL (1 +..::13(p2))by"a

+ (L~lQ+..::12'a(p2))b(

,

(7.85)

thus, we can write (7.80) as X(b)

Cih\GFbl(P) = -x(a)DR1(p)Gcaeb(P) ,

(7.86)

V~(b)

which relates the ghost propagator with the two-point Green function containing one F field. Note that it vanishes unless F is connected with a VILa or a wQ • As we have already commented, later on we will make use of this result when applying the reduction formula, in order to translate Green functions in S-matrix elements.

General Case Let us go back to our main objective which is to relate, through the gaugefixing functions, any number of longitudinal gauge bosons to their corresponding NGB. That is, we are interested in Green functions with an arbitrary number of F fields. We can obtain such Slavnov-Taylor identities from

o rrs 0 rrm bJeal (-k).J=2 01F:. (-k·) aJ J k=l

0

ojA (-Pk) k

I[J]

= O.

J=O

(7.87)

208

7. The Standard Model Symmetry Breaking Sector

In order to consider the most general physical processes with longitudinal gauge bosons, we have also taken some J A derivatives associated to spectator physical fields. In the model we are working with, these physical fields can only be gauge bosons, either longitudinal or transversely polarized. From the BRS transformations it is easy to see that the possible contributions coming from s[F], s[w] and s[c] will vanish when setting J = 0. Even more, as the A k fields are physical gauge bosons, their polarization vectors satisfy E:. k = 0, and thus the first contribution in SR[V:] = ik/.Lca + ERbc V/.LbCc will also cancel. In brief, we should only take into account those terms coming from SR[C] or S R [V:]. From now on we will call the latter "bilinear terms" , since we have seen that the linear part in SR[V:] vanishes. Thus we are left with x(all J~(all GFu,Fu2 ..·FusA, ... Am(kt, ... , ks,Pl, .. ·Pm)

+ bilinear terms =

°,

(7.88)

where Li k i = - LiPi. This is the Slavnov-Taylor identity that we needed to establish the ET and thus we have completed the fifth step in Fig. 7.3. Now we still have to translate this result to S-matrix elements, that is, to physical observables. As a last remark, remember that from the moment we started building the effective Lagrangian, we had decided not to consider fermions. Nevertheless, if we wanted to include them, by looking at their BRS transformations (that can be found in Appendix C.2), it can be seen that they would only contribute to the bilinear terms.

7.6.3 The Reduction Formula Following Chap. 2, the S-matrix elements and the Green functions can be obtained from one another by multiplying by the inverse two-point connected Green function of each field and then setting the external momenta on-shell. That is

(ft

G"A:A;(PI)) GA, ... AN(Pl .. ·PN)

1=1

==

-

(

SA~ .. A:"" Pl·· ·PN

) on-shell ---+

SA~ .. A:",,(Pl .. ·PN) ,

(7.89)

where, for further convenience, we have denoted by S the amputated Green functions, which on-shell become the S-matrix elements. Note also that there is a sum over each Af field. In the most usual cases the propagators only connect one field with itself, which means G Ai A; = 0 if Al f= Af, and then we obtain the formula given in Sect. 2.2. However, that is not the case in our model, since we have already seen that there are non-vanishing two point

7.6 The Equivalence Theorem

209

Green functions GFal, with one F field and l = V or w. Hence, if we apply the reduction formalism to the Slavnov-Taylor identity in (7.88), we get

;(;~:)

(IT

L

(IT

~

X

GAiAi(ki ))

1=1

Ij

G FUj Ij (Pj)) Sll ..lsAl ... Am (PI . .. Ps, k 1 ... km )

j=1

+bilinear terms = 0 ,

(7.90)

so we can get rid of the X / Jf, factor since the al index is not contracted. Now, in order to extract the S-matrix elements, we have to multiply by the inverse propagators, and then set all their k i momenta on-shell. Thus, we multiply the last equation by the A fields inverse propagators, which are physical and therefore diagonal. Let us now concentrate on the "bilinear terms". They will typically have the following form:

J

d4q

a

(2n)4 fR bcGCal ,Fa2 ... Fa•V;A2 ... Amcc(Pl,PZ, ... ,Ps, k1 - q, kz ... km , q) ,

(7.91 )

kr

where we have taken Al = V;:. But when we set --+ m~i' the V;: field remains off-shell, and therefore there will not be a pole to compensate for GA~Al (kd --+ 0 when kr --+ k~l· The same happens for any other bilinear term, so that all them vanish. Thus we are left with "L......t Ij

J~(aj)

s

(

.

II--X(Cj)Gc-a(p·)DcJ x(aj) cJcJ J Rl j (p.) J

)

j=1

= 0,

XS11 .. I,Al ... Am(Pl ... Ps,k 1 ... km)1

(7.92)

k~=m~i

where we have included in Ai their polarization vectors, and we have made use of (7.86) which is the Slavnov-Taylor identity for the F field two-point Green function. Again we can get rid of Jf,/ X and, multiplying by G-} _a. (k ·), we J c Jc J can also eliminate the last X factor, thus arriving to

L

[I

1] ... IT 1-1

D';ili(Pi) Sl l ..I"Al .. Am(Pl ..Ps,k 1 .. km )1

=

o.

(7.93)

k2i -m2 Ai

Note that the F fields have disappeared, but instead we have the D'ju(p) operators which relate the V and NGB. We cannot set the V momenta on shell since they are not the physical gauge fields. Hence, we have to write the last equation in terms of W;, Zo and the electromagnetic field AIL" This can be achieved by using the transformation V;: = RabV:, which is defined as follows:

210

7. The Standard Model Symmetry Breaking Sector

( ~) (~") =

-4 VJ.L

Aphys

J.L

=

(

1/..;2 i/..;2 1/..;2 -i/..;2

o o

0 0

o o cosO sinO'

oo )

- sinO cosO'

J.L

VJ.L2 (VI)

VJ'

(7.94)

V:

Thus we redefine L~~b = L~~a(R-I)ba as well as the other quantities as ..1~a' etc. Only now we are allowed to set = M;phyS and we obtain

PI

s

L IT D'fidpi)SI, ..I.,A, .. A",(PI ..Ps,kl ..km ) = 0,

(7.95)

l, .. .lr i=1

where now

DRI(P) = ipJ.L

(1 + ..13(M;hys) ) b~J:a + (L~lo. + ..1~a(M;hYs)) b( . (7.96)

The important point is that the DRI (p) operator relates the contraction of each external VJ.L with its own momentum and its corresponding NGB. As we will only be interested in the momenta, we can group all the other factors in the following constants K o.

-

Ra -

£CO)o.

+ ..10.

(M(a)2)

(1 +..13 phys

Ra M(a)

2a

phys phys

(M(a)2)) ,

(7 97) .

which do not depend on the external momenta and that in the most usual renormalization schemes are K ::::::: 1 + "loop corrections". From now on and in order to follow the most usual phenomenological notation, we will work with scattering amplitudes T instead of S-matrix elements, which are related by (7.98)

However, up to this level our gauge fields do not have any definite polarization. Our next step will be to use the PJ.L momentum factors to obtain longitudinally polarized gauge bosons. 7.6.4 The Generalized Equivalence Theorem Let us summarize what have we done up to this point, for those readers who have not gone through the technical details. We have already obtained the Slavnov-Taylor identities relating renormalized Green functions containing gauge bosons with those containing NGB. The reduction formula has allowed us to translate these relations into equations containing amplitudes. We want

7.6 The Equivalence Theorem

211

V;,

to notice that the gauge bosons, that we ~re denoting by always appear contrac~ed with their own momenta as pIJ.V;, because they are obtained from the 8IJ.V; term in the gauge-fixing function. Nevertheless, the longitudinal polarization vector satisfies

cl{

= ~ + vIJ.

(7.99)

.

Mphys

Due to the fact that the v term is O(MphYs/ E), we would then expect to replace, at high energies, each momentum factor by a longitudinal polarization vector and therefore relate the NGB only with longitudinal gauge bosons. Unfortunately this step is not straightforward, because replacing pJ1. = Mphys(cl{ - vIJ.) in (7.95) yields many "crossed" terms, whose highenergy behavior is not obvious at all, since they contain longitudinal polarization vectors as well as v factors. Indeed, in order to extract the leading order at high energies, it is very convenient to derive from (7.95) a relation where the v factors do not appear mixed with longitudinal polarization vectors, or in other words, where the negative energy powers are not mixed with the positive ones. We will not give the details here [24, 26], but such an expression can be obtained after some algebra, taking into account all the possible permutations of similar indices and summing over them. The result is

T(Vt" ... ,vt

n

;

A)

_~(_')l(rrl Kaj)

- L...J 1=0

t

O<j

j=l

(rr

(7.100)

n

V

J.ti)T-(

WO
WO
V-a,+1 IJ.l+l

•••

V-an-A) IJ.n' ,

i=l+l

which is known as the generalized equivalence theorem (GET), and is valid either for an effective Lagrangian or for the MSM. Notice that on the right hand side we have a sum over all.!;he amplitudes T where we have replaced one, two, ... up to n longitudinal V bosons by their corresponding NGB; the bar over the amplitudes means that we are considering all the permutations of the indices. Observe that the only energy dependent factors that multiply the amplitudes are the vIJ.' one per each V which has not been replaced by its NGB. We only have one step left to obtain the ET, which is to extract from the above equation the leading order at high energies. The explicit expression of the K constants can be found in (7.97), but for our main purposes it is enough to know that in most of the usual renormalization schemes they are K ~ 1 + "loop corrections" . 7.6.5 The Equivalence Theorem There is just one final step before obtaining the ET and it is precisely here where the lack of strict unitarity in the effective Lagrangian approach invalidates the usual derivation. Thus we are going to present two formulations: one for well-behaved models and another one for effective Lagrangians.

212

7. The Standard Model Symmetry Breaking Sector

Models Respecting U nitarity Let us suppose that we are working with a model whose amplitudes satisfy the unitarity constraints and thus we know that they cannot grow with the energy. In such case, it is very easy to obtain the leading term from the GET, since we only have to keep in its right hand side the term without any vI-' factor. The result is

T(Vt',···, Vt'; A) '"

(n K~;)

T(w o ,

.. W o .;

A)

+0

(~),

(7.101)

which is the most common version of the ET. It relates, at energies E » M, any amplitude involving longitudinal gauge bosons VL, with the amplitude where they have been replaced by the corresponding NGB. As we have already commented, this approach considerably simplifies the calculations, and most of the phenomenological work on the strongly interacting SBS has been carried out using the ET. It is important to remark that we can calculate the right hand side with the NGB momenta on-shell; that is, = 0, which is called the chiral limit. This is due to the fact that although in the proof = Mfai)' the difference between this on-shell condition and the we had set chirallimit is O(MjE).

p;

p;

Example: In order to clarify how this theorem works, let us illustrate it with the process wtwi -> ZLZL at tree level and within the MSM. This reaction is simple enough so that we do not have to fill pages with formulae, but it displays some relevant features of the ET. Thus, to obtain the left hand side of the theorem, we have to calculate, at tree level, the diagrams in Fig. 7.4. (We are also setting 9' = 0 for simplicity). Notice that, even though most diagrams in Fig. 7.2 and 7.4 may look the same, their Feynman rules are different. After a lengthy but straightforward calculation using the Feynman rules coming from the Lagrangian in Sect. 7.4 (plus the standard gauge fixing and Faddeev-Popov terms) and projecting out the longitudinal components, we arrive to the following result:

(7.102)

7.6 The Equivalence Theorem

213

w~z

w~z

Fig. 7.4. Diagrams contributing to the W+W- -> ZZ reaction at tree level in the MSM with g' = O. The continuous and dashed lines represent the NCB and the Higgs respectively where E is the gauge boson energy and x = cos (), with () being the angle between the initial W+ and one of the final Z. The right hand side of the ET is obtained, at tree level, from the amplitude for the process w+w- -> zz (see Fig. 7.5). These NGB states are defined as z = w 3 and w± = (wI =f w 2 )/"fi. The result is given by T(w+w-

--+

zz) =

4E 2 4

E2

1

M2

-

H

4(1 - x 2 )

M2

x [ -4-.lf(1 - x2) 2

v

+12

2

2

M2

+ 2g 2 (3 + x 2 ) 2

4

v't ME"i - 4 MJ! ME"":

M

M4

+ 4 Ef + Ef

]

.

M 2 M2

2-.lf--!£.(5 2

v

E2

+ x2)

(7.103)

As expected, it can be easily checked that these two amplitudes satisfy (7.101), because we are working with the MSM. In addition, as far as we are working at tree level, the K factors are trivial. We can also observe that at low-energies both amplitudes behave as s/v 2 as it has been argued at the beginning of this section For the discussion that follows, it will be important to notice that we are allowed to expand the expression of the ET in powers of g, A or even n (loops) and then both sides of the theorem should be the same order by order. Indeed, by choosing to work at tree level we are making an expansion in n and keeping the lowest order. But that does not mean that we are working at the lowest order in 9 or A. As a matter of fact, our expressions include orders of 9 which are not just the lowest. When dealing with external NGB, those 9 contributions come only from the diagrams in Fig. 7.5 with intermediate gauge bosons, whereas on the left hand side of the theorem such terms can also be found in other diagrams. The important fact is that (7.101) implies

214

7. The Standard Model Symmetry Breaking Sector

~><: ~>- -<' w~' ~I: z

w+

z

w

z

Fig. 7.5. Diagrams contributing to the w+wMSM with g' = 0

-+

z

w

zz reaction at tree level in the

that, when working at a given loop order, both sides are the same, up to That will not be the case when dealing with chiral Lagrangians, where both sides will be the same only at lowest order in the electroweak couplings.

O(MjE).

The ET and Effective Lagrangians As we have already stressed several times, the effective Lagrangian formalism is, by construction, a low-energy limit. Hence, it is not clear that it can be used together with the ET which, as we have just seen, is a high-energy limit. We will probably need some compatibility conditions. Moreover, the amplitudes are obtained from the effective Lagrangians as a power expansion, up to order N in the external momenta and they do not satisfy the unitarity constraints. Therefore, we have to perform a more subtle power counting to extract the leading order from the GET and, in particular, we have to take into account that the amplitudes are obtained as formal series in the energy. That is _ _

T(V~l

_

...

