Field Theory: A Modern Primer [2nd ed.] 0201157721

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Second Edition

Field Theory: A Modern Primer

Second Edition

Field Theory: A Modern Primer

Pierre Ramond Physics Depa1'lment University of Florida Gainesville, Florida

Addison-Wesley Publi'shing Company, Inc. The Advanced Book Program Redwood City, California • Menlo Park, California Reading, Massachusetts • New York• Amsterdam Don Mills, Ontario • Sydney • Bonn • Madrid Singapore •Tokyo• San Juan • Wokingham, United Kingdom

Publisher: Allan M. Wylde Production Administrator: Karen L. Garrison Editorial Coordinator: Alexandra McDowell Electronic Production Consultant: Cheryl Wurzbacher Promotions Manager: Celina Gonzales

Library of Congress Cataloging-in-Publication Data Ramond, Pierre, 1943Field theory. (Frontiers in physics : V.74) Bibliography: p. Includes index. 2. Perturbation (Quantum dynamics). 1. Quantum field theory. Path. I. Title. II. Series. 88-16612 530.1 '43 QCl 74.45R35 1989 ISBN 0-201-15772-1

3. Integrals,

This book was prepared under the supervision of the author, using the T£X typesetting language. The final output is from an Apple Laser Writer. Copyright

©

1989 Addison-Wesley Publishing Company, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers. Printed in the United States of America. Published simultaneously in Canada. ISBN 0-201-15772-1 ABCDEFGHIJ-AL-898

TO MY GIRLS

Frontiers in Physics

David Pines/Editor Volumes of the Series published from 1961 to 1973 are not officially numbered. The parenthetical numbers shown are designed to aid librarians and bibliographers to check the completeness of their holdings. Titles published in this series prior to 1987 appear under either the W. A. Benjamin or the Benjamin/Cummings imprint; titles published since 1986 appear under the Addison-Wesley imprint. (1)

N. Bloembergen

(2)

G. F. Chew

(3)

R. P. Feynman

(4)

R. P. Feynman

(5)

(6)

L. Van Hove, N. M. llugenholtz, and L. P. Howland D. Pines

(7)

H. Frauenfelder

(8)

L. P. Kadanoff G. Baym

(9)

G. E. Pake

Nuclear Magnetic Relaxation: A Reprint Volume, 1961 S-Matrix Theory of Strong Interactions: A Lecture Note and Reprint Volume, 1961 Quantum Electrodynamics: A Lecture Note and Reprint Volume, 1961 The Theory of Fundamental Processes: A Lecture Note Volume, 1961 Problem in Quantum Theory of Many-Particle Systems: A Lecture Note and Reprint Volume, 1961 The Many-Body Problem: A Lecture Note and Reprint Volume,1961 The Mossbauer Effect: A Review-with a Collection of Reprints, 1962 Quantum Statistical Mechanics: Green's Function Methods in Equilibrium and Nonequilibrium Problems, 1962 Paramagnetic Resonance: An Introductory Monograph, 1962 [er. (42)-2nd edition]

vii

viii

FRONTIERS IN PHYSICS

(10)

P. W. Anderson

(11) (12)

S. C. Frautschi R. Hofstadter

(13)

A. M. Lane

(14)

R. Omnes M. Froissart E. J-. Squires

(15)

(19) (20)

H. L. Frisch J. L. Lebowitz M. Gell-Mann Y. Ne'eman M. Jacob G. F. Chew P. Nozieres J. R. Schrieffer

(21)

N. Bloembergen

(22) (23)

R. Brout I. M. Khalatnikov

(24) (25) (26) (27) (28)

P. G. deGennes W. A. Harrison V. Barger D. Cline P. Choquard T. Loucks

(29)

Y. Ne'eman

(30)

S. L. Adler R. F. Dashen A. B. Migdal J. J. J. Kokkedee A. B. Migdal V. Krainov R. Z. Sagdeev and A. A. Galeev J. Schwinger

(16) (17) (18)

(31) (32) (33) (34) (35)

Concepts in Solids: Lectures on the Theory of Solids, 1963 Regge Poles and S-1\fatrix Theory, 1963 Electron Scattering and Nuclear and Nucleon Structure: A Collection of Reprints with an Introduction, 1963 Nuclear Theory: Pairing Force Correlations to Collective Motion, 1964 Mandelstam Theory and Regge Poles: An Introduction for Experimentalists, 1963 Complex Angular Momenta and Particle Physics: A Lecture Note and Reprint Volume, 1963 The Equilibrium Theory of Classical Fluids: A Lecture Note and Reprint Volume, 1964 The Eightfold \iVay: (A Review-with a Collection of Reprints), 1964 Strong-Interaction Physics: A Lecture Note Volume, 1964 Theory of Interacting Fermi Systems, 1964 Theory of Superconductivity, 1964 (revised 3rd printing, 1983) Nonlinear Optics: A Lecture Note and Reprint Volume, 1965 Phase Transitions, 1965 An Introduction to the Theory of Superfluidity, 1965 Superconductivity of Metals and Alloys, 1966 Pseudopotentials in the Theory of Metals, 1966 Phenomenological Theories of High Energy Scattering: An Experimental Evaluation, 1967 The Anharmonic Crystal, 1967 Augmented Plane \Vave Method: A Guide to Performing Electronic Structure CalculationsA Lecture Note 'and Reprint Volume, 1967 Algebraic Theory of Particle Physics: Hadron Dynamics in Terms of Unitary Spin Currents, 1967 Current Algebras and Applications to Particle Physics, 1968 Nuclear Theory: The Quasiparticle Method, 1968 The Quark Model, 1969 Approximation Methods in Quantum Mechanics, 1969 Nonlinear Plasma Theory, 1969 Quantum Kinematics and Dynamics, 1970

FRONTIERS IN PHYSICS

(36) (37) (38) (39) (40) (41) (42)

ix

R. P. Feynman R. P. Feynman E. R. Caianiello

Statistical Mechanics: A Set of Lectures, 1972 Photo-Hadron Interactions, 1972 Combinatorics and Renormalization in Quantum Field Theory, 1973 G. B. Field, H. Arp, The Redshift Controversy, 1973 and J. N. Bah call Hadron Physics at Very High Energies, 1973 D. Horn F. Zachariasen S. Ichimaru Basic Principles of Plasma Physics: A Statistical Approach, 1973 (2nd printing, with revisions, 1980) G. E. Pake The Physical Principles of Electron Paramagnetic T. L. Estle Resonance, 2nd Edition, completely revised, enlarged, and reset, 1973 [cf. (9)-lst edition]

Volumes published from 1974 onward are being numbered as an integral part of the bibliography. Theory of Nonneutral Plasmas, 1974 43 R. C. Davidson 44 S. Doniach Green's Functions for Solid State Physicists, 1974 E. H. Sondheimer P. H. Frampton Dual Resonance Models, 1974 45 S. K. Ma Modern Theory of Critical Phenomena, 1976 46 D. Forster Hydrodynamic Fluctuations, Broken Symmetry, 47 and Correlation Functions, 1975 Qualitative Methods in Quantum Theory, 1977 48 A. B. Migdal Condensed Matter Physics: Dynamic Correlations, S. W. Lovesey 49 1980 L. D. Faddev Gauge Fields: Introduction to Quantum Theory, 50 A. A. Slavnov 1980 Field Theory: A Modern Primer, 1981 [cf. 74-2nd P. Ramond 51 edition] Heavy Ion Reactions: Lecture Notes Vol. I: Elastic R. A. Broglia 52 and Inelastic Reactions, 1981 A. Winther Heavy Ion Reactions: Lecture Notes Vol. II, in R. A. Broglia 53 preparation A. \Vinther Lie Algebras in Particle Physics: From Isospin to I-I. Georgi 54 Unified Theories, 1982 Basic Notions of Condensed Matter Physics, 1983 P. "\V. Anderson 55 Gauge Theories of the Strong, Weak, and ElectroC. Quigg 56 magnetic Interactions, 1983 Crystal Optics and Additional Light Waves, 1983 S. I. Pekar 57 Superspace or One Thousand and One Lessons in S. J. Gates, 58 Supersymmetry, 1983 M. T. Grisaru, M. Rocek, and "\V. Siegel

FRONTIERS IN PHYSICS

X

59

R. N. Cahn

60 61

G. G. Ross S. W. Lovesey

62 63 64

P. H. Frampton

65

66 67 68 69 70 71 72 73 74

J. I. Katz T. J. Ferbel T. Appelquist, A. Chodos, and P. G. 0. Freund G. Parisi R. C. Richardson E. N. Smith J. W. Negele H. Orland E. Kolb M. Turner E. Kolb M. Turner V. Barger R. J. N. Phillips T. Tajima W. Kruer P. Ramond

Semi-Simple Lie Algebras and Their Representations, 1984 Grand Unified Theories, 1984 Condensed Matter Physics: Dynamic Correlations, 2nd Edition, 1986 Gauge Field Theories, 1986 High Energy Astrophysics, 1987 Experimental Techniques in High Energy Physics, 1987 Modern Kaluza-Klein Theories, 1987

Statistical Field Theory, 1988 Techniques in Low-Temperature Condensed Matter Physics, 1988 Quantum Many-Particle Systems, 1987 The Early Universe, 1989 The Early Universe: Reprints, 1988 Collider Physics, 1987 Computational Plasma Physics, 1989 The Physics of Laser Plasma Interactions, 1988 Field Theory: A Modern Primer 2nd edition, 1989 [cf. 51-lst edition]

