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Cambridge University Press 978-0-521-19702-1 — Supersymmetry, Supergravity, and Unification Pran Nath Frontmatter More Information

SUPERSYMMETRY, SUPERGRAVITY, AND UNIFICATION

This unique book gives a modern account of particle physics and gravity based on supersymmetry and supergravity, two of the most significant developments in theoretical physics since general relativity. It begins with a brief overview of the history of unification and then goes into a detailed exposition of both fundamental and phenomenological topics. The topics in fundamental physics include Einstein gravity, Yang–Mills theory, anomalies, the standard model, supersymmetry, and supergravity and the construction of supergravity couplings with matter and gauge fields as well as computational techniques for SO(10) couplings. The topics of phenomenological interest include implications of supergravity models at colliders, CP violation, and proton stability, as well as topics in cosmology such as inflation, leptogenesis, baryogenesis, and dark matter. The book is intended for graduate students and researchers seeking to master the techniques for building unified models.

pran nath is Matthews University Distinguished Professor of Physics at North- eastern University in Boston. He has made pioneering contributions to the development of supergravity theory and its applications to particle physics including the mSUGRA model. He has also contributed to the development of effective Lagrangian methods, pion physics, current algebra, solving the U(1) problem, dark matter and proton stability in unified models. He is the founding chair of the Supersymmetry and Unification of Fundamental Interactions (SUSY) con- ference and the Particles, Strings and Cosmology (PASCOS) symposium. He is a Fellow of both the American Physical Society and of the American Associa- tion for the Advancement of Science, and a recipient of the Humboldt Prize in physics.

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Supersymmetry, Supergravity, and Uniﬁcation

PRAN NATH Northeastern University, Boston

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www.cambridge.org Information on this title: www.cambridge.org/9780521197021 c Pran Nath 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Names: Pran Nath, 1939– author. Title: Supersymmetry, supergravity, and unification / Pran Nath, Northeastern University, Boston. Other titles: Cambridge monographs on mathematical physics. Description: Cambridge, United Kingdom; New York, NY : Cambridge University Press, [2017] | Series: Cambridge monographs on mathematical physics | Includes bibliographical references and index. Identifiers: LCCN 2016026402 | ISBN 9780521197021 (hardback ; alk. paper) | ISBN 0521197023 (hardback ; alk. paper) Subjects: LCSH: Supersymmetry. | Supergravity. | Unified field theories. Classification: LCC QC174.17.S9 P73 2017 | DDC 530.14/23–dc23 LC record available at https://lccn.loc.gov/2016026402 ISBN 978-0-521-19702-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To: Kedar Nath Sastri and Tara Devi and Shashi, Rishi, Anjali

Contents

Preface page xv

1 A Brief History of Uniﬁcation 1 References 17

2 Gravitation 20 2.1 The Equivalence Principle 20 2.2 The Ricci Tensor and Curvature Scalar 30 2.3 The Einstein Field Equations 31 2.4 Variation of the Einstein Action 31 2.5 The Stress Tensor 32 2.6 The Linearized Field Equations of Einstein Gravity 34 2.7 Motion of a Particle in a Non-inertial Frame 35 2.8 The Newtonian Limit 36 2.9 Gauge Transformations of the Einstein Metric 39 2.10 The Palatini Variational Method 40 2.11 The Vierbein Formalism 41 2.12 Modiﬁcations of the Einstein Theory 45 2.13 Parallel Transport 45 2.14 Problems 47 References 50 Further Reading 51