V~L, waL+I

... wan; A) c:::

I: a7 (E)k + I: a1k (M)k E N

<Xl

41rV

k=O

.

k=l

(7.104) It may be surprising that in the above expansion we had also considered negative powers of the energy, since up to now our effective Lagrangian was built as a derivative expansion and the energy only appeared in positive powers or logarithms. However, in the gauged model we also have to evaluate internal lines of gauge bosons, whose propagators are'" 1j(q2 - M 2). The high-energy expansion of the integrals where these propagators appear, yields the negative powers of the energy. These series are formal in the sense that the coefficients can also contain a logarithmic dependence in the energy, which

7.6 The Equivalence Theorem

215

has not been explicitly shown as it does not affect the general arguments in what follows. Notice that in the above expression we have momentarily set g' = 0; later on we will consider the realistic case. Now we are ready to solve the apparent contradiction of using the ET and the effective Lagrangian formalism at the same time. The ET is a high-energy limit, but here "high" means E» M, whereas the effective Lagrangian is a low energy approximation, where now "low" means E « 471"v. The relation between both reference scales is given by the factor g/471", which is the weak coupling, that allows us to expand perturbatively the coefficients as

+ a~L+l) :71" + ..... = a?z, K~':::::K~(O) + K~(1)(g/471") + .... , a? = a?L

[1 + 0 (:71" )]

(7.105)

where aL is the lowest order term in the 9 expansion of a. Finally, introducing all these approximations in the GET we are left with

(IT

C(L)J1.i)

T(iI,i,l, ... , Vt"n; A)

"(11 K~;(O») to t=1

+0 (

M E -

(a,\£il+ 0(9/ 4K)j)

supressed

)

+0

( E

(4~S

)N+1

471"v'

(7.106)

which is the precise formulation of the ET in the effective Lagrangian formalism. It is straightforward now to consider g' =f. 0, since g' < 9 and thus M~hYS '::::: M~Ys '::::: M(a) for any a. We just have to consider g' and 9 of the same order and then all the expressions remain valid. To summarize, within the effective Lagrangian approach, we can write the ET symbolically as

T(VZt,···, VZn; A) ':::::

(IT K~~(O)) TI: (WC>l .. .WC>n; A) J=1

+0

) + O(g or g' - supressed) + 0 (471"v E)N+1' (7.107) - supressed (EM

where TI: is the lowest order in the electroweak couplings (g and g') of the amplitude obtained from the effective Lagrangian up to order O(pN). Therefore, (7.107) will allow us to translate directly any NGB amplitude into another one with longitudinal gauge bosons and viceversa. On the one hand, we have thus established a way to link the physical observables with the hidden SBS; on the other hand, we now have a powerful tool to obtain longitudinal gauge boson amplitudes (which are extremely hard to calculate), from NGB amplitudes, which are much easier to write. The ET is somehow

216

7. The Standard Model Symmetry Breaking Sector

telling us that both them are the same. To what extent such an approximation is valid will be the topic of the next section.

7.7 The Applicability of the Equivalence Theorem Up to this point we have obtained the mathematical expression of the ET, which provides the high-energy leading contribution of an amplitude involving longitudinal gauge bosons. Now we are going to analyze whether it is a good physical approximation to neglect the other terms which are not given by the ET, as well as possible methods to improve the general applicability conditions. Energy Applicability Range As we have seen there are two cases, depending on the unitarity behavior of the amplitudes • First we will concentrate in the MSM. In that case the amplitudes respect unitarity, and therefore we can use the simplest version of the ET, (7.101). Hence, in order to neglect higher order terms we just have to ask for E»

M.

• When dealing with an effective Lagrangian the applicability conditions are rather different. The amplitudes are obtained from this formalism as power series in the energy, and thus they do not satisfy the unitarity conditions. Therefore we have to use the ET restricted version of (7.107). The general applicability conditions are now M

«

E

«

(7.108)

41rV .

The first one is the same as before and is due to the approximation p/M. The second inequality is typical from the effective Lagrangian approach. Finally, we have to remember that in order to keep the leading term in the right hand side of (7.107) we have to neglect the M / E-supressed and the 9 or g'-supressed contributions while keeping those of O(E /41rv)N. Thus if we want to have a consistent approximation we should ask for

cL ~

o (~ - supressed)

and O(g or g' - supressed)

«

(4~V) N

(7.109)

Note that this last applicability condition becomes more strict when considering higher energy powers N in the effective Lagrangian expansion. This fact deserves a more careful interpretation that we will give below. The above applicability conditions seem very restrictive, but that is because they refer to the general case. In practice, we have already pointed out that the most interesting processes to study the strongly interacting SBS are

7.7 The Applicability of the Equivalence Theorem

VLVL qq'

----t ----t

VLVL

ET +-----+

VLVL

ET +-----+

WW ----t -I

qq

----t

217

ww ww .

(7.110)

In all these reactions there is an even number of longitudinal polarization vectors, and therefore only even powers of the energy appear in the amplitudes. This fact can help us to obtain better applicability conditions. Indeed for the MSM they can be relaxed to E 2 » M 2 . It is usually considered that the ET is a good approximation for energies larger than::: 300 GeV. In case we were dealing with effective Lagrangians we would be left with

M2

«

E2

«

(47fV)2 .

(7.111)

For VL scattering, the biggest neglected contribution is O(g or g' surpressed) ::: 0(E 2/v 2 x M 2/ E 2), whereas the 0(E 4 ) term behaves as 0(E 2/v 2 x E 2/(47fv)2), so that (7.109) becomes

E

VB;

> 47fV (9 ::: 0.5 TeV.

(7.112)

Thus, in order to see the 0(E 4 ) effects, we need E > 500 GeV. Naively, we would set the upper limit at E < 47fV ::: 3 TeV, but that is too optimistic. We can see why by turning to ChPT and pion physics, which is an analogous formalism for NGB but scaled down to energies of the order of 1 GeV. As a matter of fact, in that approach 47f F 7r ::: 1 GeV plays the same role as 47fv ::: 3 TeV in the present case. However, we have already seen in Chap. 6 that 0(p4) ChPT, in the analogous processes, only works reasonably well up to 500 MeV::: 27f F 7r , at most. Therefore, it seems that the effective formalism is indeed limited at about 27fV ::: 1.5 TeV, probably even below. Throughout this section we have to keep in mind that the numbers we are giving are estimates, not strict mathematical bounds. They are calculated assuming that the coefficients of the energy expansion are of the same order, i.e. none of them is considerably bigger than the others.

Higher {) (g) Effects The two constraints 300 GeV < E < 1.5 TeV were naturally expected since one comes from the usual approximation of momenta by eLand the other from the energy expansion in the effective Lagrangian. As we have seen, the low energy bound in (7.112) depends on what terms we want to keep in the energy expansion. But in case we used the ET for effective Lagrangians, we are also doing another approximation when neglecting higher orders in the electroweak couplings 9 and g'. As a matter of fact, it has been shown that the higher order 9 and g' effects do not alter qualitatively the results, but their contribution is quantitatively significant, around 10% [21]. Unfortunately, when using the effective Lagrangian formalism, just to include more terms in the electroweak couplings is not enough, since we cannot directly apply the ET to the amplitude calculated to next order in 9

218

7. The Standard Model Symmetry Breaking Sector

or g'. It is easy to see from the GET that to the next order we have to take into account the following contributions: • The amplitude T(w, ... w) itself, up to the following order in 9 or g'. • The renormalization constants Z to next order. • The Ll i terms in the K correction factors. They did not occur before in the ET since they are 0(g2). • The lowest order contribution in the electroweak couplings, 9 and g', of the T(WL , w.. ... w) amplitudes, which appear in the GET multiplied by a vp. factor. In the literature there have been proposed renormalization schemes that yield K = 1 at any order [26]' thus simplifying the GET and ET expressions. However the most commonly used is the on-shell renormalization scheme and in that case all the latter contributions have to be calculated explicitly. This has been done for some of the processes listed above and the results have been compared, at tree level, with the 0(g2) tree level predictions without using the ET, showing a perfect agreement [27]. It would not make any sense to perform these 0(g2) calculations at one loop with the ET, since they will be as complicated as without it.

7.8 Gauge Boson Scattering at High Energies Now that we have the precise formulation of the ET, let us use it to obtain the scattering of longitudinal gauge bosons from NGB elastic scattering. Throughout the derivation we have to keep in mind that our aim is to measure the most relevant chiral couplings, so that we will need the 0(p4) expressions. From now on we will work in the lowest order in 9 and g'. As we have already seen in (7.107), such an approximation is mandatory when dealing with the ET and effective Lagrangians at the same time. However, it is also widely used even with the usual ET version, which is not restricted to lowest order in the electroweak couplings. In practice, this approach amounts to turning off the gauge fields throughout the calculation. Therefore, we do not need the whole EChL in (7.10) and (7.21) but just [28] £.

v2

= -:itr (8p.U8P.Ut) + a4tr (8P.U8v Ut) tr (8p.U8 v Ut) +a5tr (8p.U8P.Ut) tr (8v U8 v Ut) ,

(7.113)

where we have used the identity U t 8p.U = -8p.UtU. Notice that only a4 and a5 appear in the above Lagrangian. Only these two parameters can be measured in longitudinal gauge boson scattering (some others could be obtained from different processes, as qij ---. VV). We now want to calculate the NGB elastic scattering amplitudes. In Appendix D.3 we have shown how, from the invariance of the above Lagrangian

7.8 Gauge Boson Scattering at High Energies

219

under global SU(2)L x SU(2)R transformations, the amplitudes in any channel can be related to one single function A(s, t, u) (incidentally, it is nothing but the T(w+w- -> zz) amplitude). Now we can use the standard procedures to obtain this function from the above Lagrangian, including the one-loop contributions coming from the two derivative term. As explained in Sect. 7.3 the latter contain divergences which can be absorbed in the renormalization of the chiral parameters a4 and as (notice that no wave function renormalization is needed in this approximation). The result is

A(s,t,u) =

s2

v

+

+

2 [4aS(J.L)S2

v4

96: v

2 4

+ a4(J.L) (S2 + t(t - u) + u(u - t))]

1 [3S 2 C6 + log ( ~:) )

+ u(u - t)

C63 + log (~:))]

.

+ t(t - u)

C:

+ log (

~:) ) (7.114)

As there is no wave function renormalization at this level, the amplitude is an observable. Indeed, it can be checked, by using the evolution equations in (7.39), that the explicit dependence on the renormalization scale J.L is exactly canceled by the dependence through the chiral couplings a4(J.L) and as (J.L). Therefore, the above amplitude, via the ET describes, to lowest order in the electroweak couplings, the scattering of longitudinal electroweak gauge bosons. In view of the above equation, we can notice that the lowest order dependence on the Mandelstam variables s, t and u is completely determined by the SU(2)L x SU(2)R symmetry. This is a consequence of the fact that, as we pointed out at the end of Sect. 7.2, the O(p2) term in the Lagrangian is universal and gives rise to the analogous of the Weinberg low-energy theorems in pion physics, but for the SBS of the SM. To O(p4) the amplitude in (7.114) depends on two unknown chiral constants which parametrize our ignorance on the underlying physics. As explained in Sect. 7.4 different SBS theories will correspond to different values of these parameters. In particular it is immediate to find the NGB low-energy amplitudes in the Higgs or QeD-like models by using the values of a4 and as obtained in that section. However, we have already discussed that the effective Lagrangian approach is especially well suited for an strongly interacting SBS and in such case it is expected that some resonances will appear in the physical spectrum. Unfortunately those resonant states cannot be properly reproduced with a truncated power expansion in the external momenta like that in (7.114). By analogy with other strongly interacting systems, the physical region where these new dynamical effects are expected to appear lies beyond the 1 or 1.5 TeV, which, as we saw is the upper applicability bound of the approach. As a consequence, we are forced to introduce some new tools to explore this high-

220

7. The Standard Model Symmetry Breaking Sector

energy domain. In fact, a similar problem was already faced in the previous chapter when we tried to extend ChPT to higher energies. As we saw in that case, there are some unitarization procedures that improve the high-energy behavior of the amplitudes and naturally accommodate new resonant states. Among them we have already discussed in some detail the use of dispersion relations and the large-N limit (N being the number of NCB). Due to the formal analogies between the effective Lagrangians describing the low-energy dynamics of two flavor massless QCD and the low-energy strongly interacting 8M 8B8, those unitarization methods can also be applied in this last context. Once more, with the help of the ET we will relate the NCB dynamics with the longitudinal electroweak boson scattering that will be studied experimentally at LHC. In the following we will give some details of this approach stressing the main features of the two methods. 7.8.1 Dispersion Relations for the SM Symmetry Breaking Sector From the above general form of the low-energy NCB amplitude it is immediate to find the partial waves corresponding to different channels which, once more, are obtained as a perturbative expansion (7.115)

where tj~(8) and tj~(8) are the 8 and the 82 terms, respectively. Of course, in this context I is not the usual isospin, but a weak isospin associated to the custodial SU(2)L+R. For physical 8 values it is possible to check that Imtj~(8) =1 tj~(8) 12 , i.e. the elastic unitary condition is satisfied perturbatively. Hence, we have a similar situation to that found in Chap. 6 for ChPT, and therefore we can follow the very same methods described in that chapter. Namely, we can use the inverse amplitude method on the NCB amplitudes to obtain the unitarized result t IJ (8 )

(0) ( ) tIJ 8

~ ---;-::-;'-"--'---'--:=-

1 - tj~(8)/tj~(8)

(7.116)

The partial waves obtained with the above equation will satisfy exactly the unitarity condition ImtIJ(8) =1 tIJ(8) 12 for physical values of 8. In addition, they can reproduce resonant states whose physical parameters (1, J, mass and width) depend on the underlying dynamics through the chiral couplings a4 and a5. But even in those channels where there are no resonances, these amplitudes present the appropriate high-energy behavior and thus they can be used with the simplest version of the ET, (7.101). As a consequence, since there is no wave function renormalization and the K factors are trivial at lowest order in 9 or g', for energies 8 >> Mfv the above partial wave can be directly understood as the longitudinal gauge boson scattering amplitude. But in contrast with the non-unitarized case, this amplitude can be used up

7.8 Gauge Boson Scattering at High Energies

221

to much higher energies, which has a direct phenomenological application to LHC physics [29]. Indeed, the whole pp -+ VL VL cross sections can be evaluated from the unitarized amplitudes following the same steps of Sect. 7.5. In this way it is possible to give a physically acceptable phenomenological description of the different symmetry breaking scenarios in terms of just two parameters, namelya4 and as, and make definite predictions for the LHC. In particular both the Higgs and the QCD-like models have been studied in great detail. In the first case a broad resonance is found in the I = J = 0 channel. Such a resonance represents the physical Higgs boson and it displays the so-called saturation property. In brief, that means the following: despite we are using the on-shell scheme, M H cannot be identified with the physical Higgs mass for large MH, since the system becomes strong and the one-loop perturbative calculation of the Higgs two point function cannot be trusted. However, it is possible to see that the physical resonance mass M phy is an increasing function of the renormalized Higgs mass, although in the M H -+ 00 limit the physical mass does not get values bigger than 1 TeV. Indeed, MH still can be used as the energy parameter that completely defines the Higgs model. This saturation property has been observed in other approaches to the heavy Higgs dynamics like the so called N 1D method applied to the tree level amplitudes [30] or the large-N limit to be discussed later. It is important to realize that this effect is not due to the chiral Lagrangian formalism, but to the heavy Higgs field. As a matter of fact, even in the MSM with a heavy Higgs we could not calculate perturbatively its physical mass nor its width, so that some unitarization procedure is still likely to improve the results. For the QCD model we find similar results than those in Chap. 6, but scaled up by a factor of vi F1r ~ 2500. Thus it is also possible to observe a J = I = 1 resonance that could be called a techni-p. After the phenomenological analysis in [29] this case turns out to be quite promising since this resonance could in principle be detected at the LHC. However, the lack of a generally accepted technicolor model makes the real scope of this prediction unclear.