Editor's Foreword

The problem of communicating in a coherent fashion recent developments in the most exciting and active fields of physics continues to be with us. The enormous growth in the number of physicists has tended to make the familiar channels of communication considerably less effective. It has become increasingly difficult for experts in a given field to keep up with the current literature; the novice can only be confused. What is needed is both a consistent account of a field and the presentation of a definite "point of view" concerning it. Formal monographs cannot meet such a need in a rapidly developing field, while the review article seems to have fallen into disfavor. Indeed, it would seem that the people most actively engaged in developing a given field are the people least likely to write at length about it. FRONTIERS IN PHYSICS was conceived in 1961 in an effort to improve the situation in several ways. Leading physicists frequently give a series of lectures, a graduate seminar, or a graduate course in their special fields of interest. Such lectures serve to summarize the present status of a rapidly developing field and may well constitute the only coherent account available at the time. Often, notes on lectures exist (prepared by the lecturer himself, by graduate students, or by postdoctoral fellows) and are distributed in mimeographed form on a limited basis. One of the principal purposes of the FRONTIERS IN PHYSICS Series is to make such notes available to a wider audience of physicists. It should be emphasized that lecture notes are necessarily rough and informal, both in style and content; and those in the series will prove no exception. This is as it should be. One point of the series is to offer new, rapid, more informal, and, it is hoped, more effective ways for physicists to teach one another. The point is lost if only elegant notes qualify. The publication of collections of reprints of recent articles in very active fields of physics will improve communication. Such collections are themselves useful to people working in the field. The value of the reprints will, however, be enhanced if the collection is accompanied by an introduction of moderate length which will serve to tie the collection together and, necessarily, constitute a brief

xi

xii

EDITOR'S FOREWORD

survey of the present status of the field. Again, it is appropriate that such an introduction be informal, in keeping with the active character of the field. The informal monograph, representing an intermediate step between lecture notes and formal monographs, offers an author the opportunity to present his views of a field which has developed to the point where a summation might prove extraordinarily fruitful but a formal mongraph might not be feasible or desirable. Contemporary classics constitute a particularly valuable approach to the teaching and learning of physics today. Here one thinks of fields that lie at the heart of much of present-day research, but whose essentials are by now well understood, such as quantum electrodynamics or magnetic resonance. In such fields some of the best pedagogical material is not readily available, either because it consists of papers long out of print or lectures that have never been published. The above words, written in August, 1961, seem equally applicable today. Pierre Ramond has made significant contributions to our understanding of quantum field theory and particle physics. As he has noted in his Preface to this volume, despite the dramatic progress which has been made during the past decade in developing both new methods and applications of quantum fie ld theory, few textbooks have been written which cover this material. The present volume should go a long way toward bridging this gap, since it emphasizes the structure and methods of perturbation field theories while discussing both renormalization theory and the evaluation of Feynman's diagram for gauge theories at a level suitable for the beginning graduate student, with no important steps left out of the key derivations and problems at the end of each chapter. It is a pleasure to welcome him as a contributor to this Series. 1

Editor's Foreword to Second Edition During the past eight years, Pierre Ramond's informal text-monograph has been an invaluable pedagogical resource for the particle physics community, as well as for many physicists working in other fields, who have used it as a primer in the basic techniques of modern quantum field theory. The appearance of a second edition, with significant new material on finite temperature field theory, gravity as a gauge theory, and field theoretic anomalies, further enhances its usefulness, and will be warmly received by his readers, old and new.

David Pines Urbana, Illinois June, 1988

Preface

Since 't Ilooft proved the renormalizability of Yang-Mills theories, there has been a consequent dramatic increase in both the methods and the applications of Quantum Field Theory. Yet, few textbooks have been written in the intervening years, so that the student of Field Theory is left to depend on the original literature or on one of the many excellent reviews on the subject. Unfortunately these are often written for the specialist rather than for the neophyte. These notes, based on a one-year graduate course in Field Theory offered at Caltech between 1978 and 1980, aim to fill this gap by introducing in a straightforward, calculational manner some of the tools used by the modern Field Theorist. It is no longer possible to present perturbative Field Theory pedagogically in one year-the days of the standard one-year course based on QED are gone forever. Thus these notes cover only a selected set of topics. A modern presentation must consist of at least three parts-a first course emphasizing the structure and methods of perturbative Field Theories, with the intent of acquainting the student with renormalization theory and the evaluation of Feynman diagrams for gauge theories,-a second course dealing with applications of gauge theories, centering around perturbative calculations in Quantum Chromodynamics (QCD), Flavor Dynamics (Glashow-Weinberg-Salam model), and possibly Grand Unified Theories, and finally a third course on non-perturbative techniques. These notes address themselves to the first part, concentrating at an elementary level first on Classical Field Theory including a detailed discussion of the Lorentz group, Dirac and Majorana masses and supersymmetry, followed by a presentation of regularization methods, renormalization theory and other formal aspects of the subject. The approach is calculational-no proof of renormalizability is given, only plausibility arguments. Renormalization is treated in great detail for >.¢, 4 theory, but only lightly for gauge theories. The passage from Classical to Quantum Field Theory is described in terms of the Feynman Path Integral, which is appropriate to both perturbative and non-perturbative treatments. Also the (-function technique for evaluating functional determinants is presented for simple theories.

xiii

xiv

PREFACE

The material is discussed in sufficient detail to enable the reader to follow every step, but some crucial theoretical aspects are not covered such as the Infrared Structure of unbroken gauge theories, and the description of calculations in broken gauge theories. Still it is hoped that these notes will serve as an introduction to the perturbative evaluation of gauge theories. Problems are included at the end of each section, with asterisks to denote their degree of difficulty. A bibliography is included to provide a guide for further studies. I would like to express my gratitude to Professor S. Frautschi and J. Harvey for their diligent reading of the manuscript and constructive criticisms. Also special thanks go to E. Corrigan and J. Harvey for teaching me (-function techniques, as well as to the students of Phys. 230 for their patience and their numerous suggestions. Finally, these notes would not have seen the light of day had it not been for the heroic efforts of two modern day scribes, Roma Gaines and Helen Tuck, to whom I am deeply grateful.

P. Ramond Pasadena, Summer 1980

Preface to the Second Edition

In this second edition I have added three more topics. First an elementary discussion of finite temperature field theory using path integral techniques. I have deemed this step necessary because of the confluence between cosmology and particle physics. Secondly, in response to modern(?) trends in unifying all interactions, and because of the lack of similar treatments in books, I have included a section which introduces classical gravity as yet another (special) type of gauge theory. Finally I have rectified a great omission in the first edition by adding a discussion of anomalies using both perturbative and path integral techniques. I hope that these additions will serve to enhance the value of this book as a primer in the methods of quantum field theory. I have also made many mini-changes to improve the clarity of the presentation. I wish to thank the many who sent me comments, corrections, and suggestions, particularly E. Braaten, T. DeGrand, W. Dittrich, P. Ensign, D. Harari, J. lpser, G. Kleppe, L. Heck, W. Moreau, D. Murdoch, L. Ryder, P. Sikivie, R. Viswanathan, D. Zoller, and M. \Volf. Also thanks are due to l\frs. Y. Dixon for typing the new sections, and to Roger "('JEX)" Gilson for retyping the original manuscript.

P. Ramond Gainesville, Spring 1988

xv

Contents

1

How to Build an Action Functional 1.1 1.2 1.3 1.4 1.5 1.6 1. 7 1.8

2

The Action Functional in Quantu1n Mechanics 2.1 2.2 2.3

3

The Action Functional: Elementary Considerations The Lorentz Group (A Cursory Look) The Poincare Group Behavior of Local Fields under the Poincare Group General Properties of the Action The Action for Scalar Fields The Action for Spinar Fields An Action with Scalar and Spinar Fields and Supersymmetry

1 1 4 10 13 23 29 34 38

45

Canonical Transformations in Classical and Quantum Mechanics 45 The Feynman Path Integral 51 The Path Integral and the Forced Harmouic Oscillator 56

The Feynman Path Integral in Field Theory

63

3.1 3.2 3.3 3.4 3.5 3.6 3. 7

63

The Generating Functional The Feynman Propagator The Effective Action Saddle Point Evaluation of the Path Integral First Quantum Effects: (-Function Evaluation of Determinants Scaling of Determinants: Scale Dependent Coupling Constant Finite Temperature Field Theory

xvii

66

70 74 81 85 88

xviii

4

CONTENTS

Perturbative Evaluation of the FPI: '-~(m:i-i(),:i)

(3.2.8) We observe that the '-dependent term is just the same as the ef, term in (3.2.1) with J = 0, allowing us to write

Wo[J] = W0 [0] exp {--2i

j d4p {(p)j~-p~ } +u p -m

.

(3.2.9)

=

1. Note that W0 [0] can be formally calBy adjusting N, we can take Wo[O] culated using formulae of Appendix A. The important thing is that we have succeeded in finding the explicit dependence of W0 [J] on J. The use of the inverse Fourier transform yields

(3.2.10) where ~F12 stands for ~F(x1 - x2): (3.2.11)

It is the Feynman propagator. We now interpret the Green's functions obtained from Wo. From (3.1.10) we find G~2)(x1, x2) G~4 )(x1, X2, X3, X4)

= i~F(x1 -

x2)

= -[~F(X1 -

(3.2.12)

x2)~F(X3 - X4)

+ ~F(X1 -

+ ~F(X1 -

X4)~(x2 - x3)) ,

X3)~F(X2 - X4) (3.2.13)

etc. · · · together with the vanishing of the G's with odd number of variables. This fact is easy to understand since W0 [J] depends only on J 2 • In passing, note that all G's are functions of only the difference of coordinates, reflecting the translation invariance of the theory. Another lesson is that the higher Green's functions can all be understood in terms of G~2). Hence it would appear more convenient to set W[J] = eiZ(J) , (3.2.14) and define new Green's functions in terms of Z[J] (3.2.15) Then, at least in the case of W0 , we see that Ge is much simpler than G.

