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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Advanced Topics in Quantum Field Theory A Lecture Course
Since the advent of Yang–Mills theories and supersymmetry in the 1970s, quantum field theory – the basis of the modern description of physical phenomena at the fundamental level – has undergone revolutionary developments. This is the first systematic and comprehensive text devoted specifically to aspects of modern field theory at the cutting edge of current research. The book emphasizes nonperturbative phenomena and supersymmetry. It includes a thorough discussion of various phases of gauge theories, extended objects and their quantization, and global supersymmetry from a modern perspective. Featuring extensive cross-referencing from more traditional topics to recent breakthroughs in the field, it prepares students for independent research. The side boxes summarizing the main results, and over 70 exercises, make this an indispensable book for graduate students and researchers in theoretical physics. M. Shifman is the Ida Cohen Fine Professor of Physics at the University of Minnesota. He was awarded the 1999 Sakurai Prize for Theoretical Particle Physics and the 2006 Julius Edgar Lilienfeld Prize for outstanding contributions to physics.
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Advanced Topics in Quantum Field Theory A Lecture Course M. SHIFMAN University of Minnesota
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521190848 © M. Shifman 2012 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 2012 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Shifman, Mikhail A. Advanced topics in quantum field theory : a lecture course / M. Shifman. p. cm. Includes bibliographical references and index. ISBN 978-0-521-19084-8 (hardback) 1. Quantum field theory. I. Title. QC174.46.S55 2011 530.14 3–dc23 2011029847 ISBN 978-0-521-19084-8 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.
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
To Rita, Julia, and Anya
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Contents
Preface
References for the Preface
Acknowledgments Conventions, notation, useful general formulas, abbreviations Introduction
References for the Introduction
page xi xii
xiv xv 1
7
Part I Before supersymmetry 1 Phases of gauge theories
11
2 Kinks and domain walls
40
3 Vortices and flux tubes (strings)
90
1 Spontaneous symmetry breaking 2 Spontaneous breaking of gauge symmetries 3 Phases of Yang–Mills theories 4 Appendix: Basics of conformal invariance References for Chapter 1
5 Kinks and domain walls (at the classical level) 6 Higher discrete symmetries and wall junctions 7 Domain walls antigravitate 8 Quantization of solitons (kink mass at one loop) 9 Charge fractionalization References for Chapter 2
10 11 12 13 14
Vortices and strings Non-Abelian vortices or strings Fermion zero modes String-induced gravity Appendix: Calculation of the orientational part of the world-sheet action for non-Abelian strings References for Chapter 3
12 19 25 34 38
41 57 66 72 81 88
91 99 110 116 120 122
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Contents
viii
4 Monopoles and Skyrmions
123
5 Instantons
171
6 Isotropic (anti)ferromagnet: O(3) sigma model and extensions, including CP(N − 1)
248
7 False-vacuum decay and related topics
274
8 Chiral anomaly
298
9 Confinement in 4D gauge theories and models in lower dimensions
330
15 Magnetic monopoles 16 Skyrmions 17 Appendix: Elements of group theory for SU(N) References for Chapter 4 18 Tunneling in non-Abelian Yang–Mills theory 19 Euclidean formulation of QCD 20 BPST instantons: general properties 21 Explicit form of the BPST instanton 22 Applications: Baryon number nonconservation at high energy 23 Instantons at high energies 24 Other ideas concerning baryon number violation 25 Appendices References for Chapter 5
26 O(3) sigma model 27 Extensions: CP(N − 1) models 28 Asymptotic freedom in the O(3) sigma model 29 Instantons in CP(1) 30 The Goldstone theorem in two dimensions References for Chapter 6 31 False-vacuum decay 32 False-vacuum decay: applications References for Chapter 7
33 Chiral anomaly in the Schwinger model 34 Anomalies in QCD and similar non-Abelian gauge theories 35 ’t Hooft matching and its physical implications 36 Scale anomaly References for Chapter 8
37 Confinement in non-Abelian gauge theories: dual Meissner effect 38 The ’t Hooft limit and 1/N expansion 39 Abelian Higgs model in 1 + 1 dimensions 40 CP(N − 1) at large N 41 The ’t Hooft model 42 Polyakov’s confinement in 2 + 1 dimensions 43 Appendix: Solving the O(N) model at large N References for Chapter 9
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124 148 167 169 172 180 183 187 221 229 238 240 244
249 252 256 265 268 272
275 283 296 299 317 324 327 329
331 333 357 361 367 381 392 398
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Contents
ix
Part II
Introduction to supersymmetry
10 Basics of supersymmetry with emphasis on gauge theories
403
11 Supersymmetric solitons