7.8.2 The Large-N Limit: The Higgs and the General Case Once again, an alternative approach to the description of the high-energy gauge boson dynamics is the large-N limit. In the context of the electroweak theory, this method is indeed an SU(2)L x U(l)y gauged version of the NLSM SU(2)L x SU(2)RISU(2)L+R ~ 0(4)/0(3) generalized to the coset O(N + l)/O(N) that we already met in Chap. 6. However, to start with, we will first consider the related linear sigma model (LSM). The classical Lagrangian is then given by (see Chap. 2) L = LYM

+ ~D!J.
V(

(7.117)

222

7. The Standard Model Symmetry Breaking Sector

with iJ>T = (71"1, 71"2, ... , 7I"N, (J) and iJ>2 = iJ>TiJ>. L YM is the standard SU(2)L x U(l)y Yang-Mills term and the corresponding covariant derivatives are defined as

DlJotP = olJoiJ> - igT:W:iJ> + ig'T Y BIJo ,

(7.118)

= -(i/2)M!:

where the SU(2)L and the U(l)y generators are T!: -(i/2)M Y with 0 00

_) 0

.~.+O

O

+ 00

0

0+0 - 00

0) 0

.~. 00

+

o0-

0

o0-

Mf=

Mf

=

(

(

Mf = (

MY =

(

00+ o 00 - 00

0) 0

0+0

0

0 +0 - 00 0 00

0) 0 -

o 0+

0

and T Y =

and the potential is given by

2 V(iJ>2) = _J-L iJ>2

+ ~(tP2)2 . (7.119) 2 4 At tree level, the minimum of the potential is given by iJ>2 = v 2 = N p2 = J-L2 / >.. thus spontaneously breaking the original O(N + 1) symmetry to O(N). By choosing the vacuum state as iJ>~ac = (0,0, ... ,0, v) and defining the Higgs field as H = (J - v the potential can be written as 1 >.. V(7I", H) = 2M'JIH2 + '4(71"2 + H 2)2 + >"VH(7I"2 + H 2) ,

(7.120)

where the tree level Higgs mass is given by M'JI = 2>..v 2. Notice that for N = 3 the above model is just the SBS of the MSM discussed in Sect. 7.4. Then the simplest way to obtain the gauge boson scattering amplitude is by making a perturbative expansion on the coupling constants g, g' and >... For simplicity, in the following we will take g' = 0 but the conclusion will apply also to the general case. However, the tree approximation is not appropriate at the phenomenological level in the heavy Higgs case, since it does not take into account the Higgs width, which becomes very large with M H . A possible solution is to include it by hand in the propagators in the s channel. Nevertheless, this method is not completely consistent since it changes the low-energy behavior predicted by the Weinberg low-energy theorems [31]. The tree level NGB amplitudes could also be improved with the one-loop corrections [32]. Unfortunately, the one-loop mass is not a good approximation to the heavy Higgs physical mass even working in an on-shell scheme. In contrast, the large-N limit of the MSM [33] yields a consistent treatment of the width. Within that

7.8 Gauge Boson Scattering at High Energies

223

approach, we consider the SU(2)L x U(l)y gauged LSM based on the coset

O(N + l)jO(N) that we have just defined [34]. Then, the amplitudes can be

expanded in powers of 1 j N. More precisely we can take the limit N --+ 00 with A --+ 0 and 9 --+ 0 (and g' --+ 0) so that N A and N g2 (and N g'2) remain constant. In particular, that means that F 2, M H and Mw (and Mz) are also constant. A simple way to study the main properties of the Higgs resonance in the large-N limit is to start with the simpler case where 9 = g' = 0, i.e. we will turn off the gauge fields. Then we only have to consider the N NGB 1r a and the Higgs H. Let us remember (see Appendix D.3) that the 1r a 1r b --+ 1r c 1r d scattering amplitude can be written in terms of a single function A(s, t, u), although now it only depends on s. The tree level contributions to the A function (A o) correspond to the diagrams in Fig. 7.6a 2

) M S 1 A o(s)=-2A ( 1+ s-Zt'f.J = NF21-sjM2'

c)

H-e+ .ll...

H

---<JO-

H

_H

+ + .ll...

H-o--(XX)-

(7.121)

H

_H

+ ...

Fig. 7.6. Diagrams contributing to (a) the tree level NGB scattering amplitude (b) the leading order in the liN expansion for the same process (c) the Higgs propagator at leading order in the 1 IN expansion

In the large-N limit, the relevant diagrams are those shown in Fig.7.6b which yield s 1 A(s) = NF2 1 _ sjM'f.J + s1(s)j2F2 ' (7.122) where the one-loop integral 1(s) was defined in (6.102). Its divergences can be absorbed in the renormalized Higgs mass Mk, which can be defined as I I I

Mk = M'f.J + 2(41r)2 F2 (Ne + 2) ,

(7.123)

7. The Standard Model Symmetry Breaking Sector

224

so that we find s A(s)=NF21

1

- m + 2(4'1l-)2F2 1og Ji2 8

8

-8

(7.124)

In this approach the Higgs mass is the only parameter that needs to be renormalized and in particular there is no wave function renormalization. Thus the above amplitude is an observable and IL-independent quantity. This fact can be used to find the dependence of the renormalized Higgs mass MR on the renormalization scale IL which turns out to be M2 ( R

IL

)

Mk{ILO)

=

1-

M&(I'O) 2(41r)2F2

1'2'

log ~

(7.125)

The renormalized coupling >"R can be defined so that it keeps the tree level relation Mk = 2>"RN p2 and then its running can be obtained from the above evolution equation. In practice it is useful to introduce the mass parameter M 2 defined by the equation M2

= Mk{M 2 )

and then we find Mk(lL)

=

1-

(7.126)

,

M2

M2

2(41r)2F2

log

1'2

w

(7.127)

as well as >"R(IL) = 1

>"(M) 1 1'2 - (41i12 og W N>'(M)

(7.128)

Therefore, for 9 = g' = 0, the observables will depend on the only free parameter of the model, which is M. However, it should not be confused with the physical Higgs mass, which is obtained as the real part of the position of the pole that appears in the scattering channel with the Higgs quantum numbers. For the interesting case where N = 3 and the coset space is 0(4)/0(3) = SU(2)L x SU(2)R/SU(2)L+R the interactions are SU(2)L+R symmetric (weak isospin group). There are three NCB and the scattering channels can be labeled by the third component of the isospin which can take the values I = 0,1,2. The partial waves tJj can be defined as in (6.122) and thus too is the dominant channel in the large-N limit. This partial wave has some properties which make the large N limit a sensible approximation to the Higgs physics. First, at low energies we find s too(s) ~ 321l"p2 (7.129) in agreement with the Weinberg low-energy theorems. Second, it has the proper unitarity cut along the positive real axis of the s variable. Moreover, for s values over the unitarity cut, where loge -s) = log s - i1l", we have Imtoo

=1 too 12 +O(l/N)

,

(7.130)

7.8 Gauge Boson Scattering at High Energies

225

which is the elastic unitarity condition. Finally, as we have already commented, the too partial wave has a pole in the second Riemann sheet. This pole can be understood as the physical Higgs resonance, which is narrow for low M values where the standard Breit-Wigner description of a resonance is possible. The physical mass is given just by M and the width is

M3

r=

(7.131)

321rF2 '

which is the tree level result. However, when M increases, the Higgs resonance becomes broader. The pole moves in the complex plane away from the real axis and the Breit-Wigner description cannot be used any more. Nevertheless, the real part of the pole position remains bounded even for very large M. This is the above mentioned saturation property which has been observed in other non-perturbative approaches to the Higgs dynamics. Once the main properties of the physical Higgs has been established in this large-N approximation, we can turn on again the gauge fields. It is not difficult then to show that, at the leading order considered here, i.e. to O(l/N, A, g2) (or g'2), the elastic gauge boson scattering amplitudes are obtained just by replacing in the tree level calculation the Higgs propagator D(q2) = 1/(q2 - M'k) by the following modified Higgs propagator -

2

1

D(q ) = q2 _ M'k( _q2) ,

(7.132)

which is obtained as the sum of the bubble graphs in Fig. 7.6c, and we have defined

M'JT(_q2) = _ _---:-:-;M,...-H~2

MiI

I

_ _q2

1 - 2(41r)2F2 og 7""

(7.133)

MH being the Higgs mass renormalized at the scale /-L. Thus, for example, the W+W- -+ ZZ scattering amplitudes are given by the diagrams in Fig. 7.4 but with the Higgs propagator defined as in Fig. 7.6c, that is, using (7.132). Once again, the amplitudes obtained in this large-N approximation have improved in several aspects. First they satisfy the ET, but now there is no wave function renormalization and the K factors are trivial. Second, the low energy behavior is also that predicted by the Weinberg low-energy theorems, although, and this is the important point, the amplitudes have the good analytical and unitarity properties that we had already remarked when considering the large-N limit in Chap. 6. In particular, the above Higgs propagator has an unitarity cut along the real axis. Moreover, it is able to describe the behavior of the physical Higgs resonance which is also expected from other non-perturbative approaches to the strongly interacting case. Therefore, the large-N limit yields a simple phenomenological non-perturbative description of gauge boson dynamics that takes into account the Higgs width in the heavy Higgs case, consistently with all the known theo-

226

7. The Standard Model Symmetry Breaking Sector

retical constraints. For these reasons this approximation will be quite useful to compare the LHC results with the predictions of the MSM. Finally we would like to comment that the large-N limit approach can be extended to the NLSM too. This will require gauging (with the 5U(2) x U(l)y group) the results obtained in Sect. 6.6.2. However the simplest way to do that is to start from the results obtained from the LSM described in this section. By working with the gauge boson amplitudes it is possible to write them in terms of the function

CR(s; J.L)

=

1

s

1- M~{IJ-}

(7.134)

and log( -sf J.L2) so that the amplitudes are J.L independent. Then the most general NLSM result can be obtained by taking an arbitrary generating function CR(s; J.L) as those defined at the end of Sect. 6.6.2. In particular the dependence of this generating function on the renormalization scale J.L should be given by (6.114).

7.9 References [1] [2] [3] [4] [5]

[6] [7] [8]

[9] [10]

F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321 P.W. Higgs, Phys. Rev. Lett. 13 (1964) 508; Phys. Rev. 145 (1966) 145 P.W. Anderson, Phys. Rev. 130 (1963) 439 G. t'Hooft, Nucl. Phys. B35 (1971) 167 J. Goldstone, Nuovo Cimento 19 (1961) 154 J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965 S.L. Glashow, Nucl. Phys. 22 (1961) 579 S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264 A. Salam, Proc. 8th Nobel Symp., ed. N. Svartholm, p. 367, Stockholm, Almqvist and Wiksells (1968) Y.A. Golfand and E.P. Likhtman, JETP Lett. 13 (1971) 323 D.V. Volkov and V.P. Akulov, Phys. Lett. B46 (1973) 109 J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39 S. Weinberg, Phys. Rev. D19 (1979) 1277 S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237 E. Farhi and L. Susskind, Phys. Rep. 74 (1981) 277 Y. Nambu, Proceedings of the 1988 International Workshop on New Trends in Strong Coupling Gauge Theories, Nagoya, 1988, editors M. Bando, T. Muta and K. Yamawaki, World Scientific, Singapur, 1989, p.3. Y. Nambu, Proceedings of the 1989 Workshop on Dynamical Symmetry Breaking, Nagoya, 1989, editors T. Muta and K. Yamawaki, Nagoya University, Nagoya, 1990, p.l. V.A. Miransky, M. Tanabashi and K. Yamawaki, Mod. Phys. Lett. A4 (1989)1089; Phys. Lett. B211 (1989) 177 W.J. Marciano, Phys. Rev. Lett. 62 (1989) 2793 W. A. Bardeen, C.T. Hill and M. Lindner, Phys. Rev. D41 (1990) 1647 V.A. Miransky, Int. J. Mod. Phys. A6 (1991) 1641 P. Sikivie et al., Nucl. Phys. B173 (1980) 189 M.S. Chanowitz, M. Golden and H. Georgi, Phys. Rev. D36 (1987) 1490 M.S. Chanowitz, M. Golden and H. Georgi Phys. Rev. Lett. 57 (1986) 2344