68

FPI IN FIELD THEORY

We now turn to the physical meaning of the Green's functions generated by Wo. By direct computation, we find (3.2.16) thus identifying t:lF with the Green's function of the operator D+ m 2 • Its boundary conditions are determined from the -fr procedure by the path integral. We can therefore identify flF(x - y) with the propagator of a signal from x to y. The signals it propagates are single particle and antiparticle states, since those are solutions of the Klein-Gordon equation (3.2.17) The -fr procedure tells us which of the solutions are propagated. One finds that positive energy solutions of the Klein-Gordon equations are propagated forward in time while negative energy solutions are propagated backwards in time (see problem). Since these solutions are to be identified with particle (antiparticle) states of energy E

= p 0 = ✓·r? + m 2 (- ✓p2 + m 2) , we arrive at the very nice physical

picture that information propagates forward in time with particle and backward in time with antiparticle. As an example, ask how many ways a quantum number can be carried from x to y given that we have a particle which carries one unit and an antiparticle with minus one unit. The quantum number could be the electric charge and the particle a 1r+ meson. There are two ways: by propagating a 1r+ meson from x toy thus destroying a charge +1 at x and depositing it at y or by using a 71"-, the antiparticle of 1r+, to carry a negative charge from y to x. The lesson is: 1) we have recognized the Green's function as the propagator of certain signals, and 2) we know which signals it propagates. Then it naturally follows that the states are, in our example, going to be particles of mass m 2 , and we interpret G~2 y) as the amplitude for this particle to go from toy. \Ve can invent a diagrammatic representation in x-space, associating with t:lF(x -y) a line connecting the two space-like points x and y:

\x -

x

X

y

For higher Green's functions we just diagrammatically add the contribution of, say (3.2.13),

It is clear that G~4 ) is a rather disconnected object. It can be interpreted as the amplitude for, say, a transition from x 1 , x2 to X3, x4. In this approximation there

3.3

69

THE FEYNMAN PROPAGATOR

are only so many ways for signal propagation, all expressed diagrammatically in the above picture. A much more transparent interpretation is obtained in the Fourier transformed space. We have seen that the nature of fl.F impels us to interpret Pµ as the four-momentum of a particle state. This is consistent with translation invariance, leading to p-conservation. Indeed, since G depends only on differences of x's, the naive Fourier transform

necessarily contains a c5-function of {p1 + • · · + PN)- So, instead we set c;(N)(P1, · · · ,PN)(21r) 4 c5( 4 )(p1

=

J

d4x1 ... d4XN

+ · · · + PN)

e-i(p1X1 +.. ·+PNXN)c(N)(x1'

with c;(N)(p1, · · · ,PN) defined only when Pl+···+ PN -c2) _ 1 Go (p, -p) - p2 - m2 + if

... 'XN) ,

(3.2.18)

= 0. For example, {3.2.19)

gives the amplitude that a particle of momentum P and mass m 2 propagates. We can represent this diagrammatically as well

p

(3.2.20)

In general, however, we will represent the Green's function c;(N)(p1, · · · ,PN) as a blob with N lines, labeled by P1,P2, · · · ,PN, entering it and with P1 + P2 + •••+ PN 0, which reflects the conservation of momentum:

=

(3.2.21)

It will be interpreted, say, as the scattering amplitude of states of momenta p 1 , •••,Pi into states of momenta P;+i, · · · ,PN, if we take the lines j + 1, · · ·, N to be outgoing. Again, note that the form of G~2) is what suggests the nature ~f the external states. We will later discuss the unitarity constraints imposed on G.

70

FPI IN FIELD THEORY

PROBLEMS A. Starting from the Euclidean formulation for WE[JJ, work out the corresponding expression the Feynman propagator and show that by analytic continuation in Minkowski space it reduces to the usual one. B. Given ~F(x ), show that is propagates positive energy signals forward in time, and negative energy signals backward in time. C. Find the real and imaginary parts of ~F(x); interpret physically. Can you express ~F in terms of Im~F?

3.3 The Effective Action Out of the generating functional we can construct local quantities which lend themselves to familiar interpretations. For instance, (3.3.1) so that (o]

-½(m2-if)cf>2+Jcf>)

(3.3.10) (3.3.11)

Now comes the trick: observe that

!

_6_ ei(Jcf>) i 6J(x)

= A.(x) ei(Jcf>) y,

'

(3.3.12)

and since J and 0 (it's a Gaussian), neglecting the higher derivatives. The success of this approximation rests on the fact that the integrand is largest when a( x) is smallest and that the points away from the minimum do not significantly contribute, as in the figure

a(x)

V (good)

a(x)

(bad)

In this section we apply this technique to the Euclidean space generating functional. We start from the Euclidean space definition of the generating functional

(3.4.4) where

(3.4.5)

3.4

75

SADDLE POINT EVALUATION

We then expand the action around a field configuration < 1) + .. ·) 4J3.4.15)

-½ j d4xJq,(o) - ~ j

d4x

[Jq, +

1124>i2) .

(3.4.16)

If we define the Euclidean Green's function (in an obvious notation)

(3.4.17) if follows that

(O)(x)

= (GxaJa)

(1) (x ) --

0

- !6 (GxyGyaGybGycJaJbJc)abcy

(3.4.18)

, etc . .

Thus,

SE(J]

= - 21 (JaGabJb)ab + 4>i! (GxaGxbGxcGxdJaJbJcJd)abcdx .X 2 - 3 . 41 (GxaGxbGxcGryGydGyeGyjJaJbJcJdJeJJ)abcdefxy

+ O(A 3 ) (3.4.19)

Correspondingly, the ( connected) Euclidean Green's functions are given by

(3.4.20) where

(3.4.21)

3.4

77

SADDLE POINT EVALUATION

In this classical approximation we find the connected Green's functions to be (3.4.22)

G~\x1,X2,X3,x4)

= -,\

J

d4yG(x1,y)G(x2,y)G(x3,y)G(x4,y) (3.4.23)

(3.4.24) where

P(x,y,{xi}) =

L G(x,xi)G(x,ij)G(x,x1;)G(y,it)G(y,xm)G(y,xn)

'

(ij k)

(3.4.25) where the sum runs over all the following values of the triples, (ijk) = (123), (124), (125), (126), (135), (136), (145), (146), (156), with (lmn) assuming the complementary value [e.g., (fmn) = (456) when (ijk) = (123)]. Note that ijk runs only over half of the possible values. This is because the expression for P is symmetric under the interchange x --+ y. In this classical approximation and to order ,\ 2 these are the only non-zero Green's functions. The momentum space Green's functions, defined by

(3.4.26)

are easily seen to be given at - E(2) (- - ) G Pt , P2 - -p1 -

-2 +1

P1

m

2

+ O(n)

(3.4.27)

-2

1

p4+m

2) -' + o (n) (3.4.28)

6

-cs) - _ GE (p1, .. · ,Ps) -

,2

IT (p-? +1 m2) "~' - - " - - - O(n) (-· + -. + - )2 + 2 +

i=l

•

(ijk) P,

PJ

Pk

m

(3.4.29)

78

FPI IN FIELD THEORY

where again the sum is over the same triples as in (3.4.25). In the above expressions the G's are to be evaluated only when the sum of their arguments vanishes. In the perturbative evaluation of ¢, 0 , we note that ¢,(k) always depends on J 1+2k. Hence the _xk order contributes solely to G( 2(k+l)). This is an artifact of the approximation, which neglects contributions of order n. Following Feynman, we develop a pictogram for these Green's functions. We represent (fi1, · · · , PN) by a blob with N external lines

cl;n

~

,, . . . --- ... ,,

PN

P.

each line carrying a p label, and with 1all arrows pointing into the blob. This blob is them represented diagrammatically by the following rules: means _2 p

X

1

+m 2

,

the propagator factor

means - ..X, the vertex .

It is understood that the net amount of p flowing through the vertex is conserved. The vertex has no arrows on its lines, thus indicating that the propagator for the lines is not included. Comparison with (3.4.27 - 3.4.29) leads to

X

r-0t ~

)2 I

+

;y I

O(lO

+

p

p4

3*-4

2

=

+

0(1')

P4

~~ 6

2;) ~~ 2;> ~~ 4H2 5H2 5

4

+

+

I

3 ~4 + 3 5 I

I

6

5 + 3 6 I

4 6

6

3.4 SADDLE POINT EVALUATION

+

:;► I

+

~;> I

5H2 ~; + 4 I

~:

+

79

6

2

3 + 4H3 6 I 5

0 (li)

It appears then that the Feynman rules, to this order in Ti, are to draw all possible arrangements using - - - and ~ as basic building blocks with no closed circuits ( also called loops). Such diagrams are called tree diagrams. It is not hard to see that this approximation represents the Green's functions by their tree diagrams. Consider a diagram with E external lines, I internal lines and Vn vertices, each with n lines (in our case n 4). Each internal line hooks on the two vertex lines. Hence the number of external lines is just equal to the number of vertex lines minus twice the number of internal lines

=

= nVn -21

E To 0(1i 0 ), we have E

= 2k + 2, V = k hence 4

0(1i 0 ) only ,

so that there cannot be any closed loop in these Feynman diagrams: they are called tree diagrams. The rule for an arbitrary (tree) diagram is to draw all topologically inequivalent diagrams with the external lines identified. For example, in G( 6) when the same lines emanate from the vertex, the diagram enters only once because their rearrangements would give topological equivalent diagrams

, etc ....

These rules make it easy to write the expression for G(s), · · •: we first draw all possible inequivalent tree arrangements with the external lines identified, and then use the rules in reverse to get the analytical expression. This use of the pictorial representation has proved to be an essential tool in the perturbative evaluation of the Green's functions.

80

FPI IN FIELD THEORY

It is interesting to rewrite these results in terms of the classical field ¢>e1 (x) and to see the form of the resulting effective action. In Euclidean space we define the classical field as (3.4.30) which, using (3.4.19) gives us ¢>e1(x) as a functional of J, order-by-order in ..\. Then, we invert the equation in perturbation theory, and find J(x) as a functional of ¢>e1. The result is (3.4.31) The remarkable thing is that there are no terms of higher order in ..\ in this equation. By comparing with (3.4.7), we conclude that

¢>e1(f)

= o(f) + O(h)

.

(3.4.32)

Integration of (3.3.9) gives us immediately the effective action to this order (3.4.33) Thus, r E is the classical action. We can therefore derive the expression for the proper vertices. In this approximation, we see that f( 2 ) is minus the inverse propagator and that f(n) for n > 4 vanish. The higher order diagrams do not appear. The reason is that the f(N) generate only Feynman graphs that cannot become disconnected by cutting off one of their internal lines. Such graphs are called one-particle irreducible. As we have seen, all the tree graphs are oneparticle reducible except for the lowest one.

PROBLEMS A. Show that the saddle point evaluation of the path integral corresponds to an asymptotic expansion in h. B. Solve the equation

(a2 _ m2 _ ;,2) = -J order-by-order about ..\ =0. If we set ¢> = ¢>< 0 ) + ..X¢,( 1 ) + ..\ 2 ( 2 ) and

q,( 3 ).