560
44 Introduction 45 Spinors and spinorial notation 46 The Coleman–Mandula theorem 47 Superextension of the Poincaré algebra 48 Superspace and superfields 49 Superinvariant actions 50 R symmetries 51 Nonrenormalization theorem for F terms 52 Super-Higgs mechanism 53 Spontaneous breaking of supersymmetry 54 Goldstinos 55 Digression: Two-dimensional supersymmetry 56 Supersymmetric Yang–Mills theories 57 Supersymmetric gluodynamics 58 One-flavor supersymmetric QCD 59 Hypercurrent and anomalies 60 R parity 61 Extended supersymmetries in four dimensions 62 Instantons in supersymmetric Yang–Mills theories 63 Affleck–Dine–Seiberg superpotential 64 Novikov–Shifman–Vainshtein–Zakharov β function 65 The Witten index 66 Soft versus hard explicit violations of supersymmetry 67 Central charges 68 Long versus short supermultiplets 69 Appendices References for Chapter 10
70 Central charges in superalgebras 71 N = 1: supersymmetric kinks 72 N = 2: kinks in two-dimensional supersymmetric CP(1) model 73 Domain walls 74 Vortices in D = 3 and flux tubes in D = 4 75 Critical monopoles References for Chapter 11
Index
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404 406 413 415 422 428 445 446 450 453 456 459 470 475 477 482 497 498 506 528 531 533 538 541 546 547 555
561 567
582 593 602 608 613
616
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Preface
Announcing the beginning of a Big Journey. — Outlining the roadmap.
Quantum field theory remains the basis for the understanding and description of the fundamental phenomena in solid state physics and phase transitions, in high-energy physics, in astroparticle physics, and in nuclear physics multi-body problems. It is taught in every university at the beginning of graduate studies. In American universities quantum field theory is usually offered in three sequential courses, over three or four semesters. Somewhat symbolically, these courses could be called Field Theory I, Field Theory II, and Field Theory III although the particular names may (and do) vary from university to university, and even in a given university, as time goes on. Field Theory I treats relativistic quantum mechanics, spinors, and the Dirac equation and introduces the Hamiltonian formulation of quantum field theory and the canonical quantization procedure. Then basic field theories (scalar, Yukawa, QED, and Yang–Mills theories) are discussed and perturbation theory is worked out at the tree level. Field Theory I usually ends with a brief survey of the basic QED processes. Frequently used textbooks covering the above topics are F. Schwabl, Advanced Quantum Mechanics (Springer, 1997) and F. Mandl and G. Shaw, Quantum Field Theory, Second Edition (John Wiley and Sons, 2005). Field Theory II begins with the path integral formulation of quantum field theory. Perturbation theory is generalized beyond tree level to include radiative corrections (loops). The renormalization procedure and renormalization group are thoroughly discussed, the asymptotic freedom of non-Abelian gauge theories is derived, and applications in quantum chromodynamics (QCD) and the standard model (SM) are considered. Sample higher-order corrections are worked out. The SM requires studies of the spontaneous breaking of the gauge symmetry (the Higgs phenomenon) to be included. A typical good modern text here is M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, 1995). Some chapters from A. Zee, Quantum Field Theory in a Nut Shell (Princeton, 2003) and C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, 1980) can be used as a supplement. Field Theory III has no canonical contents. Generically it is devoted to various advanced topics, but the choice of these advanced topics depends on the lecturer’s taste and on whether one or two semesters are allocated. Sample courses which I have given (or have witnessed in other universities) are: (i) quantum field theory for solid state physicists (for critical phenomena conformal field theory is needed); (ii) supersymmetry; (iii) nonperturbative phenomena (broadly understood). In the first two categories some texts exist, but I would not say that they are perfectly suitable for graduate students at the beginning of their career, xi
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Preface
xii
nor that any single text could be used in class in isolation. Still, by and large one manages by combining existing textbooks. In the third category, the set of books with pedagogical orientation is slim. Basically, it consists of Rubakov’s text Classical Theory of Gauge Fields (Princeton, 2002), but, as can be seen from the title, this book covers a limited range of issues. A few topics are also discussed in R. Rajaraman, Solitons and Instantons (North-Holland, 1982). I moved to the University of Minnesota in 1990. Since then, I have lectured on field theory many times. Field Theory III is my favorite. I choose topics based on my experience and personal judgment of what is important for students planning research at the front line in areas related to field theory. The