7.9 References [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

227

T. Appelquist and C. Bernard, Phys. Rev. D22 (1980) 200 A. Longhitano, Phys. Rev. D22 (1980) 1166; Nucl. Phys. Bl88 (1981) 118 T. Appelquist and G. Wu, Phys. Rev. D48 (1993) 3235 M.J. Herrero and E. Ruiz Morales, Nucl. Phys. B418 (1994) 431; Nucl. Phys. B437 (1995) 319 B. Holdom and J. Terning, Phys. Lett. B247 (1990) 88 A. Dobado, D. Espriu and M.J. Herrero, Phys. Lett. B255 (1991) 405 M. Golden and L. Randall, Nucl. Phys. B361 (1991) 3 M. Bohm, H. Spiesberger and W. Hollik, Fonschr. Phys. 34 (1986) 687 W.J. Marciano and A. Sirlin, Phys. Rev. D22 (1980) 2697 K. Hagiwara, RD. Peccei and D. Zeppenfeld, Nucl. Phys. B282 (1987) 253 S. Dawson, Nucl. Phys. B249 (1985) 42 C. Weizsiicker and E.J. Williams, Z. Phys. 88 (1934) 612 E.W.N. Glover and J.J. Van deer Bij, Nucl. Phys. B321 (1989) 561 A. Dobado, M.J. Herrero, J.R Pelaez, E. Ruiz Morales and M.T. Urdiales Phys. Lett. B352 (1995) 400 A. Dobado and M. Urdiales. Z. Phys. C71 (1996) 659 J.M. Cornwall, D.N. Levin and G. Tiktopoulus, Phys. Rev. DIO (1974) 1145 C.E. Vayonakis, Lett. Nuovo Cimento 17 (1976) 383 C. Becchi, A. Rouet and R Stora, Comm. Math. Phys. 42 (1975) 127 M.S. Chanowitz and M.K. Gaillard, Nucl. Phys. B261 (1985) 379 G.K. Gounaris, R Kogerler and H. Neufeld, Phys. Rev. D34 (1986) 3257 Y.P. Yao and C.P. Yuan, Phys. Rev. D38 (1988) 2237 J. Bagger and C. Schmidt, Phys. Rev. D41 (1990) 264 H.J. He, Y.P. Kuang and X. Li, Phys. Rev. Lett. 69 (1992) 2619 H.J. He, Y.P. Kuang and X. Li, Phys. Lett. B329 (1994) 278 A. Dobado and J.R Pelaez, Phys. Lett. B329 (1994) 469; Nucl. Phys. B425 (1994) 110 D. Espriu and J. Matias, Phys. Rev. D52 (1995) 6530 A. Dobado and M.J. Herrero, Phys. Lett. B228 (1989) 495; B233 (1989) 505 J.F. Donoghue and C. Ramirez, Phys. Lett. B234 (1990) 361 A. Dobado, M.J. Herrero and J. Terron, Z. Phys. C50 (1991) 205 and 465 B.W. Lee, C. Quigg and H. Thacker, Phys. Rev. Dl6 (1977) K. Hikasa and K. Igi, Phys. Rev. D48 (1993) 3055 G. Valencia and S. Willenbrock, Phys. Rev. D46 (1992) 2247 S. Dawson and S. Willenbrok, Phys. Rev. Lett. 62 (1989) 1232 M.J.G. Veltman and F.J. Yndurain, Nucl. Phys. B325 (1989) 1 R. Casalbuoni, D. Dominici and R Gatto, Phys. Lett. Bl47 (1984) 419 M.B. Einhorn, Nuc. Phys. B246 (1984) 75 A. Dobado, J. Morales, J.R. Pelaez and M. Urdiales, Phys. Lett. B387 (1996) 563

8. Gravity and the Standard Model

In this chapter, we deal with the standard model (8M) of elementary particles interacting with classical gravity. Within this framework, we study the cancellation of anomalies in gauge and Lorentz currents as well as lepton and baryon number violation. We also consider the effect of the matter fields on gravitation. Finally, the pure gravitational sector is also described within the effective Lagrangian approach as the low-energy limit of some more fundamental theory of gravity.

8.1 Introduction As we have already seen in Chap. 5, the 8M successfully describes the electroweak and strong interactions of quarks and leptons. At the classical level, the gravitational interaction is also well described within the framework of General Relativity (GR) by the Einstein-Hilbert action Be = -

2

/'i,2

J

4

d xygR,

/'i,

2

321l"

= M~ ,

(8.1)

where R is the scalar curvature (see Appendix B.1), the Planck mass is given by Mp = J1/G ~ 1.2 x 10 19 GeV and G is the Newton constant. However, the attempts to build a quantum theory of gravitation have not driven to any successful solution (see [1] for a review). One of the reasons is the non-renormalizability of GR. Thus, the problem of finding a unified theory of all interactions still remains open. Nevertheless, the above discussion suggests that the phenomenological information about all those interactions could be encoded in the 8M built on a classical gravitational background and the Einstein-Hilbert Lagrangian, at least at the presently accessible energies. This is not commonly considered as the most fundamental 'approach, but at least it is the minimal one compatible with all the experimental data. The underlying theory, if it was known, would allow the calculation of the undetermined parameters present in the 8M which nowadays are obtained from the experiment. In absence of such a theory, the form of the low-energy effective Lagrangian interacting with gravity is determined by symmetry considerations. In this sense it is the most general expression that includes all the known matter fields, which is scalar under coordinate changes and invariant A. Dobado et al., Effective Lagrangians for the Standard Model © Springer-Verlag Berlin Heidelberg 1997

230

8. Gravity and the Standard Model

under the SU(Nc ) x SU(2)L x U(l)y as well as local Lorentz transformations. We will see that by the minimal coupling and following the Einstein equivalence principle (EP), it is possible to build such a Lagrangian. However there are many other terms satisfying the above symmetry requirements apart from those obtained from the minimal coupling. Nevertheless, they will be negligible at low energies, which for the gravitational field means small curvature. The reason is that such higher order operators will be suppressed by the scale of the new physics (typically the Planck mass M p ) that eventually might appear above the SM energies, i.e, ITeV. Besides the above considerations, the quantum aspects of a theory in curved space-time present some rather technical subtleties, which are not present in Minkowski space-time, and that we will only enumerate: • First of all in defining the fields themselves. Dirac spinors can only be defined if the space-time manifold satisfies certain technical conditions, namely, that a topological invariant called the second Stiefel-Withney class vanishes [2]. Such manifolds admit a spin structure and accordingly they are called spin manifolds. In particular all spheres are spin manifolds. • Second, in the canonical quantization procedure. In an arbitrary curved manifold, Poincare invariance is no longer a symmetry and B/Bt, in general, is not a Killing field. The existence of such a Killing vector provides a natural definition of positive energy modes and therefore of creation and annihilation operators. As far as the vacuum is defined using annihilation operators, in curved space-time the vacuum is not unique. In this sense a given state which for certain observer is empty, may have some particles for a different (accelerated) observer. These processes of particle creation are typical of quantum field theory (QFT) in curved space-time, see [3]. • Third, the definition of an S-matrix requires a time parameter with respect to which we can define asymptotically free states in the remote past and future. In fact, in Minkowski space-time, particles can be well separated before and after the interaction. However in curved space-time it is not obvious that this situation can always take place. Indeed if all spatial sections of space-time are closed, particles cannot be infinitely separated and therefore we do not have free states. Mathematically, the conditions to have a proper S-matrix formalism are: a globally hyperbolic space-time with Cauchy hypersurfaces which are either non-compact or compact without boundary, but with an infinite volume in the remote past and future [4]. However it also possible to relax these conditions when we are not concerned with scattering processes, as it happens when dealing with anomalies. • Finally it is also interesting to consider the problem of renormalizability in curved space-time. Naively one does not expect any change concerning this issue, since the ultraviolet divergences are related to short distances, and the EP tells us that curved space-time looks locally like a flat spacetime. However, the explicit calculation of the short-distance behaviour of the Green functions shows that new terms which were not present in the

sn

8.2 The Standard Model in Curved Space-Time

231

flat case may appear in curved space-time. In any case, if the space-time is asymptotically flat and the gravitational field is weak, a renormalizable theory in flat space-time can be reformulated to be renormalizable in curved space-time [5]. In the following three sections we will concentrate in two different points: first, the effect of the gravitational interaction at the classical level in the 8M and its anomalies; second, the influence of the quantum fields on gravitation.

8.2 The Standard Model in Curved Space-Time The Equivalence Principle Classically, the derivation of physical laws in presence of gravitational fields is obtained by means of the EP (see [6] for a review). This principle has two possible formulations: • The weak equivalence principle (WEP) refers only to the effect of gravitation on test particles and states that at each point of space-time it is always possible to find a privileged reference system in which locally the gravitational interaction is switched off. This statement is equivalent to the proportionality between inertial and gravitational masses. As far as in absence of gravity the space-time should be minkowskian, the WEP implies that gravitation can be suitably formulated using a semi-Riemannian manifold. • The strong equivalence principle (8EP) refers to all the interactions and states that in the privileged reference frame all physical laws are locally the same as in absence of gravitational fields. As we will see later, the 8EP is related to the minimality of the 8M Lagrangian coupled to gravity. Both formulations of the EP together with the principle of general covariance (physical laws do not depend on the specific coordinates we choose to write them) make it possible to write a Lagrangian for the 8M interacting with gravity at the classical level. The 8EP yields a simple procedure to couple gravity to any field theory built in a flat space-time, starting from the Lorentz invariant action. It is enough to identify the coordinates appearing in it with those of the locally inertial system, then perform a coordinate change to a general coordinate system and the gravitational interaction will appear automatically. In the following, we will first apply this recipe in detail to work out the gravitational interaction of Dirac spinors. At the end of this section we will also obtain the Lagrangians for scalar and gauge fields interacting with gravity. The Dirac Lagrangian in Curved Space-Time In Minkowski space-time the hermitian form of the Dirac Lagrangian is given by

232

8. Gravity and the Standard Model

(8.2) In order to work in the path integral formalism, we will be interested in the Euclidean Lagrangian (see Appendix A) (8.3) Before coupling gravity to this Lagrangian let us introduce some notation. We will use latin indices m, n . .. for objects referred to the locally inertial coordinate frame and Greek indices 11, 1/ . . . for any other. If {~m} are the coordinates in the privileged system and {xJ1.} the coordinates in any other, then

= e~(x)e~(x)7]mn , (8.4) where 7]mn = (-, -, -, -) is the Euclidean flat metric and e~(x) = 8xJ1. /8~m gJ1.V(x)

is the vierbein, which at each point gives the change of coordinates to the privileged system. It is also possible to define an inverse vierbein by: e~ e~ = 8;;:' and e~e;;' = 8~ (see Appendix B.l). Finally let us introduce the volume form written in terms of vierbeins d4~ =..;g d4x = (dete;)d 4x (8.5) with 9 = I detgJ1.vl· In flat space-time, Dirac spinors change in the following way under Lorentz transformations: ¢(p)-+U¢(p)

= e~·",nE",n¢(p)

~(p) -+~(p)Ut = ~(p)e-~·rrm E",n ,

(8.6)

where E mn = ~bm,'Yn] are the hermitian generators of the 80(4) group in the spinor representation. Notice that, in Euclidean space ¢ and ~ are independent variables and the transformation rule of ~ is taken such that ~¢ is invariant [7]. Therefore, the flat space-time Dirac Lagrangian in (8.3) is invariant under those global transformations. In flat space-time it is always possible to integrate by parts and write the Lagrangian in the more usual form (8.7)

The EP requires the invariance of the Dirac Lagrangian under Lorentz transformations to be not only global but also local. With that purpose, let us introduce a covariant derivative \7 m, so that we can write the gauged hermitian Dirac Lagrangian in the following way:

(8.8) The above expression is written in the locally inertial frame. Now we can write it in any other coordinate system by means of the coordinate change given by the vierbein (8.9)

8.2 The Standard Model in Curved Space-Time

233

where we have defined the Dirac matrices in curved space-time ')'Jl(x) = e~(xhm. These matrices satisfy: {')'Jl(x),')'V(x)} = -2g JlV (x). The covariant derivative is defined as usual by (8.10) where nJl is known as the spin connection. In order to keep the invariance under local Lorentz transformations, nJl should transform as follows:

nJl

->

n~

= U(x)nJlU-1(x)

- (aJlu)U-l(x)

(8.11)

or infinitesimally

nJl

->

nJl

+ ~Eab(x)[Eab' nJl]

-

~(aJlEab(x))Eab .

(8.12)

In Appendix B.1 it is shown that the connection components in (semi-)Riemannian geometry (denoted i~ b) do have precisely the above transformation rule, so that we can identify i,

b

(8.13)

nJl == -2r~ Eab · As a consequence the derivative acts on spinors as i

'a

b

-

-

i,a

b-

'V Jl'lj; = (aJl - -2 r Jl Eab)'Ij;, 'V Jl'lj; = aJl'lj; + -r Jl 'lj;Eab . . 2

(8.14)

Note that, depending on the object this derivative acts on, the generators will appear in the corresponding representation (vector, tensor, etc) of the Lorentz group. It is easy to see that the gauge covariant derivative is nothing but the ordinary geometric covariant derivative but referred to the privileged coordinate system (see Appendix B.1). It is important to note that this gauge formulation based on the Lorentz group allows us to introduce spinors in curved space-time, which otherwise would be impossible, since the GL(4) group does not posses spinor representations. Following with the previous discussion, notice that {if; b} does not have to be a Levi-Civita connection (that is, torsion free and metric), which we will denote {r~ b}. As we will see, in the first-order Palatini formalism, the torsion-free condition is a dynamical constraint which disappears when fermions are present [8] and in that case, the vierbein and the connection are considered as independent objects. This is in contrast with the Levi-Civita case in which it is possible to obtain the connection from the vierbein in terms of Christoffel symbols. Using the decomposition of the metric connection in (B.25), we can write the Dirac Lagrangian in (8.9) in terms of the Levi-Civita connection plus an additional term depending on the torsion

.c = where

( aJl - 2r: i bE 1 ) 'Ij;, _'Ij;')'Jl ab - gSJl')'5

(8.15)

234

8. Gravity and the Standard Model

(8.16)

where TJ.Lv>' is the torsion tensor and accordingly SJ.L is its axial part. In conclusion, the Lagrangian for Dirac fermions in a curved space-time with torsion is that of a fermion in a curved space-time without torsion plus an axial interaction with Sw For simplicity, in the rest of the chapter we will not consider the effect of the torsion. Hence, we will work with a Levi-Civita connection. In particular, we will use such a connection for the calculation of the SM anomalies in next section, although the more general case including the torsion has also been studied [9]. Only at the end of the chapter we will discuss other approaches in which the vierbein and the connection are independent.

The SM Lagrangian in Curved Space-Time In Chap. 5 we have written the matter Lagrangian for the SM in a flat spacetime. Following the above arguments we can now write the corresponding expression in curved space-time, which reads (8.17)

where the Dirac operators for quarks and leptons are defined as

[\7 J.L + GJ.L + WJ.LPL + ig'(yLQPL + yRQPR)BJ.L] WL. =,J.LD; = ,J.L [\7 J.L + WJ.LPL + ig' (Yf PL + Yi PR)BJ.L] .