C. For ..\(x1,x2,x3,x4) = -A j d4xll.(x1 -

x)ll.(x2 - x)ll.(xa - x)ll.(x4 - x)

J

A2 +2 d4xd4yll. 2 (x - y)[ll.(x1 - x)ll.(x2 + ll.(x1 - x)ll.(xa - x)ll.(x2 - y)ll.(x1 - y) + ll.(x1 - x)ll.(x4 - x)ll.(x2 - y)ll.(xa - y)] A2 +2

x)ll.(xa - y)ll.(x4 - y)

J

d4xd4yll.(y- y)ll.(x - y) [ll.(x1 - x)ll.)x2 - x)ll.(x 3 - x)

ll.(x 4 - y) + cyclic permutations)+ 0(A) 3 )

,

(4.1.22)

and finally

c< 6>(x 1, • • • ,x6) = A2 jd4xd4yll.(x -

y)

L ll.(x1 -

x)ll.(x; - x)ll.(x1; - x)

(ij k)

ll.(xl - y)ll.(xm - y)ll.(xn - y)

+ 0(A3 )

(4.1.23)

4.1

105

FEYNMAN RULES

where the sum in the last expression runs over the triples (ijk) = (123), (124), (125), (126), (134), (135), (136), (145), (146), (156), with (lmn) assuming the complementary value, i.e., (lmn) = (456) when (ijk) = (123), etc. The remaining Green's functions get no contribution to this order in A. Note that the A0 contribution to G( 2 ), the A contribution to 0(4 ) and the A2 contribution to G( 6 ) were all previously obtained in the classical approximation of the last chapter. It is straightforward to derive the p-space Green's functions, using (3.4.26). We find

a(2)(

J

)_ 1 A 1 P, -p - p2 + m2 - 2 (p2 + m2)2

A2

1 (p2 + m2)2

+6 A2

1

+4

(p2

+ m2) 2

(p2

+ m2)2

A2

1

+4

j

d4q

1

(21r)4 q2

+ m2

o(p - q1 - q2 - q3)(21r )4 (2,r)4 (2,r)4 (2,r)4 (qr+ m2)(q~ + m2)(q~ + m2)

J J

d4 q1

d4q2

d4q

1

d4 q3

(21r) 4 q2 + m 2

d4q

J

d4l2 o(p- l1 - l2)(21r) 4 (21r) 4 (21r) 4 (lr + m 2 )(l~ + m 2)

1

(21r)4 q2

d4 l1 1

+ m2

p2

+ m2

J

d4l

1

(21r)4 £2

+ m2

+ 0(A 3 ) -(4)

G

-

(P1,P2,p3,p4) -

(4.1.24)

II 1 { . ( ~ + m 2) -

!

4

•=l

A2 +2

P,

J

2

A+ 2 A

J

d4 q ( 2,r)4

q

1 ~ 1 2 + m2 ~ ~ + m2 •=1 P,

d4 q1 d4q2 1 " (21r)4 (21r)4 (qr+ m2)(q~ + m2) &no(q1

+ 0(A 3 )

+ q2 -

Pi - Pj)(21r)

(4.1.25)

.

=

(12), (13), (14) only. Finally In the last expression, the sum ij runs over (ij) 6 G( ) is given by (3.4.29). These expressions are clearly unwieldy. One needs to devise a clever way of remembering how to generate them. This is exactly what the Feynman rules achieve. We now proceed to state them: 1. For each factor

P2

im

2

draw a line with momentum p flowing through it:

[)

1

p

p2

+ m2

·

2. For each factor of -A/4! draw a four-point vertex with the understanding that the net momentum flowing into the vertex is zero:

~.

P3

A 4!

(Pi+ P2

~

+ p3 + P4

= 0)

.

4}

106

4 THEORY

3. In order to get the contributions to {;(N)(p 1, · · · , PN), draw all the possible arrangements which are topologically inequivalent after having identified the external legs. The number of ways a given diagram can be drawn is the topological weight of the diagram. 4. After having conserved momentum at every vertex, integrate over the internal loop momenta with J (;;~ 4 • The result gives the desired Green's function. Perhaps a more systematic way to describe these rules is to attach to each vertex the factor -¾i6(Ep)(21r)4, where Ep is the net incoming momentum at that vertex. The one integrates over all internal momenta. In this manner one obtains an overall (21r) 46(Ep) where Ep is the net momentum flow into the Green's function. For instance, the expression (4.1.24) is diagrammatically rewritten as

+ From the rules we can easily obtain the analytical expressions corresponding to these diagrams:

•

0 •2

• 0 •2

a) We need one vertex and three propagators. There are four ways to attach the first let of the vertex to 1, three ways to attach the second let to 2. Hence the weight -:h 4 • 3 = ½The vertex counts for -A. Let q be the momentum circulating around the loop. The rules then give

b) We need two vertices. There are four ways to attach the first leg of the first vertex to 1, four ways to attach the first leg of the second vertex to 2, three ways to sew the second let of the first vertex to the second, and two ways to sew the third leg of the first vertex to the second. Hence the weight -;h4 • 4 • 3 • 2 = ¼. Note that we did not count that we initially had two vertices to play with. This is because the diagrams are the same irrespective of which vertex was used. The strength of this diagram is (-A ) 2 A2 . If we call the momenta flowing in the internal legs q1, q2, q3 , the Feynman rules give

J, •

=

A2

6

1

(p2

+ m2)2

J

d"q1

d4q2

d"q3 ( 2

(21r)4 (21r)4 (21r)4

s

,r)

6(p1 - q1 - q2 - q3)6(p2

+ q1 + q2 + q3)

X -----,,--------,,------,,------

(qt+ m2)(qi

+ m2)(q~ + m2)

4.1

107

FEYNMAN RULES

c) We need two vertices: four ways to attach the first leg to 1, four ways to attach the third leg of the tied vertex to the other, three ways to tie the fourth leg of the first vertex to the second one. Hence 3 • 4 • 3 =¼-Strength (-;\) 2 •

_8_

l, l,4 •

2

d) We need two vertices; four ways to attach one vertex to 1, four ways to attach the other vertex to 2. This leaves three legs from each vertex free to be tied together in one way. For each vertex there are three ways to close the buckle. Hence 3 · 3 = ¼- Strength (-;\) 2 .

l, J,4 ·4·

Thus we see that the Feynman rules have reduced the problem to that faced by a child assembling a "Leggo" set. The basic tools are the propagator (line and the vertex. With a bit of skill one can read those directly from the Lagrangian. The same applies for the four-point function:

(4.1.26)

+

+

+

corresponding to the analytical expression ( 4.1.25). We leave it to the reader to verify the correctness of the numerical factors in (4.1.25). Let us remark that in the expression for c;(N)(p 1 , • • •, PN ), we will have the multiplicative factor TI~ 1 (Pi+ m 2)- 1 corresponding to the propagation of the external legs. It is much simpler to deal with the Green's functions generated by the effective Action. Their relation to c;(n) is very simple: while c;(n) are connected, the f(n) are one particle irreducible. In particular f( 2)(p) is minus the inverse propagator. We have already come in contact with this result in conjunction with the tree diagram, but the result holds true in all orders of perturbation theory. As a result r( 4) contains only the diagrams

r(4)

=

2x3 203· +

I

4

I

4

2 3

+

+ 4

with no propagators for the external legs. In order to show that the f(n) are one particle irreducible, it is convenient to use as a starting point the defining

108

4 THEORY

equations:

(4.1.27) and keep on differentiating them. After all this work, let us go back to the Lagrangian where all the ingredients of the Feynman rules are easily identified: the four-particle vertex coming from the Aq, 4 term, and the propagator coming from the kinetic and mass terms. Hence, after a little bit of practice, one can just read off the Feynman rules from .C. The difficult part is to get the right signs and weight factors in front of the diagram.

PROBLEMS A. Draw the contributions in the two point function B. Draw the contributions to

6( 2) that are of order A3 •

6( 4 ) of order A3 .

C. Write the analytical expressions for the diagrams of problem A, including the weights. •D. Derive the Feynman rules for V

= ;!q, + frq, 4

3•

.. E. Show that Z[J] generates only connected Feynman diagrams.

•••F. Derive the form of the effective Action sefl'[4'cd to order A2 , and show that the one particle reducible diagrams do not appear in the Green's functions it generates.

4.2

Divergences of Feynman Diagrams

No sooner has the beauty of the Feynman rules sunk in that one realizes that most of the loop integrations diverge! For instance the O(A) contribution to G( 2 ) involves the integral d4q 1

J

(2,r)4 q2 + m2 i

it clearly diverges when q -+ oo since the integrand does not have enough juice to make up for the measure. Such a divergence is called an ultraviolet divergence. It occurs for large momenta, or equivalently, small distances [in x-space it comes from ~(O)), and clearly has to do with the fact that one is taking too many

4.2

109

DIVERGENCES

derivatives with respect to J at the same point. Another example occurs in the "fish" diagram

Here as q--+ oo, the integral behaves as~' which is a logarithm: logq. It also diverges! In passing, we note that when m 2 = 0, (p 1 + p 2 ) 2 = 0 it also diverges for low q. Such a divergence is called an infrared divergence. Typically such divergences occur only in the massless case and for special value of the external momenta [in this case Pl + p 2 0 because we are in Euclidean space]. For the moment we take m 2 ;/; 0 and concentrate on the ultraviolet aivergences. On the face of it, it is catastrophic to our program to find that our carefully constructed Green's functions diverge. Still, with the hindsight of History we do not get discouraged, but try to learn more about these divergences. We will find that they appear in a very traceable way, and that by a suitable redefinition of the fields and coupling constants, they will disappear! This is the miracle of renormalization which, as we shall see, occurs only for certain theories. By a mixture of topology and power counting we now show how to tell where the divergences are. Consider a Feynman diagram with V vertices, E external lines and I internal lines. For a start we assume only scalar particles are involved. The number of independent internal momenta is the number of loops, L, in the diagram. The I internal momenta satisfy V - 1 relations among themselves (the -1 appears here because of overall momentum conservation), so that

=

L=l-V+l.

( 4.2.1)

This relation enables us to compute the naive count of powers of momenta for the diagram. This yields the superficial (apparent) degree of divergence of the diagram D. To compute it, we note that there are

-

L independent loop integrations, each providing, in d dimensions, d powers of momenta. I internal momenta, such providing a propagator with two inverse powers of momenta. Hence (4.2.2) D = dL- 21 .