two-semester lecture course goes on for 30 weeks. Lectures are given twice a week and last for 75 minutes per session. The audience is usually mixed, consisting of graduate students specializing in high-energy physics or in condensed-matter physics. This “two-phase” structure of the audience affects the topic selection process too, shifting the focus towards issues of general interest. The choice of topics in this course varies slightly from year to year, depending on the student class composition and their degree of curiosity, my current interests, and other factors. Usually (but not always) I keep notes of my lectures. This book presents a compilation of these notes. The reader will find discussions of various advanced aspects of field theory spanning a wide range – from topological defects to supersymmetry, from quantum anomalies to false-vacuum decays. A few words about other relevant textbooks are in order here. None covers the full spectrum of issues presented in this book. Some parts of my course do overlap to a certain extent with existing texts, in particular [1–15]; however, even in these instances the overlap is not complete. The chapters of this book are self-contained, so that any student familiar with introductory texts on field theory could start reading the book at any chapter. All appendices, as well as sections and exercises carrying an asterisk, can be omitted at a first reading, but the reader is advised to return to them later. A list of references can be found at the end of each chapter.
References [1] M. Shifman, ITEP Lectures on Particle Physics and Field Theory (World Scientific, Singapore, 1999), Vols. 1 and 2. [2] R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1982). [3] V. Rubakov, Classical Theory of Gauge Fields (Princeton University Press, 2002). [4] Yu. Makeenko, Methods of Contemporary Gauge Theory (Cambridge University Press, 2002). [5] A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press, 2003). [6] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, 1994). [7] N. Manton and P. Sutcliffe, Topological Solitons (Cambridge University Press, 2004). [8] T. Vachaspati, Kinks and Domain Walls (Cambridge University Press, 2006). [9] J. Wess and J. Bagger, Supersymmetry and Supergravity, Second Edition (Princeton University Press, 1992).
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
References for the Preface
xiii
[10] J. Terning, Modern Supersymmetry (Clarendon Press, Oxford, 2006). [11] M. Srednicki, Quantum Field Theory (Cambridge University Press, 2007). [12] T. Banks, Modern Quantum Field Theory: A Concise Introduction (Cambridge University Press, 2008). [13] Y. Frishman and J. Sonnenschein, Non-Perturbative Field Theory (Cambridge University Press, 2010). [14] A. Smilga, Lectures on Quantum Chromodynamics (World Scientific, Singapore, 2001). [15] A. S. Schwarz, Topology for Physicists (Springer-Verlag, Berlin, 1994).
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Acknowledgments
This book was in the making for four years. I am grateful to many people who helped me en route. First and foremost I want to say thank you to Arkady Vainshtein and Alexei Yung, with whom I have shared the joy of explorations of various topics in modern field theory, some of which are described below. Not only have they shared with me their passion for physics, they have educated me in more ways than one. I would like also to thank my colleagues A. Armoni, A. Auzzi, S. Bolognesi, T. Dumitrescu, G. Dvali, A. Gorsky, Z. Komargodski,A. Losev,A. Nefediev,A. Ritz, S. Rudaz, N. Seiberg, E. Shuryak, M. Ünsal, G. Veneziano, and M. Voloshin, who offered generous advice. Dr Simon Capelin, the Editorial Director at Cambridge University Press, kindly guided me through the long process of polishing and preparing the manuscript. I am very grateful to Susan Parkinson – my copy-editor – for careful and thoughtful reading of the manuscript and many useful suggestions. I would like to thank Andrey Feldshteyn for the illustrations that can be seen at the beginning of each chapter. Alexandra Rozenman, a famous Boston artist, made her work available for the cover design. Thank you, Alya! Maxim Konyushikhin assisted me in typesetting this book in LATEX. He also prepared or improved certain plots and figures and checked crucial expressions. I am grateful to Sehar Tahir for help and advice on subtle aspects of LATEX. It is my pleasure to thank Ursula Becker, Marie Larson, and Laurence Perrin, who handled the financial aspects of this project. In the preparation I used funds kindly provided by William I. Fine Theoretical Physics Institute, University of Minnesota, and Chaires Internacionales de Recherche Blaise Pascal, France. Without the encouragement I received from my wife, Rita, this book would have never been completed.
xiv
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Conventions, notation, useful general formulas, abbreviations ∂L and ∂R ← ∂) ∂( ↔
→
2D chiral derivatives, p. 116 The partial derivative differentiates everything that stands to the right (left) of it.