WQ=,J.LD; =,J.L

(8.18)

Here we have followed the same notation introduced in Sect. 5.1 for the GJ.L' WJ.L and BJ.L fields, quarks and lepton doublets (Q, £), as well as for the hypercharge matrices Y. As in the flat space-time case, these operators are not hermitian due to the chiral couplings of SU(2)L and hypercharge fields. Thus the adjoint operators are

(i WQ)t =iTJ.L D; = iTJ.L(\7J.L (i .fbL.)t =iTJ.L D~ =

+ GJ.L + WJ.LPR + ig'(Yf-P L + yLQ PR)BJ.L) iTJ.L(\7J.L + WJ.LPR + ig'(Yi PL + Yf PR)BJ.L) , (8.19)

where \7 J.L = oJ.L + [l w Notice that, since there is no right neutrino, the spin connection can be written as follows for leptonic operators: [l

J.L

= _ira b (PLEab

2 J.L

Eab

)

[l

'J.L

= _ira b (PRE ab

2 J.L

) Eab'

(8.20)

where the matrices act on flavor space. For quark operators the spin connection is the same as for leptons but without the PL,R projectors. Finally we will give the Lagrangians in curved space-time for the rest of fields present in the minimal standard model (MSM) • The symmetry breaking sector in flat space-time is given in (5.18). As the scalar fields do not change under Lorentz transformations, their covariant derivative is just an ordinary derivative. Then, according to the prescription

8.2 The Standard Model in Curved Space-Time

235

based on the EP, we simply have to use the vierbein to perform an arbitrary coordinate transformation. The final expression for the action reads

SSBS =

J xyg d

4

(gJ.LV(DJ.L¢)t (D v ¢) - V(¢) + .c YK ) ,

(8.21 )

where DJ.L is given in (5.19) . • The Yang-Mills Lagrangian in flat space-time is given by

.c YM

=

-41 Famn F amn .

(8.22)

We consider the strength tensor Fmn as defined in a locally inertial coordinate system. Fmn is a Lorentz tensor and F::'nF::n is invariant under global and local Lorentz transformations. Therefore we only have to transform it to an arbitrary coordinate system using the vierbein (8.23) Thus the action reads

SYM

=

Jdxyg ( _~gJ.LPgVC1 F: F;(1) 4

v

(8.24)

The above Lagrangians have been obtained by the minimal coupling. However, there is no reason to assume that the non-minimal couplings have to be discarded. In fact, we are going to show that some of these terms are necessary for renormalizability. Up to now we have considered the classical theory. The quantum theory presents some subtleties, as those commented in the introduction. Moreover, some of the above minimal Lagrangians are not renormalizable. In fact, the one-loop calculations require counterterms which were not present in the original Lagrangian [5J. For instance, for the scalar sector one needs to introduce the counterterm R¢2 where R is the scalar curvature. In addition, one should include in the pure gravitational sector some counterterms that absorb the vacuum divergences (see Sect. 8.4), which cannot be discarded by the procedures used in flat space-time (such as normal ordering). Furthermore, since symmetry is our only guiding principle in constructing the SM Lagrangian in curved space-time, any other non-minimal term could be included provided it respects the symmetries of the theory. Such terms are different from the minimal ones in the sense that they violate the SEP. The reason is that a term like R¢2 vanishes in flat space-time, but that is not the case in a free-falling reference frame due to the presence of the scalar curvature. In contrast, the minimal couplings are the same either in a flat space-time or in a free falling frame. Non-minimal terms also arise in the low-energy effective action for the light fields obtained by integrating out a heavy field (see Chap. 1). Thus, for instance, they have been explicitly obtained in the curved space-time version of the Euler-Heisenberg Lagrangian [lOJ. The violation of the SEP does not mean a breaking of Lorentz invariance (provided the non-minimal terms are

236

8. Gravity and the Standard Model

Lorentz scalars). Nevertheless, we will see in Sect. 8.3.3 that the anomaly effects may also violate Lorentz invariance, although for consistency we will require its conservation. According to this discussion we conclude that the SEP is only a low-energy effect which will not be satisfied when higher order corrections are included in the effective Lagrangian.

8.3 Anomalies in the Standard Model In Chap. 5 we studied the cancellation of gauge anomalies in flat space-time and obtained the expressions for the anomalies in the baryonic and leptonic currents. We showed that the cancellation was possible with the hypercharge assignment in the SM. In addition, we have also found that the baryonic and leptonic currents were not conserved although their difference B - L indeed was. In this chapter we are interested in the effect of gravitation on these anomalies. Concerning gauge anomalies, it is necessary for the consistency of the theory that the gravitational contributions do not affect their cancellation. Moreover, the possible gravitational terms in the baryonic and leptonic anomalies could have some relevance to explain the baryon asymmetry in the Universe. Let us analyze all these symmetries in a curved space-time. In order to calculate the SM anomalies we will follow the same method used in Sect. 5.3. Accordingly, we introduce the curved space-time version of the operators H..p and H-:;p considered in (5.42) Hi'L. = (i flJQ,L.) t (i flJQ,L.) = D;'L. D~,L. -

~ h'JL, /,vj[D;'L., D~,L.l

HE'L. = (iflJQ,L.) (iflJQ,L.)t = DQ,L. D JL _ ~ [JL Vj [DQ,L. DQ,L. ..p JL Q,L. 4 /' , /' JL' v J ,

(8.25)

where =RQ,L. [DQ,L. JL ' DQ,L.J v JLV

+ (G JLV ) + WQ,L. JLV + BQ,L. JLV

(8.26)

and analogously for [D;'L., D;'L.j, with DJL and DJL given in (8.18) and (8.19) respectively. The parenthesis in G JLV mean that it does not appear in the leptonic case. The curvature terms, due to the absence of right neutrinos, are written for leptons as follows: RL. = _!..Rab (PLEab ) IiL. = _!..Rab (PREab ) (827) JLV 2 JLV E ab ' JLV 2 JLV E ab "

For quarks they are the same but without the P L and PR projectors. The second coefficient in the heat-kernel expansion in curved space-time for the operators in (8.25), which as we saw in Chap. 4 is the only relevant for the anomaly calculation, reads [3, 11, 12J

8.3 Anomalies in the Standard Model

a2(H1/J'x) = 11 [DJ.L,Dv][DJ.L,D V] + 2

~[DJ.L' [DJ.L,X]] + ~X2 - ~RX

-~R. J.L + ~R2 + _1_(R 30

,J.L

1 -

-

180

72

237

J.Lvpa

RJ.Lvpa _ R

J.LV

(8.28)

RJ.LV)

and

a2(H1J ,x)

1-

-J.L - v

-J.L -

1-2

= 12 [DJ.L' Dv][D ,D] + 6 [DJ.L' [D ,X]] + 2X -~R J.L 30 ;J.L

~R2

+ 72

_1_(R

+ 180

J.Lvpa

RJ.Lvpa _ R

-

1--

6R X

RJ.Lv) J.LV'

(8.29)

where according to (8.25)

X=-~bJ.L"V][DJ.L,Dv] 1 X =-4 b J.L"V][DJ.L' Dv] .

(8.30)

For the sake of simplicity we have omitted the Q, £ indices. Notice that, for quarks, the curvature terms are the same either with or without bar. With the above results, we can already obtain the explicit expression for the different anomalies. 8.3.1 The Leptonic and Baryonic Anomalies In curved space-time the absence of right neutrinos implies that, in some sense, gravity couples chirally and, as we will see, the anomaly in the leptonic current acquires a gravitational contribution. Nevertheless, these gravitational terms are not present in the baryonic sector, thus yielding a B - L non-conservation. In the calculation of these anomalies we will follow the same steps as in Sect. 5.4. We will calculate the baryonic case but the procedure is completely analogous for leptons. In order to obtain the Ward identities corresponding to these symmetries, first we perform the infinitesimal local transformations in (5.55). Under these transformations, the matter Lagrangian in (8.17) changes as follows:

j d xy'g£m ~ j d xy'g[£m + ia(x)'VJ.Lj~] . 4

4

(8.31 )

As it happened in Sect. 5.4, we have

e-r[G,n,e, ... j

4

= j[dQdQ ... ]eJ d xy'9.c m (Q,"Q ... ) = j[dQdQ ... J X

4 4 4 e - J d xy'9io:(x)A(x) e J d xy'9io:(x)\1 "j~ e J d xy'9.c m

(8.32) (Q,"Q ... )

.

From this equation we find the anomalous Ward identity for the baryonic current, which reads

'V J.Lj~(x) = A(x) with A(x) regularized as

(8.33)

238

8. Gravity and the Standard Model 1

A(x) = (471")2 tr[a2(H~·.c, x) - a2(Hi'.c, x)] ,

x)

(8.34)

x)

where a2 ( H~·.c, and a2 (Hi'.c, are given in (8.28) and (8.29). The explicit evaluation of the traces in the above equations yields for the anomalies in the baryonic and leptonic currents the following results:

V I-'J'1-'B

-

1 EI-'vpu (g2 3271"2 2 Wa1-'1.1 wapu

+ 9 /2 B 1-'1.1 B pu ""' ~

(2 YL

-

2))R(835) Y ·

all quarks

and V

'1-' _ I-'JL -

1

1

pU'"l6 {

3271"2 E

-

24

R

RI-'v I-'Vpu

+g/2 B'"I6Bpu

'"16

g2 W a Wa

+2

L all leptons

'"16

(yI -

pu

Yh)} .

(8.36)

Therefore the baryonic anomaly coincides with the flat space-time result, that is, there are no contributions from the curvature. In the leptonic case we see that, due to the non-existence of right neutrinos, some terms depending on the curvature appear in the anomaly and B - L is no longer conserved [13]. This violation would occur through the so called gravitational instantons [14], although this issue is beyond the scope of this book. In contrast, if there were right neutrinos, B - L would be conserved, as it happens in flat space-time (prOVided (5.60) is satisfied). 8.3.2 Gauge Anomalies

We have just seen how gravity could spoil the conservation of global currents as B - L, which are conserved in flat space-time. In this section we will study whether something similar happens to gauge currents. The non-conservation of gauge currents would spoil the consistency of the model and then it is necessary that the new terms depending on the curvature cancel. This could impose new constraints to the SM hypercharges. The calculation of these anomalies in curved space-time is completely analogous to the flat case studied in Sect. 5.3. We will explicitly write the expression for the Ward identities in the SU(Nc ) case. The SU(2)L and U(l)y results are obtained in a similar way. The effective action for the SM matter sector is given by

e-r[G..a,e.... j

=

J[dQdQ]eJ d xyg.c= , 4

(8.37)

where the dots stand for the rest of the SM fields which are not relevant in the present calculation. Under the SU(Nc ) gauge transformations given in (5.3) and (5.11), the effective action may have an anomalous variation given by

8.3 Anomalies in the Standard Model

F[G - De, il, e, ...] - F[G, il, e, ...] = -

Jd4x...;gieb(DIJ.(j~))b,

239

(8.38)

where DIJ. = \71J. + [GIJ.,·] and we follow the notation introduced in Sect. 5.3. This transformation comes form the change in the integration measure and can be obtained in the standard fashion, it yields [dQdQ]

->

[dQdQ] exp

(i J

d4x...;ge a(X)A a(x)) .

(8.39)

The anomaly Aa(x) in the above equation is obtained from (5.41) just by and given in (8.25). In (5.43) we gave considering the operators the gauge anomalies in terms of the heat-kernel coefficients, which now have to be replaced by those in (8.28) and (8.29). Therefore transforming the effective action we obtain the anomalous Ward identity

H;;'.c.

Hi'.c.

(8.40) Finally taking the traces in Lorentz and internal indices we obtain the explicit expressions for the gauge anomalies

(8.41 ) Again we find the same result as in flat space-time, in (5.46). There are no new contributions from curvature [15, 16, 17] and thus the cancellation conditions are given in (5.47). - SU(2)L 1 I IJ.IIPO"W a B (D IJ. (·IJ.))a JL - - 327r 2gg E IJ.II pO"

x

(Nc L

YL

all quarks

+

L

YL).

(8.42)

all leptons

As in the previous case the result is the same as in flat space-time (5.48). Hence, the cancellation condition is given in (5.49)

- U(l)y 'IJ.) _ 1 ( D IJ. (Jy - 327r 2

x

-

1 PO"-Y0R RIJ.II 24 E IJ.IIPO"-yO

[Nc L

all quarks

2

(YL - YR) +

a +g8 ElJ.lIPO"G a 2 IJ.II G pO"

""' ~

all quarks

L

all leptons

(y L _ YR )

(YL - YR)] 2

a Wa + fL4 EIJ.lIPO"wIJ.II pO"

240

8. Gravity and the Standard Model

L

[NC

x

YL

all quarks

[Nc L

X

L

+

YL]

+ g,2€I'V p a Bl'vBpa

all leptons

(yi -

all quarks

Y~) + L

(yi -

all leptons

Y~)] ) .

(8.43)

Notice the appearance of terms depending on the curvature, which did not occur in the case of non-Abelian gauge fields. The new terms that were not present in flat space-time impose a new cancellation condition, namely, the vanishing of the sum of all hypercharges (8.44) all quarks

all leptons

In addition, we have that the cancellation of the terms already present in flat space-time gives the same conditions as in (5.51). 8.3.3 Gravitational Anomalies

As we have mentioned in Sect. 8.2, the EP states that any theory in curved space-time should be invariant under local Lorentz transformations. In this section, we consider the possible violation of this symmetry due to quantum effects when chiral fermions are present. We will conclude that whenever Abelian chiral gauge fields are present, as it is the case of the hypercharge field, local Lorentz invariance is broken [16]. However, due to the specific hypercharge assignment in the SM this anomaly is exactly cancelled. The condition for the cancellation of the Lorentz anomaly is the same as that in (8.44) for the cancellation of terms depending on curvature in the U(l)y anomaly. As we saw in Sect. 8.2, the classical Dirac Lagrangian in curved space-time is invariant under the 50(4) transformations in (8.6) and (8.12). Therefore we can calculate the gravitational anomalies as gauge anomalies of the Lorentz group, the only difference is the appearance of an additional field, the vierbein, which also transforms under this group. Hence, the changes in all these fields are

1/;(p) ->e1emnex)Emn1/;(p) 1[;(p) ->1[;(p)e-1emnex)Emn ea -> ea _ €a (x)eb I'

I'

b

I'

r~ b->r~ b + €ac(x)r~b _ €cb(x)r~ c

-

0l'€ab(x) .