We need one more relation among V, E and I. Suppose VN stands for the number of vertices with N legs. In a diagram with VN such vertices, we have NVN lines

110

4 THEORY

which are either external or internal. An internal line counts twice because it originates and terminates at a vertex, so that NVN

(4.2.3)

= E+2I

These relations allow us to express in D terms of the number of external lines and vertices

(4.2.4) In four dimensions, this reduces to

D

= 4- E + (N -

4) VN

Furthermore, in the theory of interest to us, N D=4-E

(4.2.5)

[four dimensions]

= 4. Hence (4.2.6)

[for >...4 theory, diagrams can be at most three-particle reducible because any vertex attached to an external line can be disconnected from the diagram by cutting its remaining three legs. The procedure of hunting for diagrams which have a negative D and yet are divergent is now clear. Take any Feynman diagram and catalog it in terms of its reducibility: in our case it is either one-, two-, or three-particle reducible. It if is three-particle reducible, decompose it into units, and look for breakups of the form

N~5 N

Then in this type of diagram, we have a primitively divergent four-point function, where the second blob can be decomposed in the same way. Similarly, hidden divergences in two-particle reducible diagrams will arise in the diagrams of the form

with the same breakup to be repeated in the second blob. Finally, one-particle reducible diagrams which can be decomposed in the form

~I N

N> 5

~ I

N>3

N

will have hidden divergences. The same decomposition can be carried out for the second blobs until all such structures have been uncovered. This exhaustive catalog shows that truly convergent diagrams do not contain hidden two- and four-point functions. One can understand the origin of hidden ultraviolet divergences in any diagram in a more pedestrian fashion. Consider any loop residing inside a diagram. Integration over the loop momentum in four dimensions will lead to a UV divergence if the loop is bounded by one or two propagators (internal lines) at most.

4.2

DIVERGENCES

113

Any more will give UV convergence. A loop bounded by one propagator involves only one vertex,

0 leaving two free legs, which in turn may be attached to the rest of the diagram (or one external, one attached). In this case one can isolate this two-point function from the insides of the diagram. A loop bounded_ by two propagators involves two vertices and therefore four free legs

which can be attached to the rest of the diagram, or else up to three can serve as external lines. In all these cases, one is led to isolate from the diagram a four-point function, or a two-point function if two of the four legs are attached together (in this case the divergence becomes quadratic). In this way, one sees that UV divergence inside a diagram originate from such loops and that such loops appear in two- and four-point functions nested inside the diagram. Thus, for the >.¢, 4 theory in four dimensions, a Feynman diagram is truly convergent if its superficial degree of convergence D is positive and if it cannot be split up into three-, two- or one-particle reducible parts of the kind just described which can contain isolated two- and four-point function blobs. Stated more elegantly, a Feynman diagram is convergent if its superficial degree of convergence and that of all its subgraphs are positive. This is known as Weinberg's theorem, and it holds irrespective of the field theory. This means that the generic sources of the divergences are the two- and fourpoint functions and nothing else. They are the culprits! So, if we control them in G( 2 ) and G( 4 ), we have the possibility of controlling the divergences of all the other b(N) 's! The graphs which contain the generic divergences are said to be primitively divergent. The fact in >.¢, 4 theory that the primitively divergent interactions are finite in number (two- and four-point interactions) and are of the type that appears in the Lagrangian, is a necessary ingredient for the successful removal of the ultraviolet divergence by clever redefinitions. A theory for which this is possible is said to be renormalizable. We can see from (4.2.4) that very few theories of interacting scalars satisfy these requirements (see problem).

=

When d 4, we see that D grows with the number of vertices for which N > 4. Hence ¢, 5 , .¢,3 in six dimensions, where >. is dimensionless, since ¢, now has dimension -2. There the primitively divergent diagrams are few since V does not appear in the expression for D D = 6 - 2E ( ¢, 3 in six dimensions) , (4.2.9) so that the one-, two- and three-point functions are primitively divergent ( quartic, quadratic and logarithmic, respectively). This theory, although having an unsatisfactory potential unbounded from below, is interesting in that it shares with the more complicated gauge theories the property of asymptotic freedom. To summarize this section, we have noticed the appearance of ultraviolet divergences in Feynman diagrams with loops (the bad news), but we have seen that we can, at least in our theory, narrow them down as coming only from two primitively divergent Green's functions (the good news). Hence, if we can arrange to stop divergences from appearing in 6( 2 ) and 6( 4 ), we have a hope of stemming the flood and obtaining convergent answers!

PROBLEMS A. In four dimensions, find all primitively divergent diagrams for the ¢, 3 theory. For each, give examples in lowest order of perturbation theory.

=

3), list all theories of interacting scalars with a B. In three dimensions (d finite number of primitively divergent graphs. Give graphical examples for each.

4.3

115

DIMENSIONAL REGULARIZATION

C. Repeat B, when = 5, and show that when d > 1 there are no theories with a finite number of primitively divergent graphs.

=

D. For d 2 3, 5, 6, find the dimensions of the various coupling constants in the theories where there is a finite number of primitively divergent graphs.

4.3

Dimensional Regularization of Feynman Integrals

In the following, we proceed to evaluate the Feynman diagrams. The procedure is straightforward for the UV convergent ones, while special measures have to be taken to evaluate the divergent ones. In those we are confronted with integrals of the form

(4.3.1) where for large£, F behaves either as e- 2 or e- 4 • The basic idea behind the technique of dimensional regularization is that by lowering the number of dimensions over which one integrates, the divergences trivially disappear. For instance, if F--+ e- 4 , then in two dimensions the integral (4.3.1) converges at the UV end. Mathematically, we can introduce the function

I(w, k)

= j d2w lF(l, k)

,

(4.3.2)

as a function of the (complex) variable w. Evaluate it in a domain here I has no singularities in thew plane. Then invent a function I'(w, k) which has welldefined singularities outside of the domain of convergence. We say by analytic continuation that I and I' are the same function. A nice example, which is the basis for the method of analytic continuation, is the difference between the Euler and Weierstrass representations of the ffunction. For Re z > 0, the Euler representation is

r(z)

= fo

00

(4.3.3)

dt e-ttz-l .

As such, it diverges when Re z < 0, because as t approaches zero, the integral behaves as dt/t 1+1 Rezl, which leads to an infinity. Starting from ( 4.3.3) we can split up the troublesome integration limit

f(z)

=

t (-Irn. lo

fa dttn+z-t

n=O

+ /

00

la

dt e-ttz-t ,

(4.3.4)

where o is totally arbitrar'y. The second integral is well-defined even when Re 2 < 0 as long as o > 0. The first integral has simple poles whenever z is a negative integer or zero. We find

=L 00

f(z)

n=O

(-1)" n.I

(4.3.5)

116

4 THEORY

This form is valid everywhere in the z-plane. Furthermore, it should not depend on the arbitrary coefficient a (you can check that ~~ = 0). When a = 1, it is the Weierstrass representation of the r-function. Still, to isolate the singularities we did introduce an arbitrary scale in the process, although the end result is independent of it. Our problem is that integral expressions like ( 4.3.2) are like Euler's. We want to find the equivalent of Weierstrass' representation. Our procedure will be as follows: 1) establish a finite domain of convergence for the loop integral in the w-plane. For divergent integrals, it will typically lie to the left of the w = 2 line; 2) construct a new function which overlaps with the loop integral in its domain of convergence, but is defined in a larger domain which encloses the point w = 2; 3) take the limit w -+ 2. We now show how this is done in the case of one-loop diagrams, following the procedures of 't Hooft and Veltman, Nucl. Phys. 44B, 189 (1972). Let us split up the domain of integration as

d2w f,-+ d4 f d2w-4f Next in the 2w - 4 space, introduce polar coordinates, and call L the length of the 2w - 4 dimensional £-vector. The integral now reads (4.3.6) Integration over the angles can be performed (see Appendix B), with the result 211'w-2

l=r(w-2)

Jd4£ 1

00

o

1

(4.3.7)

dLL2w-s(L2+f2+m2).

This expression is not well-defined because it is UV divergent for w ~ 1, and the integration over L diverges at the lower end ("infrared") whenever w ~ 2. Thus, there is no overlapping region in the w plane where I is well-defined. The IR divergence is, however, an artifact of the break up of the measure. Observe that by writing (4.3.8) and integrating by parts over £ 2 , and throwing away the surface term, we obtain

I=

r:~\) Jd4f.1

00

t- 2 (- d12 )

dL 2 (L 2

£2

+ £~ + m2

,

(4.3.9)

where we have used f(w - 1) = (w - 2)f(w - 2). Now the representation ( 4.3.9) has an infrared divergence for w ~ 1 and the same UV divergence for w ~ 1, that is, still no overlapping region of convergence. So we do it again, and obtain (4.3.10)

4.3

117

DIMENSIONAL REGULARIZATION

an expression which is well-defined for O < w < 1. Note that we had to move the IR convergence region two units to obtain a nonzero region of convergence. Had the loop integral been logarithmically divergent, one such step would have sufficed. Having obtain an expression for I convergent in a finite domain (in this case 0 < w < 1), we want to continue I of ( 4.3.10) to the physical point w = 2. It goes as follows: insert in the integrand the clever expression (4.3.11) and integrate by parts in the region of convergence. We obtain

A little bit of algebra gives, after reexpressing the right-hand side in terms of I, (4.3.13) This expression displays explicitly a pole at w = l. The integral now diverges at the upper end when w ~ 2. So we repeat the process and reinsert (4.3.11) in (4.3.13). The result is predictable:

2. 3. 4m4 7rw-2 I - - - - - - - ---

- (w - l)(w - 2) f(w)

J loo d4 f

(L2t-1 dL 2 ---------=(£2 +e2 + m2) 5 0

(4.3.14)

This is the desired result. The only hint of a divergence comes from the simple pole at w = 2, since the integral itself now converges. Let us summarize. We first define a finite integral in the w-plane to be what we mean by f d2wfF(f,k): in this case it is the expression given by (4.3.10), and cons ti tu tes our starting point. Then if the region of convergence does not include w = 2, we continue analytically by means of the trick (4.3.11). It would be nice to show that for a convergent integral, the procedure that leads to (4.3.10) indeed gives the right answer. Take as an example the convergent integral

It is easy to see that the same procedure leads to the expression

118

4 THEORY

which is perfectly finite at w

= 2. We find

as desired. Therefore, the procedure is entirely consistent. After all this ponderous work, let us turn to a more cavalier interpretation of integrals in 2w-dimensions, as derived in Appendix B. If we were to blindly plug in the formulae of Appendix B, we would obtain ( 4.3.16)

We expand around w = 2, using for n = 0, 1, 2, • • • and€--+ 0 the formula f(-n

+ €)

= (-:?n [¼ + 1/,(n + 1) + ½€ [ ,r; + v, 2 (n + 1) -1/,'(n + 1)] + 0(€2 )] ( 4.3.17)

where

2

v,' (n + 1) = : -

n

I:

1 k2 ,

2

= ~6

( 4.3.19)

= -, = -0.5772 · · ·

( 4.3.20)

k=l

1/,'(1)

, being the Euler-Mascheroni constant 1/,(1)

The result is (4.3.21)

It can be shown that this is the same result as that obtained by integrating (4.3.14). From now on, we are going to use the naive formulae of Appendix B, and not concern ourselves with their justification since we know they are legitimate, thanks to 't Hooft and Veltman.