3 Non-abelian Gauge Theory 52 3.1 Yang–Mills Gauge Theory 52 3.2 Canonical Quantization of the Maxwell Theory 55 3.3 Canonical Quantization of the Yang–Mills Theory 57 3.4 Gribov Ambiguity 59 3.5 Path Integral Quantization of Yang–Mills Theory 62 3.6 BRST Transformations 66 3.7 Quantum Yang–Mills Theory 68 3.8 Problems 69 References 71 Further Reading 71

x Contents

4 Spontaneous Breaking of Global and Local Symmetries 72 4.1 Spontaneous Breaking of Abelian Symmetries 72 4.2 Spontaneous Breaking of Non-abelian Symmetries 75 4.3 Problems 78 References 78 5 The Standard Model 79 5.1 Electroweak Sector of the Standard Model 79 5.2 The Strong Interaction Sector of the Standard Model 86 5.3 Problems 87 References 89 Further Reading 89 6 Anomalies 90 6.1 Introduction 90 6.2 The Path Integral Approach 92 6.3 The Wess–Zumino Consistency Conditions 96 6.4 The BRST Approach and the Wess-Zumino Consistency Conditions 97 6.5 Application of Consistency Conditions to Determine Anomalies 99 6.6 Mixed-gauge and Gravitational and Purely Gravitational Anomalies 104 6.7 Anomalies and Differential Geometry 105 6.8 t’Hooft Matching Conditions 107 6.9 Anomaly Constraints for Unified Models 107 6.10 The Standard Model Anomalies 110 6.11 Problems 111 References 113 7 Effective Lagrangians 116 7.1 The Effective Lagrangian and Current Algebra 123 7.2 The Chiral U(3) × U(3) Current Algebra 126 7.3 The UA(1) Anomaly 127 7.4 The θ Dependence of the Effective Lagrangian 130 7.5 Appendix: The θ Dependence of the UA(1) anomaly 131 7.6 Problems 136 References 137 Further Reading 138 8 Supersymmetry 139 8.1 The Coleman–Mandula Theorem 139 8.2 Graded Lie Algebras 140 8.3 Two-component Spinors 142 8.4 Left Chiral Superfields 145 8.5 Right Chiral Superfields 148

Contents xi

8.6 Products of Chiral Superfields 149 8.7 Products of Chiral Left and Chiral Right Superfields 151 8.8 Construction of Supersymmetric Action Using Superfields 152 8.9 Vector Superfields 155 8.10 Vector Field Strength 158 8.11 The Fayet–Iliopoulos D Term 161 8.12 Superfields for Non-abelian Gauge Theories 161 8.13 Supersymmetric Non-abelian Gauge Field Strengths and Lagrangians 161 8.14 The Interaction between Matter and Gauge Fields 166 8.15 Supersymmetry Breaking 168 8.16 F-type Breaking of Supersymmetry 170 8.17 D-type Breaking of Supersymmetry 172 8.18 Spurion Fields 176 8.19 R-Symmetry 177 8.20 Linear Multiplets 179 8.21 Problems 180 References 182 Further Reading 183 9 Grand Unification 184 9.1 SU (4) ⊗ SU (2)L ⊗ SU (2)R Unification 184 9.2 SU (5) Grand Unification 186 9.3 Supersymmetric Grand Unification 196 9.4 SO(10) Grand Unification 201 9.5 Unification with Exceptional Groups 219 9.6 Quark-Lepton Masses 220 9.7 Vector Multiplets 223 9.8 Appendix 224 9.9 Problems 230 References 233 Further Reading 235 10 The MSSM Lagrangian 236 10.1 The Scalar Potential 240 10.2 The Fermion Sector 241 10.3 Gaugino–Higgsino Sector 242 References 246 Further Reading 246 11 N =1Supergravity 247 11.1 Supergravity with Auxiliary Fields 250