(8.45)

Following the same steps as in the previous section, we see that the effective action may have an anomalous variation under those transformations, given by

8.3 Anomalies in the Standard Model

W[st - DE, e - Ee, ... J = W[st, e, ... J + X

J

8W r c 8W c 8W ( \71-' 8r a b + al-' 8r c b - r I-'b 8r a c I-'

I-'

241

d4 xvgE ab (X)

-

Tab

)

(8.46)

,

I-'

r

where we have used W instead of to denote the effective action (in order to avoid confusion with the components of the spin connection r~ b) and Tab = ebl-'8W/ 8e aI-' is the expectation value of the energy-momentum tensor in the presence of background fields. Notice that, in general, this energymomentum tensor is not symmetric and therefore it may not coincide with that defined in Sect. 2.1. We can write this result more conveniently using

8W i 8r a b = -"'4 (1/;(/1-' E ab

i

.

+ E ab,I-')1/;) = -"2 (Jab) I-'

I-'

(jl-') = (jab E ab ) .

(8.47)

Therefore (8.46) reads

W[st - DE, e - Ee, ... J = W[st, e, ... J +

J

d4 xvgE ab(X)

(-~(DI-'(jl-') )ab -

(8.48)

Tab) ,

where now the explicit form of the SO( 4) gauge covariant derivative is DI-' = \71-' + [stl-" 1 In addition, the change in the effective action due to the change in the integration measure is given by e- w[n' ,e' ,... J

=

J

[d1/;d7Jj .. .leI d4X,j§.c.~e ~ I

ab 4 d x,j§(€ab(X)A (x»

,

(8.49)

where we have denoted by 1/; all the SM fermions and the anomaly yields (8.50) n

Here ¢>n and ~n are the eigenfunctions of the operators Hi'£. and H~'.c. respectively, which are given in (8.25). Finally, from (8.48) and (8.49), we find the anomalous Ward identity (8.51 ) The regularized result obtained following the same methods as in Chaps. 4 and 5 is (8.52) After a lengthy calculation we arrive to the final expression for the Lorentz anomaly

242

8. Gravity and the Standard Model

(8.53) Notice that pure gravity terms do not occur in agreement with the result that there are no pure gravitational anomalies in four dimensions [15, 18]. Observe also that all the terms depend on the Abelian B ab field, whereas there is no contribution from non-Abelian gauge fields. Finally, the cancellation condition agrees with that of (8.44), which ensures the vanishing of the gravity terms in the U(I)y anomaly and, as we have already commented, is satisfied in the SM.

8.3.4 Charge Quantization in the 8M We will finish this section by discussing the consequences of imposing the cancellation of the above gauge and gravitational anomalies for one family. As we saw in Sect. 5.3, the conditions for the cancellation of gauge anomalies in flat space-time, (5.51), together with the gauge invariance ofthe MSM Yukawa sector, (5.52), allows us to fix all the hypercharges up to a normalization constant. However, we have just seen that, in curved space-time, we have an additional constraint on the hypercharges, (8.44), coming both from the curvature terms in the U(I)y anomaly and from the local Lorentz anomaly. Within the MSM, this condition is compatible with the others. But we can take a different point of view and, without assuming any specific symmetry breaking sector, try to fix the hypercharges. Then, the three conditions coming from gauge invariance in flat space-time in (5.51) together with (8.44) form a set offour equations for five unknowns YL = y'L, Y't = yt, YR., Yn , y~. Let us try to solve the system explicitly and check whether they determine all the hypercharges up to a normalization factor. First, we note that the four equations can be reduced to a single one for two unknowns, namely (8.54) which, in turn, can be expressed in terms of one variable for y~ =J 0 (if y~ = 0 all the hypercharges vanish, which is unphysical) 1+

U) 3 + _21 ( YR y~ 6

( YR U) 2 y~

21 YR U= +_ 6 y~

0.

(8.55)

It is not difficult to see that there are three real solutions,

yU

1 ~ = -1, -2, -- . YR 2

The rest of the hypercharges can be obtained as follows:

(8.56)

8.4 The Effect of Matter Fields on Gravitation

1L

d

"2l(1L+d) YR YR,

e

v

-"23(U+d) YR YR

YL =YL = YL=YL =

Y~ = -3(YR

+ y~) .

243

(8.57)

Hence, there are only three possible sets of hypercharges, up to a global normalization factor. We have listed them in Table 8.1. The first solution, whose normalization is arbitrary, together with the usual weak isospin assignment Q = T 3 + Y implies that the right component of the electron is chargeless. The second set is the usual hypercharge assignment in the 8M, normalizing as usual Q£ = -1. The third solution, keeping the same normalization, leads to different electric charges for the left and right components of the quark fields and therefore to chiral electromagnetism. Table 8.1. Hypercharge assignments 1sf set 2nd set 3rd set

UR y 2/3 -1/3

UL

0 1/6 1/6

'OR -y -1/3 2/3

'OL

0 1/6 1/6

NL

0 -1/2 -1/2

t:R

0 -1 -1

t:L

0 -1/2 -1/2

To summarize, gauge invariance in flat space-time in the minimal 8M with one Higgs doublet is enough to fix all the hypercharges (up to a normalization factor). If we had not made any specific choice of symmetry breaking sector, we could have not determined the hypercharges completely. In curved spacetime, if we do not choose a specific symmetry breaking mechanism, gauge invariance implies that there are only three possible ways to assign hypercharges. The requirements of a charged electron or vector electromagnetism can be invoked to remove the two unphysical solutions.

8.4 The Effect of Matter Fields on Gravitation Up to now, we have considered the effect of gravitation on the 8M fermionic sector. In particular we have seen that some of the flat space-time anomalies are modified by curvature terms. In this section, we will take the inverse point of view and will try to account for the effect of the 8M matter fields on gravitation. Throughout this chapter, we have been working in the semi-classical approximation in which gravitation is considered as a classical background on which we define the quantum matter fields. Following this line, and as we did in Chap. 1, we can calculate the effective action for the metric tensor in our case, integrating out the matter fields, i.e, they will appear inside the loops, whereas the metric tensor is considered classically and only appears in the external legs. In general, it is not possible to obtain the complete expression

244

8. Gravity and the Standard Model

for the effective action. However, we can find an expansion in the number of derivatives of the metric tensor. By covariance, this expansion has to be a series in powers of the curvature and its derivatives over some dimensional parameter. For simplicity, we will study in this section the case of a scalar field with a non-minimal coupling ~R¢2 to the scalar curvature, although the procedure can be equally applied to higher spin fields. At the end of the section, we will consider the example of a conformally invariant theory, that corresponds to ~ = 1/6. The object we are interested in is the effective action for the metric tensor obtained by integrating out a massive scalar field e-r[g,.v] = J[d¢] exp( - J

where defining 0 = 0

1J

,

(8.58)

+ m 2 + ~R we can write

r[9I'v] =log(detO)-1/2

= -"2

d4x..j9~¢(O + m 2 + ~R)¢)

= -~Tr logO = -~ J

(-1

d 4 x..j9tr (47r)D/2

~m f::o

D-2n

d 4x..j9tr(xllogOlx)

T(n - D/2)an (x)) .

(8.59)

In the last step we have used the heat-kernel expansion in (C.93) in dimensional regularization with D = 4 - to. The first three heat-kernel coefficients for the scalar field case read [3]

(8.60)

The corresponding first three terms in the effective action are divergent in four dimensional space-time. Higher order terms, with six or more derivatives are finite. Note that a2(x), with four derivatives, is not present in the Einstein-Hilbert action in (8.1). That is, in order to absorb the above divergences, it is necessary to add a (cosmological) constant as well as other counterterms depending on R 2 and Rl'vRl'v. The Rl'vp"RI'VP" contribution can always be expressed in terms of the other two for asymptotically flat space-times, due to the fact that -4Rl'vp"RI'VP" + 16RI'vRI'V - 4R 2 is a total derivative in four dimensions. That is also the case of OR. The divergence proportional to ao(x) will be renormalized in the cosmological constant. That corresponding to al(x), which is proportional to R, will renormalize the Newton constant and finally, the a2(x) divergences will be absorbed in the coefficients of the two new counterterms. The numeric value of the renormalized constants can only be obtained from the experiment, in absence of

8.4 The Effect of Matter Fields on Gravitation

245

a more fundamental theory of gravity. According to the above discussion, a finite number of counterterms is enough to renormalize the theory. This is in contrast with the effective action for pure gravity, that we will discuss in next section. In that case, we expect an infinite number of divergences that cannot be absorbed in the redefinition of a finite number of parameters. Let us consider N v neutrinos, N Df Dirac fermions, N v vector fields, N gh ghosts and N sc scalar fields. For simplicity, we deal with free massless fields minimally coupled to gravity, except the scalars, which are still coupled as in (8.58). The only relevant contribution is now a2(x), which reads [3]

(8.61) where the coefficients corresponding to the vector gauge fields are given in the Feynman gauge (in general they are gauge dependent). Customarily the divergent part of the effective action, that we have just obtained, modifies the left hand side of the Einstein equations, due to the inclusion of O(R 2 ) terms. In addition, there is also a finite contribution to the effective action Ttin, which in general will be non-local and difficult to calculate. In this context it is useful to define a renormalized energy-momentum tensor 8Tfin (T/l-v)ren == 2-,,. (8.62) ug/l- V For a general space-time metric only approximate methods can be used to find these finite terms. Nevertheless, in some particular simple cases, as that of conformally invariant quantum fields on a conformally flat space-time, it is possible to find the exact finite contribution to the energy-momentum tensor from the knowledge of the trace anomaly [3]. This example may have a physical meaning in the high-energy limit in which all the 8M physical scales are negligible. In conclusion, we have seen that the effect of integrating out the matter fields on a curved space-time is the appearance of higher-derivative terms in the effective action for gravity. These new contributions to the effective action could affect the singularities appearing in the solutions of GR and even avoid them in some cases [19].

246

8. Gravity and the Standard Model

8.5 The Effective Action for Gravity 8.5.1 The Background Field Method in Quantum Gravity In this section we make a brief review of the background field method applied to quantum gravity [20]. We will follow the same steps as in Appendix C.3. This method will be useful to calculate the divergences in one and two-loop diagrams. In order to deal with the Einstein-Hilbert action in (8.1), we will use the so called second order formalism, where the basic field is the metric tensor gJ1.v. The scalar curvature R contains two derivatives of gJ1.V and therefore we will say that it is O(p2). Making variations with respect to the metric, we obtain the equations of motion, which are nothing but the Einstein field equations. As we will see in the next section, there is an alternative approach, in which the basic fields are the vierbein e~ and the connection r~ b, known as the Palatini formalism. In such case the Einstein-Hilbert action can be written in terms of first derivatives of the connection. In the background field method, we split the metric tensor in a background gp.v and a quantum fluctuation hJ1.v (the graviton). Since gJ1.V is dimensionless, in order to give dimensions to hJ1.v we have to write (8.63) The action in (8.1) is invariant under the following general coordinate transformations

8gJ1.V = fP(X)gp.v,p + gJ1.P
(8.64)

where 'V J1. is the covariant derivative corresponding to the full metric gJ1.v. These transformations can be distributed in background and quantum fields in an arbitrary manner. However there are two important cases known as • The quantum variations

8gJ1.v=0 8hJ1.v = (gpJ1. 'V v + gpv 'V J1.)fP .

(8.65)

• The background variations

8gJ1.v = 8hJ1.v =

fP gJ1.V,p

+ gJ1.P
fP hJ1.v,p

+ hJ1.p
.

(8.66)

The action in (8.1) is invariant under both transformations. In order to obtain a propagator for the graviton we have to break the invariance of the action under quantum variations. Analogously to the gauge case considered in Appendix C.3, we now introduce the background gauge-fixing term, which respects the invariance under background variations. A suitable gauge-fixing condition reads

8.5 The Effective Action for Gravity A

\l J1.h~ =

1

2\l v h J1.J1. A

247

(8.67)

'

where VJ1. is the covariant derivative associated to the background metric gJ1.v, In this gauge (called the harmonic gauge) and for a flat background metric 7]J1.V, the graviton propagator takes a very simple form t

DJ1.v,pu = 2 PJ1.V,PU q 1

PJ1.V,PU = 2(7]J1.p7]vu

+ 7]J1.u7]vp -

(8.68)

7]J1.v7]pu) .

The gauge-fixing action can be written as SCF

=

j

4 d xvg gJ1.V

(V phPv - ~ VvhPp) (V phPJ1. - ~ VJ1.hPp) .

(8.69)

Note here that indices are raised and lowered with the background metric. In addition, we have to write the Faddeev-Popov action (see Appendix C.2), which is given by S FP --

j

d4XV10. 1 gAJ1.Vn 9 c-P (A>.vn 9 v >.gAJ1.P - 2 v P) ( guJ1." v v + gUY "v J1. ) Cu ,

(8.70)

where cP and CU are the ghost fields. Now we define the generating functional of connected Green functions in the presence of the background fields eiZ[§,j] = j[dh][dc][dc]ei(SG(§+l
Jd

4

xh,.vj"v) .

(8.71)

Note that only the quantum fields couple to the external sources jJ1.v. Finally the generating functional of IPI Green functions is obtained by means of the Legendre transform r[g, It] = ZIg, j] -

j d4xltJ1.vjJ1.V ,

(8.72)

where we have defined the classical field as

It

-

J1.V -

bZ[g,j] bjJ1.V

(8.73)

Finally, r[g,O] is the gauge invariant (with respect to background transformations) effective action and generates IPI graphs with external background fields. The method just presented will be applied in the rest of this chapter.