PROBLEMS A. Show that the naive procedure of Appendix B and the more careful one outlines in this section lead to the same results for the integrals

4.4

119

EVALUATION OF FEYNMAN INTEGRALS

B. Prove the last four formulae of Appendix B.

4.4

Evaluation of Feynman Integrals

Using the techniques of the previous section and the formulae of Appendix B, we proceed to evaluate the low order Feynman diagrams for A

..), we find (remember that the correction to the inverse propagator picks up a minus sign for inverting)

----0-

o___ +

- - - - + __

(4.5.5)

This very naive procedure makes the theory finite to order >... The extra term (4.5.3) is called a counterlerm. It is crucial to note that its dependence on the

128

4 THEORY

field ( and its derivatives) is the same as that of a term already appearing in £ (in this case the mass term). Now we proceed to order A2. The one particle irreducible (IPI) four-point function is given by

4

+

+ 3

(4.5.6) where

A(s,t,u)=

4m2)1/2 J1+47:i+l ( 1+In"'""""";:::==-, 472 - 1 z=.,,t,u Z

I:

JI+

(4.5.7)

and where s, t and u are Man deist am variables

(4.5.8) As it stands f'( 4 ) is divergent as £ another term to £

0. To remedy this situation, we add yet

(4.5.9) where G1 is an arbitrary dimensionless function of£, analytic as £ - 0. This new counterterm results in an additional Feynman rule denoted by

(4.5.10)

129

4.5 RENORMALIZATION

It is added in the calculation of the new

~{4)

rnew

X

f'( 4 ),

l~

+ ~ +

=

= -µ 2 f A [ 1 -

;j (-G + 1

yielding a finite result as

f --+

0:

+

tf,(1) + 2 - In m2

½A(s, t,

-

u))] + 0(A

3)

•

(4.5.11) The contribution to f( 2 ) in 0(A 2 ) can be handled in a similar way, but there the additional Feynman rules (4.5.4) and (4.5.10) must be used. Hence we have diagrammatically for the inverse propagator

-@-

= +

_8_

+

0

+

+

0

+

~

0

0

+

(5.12)

-168-

where the extra Feynman rules have induced to this order two new diagrams. These are easily calculated to be (see problem)

m 2 2 [ -1 + -1 [ tJ,(1) + F1 - In m 21 + .. ·] -A A

4

A

f2

were we show only the poles in

Q

f

f.

,

(4.5.13)

We also have

(4.5.14)

130

4 THEORY

Comparing with the "double scoop" diagram of the previous section

_8_

(4.5.15)

we observe that its double pole is exactly canceled by the counterterm diagram (4.5.13) although the simple pole remains. This is obvious from the point of view of diagrams since the combination O + X is by definition finite. Adding all the diagrams, we find

A2

24{

[1 1

2

Ap ~ =--

2

]

m 2 -+-(F1+3G1)+··· +O(A), 3 +-A 2 {2 2{

(4.5.16) where, again, we have not shown the finite part. We are faced again with a divergent expression as f --+ 0. Lo and behold! The In m2 term present in the simple poles of individual diagrams have disappeared as well as the Euler constant present in each tp(n) but absent in the difference

tp(n + 1) -1/;(n)

= -n1

(4.5.17)

In order to cancel the poles in ( 4.5.16) we introduce a new mass counterterm so that the mass counterterm Feynman rule becomes m A 1 A A2 (F1+3G1-l) ) +A 2 F2 +AF1] =- 2 2 [A2 7+; ( A+ 4 A

where F2 is an arbitrary function off and generated by the additional counterterm

A

m

2,

A

(4.5.18) but finite as f--+ 0. This term is

( 4.5.19) This takes care of only one type of infinity in f'( 2 ). The other one is canceled by adding yet another term to our ever expanding Lagrangian of the form ( 4.5.20) J/2 being arbitrary and analytic as f --+ 0. Thus will all this patching up, we have been able to eliminate the ultraviolet divergences to order A2 • It is clear that we can continue this little game ad nauseam: calculate diagrams to order A3 , with the original ,C and the counterterms ( 4.5.19) and ( 4.5.20); then invent new

4.5

131

RENORMALIZATION

counterterms which to O(.X 3 ) are chosen to cancel the new divergences, etc. So far in this process, the remarkable thing has been that the counterterms needed to remove the divergences all generate new interactions of the same type as those that were present in the original Lagrangian; we have not been forced (to O(.X 2 )) to introduce counterterms that correspond to terms of a type absent in .C. If it can be shown that this noteworthy matching feature continues to all orders in .X, we will say that the theory is renormalizable. Here we do not attempt a proof for .X¢, 4 theory, but rather point out where the procedure might fail. Consider a generic two loop diagram

where all external legs have been removed. The diagram is divergent due to the various loop integrations. In accordance with our new rules for the counterterms, it must be added to the three counterterm diagrams, (presented here schematically)

0

0 0

Here each • represents the lowest order counterterm vertex needed to cancel the infinity coming from one loop diagrams. These counterterm diagrams will therefore contain a ~ coming from • which multiplies a In p 2 where p is some momentum coming from the loop integration. It would therefore seem that this would generate counterterms of the form lnp 2 {

which do not correspond to any term in .C because of the In p 2 residue, which is highly nonlocal in position space. Such terms would clearly throw a monkeywrench in the works. This is the famous problem of overlapping divergences. A close study of these diagrams shows that the sum of all the diagrams shown above does not contain any poles with logarithmic residue: they cancel against similar poles contained in the two loop diagram. We have in fact seen as example of this miracle when the In m2 terms canceled from the residue of the simple pole in (4.5.16). [For more details, the reader is referred to the paper of 't Ilooft and Veltman, Nucl. Phys. 448, 189 (1972).] In the proof of renormalization, it is crucial to be able to prove that these overlapping divergences do, in fact, cancel.

132

4 THEORY

Let us assume that this is so that that to an arbitrarily high order in .X only counterterms which match the original £, are needed to render the theory finite. This means that the Lagrangian which gives finite answers has the form r,ren

= [, + Cc. t.

(4.5.21)

,

where £, is our original Lagrangian (4.5.22)

and £c.t. is the counterterm Lagrangian (4.5.23)

It is (by assumption, but verified to O(.X 2 )) exactly of the same form as £, but with specially designed A, B and C so that the Green's functions generated by r,ren are finite as c - 0. By defining new fields and parameters, we can rewrite r,ren in the form [, ren

2,J.2 = 21 {)µo 0µ o + 2l mo'f'o + 41A 3 theory, verify that the overlapping divergences from the diagram

---0)--

136

4 THEORY

are, in fact, canceled by the counterterm diagram that regulates the one loop diagram

-04.6 Renormalization Prescriptions In the previous section, we carried out in detail the renormalization procedure in >.... and ..A

at p 2 = 0

(4.6.2)

at Pi= 0 ·

(4.6.3)

137

4.6 RENORMALIZATION PRESCRIPTIONS

=

In the absence of infrared divergences which occur when m 2 0, this prescription is well-defined. We have substituted the input parameters to indicate how they were defined. Note that (4.6.2) embodies two conditions since it fixes the mass as well as the field normalization. These fix the finite parts of the counterterms. In particular, we find that

Ff

= 1/,(2) -

In m~

;

Cf

= 1/,(1) -

In m~

;

Ht= 0 ,

etc. . (4.6.4)

Ideally, one would prefer to identify the coupling constant with f( 4 ) at a physical point where the particles are in Minkowski space and on their mass-shell (PX!

=

m2).

B. One can change the subtraction point at will provided it does .not interfere with the continuation to Minkowski space or with infrared singularities. We should add that as long as the subtraction procedure is performed on Euclidean Green functions, it will result in a spacelike subtraction in Minkowski space. Thus our second prescription [H. Georgi and H. D. Politzer, Phys. Rev. D14, 1829 (1976)] is the same as A but carried out in an arbitrary value of p:

f( 2>(p, mB) = p 2 + mt f( 4)(P1 ,P2,p3,p4) =

at p2 = M 2

-µ 2c >..B at PiPi = M 2

,

(4.6.5)

(oi; - ¼) ,

(4.6.6)

the later point chosen to that s = t = u = M 2 • One can, of course, choose any value for s, t and u, and the p 2 at which f( 2 ) is normalized. In this prescription, the unknown functions are now fixed to be

Ff

= 1/,(2) -

In mt

;

(4.6.7) (4.6.8)

In this case the scale µ has been totally absorbed and replaced by the scale M 2 , which is equally arbitrary. The numerical value of M 2 /mt now clearly becomes relevant in choosing M. The trouble with the type of prescription outlined above is that the renormalization group equation of the last section is not easily solved except in the deep Euclidean region where all masses can presumably be neglected. Yet equating a coupling constant with the value of an amplitude at some scale has some physical appeal even though the identification may take place at an unphysical point. The reason is that it brings in the masses explicitly in the calculation and allows for a ready identification of various physical thresholds. C. We now present a very beautiful prescription invented by 't Hooft and Weinberg [Nucl. Phys. B61, 455 (1973) and Phys. Rev. D8, 3497 (1973), respectively]. It is very simple to state and allows for a simple solution of the "renormalization group equation" (4.5.36) of the last section. In it one simply

138

4 THEORY

sets all the finite parts of the counterterms equal to zero, order by order in the coupling, i.e. Ff= Hf= =Ff= 0 , etc. . (4.6.9)

Gf

Then, by comparing with Eqs. ( 4.5.40) - ( 4.5.46) of the last section, we notice that all the a, band c coefficients are independent of m. This prescription is aptly called "mass-independent" renormalization. This mass independence survives to arbitrarily high order; that this is, in fact, true is not hard to understand by means of the following heuristic argument: when the counterterms have no finite part, they just have the "bare bones" structure needed to cancel the infinite behavior at very short distances and no more, but in this regime, i.e. at infinite momenta, all masses can presumably be neglected, provided the amplitudes are well-behaved as p --+ oo - hence the mass independence. It is true that in prescriptions A and B the mass dependence appeared only through the finite part of the counterterms. This enormous simplification enables us to compute the (J, -y and 'Ym coefficients appearing in (4.5.36) in a straightforward way. For instance, we now write (4.6.10) Differentiate with respect to µ at fixed Ao to obtain

0 = 2{ (A+

f: a1:{~A)) +µ:A (i + f: a~{~A))

k=l

µ

1

(4.6.11)

k=l

with the prime denoting differentiation with respect to A. In this formula A and ~; are analytic at ! 0. It follows that

=

(4.6.12) or taking the limit

! --+

0 (4.6.13)

showing that the (J-function appearing in the "renormalization group equation" depends only on A and is determined by the residue of the simple pole on c Using (4.6.12) and the fact that the residues of the various poles in ! must vanish in (4.6.11), we find that

( 1 - Ad~) a1:+1(A)

= a~(A) ( 1 -

Ad~) a 1 (A) .