←
∂ =∂ − ∂ Dα , D¯ α˙ Spinorial derivatives, p. 459 Dµ = ∂µ − i g Aaµ T a ε αβ , εabc 2D and 3D Levi–Civita tensors, pp. 124, 407 µναβ ε Levi–Civita tensor in Minkowski space, p. 409; ε 0123 = 1 ηaµν , η¯ aµν ’t Hooft symbols, p. 185 η¯ α˙ , ξα Weyl spinors in 4D, p. 407 ˙ Fαβ , F¯ α˙ β Gauge field strength tensor in spinorial notation, p. 409 g µν = diag {+1, −1, −1, −1} Metric in Minkowski space γ µ, γ 5 Dirac’s 4D gamma matrices, p. 410 0,1 t,z γ or γ 2D gamma matrices, p. 412 Gaµν Gluon field strength tensor, p. 148 0 1 σ1 = , 1 0 0 −i Pauli matrices σ2 = , i 0 1 0 σ3 = 0 −1 (σ a )ij (σ a )pq = 2δiq δjp − δij δpq Completeness for the Pauli matrices εabc v c (σ a )ij σ b pq = −i[( v σ )ij δpq − 2 ( v σ )iq δjp + ( v σ )pq δij ], Useful relation for the Pauli matrices; v is an arbitrary 3-vector ˙
(σ µ )α β˙ , (σ¯ µ )βα sign p = ϑ(p) − ϑ(−p) τµ± ( τ )αβ , ( τ )α˙ β˙
4D chiral σ matrices, pp. 408 Step function ˙ Euclidean analogs of (σ µ )α α˙ and (σ¯ µ )αα , p. 185 Symmetric τ matrices for representations (1, 0) and (0, 1), p. 409
xv
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Cambridge University Press 978-0-521-19084-8 - Advanced Topics in Quantum Field Theory: A Lecture Course M. Shifman Frontmatter More information
Conventions, notation, useful general formulas, abbreviations
xvi
Ta
Generator of the gauge group; C2 (R), T (R), and TG are defined on p. 471 Table 10.10, p. 536 Supergeneralization of the gauge field strength tensor, p. 438 Non-Abelian superstrength tensor, generalizing Gaαβ , p. 473 Superpotential Coordinates in the chiral superspaces, p. 423
TG Wα , W¯ α˙ Wαa W µ xL,R
Abbreviations ADHM ADS AF ANO ASV BPS BPST CC CFIV χ SB CMS CP DBI DR FI GG GUT IA IR LSP NSVZ PV QCD QED QFT QM SG SM SPM
Atiyah–Drinfel’d–Hitchin–Manin Affleck–Dine–Seiberg asymptotic freedom Abrikosov–Nielsen–Olesen Armoni–Shifman–Veneziano Bogomol’nyi–Prasad–Sommerfield Belavin–Polyakov–Schwarz–Tyupkin central charge Cecotti–Fendley–Intriligator–Vafa chiral symmetry breaking curve(s) of the marginal stability CP-invariance; also complex projective space Dirac–Born–Infeld dimensional regularization Fayet–Iliopoulos Georgi–Glashow grand unified theory instanton–anti-instanton infrared lightest supersymmetric particle Novikov–Shifman–Vainshtein–Zakharov Pauli–Villars quantum chromodynamics quantum electrodynamics quantum field theory quantum mechanics sine-Gordon standard model superpolynomial model
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Conventions, notation, useful general formulas, abbreviations
xvii
SQCD SQED SSG SUSY SYM TWA UV VEV WKB WZ WZNW
supersymmetric quantum chromodynamics, super-QCD supersymmetric quantum electrodynamics, super-QED super-sine-Gordon supersymmetry, supersymmetric supersymmetric Yang–Mills (theory) thin wall approximation ultraviolet vacuum expectation value Wentzel–Kramers–Brillouin Wess–Zumino Wess–Zumino–Novikov–Witten
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