8.5.2 General Effective Formalism As we mentioned at the beginning of this chapter, one of the main problems to find a quantum theory of gravity is that, as we will see, GR is not renormalizable. Traditionally this fact has been considered as an obstacle to make quantum predictions in GR. However, the effective Lagrangian formalism

248

8. Gravity and the Standard Model

turns out to be an appropriate technique to study the quantum behaviour of gravity at low energies. In fact, we have seen that whenever there is decoupiing, we do not need to know the high-energy dynamics in order to obtain low-energy predictions. When the theory does not decouple, the low-energy measurements provide useful information on the underlying theory which should also be valid at higher energies, although its effective Lagrangian is not renormalizable in the usual sense. As we will see in more detail below, one can consider GR as an effective· theory, only valid at low energies, in which the graviton is the relevant degree of freedom. The Divergences of Einstein Gravity As we have shown in Sect. 8.4, in order to absorb the divergences coming from matter loops in a gravitational background, we have been forced to introduce a finite number of higher order terms in the gravitational Lagrangian. With these new terms the theory is renormalizable. However when the gravitational field itself is considered as a quantum field, loops of gravitons and their corresponding ghosts come into play, giving rise to higher order terms which are not present in the Einstein-Hilbert action. Moreover, the more loops the higher the order of the terms. In fact, the one loop corrections to Einstein gravity are O(p4), analogously to those obtained from the two derivative term in the low-energy pion Lagrangian in Sect. 6.4.2. The divergences in Einstein gravity were first obtained by 't Hooft and Veltman [20] using the background field method and dimensional regularization. In the following we give a brief review of their calculation. We want to obtain the divergences of the Einstein-Hilbert action in (8.1), which as we have just seen, is invariant under the gauge transformations given by (8.64). Moreover, using the background field method the whole effective action is invariant under the background field transformations, (8.66). Therefore, the counterterms should also be invariant under the latter transformations. In what follows we will study the one-loop divergences and thus we are only interested in O(p4) operators. With these two restrictions the most general form of the regularized counterterms is

V9 (al R'2 + a2 R'J.LV R'J.LV + a3 R'J.Lvpa R'J.Lvpa) . S div -- Jd D x-f.-

(8.74)

Notice that, in contrast with Appendix A, we have not parametrized the dimensional regularization divergences by N f but by l/f., where D = 4 - f.. Taking into account that -4RJ.LvpaRJ.Lvpa + l6RJ.LvRJ.LV - 4R2 is a total divergence in four dimensions, then, up to its renormaIization, the counterterms can be written in terms of R 2 and RJ.LvRJ.LV only. Once we have the general form, let us obtain the precise value of the coefficients. With such purpose, we will use a result due to 't Hooft (see [20]), valid for the quadratic action

S

=

J

d4 xJ9 (-0J.L¢*gJ.LV ov¢

+ 2¢* NJ.L0J.L¢ + ¢* M¢)

,

(8.75)

8.5 The Effective Action for Gravity

249

where the 1> field is a column matrix and M and NJ-L are matrix-valued functions of space-time. As far as in the background field method we only have to consider diagrams without external 1> fields, the corresponding counterterms can be written as Sdiv

=~

J xyg d

xtr

4

(8.76)

[~2 (Q - ~R)2 + ~G GJ-LV + ~ (R RJ-LV _ ~R2)] 6 12 J-LV 60 J-LV 3 '

where,

GJ-Lv=8J-LNv - 8 v NJ-L + [NJ-L,NvJ Q=M - NJ-LNJ-L - 8J-LNJ-L .

(8.77)

In our case, the role of 1> is played by the graviton field hJ-Lv and the ghosts. The full action Sc + SCF + SF? contains linear terms in h which generate tadpoles, although they can be removed by shifting hJ-Lv - f hJ-Lv + 8hJ-Lv, with an adequate choice of 8hJ-Lv. Such procedure leads to an action quadratic in h which has the same form as (8.75). Thus, from the Einstein-Hilbert and gauge-fixing actions, we obtain

S2 =2 x

J xyg d

4

(-~4 'V' J-L h Pu'V'J-Lh P + ~8 'V' J-L h PP'V'J-Lh + ~h XPJ-Lh V ) 2 P uv J-L U

U

U

U

(8.78)

with

~XPJ-L = -~g- Pg-J-L R + ~g'P g'J-L R - ~g-P RJ-L + ~g-P RJ-L + ~RPJ-L 2 uv 8 U v 4 v U 2 v U 2 U v 2 uv·

(8.79)

Adding SF? in (8.70), which is already quadratic in the ghosts fields, we can now use the result in (8.76) to calculate the divergences. The one-loop result for the divergent Lagrangian is

1 (1'R + 7-RJ-L RJ-Lv

(1)

£div = (4'Il-)2E

120

2

20

v -

)

.

(8.80)

On shell, Le. for background fields satisfying the classical equations of motion which at O(p2) are -

1 -

RJ-Lv - "2R9J-LV

= 0,

(8.81)

£~~~ vanishes. Therefore as a QFT, Einstein gravity is finite at one loop. At first, this result led to think that higher order corrections might also vanish, due to some hidden symmetry and that, in the end, quantum gravity would be a consistent theory. But this is not the case since when matter fields are taken into account, divergences are present. Even more, it has been shown

250

8. Gravity and the Standard Model

that the two-loop counterterms do not vanish on shell for pure gravitation [21]. In fact, their contribution is given by £y) div-

209

RIJ.V RP<7 k;>.{3 p<7 ex.{3 /LV·

360(47f)3M~€

(8 8 ) . 2

Thus, the first non-vanishing divergence is O(p6). An increasing number of loops presumably yields higher order divergences and therefore we expect to need an infinite number of counterterms to absorb them.

The Low-Energy Effective Action Therefore, we face a similar problem as that of the Fermi-Feynman-GellMann model of weak interactions. The Einstein theory is not renormalizable in the usual sense. Also in this case the effective formalism is a useful tool to study the low-energy dynamics of the theory. We have seen in Chap. 1 that the effective Lagrang~an could be obtained from a more fundamental theory as a low-energy limit. In our case, since such a theory is unknown, we have to rely on symmetry considerations. Thus, we can consider the Einstein-Hilbert Lagrangian, with two derivatives and one M~ factor, as the first term of an expansion in the number of derivatives over Mp [22]. In addition we would have terms with four derivatives and one M~ = 1 factor, with six derivatives and one M p2 factor, etc. The form of all these operators is determined just by the number of derivatives and general covariance. The value of the dimensionless constants multiplying them will be determined by the underlying theory of gravity. The new terms not only give information about the underlying theory, but also they absorb the possible loop divergences. Hence the low-energy expansion for the gravity effective action will have the following form: Seff =

J

d4xy'g ( -

~~ R + aR2 + j3R/LvR/LV + O(p6)) .

(8.83)

We recall that the O(p4) terms vanish for pure gravity. However, they have to be considered when matter fields are included, as in the next section, since the equations of motion are modified. In principle, the undetermined renormalized constants could be obtained from experiments or observations. Once they were fitted, this effective action could be used to make new predictions. Obviously, this approach presents many difficult problems to be solved. First, we have phenomenological problems. As the higher derivative terms are suppressed by powers of Mp they become practically unobservable. These terms can only be relevant when the curvature is very large. Typically this happens in regions close to singularities of the classical solutions to Einstein gravity. This fact explains the poor experimental bounds on the a and j3 parameters in the effective Lagrangian [23].

8.5 The Effective Action for Gravity

251

Second, from the theoretical point of view, the calculations become very hard. Note that even though we know the two-loop divergent parts of the effective action, we do not know its one-loop finite contributions. In general, these finite parts are non-local, since gravitons are massless. However, as it is usual within the effective Lagrangian approach, we do not need to know the high-energy behavior to obtain predictions at low energy. Indeed we will see in the next section that the non-local contributions give rise to the leading quantum corrections to the Newton potential at low energies, compared with M p [24]. When dealing only with the Einstein-Hilbert action we obtain, at one loop, divergences proportional to polynomials in q2 j M~ as well as some non-analytic finite terms which typically contain log( _q2) factors, coming from non-local terms in the effective action. Once again, the divergences will be absorbed in the renormalization of the parameters Q, {3, etc, which somehow encode the information on the unknown underlying theory. However, the non-analytic finite terms cannot be absorbed in any parameter (since they come from non-local terms) and, as a consequence, they can be considered as pure and predictable low-energy effects. 8.5.3 Quantum Corrections to the Newton Potential The effective field theory techniques applied to GR can be used to obtain the leading long distance quantum corrections to classical gravity [24]. These corrections are independent of the short-distance, i.e. high-energy, behaviour of the theory. They only depend on the massless degrees of freedom (the graviton field) and their low-energy couplings, which according to (8.83) are given by the Newton constant G. In the following we will concentrate in the calculation of these corrections. In GR, the Newton potential is modified by higher order terms in GMjr, with M the typical scale of the masses, as follows: V( r ) = _ Gmlm2 r

(1 +a

G(ml + m2) + ... ) 2 rc

(8.84)

with ml and m2 the masses of the particles and a depending on the precise definition of the potential. Notice that in order to identify the origin of the different corrections, we have abandoned the natural units. The terms proportional to Q and (3 in (8.83) also modify this expression, but the effect of these O(p4) terms turns out to be short-ranged, falling like an exponential at long distances [23]. Moreover, the potential receives additional corrections from quantum loops in which we are interested. Let us then study the oneloop correction to the newtonian potential between spinless particles which are described by the Lagrangian (8.85)

252

8. Gravity and the Standard Model

Pl-l~ -P~ P2-

-i-

-P;

Fig. 8.1. Vertex and propagator corrections to the one graviton exchange

Replacing the expression for the metric tensor given in (8.63) with a flat background metric, i.e. 9IJ.v = TJIJ.V + ",hlJ. v in LM' we find up to 0(",2)

2 2 '" v "2 hlJ.

m LM = "21 81J.¢81J.¢ - T¢ -

+",2

(~hlJ.),hV 2 ),

[

1 ), ¢ - m 2¢ 2] (81J.¢8v¢) - "2TJlJ.v(8),¢8 )

_ ~h),hIJ.V) 8IJ. ¢8v ¢ 2),

_~",2 (h),
(81J.¢81J.¢ - m 2¢2) .

(8.86)

From this expression we see that, at tree level, the gravitational interaction between scalar particles occurs by exchanging one graviton, as in Fig. 8.1. At one loop, this diagram receives scalar-graviton vertex corrections, Fig. 8.2, as well as graviton propagator corrections, Fig. 8.3, both of them represented as black dots in Fig. 8.1. The Feynman rule for the graviton-scalar vertex can be obtained from (8.86). From the Einstein-Hilbert, Faddeev-Popov and gauge fixing Lagrangians in (8.78) and (8.70) we get the graviton and ghost propagators, the three-graviton and the graviton-ghost vertex. The calculation of any matrix element M with the above Feynman rules shows that, in general, it will have the following form (8.87) where the dots stand for higher order terms in ",2 q 2. The divergences in the diagrams in Figs. 8.2 and 8.3 give rise to the ",2 q2 analytic terms in M. They have the same form as those coming from the tree-level contribution of

Fig. 8.2. One loop vertex corrections. The dashed line represents the scalar field

8.5 The Effective Action for Gravity

Illlllil

253

0000

Fig. 8.3. One loop propagator correction. The solid line represents the ghost field

the O(R 2 ) operators in (8.83) and, accordingly, they renormalize the Q' and /3 parameters. The finite part of the loop diagrams are non-analytic, their form being typically K,2 q2 Iog (_q2) or K,2 q2Jm 2/ - q2. These contributions are different from the analytic terms in the sense that they do not mix with any tree level contribution coming from higher order terms. In this sense, they only depend on the low-energy couplings of the theory, i.e. K,2. These are the kind of corrections we are interested in. Moreover, we see that they dominate over the analytic terms at small q2. Let us then consider the energy-momentum tensor TJ1.v defined as the terms coupled to one graviton in .eM, that is .eM =

~o 2 J1.'-I-I-' OJ1.-I'I-'

m2

2

_

VT !::.hJ1. 2 J1.V

+ O(K,2)

.

(88 . 8)

Let us parametrize the non-analytic contributions to the corrected graviton-scalar vertex as VJ1.V

=

(p'I T J1.vlp) = Fl(q2)(p~pv

+ PJ1.p~ + q2TJJ1.v) + F2(q2)(qJ1.qv

- TJJ1.vq2) ,

(8.89)

where q = p - p' is the momentum transfer and F 1(q2) and F 2(q2) are form factors (analogously to quantum electrodynamics), which read

~

q F 1 (q2) = 1 + K,2 2 (_ loge _q2) 327["2 4

+~

m 7["2 ) 16 _q2

J

+ ...

(8.90)

and q F2(q2) = K,2 2 327["2

m (_~3 loge _q2) + ~8 J_ 7["2 ) + ... 2

(8.91)

q

In a similar way the non-analytic contributions to the graviton propagator read p J1.V,a{3 IIa{3,"/6 p"/6,pa 2 K,2 q4 (21 1) = -loge -q ) 327["2 120 (TJJ1.pTJva + TJJ1.aTJvp) + 120 TJJ1.vTJpa, (8.92) where PJ1.v,a{3 is defined in (8.68) and IIa{3,,,/6 are the non-analytic terms of the diagrams in Fig. 8.3. Finally, gathering all these terms, we can evaluate the diagram in Fig. 8.1 describing the interaction of two particles of masses ml and m2. We take the nonrelativistic limit p = (m,O), q = (0, q) in order to define the potential

254

8. Gravity and the Standard Model

The potential is obtained as the Fourier transform of the above expression, and reads V(r)

=

Gli)

Gmlm2 (1 _ G(ml + m2) _ 127 r rc2 301T 2 r 2 c3

.