(4.6.14)

The meaning of (4.6.12) is clear; a successful renormalization means that the bare coupling does not depend on what µ is - a change in µ is accompanied

4.6 RENORMALIZATION PRESCRIPTIONS

139

by a change in A to as to leave ( 4.6.10) invariant. Let us now evaluate perturbation theory. Using (4.5.43) we find

/3 in

oA µ 8µ

3A 2

= l61r2 + O(A

3

(4.6.15)

) .

Neglecting the terms of 0(A 3 ), we can easily integrate (4.6.15), obtaining A= A.,

1

1-

3 I Girl

A., ln

.E..

,

(4.6.16)

µ,

where A., is the value of A at some scale µ.,. It is clear from (4.6.15) that A increases withµ. Thus, is we start with a small A., ( < < 1) at a given scale µ.,, the effective coupling constant will increase with µ. In so doing, we will have to deal with larger and larger A and will therefore leave the domain of validity of perturbation theory: A .F) is crucial. If /3'(>.F) < 0, as in the figure, µ is positive for A just below >.F, which drives>. to a larger value, i.e., towards the fixed point >.F, while µ is negative when A is above the fixed point, therefore driving>. towards >.F. then see that>. will be driven towards >.F asµ increases: such a fixed point is called ultraviolet stable, because A will approach the value >.F asymptotically as µ -+ oo, from above or from below depending on the starting point >., which can be either above or below AF. If there were a field theory for which /3 behaved as in the curve, at very short distances A would be more and more like >.F. If AF were small, it would mean that if we start from a small A < AF we never leave perturbation theory! Alternatively, if we started with A > AF then as the distance decreases, A would be driven to a perturbative

i>.

wt

4.6 RENORMALIZATION PRESCRIPTIONS

141

region. These two situations are depicted below.

I

I I I -

----------

1

I

None of the field theories in four dimensions exhibit this behavior perturbatively

(AF..F) > 0 and AF is an infrared point. This means that if at µ 6 , ). 3 < AF, >.. will be driven towards 0, but if ). 3 > AF it will be driven away from AF to larger values of>.:

I I

- -+------

We can also derive the variation of the mass withµ, starting from (4.5.41) with bo 1 and bk independent of ';; it is not hard to find that

=

(4.6.20) so that 'Ym

( >.)

= >.. db1(>..) d>..

(4.6.21)

leading to the value in >.q, 4 theory (4.6.22) We also get a recursion formula for the residues of the higher poles: dbk+l db1 dbk ( d ) >.---bk>..--- 1->.- a1(.X) d>.. d>.. d>.. d>..

k=l 2 ··· ' ' .

(4.6.23)

Finally, we can also derive from the definition of Zt/> with co = 1 and independent of '; the equation

ck

(4.6.24) leading to (4.6.25)

143

4.6 RENORMALIZATION PRESCRIPTIONS

so that in

>..) -

1) ma:

+ dn

-

n,d(>.)] f(n)(sp; m, >., µ) = 0 . ( 4.6.28)

Introduce scale dependent variables ~( s) and m( s) such that

s aX(s) 8s

= /3 (>.(s))

>.(s = 1) =

>.

(4.6.29)

m(s = 1) = m

(4.6.30)

Thus (4.6.19) is easy to integrate for it becomes a first order differential equation in s; the result is

f(n)(sp; m, >.,µ)

= sdn f(n)(p; m(s))(s), µ) exp { in

1' ~~,

,d(X(s'))}

( 4.6.31) The interpretation of this equation is that under a change of scale of the external momenta the Green's functions scale in an unexpected way: the coupling constant and mass scale in a nontrivial fashion and the Green's functions develop beside the engineering dimension dn and anomalous dimension 'Yd for each outside leg. Suppose m had been .set equal to zero initially. Then the classical theory would have been invariant under dilatations ( and conformal transformations see Chapter 1). This is no longer true in quantum theory where the regularization technique introduces a scale, either via a high momentum cutoff or via the µ-parameter of dimensional regularization, which breaks dilatation invariance [see S. Coleman's lecture in Aspects of Symmetry, Cambridge University Press, Cambridge, 1985].

144

4 THEORY

Also this shows that the Green's functions at scaled momenta give a behavior ruled by ~( s) and m( s) which therefore govern the physics at large scales. It is interesting to investigate the behavior of the Green's functions at large s. Suppose the theory has an ultraviolet stable fixed point at ,\ = >.F. At large scales ,\ will be driven to >.F. Hence >.m and ')'d will be driven to 'Ym(>.F) and -yd(>.F), respectively, so that the solution of (4.6.21)

- (s ) = me J,'1 -Ym(>.(,))dln, sm

(4.6.32)

can be integrated to (4.6.33) if we assume that the integral is dominated by the large s range. A similar assumption about the integration over ,\d leads to (4.6.34) Thus, if 1 - 'Ym ( ,\F) is positive the mass can be neglected altogether at large scales and also 'Yd(>.F) appears indeed as an anomalous dimension. We see from (4.6.23) that the naive expectation that the masses decouple from the theory at large momenta depends essentially on the integral over the anomalous dimensions which we write as (4.6.35) This form suggests that unless 'Ym(>.) and /3(>.) have simultaneous zeros, the greater contributions to the integral will come from the fixed points, which "kind of" justifies our earlier assumption. If the ultraviolet stable fixed point is at ,\F = 0 (as in Yang-Mills theories), then all is well because the large momentum behavior of the theory is given by perturbation theory, that is ')'m (0) = 0. In ,\.

=b

"

). db1 d>.

!

+2

dbn /3(>.) d>.

n

= 1, 2, .. •

.

C. Given

show that

and that ). dcn+l = ). dc1 d). Cn d).

! den /3(>.)

+2

d).

D. Verify the renormalization group equation in >...') and ,:n(>..') to,(>..), ,m(>..) and [J(>..) where the prime and unprimed system refers to two mass independent renormalization schemes [F, G and F functions are first taken independent of ';:, but can have numerical values]. Show that the value of, and ,m at >..F is prescription independent.

4.8

Continuation to Minkowski Space: Analyticity

At this stage we have obtained the finite Green's functions at the price of introducing an arbitrary scale. However, we know from the renormalization group that if we alter this scale, nothing happens to the Green's functions because the change is compensated by a concomitant change in the renormalized parameters and field. There now remains to express the Green's functions in Minkowski space in order to make contact with reality. This is achieved by means of analytic continuation. Consider an Euclidean Green's function which depends on momenta p1 , • · • , PN. We first change all the time components of the p by making them imaginary

P = (Po,Pi)-+ P = (po= ipo, Pi = pi) As an example let us examine what happens to the propagator. According to the above 1

1

p2 +m2

-+

-p2 +m 2

=

,

=

since we are using the Minkowski metric goo -gu +1. This replacement is not entirely satisfactory because the Minkowski space expression develops a pole · a t p 2 m 2 , i.e., w h en Po = ± P+ m 2 •

=

Jp ·

The continuation process can be regarded as proceeding from the imaginary to the real axis in the po plane in a clockwise fashion: indeed the ability to perform this continuation rests on the avoidance of any pole. It follows that the poles in p 0 must be taken to be just below (above) the positive (negative) real axis, that is the continuation is from

1 p2 + m2

to

-1

p2 - m2 + fr ,

where c > 0, and the limit c -+ o+ is to be taken at the end of all calculations. In more complicated cases the poles will be chosen so that their locations do not interfere with the clockwise rotation of the imaginary time axis into the real time axis. In this -fr prescription we recognize the familiar device used to make the path integral convergent which we discussed earlier. Another way to look at

149

4.8 ANALYTICITY

this is to compare the Feynman rules in Minkowski and Euclidean spaces, say for

(6.1.20)

where Risa rotation matrix (proper and improper). Since 4>T4> (the length of cl>) is O(n) invariant, the matrix R obeys

(6.1.21) Proper rotation matrices can be written in the form

(6.1.22) where wij -wji are the real n(n2- 1l parameters of the rotation group, and the Eij are the n(n2- 1 > generators of the rotation group. By considering the infinitesimal change in cl> and by requiring that the group properties be satisfied

(6.1.23) one can prove that the Eij satisfy a Lie algebra

(6.1.24) In the above we have derived the Lie algebra for SO( n) by using the n x n E;; matrices which act on the n-dimensional vector cl>. It is easy to see from (6.1.21) and (6.1.22) that they are real and antisymmetric. However, one can build many kinds of matrices which satisfy (6.1.24). This is because there are many kinds of ways of representing SO(n). We have chosen to do it in the ndimensional representation but we could as well have described it in the adjoint representation which has the same number of dimensions as there are parameters in the group. In the case of SO(n) it can be represented by an antisymmetric second rank tensor A;; = -A;i• For the adjoint, it is convenient to treat the A;;(x) as matrix elements of an antisymmetric matrix A(x). Then the rotations take the form A --+ A' = RART ; (6.1.25) AT= -A '

6.1

187

GLOBAL AND LOCAL SYMMETRIES

where R is the nxn matrix given by (6.1.22). Then it is easy to build an invariant Lagrangian with A as scalar fields (6.1.26) The symmetric "quadrupole" representation Sij = +S;i can be handled in a similar way when the trace of S is recognized to be SO(n) invariant. Starting from the n representation of SO(n), one can construct more complicated representations described by higher rank tensors. An arbitrarily high rank tensor is in general a combination of tensors that transform irreducibly (among themselves) under the group. For instance, consider a third rank tensor T;,;k and take i, j, k 1, · · ·, 10 for convenience. It is decomposed into irreducible representations as follows:

=

= 120 components. 10/l':f = 220 components,

• the totally antisymmetric part T[ijk) with \~;:

• the totally symmetric part T(ijk) has 2 but it contains, by contracting two indices, a vector Tiij with 10 dimensions. Hence T(ijk) is decomposed as an irreducible 210 dimensional representation plus a 10 dimensional representation of SO10, • tensors with mixed symmetry among the indices: antisymmetric under the interchange of two indices with 320 components (= 45 x 10 -10- 120), and symmetric in two indices only, with 320 + 10 components. Thus in summary we have obtained the SO(10) decomposition of the third rank reducible tensor with 1,000 components into its irreducible parts 1000

= 120 + 220 + 10 + 10 + 320 + 320

.