xii Contents

11.2 Tensor Calculus for Scalar and Vector Multiplets in Supergravity 252 References 255 12 Coupling of Supergravity with Matter and Gauge Fields 257 12.1 Problems 261 References 261 13 Supergravity Grand Unification 262 13.1 Spontaneous Breaking of Gauge Symmetry 263 13.2 Spontaneous Breaking of Supersymmetry 264 13.3 Mass Matrices 265 13.4 Generation of Soft Terms in Supergravity Models 267 13.5 Problems 272 References 272 14 Phenomenology of Supergravity Grand Unification 274 14.1 Radiative Breaking of SU (2)L × U (1)Y 277 14.2 The Super-GIM Mechanism 284 14.3 Higgs Bosons 285 14.4 Loop Corrections to the Higgs Boson Mass 286 14.5 Low-Energy Higgs Theorems 287 14.6 Diphoton Decay of the Higgs Boson 289 14.7 Hyperbolic Geometry: The Focus Point, Focal Curves, and Focal Surfaces 290 14.8 Non-universalities in Soft Breaking 292 14.9 Yukawa Unification 293 14.10 Supersymmetry at the Large Hadron Collider 295 14.11 Problems 298 References 299 Further Reading 303 15 CP Violation in Supergravity Unified Theories 304 15.1 Introduction 304 15.2 Violation of CP in the Standard Model 305 15.3 Evidence for CP Violation 306 15.4 CP Violation in Supersymmetric Theories 310 15.5 The EDM of an Elementary Dirac Fermion 313 15.6 The EDM of a Charged Lepton in Supersymmetry 314 15.7 The EDM of Quarks 315 15.8 The EDM Operator Contribution to the EDM of Quarks 315 15.9 Contribution of the Chromoelectric Dipole Moment to the EDM of Quarks 316 15.10 The EDM of the Quarks from the Purely Gluonic Dimension 6 Operator 317

Contents xiii

15.11 g-2 with Phases 319 15.12 Supersymmetry CP Phases and CP-Even–CP-Odd Mixing in the Neutral Higgs Boson Sector 324 15.13 Effect of CP Phases on Neutralino Dark Matter 328 15.14 Appendix: Renormalization Group Evolution of the EDM 330 15.15 Problems 332 References 332 16 Proton Stability in Supergravity Unified Theories 335 16.1 Proton Decay from Lepto-quarks 337 16.2 Proton Decay in Supersymmetric Theories 340 16.3 Effective Lagrangian for Nucleon Decay 345 16.4 Problems 349 References 350 17 Cosmology, Astroparticle Physics, and Supergravity Unification 353 17.1 Particle Properties in the Early Universe 357 17.2 Time and Temperature 363 17.3 Supersymmetry and Dark Matter 363 17.4 Inflation 372 17.5 Baryogenesis and Leptogenesis 389 17.6 Problems 399 References 401 Further Reading 403 18 Extended Supergravities and Supergravities from Superstrings 404 18.1 Supergravity in Higher Dimensions 405 18.2 Supergravities from Superstrings 409 18.3 Four-Dimensional String Models 421 18.4 Gravitino Decay 423 18.5 Problems 424 References 425 Further Reading 427 19 Specialized Topics 428 19.1 Models with Extra Dimensions 428 19.2 Gravity in Extra Dimensions 432 19.3 Stueckelberg Mechanism in Supergravity and in Strings 432 19.4 Stueckelberg Extension of the Standard Model 435 19.5 Supersymmetric Extension of the Stueckelberg Model 438 19.6 Kinetic Mixing 439 19.7 Problems 441 References 441

xiv Contents

20 The Future of Uniﬁcation 443 21 Appendices 446 21.1 Exterior Diﬀerential Forms 446 21.2 Two-component Notation 450 21.3 Grassmann Co-ordinates 451 21.4 Geometry on Supermanifolds 456 21.5 Group Theory 464 21.6 Couplings of N = 1 Supergravity with Matter and Gauge Fields 477 21.7 Further Details on the Couplings of the MSSM/Supergravity Lagrangian 484 Further Reading 488 22 Notation, Conventions, and Formulae 489 22.1 Lagrangians 489 22.2 U(3) Matrices 491 22.3 The Standard Model Relations 492 22.4 MSSM Fields and Quantum Numbers 493 22.5 SU(5) 493 22.6 The Fierz Rearrangement 494 22.7 Dimensional Regularization 495 Further Reading 498 23 Constants and Units 499 23.1 Physical Constants 499 23.2 Natural Units 500 23.3 Astrophysical Parameters 501 23.4 Cross-section Units 501 23.5 Numerical Constants 501 24 Further Reading 502 24.1 Field Theory and Gauge Theories 502 24.2 Supersymmetry and Supergravity 502 24.3 Relativity 504 24.4 Cosmology 504 24.5 Other Recommended Publications 504