(8.94)

The first term in the correction is a classical contribution as shown in (8.84) since it does not have an li factor. Only the second one is truly a quantum effect, although its numerical value is completely negligible. As we have already mentioned, the contributions from divergent parts are proportional to the local term ",2 q2, whose Fourier transform behaves like a delta function. Therefore such contributions are smaller at a finite distance than any powerlike correction and their effects can be ignored. Higher order terms in the effective action will also contribute as ",2 q 2, but proportionally to ex and (3. These parameters are unknown and thus if these terms were dominant, we could not make any prediction. However, the very same reasoning on the ",2 q2 behavior ensures that they can be safely neglected. In contrast, the dominant non-analytic terms are independent of the unknown parameters of the low-energy effective action. We remark that their Fourier transforms are proportional to 1/r 2 or 1/r 3 and therefore they are the only contributions to the long-distance corrections. 8.5.4 Perspectives and Other Approaches In the previous section we have considered the effective theory for gravity in the second order formalism. However, there is an alternative method to derive the Einstein equations which is known as the first-order Palatini formalism. From this point of view, the fundamental fields are the vierbein e~ and the metric connection r~ b, which is not assumed to be Levi-Civitao We can write the Einstein-Hilbert action in terms of these variables as

M2J

S = _-.E. 161T

where R

d4 x e e lJoa eVb R abIJoV

e= I det e~1

ab IJoV

=

ar IJo

(8095)

,

and

a b v -

av r a b + r a r cv b - r avc rc b IJo

IJoC

IJo

0

(8.96)

Making variations with respect to the connection we obtain, for pure gravity, that (8.97)

8.5 The Effective Action for Gravity

255

This equation can be solved for the connection in terms of the vierbein and its derivatives and implies the vanishing of torsion in vacuum, i.e. this equation constraints the connection to be Levi-Civita. In addition, the variation with respect to the vierbein yields Rpm -

~gPm R = 0 ,

(8.98)

which is an equation for Fe:. b. When substituting the expression of the LeviCivita connection obtained from (8.97), we derive the usual Einstein equations for the vierbein and hence for the metric tensor. In case scalar or gauge fields were present (minimally coupled to gravity) then (8.97) would not change, since they do not couple to torsion; however, in the presence of Dirac fermions (8.97) no longer holds and torsion does not vanish. The discussion above strongly suggests that a more general formalism for a phenomenological description of gravity at low energy should consider Fe:. band eJ.l.a as independent quantum fields. Given a point in space-time, the associated tangent space can be considered as some kind of internal space. Local Lorentz transformations, i.e. the 80(4) group for Euclidean signature, would be rotations on this space. Within this approach the requirement of local Lorentz invariance in GR can be naturally described as an 80(4) gauge invariance and, in fact, this was our point of view when we studied the SM gravitational anomalies. Following this philosophy, Fe:. b should be considered as a gauge vector potential where a and b play the role of internal (tangent space) indices. Moreover, e~ could be understood as a vector matter field in the fundamental representation of the Lorentz group 80(4). The next step is the definition of the action, which should be a functional of the connection and the vierbein. It will determine the classical and quantum dynamics of its corresponding theory of gravity. As a first try, we could choose the Einstein-Hilbert action, i.e. the integrated scalar curvature. At the classical level it leads to the Palatini formalism described above and it is compatible with all the present experimental data. Another possibility is based in the analogy of gravity with an 80(4) gauge theory. This analogy would suggest to try something like R abJ.l.//Ra 6// as the gravitational Lagrangian which would correspond to the Yang-Mills Lagrangian. However, there are some differences. First, the tangent space is not really an internal space, since it has a very obvious space-time geometrical meaning. Second, the Yang-Mills Lagrangian does not have any dimensional parameter as the Planck mass in the Einstein-Hilbert action, which plays a very important role setting the scale of gravitational phenomena. Remember that in the former case the mass scale is introduced through the trace anomaly, since the classical theory is scale invariant. Within the Palatini formalism one could also try to follow the effective Lagrangian philosophy. This program has not been developed in detail at the present moment. However, it is known that we should consider the most general locally Lorentz invariant Lagrangian, although written in terms of the

256

8. Gravity and the Standard Model

connection and the vierbein, instead of the metric. Moreover, as far as the system has a gauge SO( 4) symmetry, the proper definition of perturbation theory requires the introduction of ghost fields. After this step, the relevant symmetry is not the SO( 4) gauged symmetry, but the associated BRS symmetry. Therefore, the most general effective Lagrangian should include all the BRS invariant operators [25] up to a given number of derivatives. Another interesting idea for the description of gravitational dynamics is to interpret the metric tensor as the NGB associated to some spontaneous symmetry breaking. For instance, in the literature, it has been considered the coset space GL(4)jSO(4) [26,27]. Of course it is still not clear whether the proper formulation of quantum gravity will require going beyond QFT, maybe to string theory [28], or a deeper modification of quantum mechanics. Even if that is the case, we do consider that the effective Lagrangian formalism is well suited to obtain an appropriate low-energy description of many quantum gravity effects.

8.6 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

[22]

[23] [24] [25]

E. Alvarez, Rev. of Mod. Phys. 61 (1989) 561 M. Nakahara, Geometry, Topology and Physics, lOP Publishing, 1990 N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, 1982 N.D. Birrell and J.G. Taylor J. Math. Phys 21 (1980) 7 LL. Buchbinder, S.D. Odintsov and LL. Shapiro, Effective Action in Quantum Gravity, lOP Publishing Ltd, 1992 S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, 1972 P. Ramond, Field Theory, Addison-Wesley, 1989 F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393 A. Dobado and A.L. Maroto, Phys. Rev. D54 (1996) 5185 LT. Drummonds and S.J. Hathrell, Phys. Rev. D22 (1980) 343 B.S. DeWitt, Dynamical Theory of groups and fields, Gordon and Breach Science Publishers, 1965 S. Yajima, Class. Quantum Grav. 5 (1988) L207 L. Ibanez, Proceedings of the 5th ASI on Techniques and Concepts in High Energy Physics. Plenum Press (1989) T. Eguchi, P.B. Gilkey and A.J. Hanson, Phys. Rep 66 (1980) 213 L. Alvarez-Gaume and E. Witten, Nucl. Phys. B234 (1983) 269 H.T. Nieh, Phys. Rev. Lett. 53 (1984) 2219 S. Yajima and T. Kimura, Phys. Lett. B173 (1986) 154 L. Alvarez-Gaume and P. Ginsparg, Ann. Phys. 161 (1985) 423 A. Dobado and A.L. Maroto, Phys. Rev. D52 (1995) 1895 G. 't Hooft and M. Veltman, Ann. Inst. H. Poincare 20 (1974) 245 M.H. Goroff and A. Sagnotti. Nucl. Phys. B266 (1986) 709 A. Dobado and A. Lopez, Phys. Lett. B316 (1993) 250 K.S. Stelle, Gen. Rel. Grav. 9 (1978) 353 J.F. Donoghue, Phys. Rev D50 (1994) 3874, Phys. Rev. Lett. 72 (1994) 2996 D.M. Capper, J.J. Dulwich and M. Ramon Medrano, Nucl. Phys. B254 (1985) 737

8.6 References

[26]

[27]

[28]

257

W. Misner, Phys. Rev. D18 (1978) 4510 V.I. Ogievetski, Lett. Nuovo Cimento 8 (1973) 988 N. Nakanishi and I. Ojima, Phys. Rev. Lett. 43 (1979) 91 M.B. Green, J.H. Schwarz and E. Witten Superstring theory, Cambridge University Press, 1987

Subject Index

anomalous dimension 283 anomaly 59 0 - 1r --> "t"t 63,64,91 - ABJ 63 - ambiguities 76 - axial 60, 65 - baryonic 109,237 - cancellation 105,239 - consistency conditions 72, 76 - consistent 71,73,78 - counterterms 76 - covariant 73, 75, 78 - gauge 68,238 - gravitational 240 - leptonic 109,237 - non-Abelian 78 - non-linear (J" model 85 - non-perturbative 83 - QCD 63,86 - regularization methods 73 - Topology 78 - trace 93 asymptotic freedom 126,286 background field method 246,293 baryon number 55 beta function 112 - QCD 127 Bianchi identities 268, 271 boson - gauge 50 - Higgs 2 - Nambu Goldstone 16,35,36,41 - - dynamics 45,48 -- large-N scattering 224 -- pseudo 37,129 - - scattering 306 -- would be 176 BRS transformations 291 - renormalized 204

Cabibbo angle 105 Cabibbo-Kobayashi-Maskawa matrix 103 characteristic classes 275 charge - conserved 24 - electric 100 charge quantization 242 Chern-Simons form 276 chiral fermions 17 chiral Lagrangian - O(p4) 135,182 - leading order 130, 179 - renormalization 136,186 Chiral Perturbation Theory 135 - Nf = 2 149 - unitarity 151 conformal group 28 connection - affine 265 - Levi-Civita 267 - metric 266 coset 42 - topology 54 counterterms - non-linear (J" model 53 covariant - derivatives 50 - formalism 43,185 CP problem 119 CP violation - strong 113 - weak 103 current - axial 60, 128 - conserved 24 - Noether 24 - vector 60, 128 curvature 268

312

Subject Index

Dashen Conditions 37,38 decoupling - chiral fermions 17 - spontaneous symmetry breaking 14 - theorem 9 dilatations 27 dimensional regularization 6,47,262 dimensional transmutation 287 Dirac - Lagrangian 231,261 - matrices 259,261 - spinors 261 dispersion relations 153,220,303 divergences 3 - General Relativity 248 dual strength tensor 272 effective action 4 Einstein-Hilbert action 229 electroweak chiral parameters 184 - heavy Higgs model 191 - phenomenological determination 193 - QCD-like model 192 electroweak interactions 98 Equivalence Principle 230,231 Equivalence Theorem 201 - applicability 216 - - 0(9 2 ) effects 217 -- energy range 216 - effective Lagrangians 214 - unitary models 212 Euler - Euler-Heisenberg Lagrangian 11 - constant 7,263 - Euler-Lagrange equations 24 Faddeev-Popov term 52,181,290 fermions 98 fixed points 286 Fujikawa method 67 functional measure 47 gauge

-

-

R~ 52, 181,291 anomaly 68 boson - scattering 197,212,218,224 - trilinear vertex 196 Feynman 245,291 fields 50,98,259, 270 fixing 52, 181 harmonic 247 invariance 51

- Landau 183 - t'Hooft 52, 181 Gell-Mann-Oakes-Renner formula 130,134 Gell-Mann-Okubo formula 130,142 General Relativity 229 Generalized Equivalence Theorem 210 generating functional 29,46 geodesic 266 ghost fields 52,291 Goldstone Theorem 33 gravitational anomalies 240 Green functions - amputated 208,284 - connected 29,49 - definition 29 - Euclidean 29 - renormalized 30,33,280 group - chiral 36 - conformal 28 - isometry 44 - Lorentz 26 - Poincare 26 GUT 112 Heat Kernel method 298 Higgs - boson 103,190,221 - mass 221,224 - mechanism 4,49, 175 - width 225 homogeneous space 43, 269 homotopy - class 54,278 hypercharge 99 - assignments 99,243 - matrices 99 identities - Bianchi 268,271 - Slavnov-Taylor 208 Ward 32,296 index - Dirac operator 68,277 - theorem 65,68, 277 infrared stable theory 286 instantons 117,274 invariance - gauge 51 - reparametrization 48 inverse amplitude method 154,220

Subject Index isometries 269 isospin 37 Killing vectors

44, 269

Lagrange - Euler-Lagrange equations 24 large-N 161 - linear (j model 221 - pion phenomenology 168 LEP 193 leptons 97 Levi-Civita connection 267 LHC 193,197 Lie bracket 265 Lorentz - group 26 - local transformations 233 low-energy constants 136 - large-Nc 145 - phenomenological estimates 142 - renormalization 139 - theoretical estimates 145 - values 144, 151 mass - decoupling 15 - fermion 176 - gauge boson 50, 175, 180 - Higgs boson 190,221,224 - mesons 141 - physical 30 - Planck 229 - quarks 39 - renormalization 9 - skyrmion 55 metric 263 - invariant 44 - tensor 263 - uniqueness 46 minimal coupling of gravity 235 minimal subtraction 281 mixing term 51 model - Fermi Feynman Gell-Mann 2,15 - Heavy Higgs 189 - linear (j 35,38,41 - - decoupling 15 -- large-N 168,221 - non-linear (j 36,41,134 - - anomalies 85 -- decoupling 16 - - divergences 48 - - dynamics 42

fermion coupling 45 - - gauged 50,53 -- generalized 45,47 - - geometry 42 - - Lagrangian 44, 131 - - large-N 163 - - quantum 46 - QCD-like 192 - standard 97 muon decay 1 Muskhelishvili-Omnes problem

313

305

Naturalness 177 Newton potential 251 Noether Theorem 24 oblique corrections on-shell conditions

194 187, 191

Palatini formalism 254 parallel transport 265 partial wave 302 PCAC 130 phase shifts 156,157 pion - 1r K scattering 307 - anomalous decay 63, 64, 91 - Nambu Goldstone boson 36,129 - scattering -- amplitudes 306 -- large-N 163,170 - - low-energy 132 -- phase shifts 157 Planck mass 229 Poincare Group 26 Principle of General Covariance 231 QCD 98 - axial anomaly 63 - anomalous processes 86,91 - chiral Lagrangian 130,135 - decoupling 10 - Lagrangian 125 - low energies 128 - skyrmions 54 - two flavor massless 36,39,54 QED - anomalies 60 - decoupling 11 quark condensate 38,39,130,134 quarks 97,126 Reduction Formula

28,49,208

314

Subject Index

relations - closure 44, 50 - commutation 42, 50 renormalization - on-shell 187 resonance - K* 159 - p 130,159 - Higgs boson 221,223 - techni-p 221 - unitarization 159 Ricci tensor 268 Riemann tensor 268 running coupling constants

112,126

S-matrix - reparametrization invariance 48 - reduction formula 30 saddle point method 4 Seeley-DeWitt expansion 67,299 self-dual configurations 272 skyrmions 54 Slavnov-Taylor Identities 205,208 solitons 54 spinors 261 Standard Model 97 - baryon anomaly 109,237 - lepton anomaly 109,237 - Symmetry Breaking Sector 101 - anomaly cancellation 105, 239 - charge quantization 242 - curved space-time 234 - heavy Higgs 189 - matter 97 - precision tests 194 - symmetries 121 - Symmetry Breaking Sector 175 strong CP problem 113 structure constants 50 symmetry - U(l)A 37 - breaking pattern 42 - BRS 53 - chiral 36 - classical 23 - explicit breaking 37 - gauge 49 - quantum 31 - spontaneous breaking 33 a models 41 - - decoupling 14

tensor - energy-momentum 27,241,245 - Ricci 268 - Riemann 268 theorem - Weinberg low-energy 48 - decoupling 9 - equivalence 201 - generalized equivalence 210 - Goldstone 33 - index 65, 277 - invariance 133 - Noether 24 theta vacuum 115 torsion 268 transformations - O(N) 16,41 - SU(2)L 99 - SU(2)L x SU(2)R 181,190 - SU(2)L x U(l)y 181,185 - SU(Nc ) x SU(2)L x U(l)y 100 - SU(Nc ) 98 - SU(Nf)L x SU(Nf)R 128 - U(l)y 99 - BRS 204, 291 - chiral 36 - conformal 28 - Lorentz 26, 232 - scale 27, 33 translations 26 triangle diagrams 61 Triviality 177,286 U(l)A problem 65 unitarity 4,151,302 unitarization 152 - 'Y'Y -+ 7m 159, 162, 170 - inelastic case 159 - inverse amplitude method - large-N 161

vacuum

-e

115

- choice 41 - classical 47 - degenerate vacua 37 - expectation value 17,34 - symmetry group 34 vertex - trilinear gauge boson 196 vielbein 264

154,220

Subject Index Ward identity 31 - anomalous 62 Weinberg - angle 100, 180, 187 - low-energy theorems

48, 132

Wess-Zumino-Witten term 89

18,57,86,

- SU(2) 92 - gauge fields 89 Wick rotation 261 winding number 79,274,278

Yukawa coupling

17,103

315

Effective Lagrangians for the Standard Model

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Mathematical perspectives on theoretical physics

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