Author Index 506 Subject Index 515

Preface

Over the past decades a remarkable evolution has occurred in the development of particle physics in relation to general relativity, astrophysics, and cosmology. Till the early 1960s, gravity was considered a force not strong enough to be a player along with the electroweak and the strong forces, and did not receive any significant attention in texts on particle theory till that time. However, these perceptions changed at a remarkably rapid pace in the second half of the last century, most directly due to the discovery of supersymmetry and supergravity and, later, string theory. Thus, supergravity theory and theories based on superstrings bring gravity into the mix in an intrinsic fashion. Models based on supergravity, specifically supergravity grand unification, indicate that electroweak phenomena are driven by supergravity-induced effects connecting in a remarkable way the Planck scale with the electroweak scale. Another way that the connection between particle physics and cosmology has deepened over the past decades is due to the experimental observations that the universe is com- posed mostly of dark matter and dark energy, and particle theory is the likely place to achieve an understanding of these phenomena. Thus, while dark energy could be sourced by a tiny cosmological term in the Einstein equations, models based on supersymmetry and supergravity provide dark matter candidates in the form of a neutral spin 1/2 particle (the neutralino) and a spin 3/2 particle (the gravitino). Further, the early history of the universe is driven by particle physics via phenomena such as baryogenesis and leptogenesis, which can provide the observed baryon excess in the universe. Thus, currently there exists an unprecedented degree of fusion of astrophysics, cosmology, and particle theory. Each of these disciplines has its own glorious history with texts that center on them. However, for a young researcher entering the field of particle theory, developing a workable knowledge of these diverse fields presents a challenge. This text is written with the hope that it may make that task a bit easier. The text is written with the assumption that the reader has a workable knowledge of field theory. Since the underlying theme of the book is unification, a number of inter- connected topics are discussed. These include Einstein and Yang–Mills theories, the standard model, supersymmetry and supergravity, and cosmology along with some specialized topics. The layout of the book is as follows. In Chapter 1 we give a brief history of unification and show that, purely on empirical grounds, one is

xvi Preface

guided towards the idea that the laws of nature are not random and disconnected but appear to flow from an underlying unified structure. The first part of the book discusses a variety of topics that are essential elements in understanding the main topics of the book – which are supersymmetry, supergravity, and the development of supergravity grand unification and its phenomenological applications. Thus, in Chapter 2 we discuss the Einstein theory of gravity because it is a suc- cessful theory of one of the fundamental interactions of physics and it is a gauge theory. In Chapter 3 we discuss the Yang–Mills gauge theory, which is a basic ingredient in the formulation of the standard model as well in the development of a grand unified theory (GUT). In Chapter 4, spontaneous breaking of global symmetries and the Goldstone phenomena are discussed. Also discussed is spontaneous breaking of local symmetries and the Higgs phenomenon. In Chapter 5 we utilize the frameworks of Chapter 3 and Chapter 4 to develop the standard model of electroweak interactions based on the gauge group SU(2)L ⊗U(1)Y and then extend it to include the SU(3)C gauge group, which describes the strong interactions involving quarks and gluons. One of the stringent constraints on model building is the constraint of anomaly cancellation, and we discuss this constraint in Chapter 6. In Chapter 7 we discuss effective Lagrangians, as they form the foundation of any field theoretical model of particle theory. The effective Lagrangians are a powerful technique and follow from simple principles, which are discussed at some length in Chapter 7. In Chapter 8 we discuss supersymmetry basics and the construction of supersymmetry-invariant Lagrangians. In Chapter 9 we discuss grand unification, which can unify the electroweak and the strong interactions within its framework. Here we consider the Pati–Salam group SU(4)C × SU(2)L × SU(2)R where we first encounter unification of quarks and leptons, with leptons treated as the fourth color. However, the Pati–Salam group is not fully unified since it is of the product form. The first fully unified group was proposed by Georgi and Glashow as the simple group SU(5). Later in this chapter we discuss SO(10), which is more unified in that it can accommodate a full generation of quarks and leptons within one irreducible representation. Calculational techniques for carry- ing out computations in SO(10) are also discussed in this chapter. In Chapter 10 we discuss the Lagrangian for the minimal supersymmetric standard model. In Chapter 11 we discuss pure N = 1 supergravity. However, in order to construct particle physics models we must couple N = 1 supergravity with N =1 matter. This is done in Chapter 12, where N = 1 matter supermultiplets are coupled to N = 1 gauge supermultiplets, which are then coupled to N =1super- gravity. The resulting Lagrangian is sometimes referred to as applied N =1 supergravity. The development of supergravity grand unification is discussed in Chapter 13. Here we first focus on the spontaneous breaking of supersymmetry and then on the generation of soft terms via gravity mediation. Resolution of the GUT and Planck scale hierarchy problem is also discussed here. Phe- nomenology based on supergravity grand unification is discussed in Chapter 14.

Preface xvii

The important topic of CP violation is discussed in Chapter 15. A hallmark of grand unified theories is that they lead to proton decay, and this topic is discussed in Chapter 16. Topics of astroparticle physics and cosmology and their overlap with particle physics are discussed in Chapter 17. In Chapter 18 we discuss supergravities in higher dimensions and supergravities arising from strings. In Chapter 19 we discuss specialized topics. These include models with extra dimensions and extensions of the standard model using the Stueckelberg mechanism. The Stueckelberg mechanism allows one to construct U(1) extensions of the conventional models without use of the Higgs mechanism. In Chapter 20, some speculations regarding the future of unification are given. In Chapter 21 many appendices with supplementary material related to the discussion in the main text are given. Notations, conventions, and some useful formulae are given in Chapter 22. Numerical values of the physical constants that enter into the discussion in the text are given in Chapter 23. In Chapter 24, a list of books and review articles is given which the reader may find useful for cross-referencing the material discussed in this text and for further reading. As mentioned at the beginning of the preface, this book is primarily geared towards young graduate students and researchers. The text grew out of several graduate-level courses that I taught over the years as well as summer school lectures that I have given at various institutions. I have benefitted greatly over the years from collaborations and discussions with a large number of colleagues. Chief among those are the late Richard Lewis Arnowitt and Ali Hani Chamseddine. I have also benefitted from many other colleagues, friends, and former students. These include K. S. Babu, Utpal Chattopadhyay, Daniel Feldman, WanZhe Feng, Haim Goldberg, Ilia Gogoladze, James Halverson, Tarek Ibrahim, Boris Kors, Zuowei Liu, Brent D Nelson, Jogesh Pati, Pavel Fileviez Perez, Raza Syed, Tomasz Taylor, and Michael Vaughn, among many others. I have also profited from discussions and communications with the late Norman Ramsay regarding the early history of CP violation discussed in Chapter 15. I would like to thank WanZhe Feng, James Halverson, Maksim Piskunov, and Brent D. Nelson for reading parts of the book and offering useful comments and suggestions. Help from Gregory Peim regarding figures in the book is acknowledged. Finally, I wish to acknowledge Dr Simon Capelin, Editorial Director (Physical Sciences) at Cambridge University Press, for his interest and patience while the book was being written.

Pran Nath