This type of construction is straightforward (if tedious); the only subtlety occurs when n is even in which case one can use the Levi-Civita tensor €ij•••k with n entries to split the n/2-rank totally antisymmetric tensor in half. Furthermore, when n/2 is even, this results in splitting the n/2-rank anti-symmetrized tensor into two real and inequivalent representations. When n/2 is odd the procedure results in two representations which are the conjugate of one another. For instance, in SO(10), the fifth rank totally anti-symmetrized tensor has 252 components, which split into the 126-dimensional representation and its conjugate, the €-symbol acting as the conjugate. The procedure of taking tensor products of vectors does not generate all representations because SO(~) has in addition spinor representations (e.g., SO(3) has half-integer spin representation). When n = 2m + 1, m = 1, 2, · · ·, SO(n) has only one real fundamental spinor representation of dimension 2m, e.g., SO(3) has a two-dimensional real representation, SO(5) a real four-dimensional spinor representation, etc., out of which all representation can be built. When n 2m, m = 2, 4, 6, SO(n) has two real and inequivalent fundamental spinor representations each with-2m-l dimensions. Finally, for n = 2m, m = 3, 5, · · ·, SO(n)

=

188

GAUGE SYMMETRIES

has two fundamental complex spinor representations conjugate to one another. For instance S0(6) has a 4 and a 4 conjugate to one another, etc. All representations can be constructed from these spinor representations, which means that they are in this sense more fundamental than the vector representation. Consider now the kinetic term for n two component spinor fields 1 ./, ta µ +-+ = 2Y-' £ U 0µ Y-'La .I,

£F

(6.1.27)

,

where a runs from 1 ton and sum over a is implied. For a= 1 we have seen that (6.1.27) is invariant under a phase transformation. When a > 1, £ is invariant under a much larger symmetry: consider the change (suppressing the a index)

(6.1.28) where U is an n x n matrix; and then

(6.1.29) Clearly, if U is x-independent and unitary

uut = utu = 1

(6.1.30)

£Fis invariant under the transformation (6.1.28). The unitarity condition implies that U can be expressed in terms of a Hermitean n x n matrix

(6.1.31) This Hermitean matrix depends on n 2 real parameters. Note that by taking H proportional to the identity matrix we recover the earlier phase invariance. The extra new transformations are then generated by the traceless part of H, expressed in terms of n 2 - 1 parameters by n:J-1

H=

I: wATA

,

(6.1.32)

A=l

where the wA are real parameters and the TA are Hermitean traceless n x n matrices. They generate SU(n), the unitary group in n dimensions, and satisfy the appropriate Lie algebra

(6.1.33) where JABC are real totally antisymmetric coefficients called the structure constants of the algebra [this relation is similar to (6.1.24), but has different f's]. Some celebrated examples are n

= 2:

n

= 3: TA = ½-XA , _xA : Gell-Mann matrices A= l, · · ·, 8 .

TA= ½uA , uA : Pauli spin matrices

A= I, 2, 3

6.1

189

GLOBAL AND LOCAL SYMMETRIES

Both Pauli and Gell-Mann matrices satisfy the normalization condition Tr (o.AuB) = Tr (.xA _xB)

= 26AB

(6.1.34)

and are given by -i

1

.X1

=

=

CD C D = (; D (~ 0

.X2

0

n 0 ~i) As=~G iJ 0 0

.X1

=

0

.X3

=

0

0

.X4

0 -1

G D 0

0

.Xs =

0

0

0

0

0

1

i

0

0 .X6

=

GD 0

(6.1.35)

1

Under SUn, then, we have the following fundamental representations

1P ~ n means that

= iwA TA 1P x ~ ii means that 6x = -iwATA•x btp

(6.1.36)

,

(6.1.37)

where the last transformation property is obtained by requiring that XT tp be invariant. The tensor structure of SU(n) is simpler than that of SO(n): associate a lower (upper) index a to a quantity transforming as the n(ii) representation of SU(n), i.e., 1Pa ~ n, 1Pa ~ ii. Starting from these as building blocks we can generate all the representations of SU(n) by taking tensor products. One representation of interest is the adjoint representation Mf where M is traceless Hermitean and contains n 2 - 1 components; as its indices indicate it is constructed out of the product of n and ii: n ®ii= (n 2 - 1) EB 1. It is convenient to express it as a matrix M which then transforms as (6.1.38) or (6.1.39) Alternatively, we could have represented the adjoint representation by an n 2 - 1 dimensional real vector in which case the representation matrices TA would have been (n 2 - 1) x (n 2 - 1) dimensional. Other types of representations can be built as tensors with arbitrary numbers of upper and lower indices. Upper and lower indices can be contracted to make

190

GAUGE SYMMETRIES

singlets but not lower or upper indices among themselves. Thus for instance Tab can be broken down to its irreducible components just by the symmetry scheme of the ab indices: symmetric and antisymmetric. For example, take SU(5): Tab

5©5

= T(ab) + 7iab)

= 15$10

.

Here(·••) means total symmetry, [· • •] antisymmetry. We have seen that by considering the kinetic terms of fermion and scalar fields, we can generate Lagrangians invariant under unitary and orthogonal transformations. It is also possible to generate symplectic group invariance by means of a kinetic term. We note that for a Grassmann Majorana field the possible candidate for a kinetic term (6.1.40) is identically zero when there is only one Majorana field. However, consider the case of an even number of Majorana fields, WMi i = 1 • • • 2n, coupled by means of an antisymmetric numerical matrix Ei; -E;i with entries

=

i>j i=j

(6.1.41)

i +m) + ...

(8.4.9)

\Ve immediately notice that the terms appearing in this expansion are not welldefined. For instance, the term linear in p +m gives

(eµ2-w)

2

J

d 2Wf 1 (2,r)2w f,4 ,

(8.4.10)

which, besides diverging at the upper end of integration when w = 2, also diverges at the lower end. This type of divergence is called an infrared divergence, and must be dealt with separately from the previously considered ultraviolet divergence. One way to deal with this problem is to give the photon a fictitious mass ,.\ and keep it in the calculation. Then the infrared divergence is avoided and all is well again. In QED this procedure is favored because it does not affect the gauge invariance of the calculation. Note that infrared divergences occur on the "mass-shell" of the external particles: (8.4.10) is at the Euclidean mass shell p +m = 0. The fact that IR divergences in QED can be circumvented by adding a mass to the photon, and arise on mass-shell of external particles indicates that they are connected with the presence of long-range forces which make the definition of asymptotic states ambiguous. Another way of remedying IR divergences is to use dimensional regularization. In this case the procedure consists of performing the parameter integrations in 2w-dimensional space and then in expanding about w 2. Infrared ( as well as ultraviolet) divergences will appear as poles in the dimension plane. More on this later. There remains to fix the vertex counterterms. Before doing so we must rewrite r p(P, ij) as calculated earlier in a recognizable form. We are going to use a prescription for whi~h the fermions are on their "mass-shell". This leads us to understand r p(p, ij) as being sandwiched between spinors on which the (Euclidean) equations of motion are satisfied. Thus it is smart to rewrite r p in the form (8.4.11)

J

=

266

PERTURBATIVE EVALUATION OF GAUGE THEORIES

where I' P is what we are after because the C and D terms will give no contribution due to the equations of motion. In this reduction process, the Gordon identities

,pi= -qp - 2i 0, we obtain our final result

N/ 2 f(A-N/2) 1 7r f(A) (a2t-N/2 .

(B.12)

We have derived this expression for N integer and Re (A - 1;) > 0, Re 1; > 0. Now we take it to be true for noninteger N by means of (B.12),by analytic continuation. Now, by Jetting f f' + p, and relabeling b2 a 2 + p 2 , we can write (B.12) in the form

=

=

(B.13) Next, by successive differentiation of (B.13) with respect to Pµ, it is easy to obtain the formula (B.14) and

317

J

dN e

o o ,{,µ,{,v - ___ 1r N/2 _ _ __ (£2 + 2p. f + a2)A - f(A)(a2 - p2/-N/2

x

[r (A-;)

PµPv

+ ½bµvf

( A-1- ; )

(a 2 -p2 )]

.

(B.15)

These formulae are derived in Euclidean space and in each case the right-hand side is taken to be correct continuation of the left-hand side to non integer values of N.

GLOSSARY OF DIMENSIONAL REGULARIZATION FORMULAE

(B.16) f(A-w) Pµ (41rt f(A) (M2 _ p2/-w

(B.17)

(B.19)

J

d 2w f fµfvlplu 1 [ f(A - w) (21r)2w (£2 + M2 + 2£. p/ = (41rt f(A) PµPvPpPu (M2 - p2/-w

1

+ 2 [6µvPpPu + bvuPµPp + bpuPµPv + bµpPvPu .

f(A- 1-w)

+ bvpPµPu + bµuPpPv] (M 2 _ p 2/-l-w 1

+ 4 [bµvbpu + bvpbµu + bµpbvu]

f(A- 2-w) (M 2 _ p2/-2-w

(B.20)

APPENDIX

C

Euclidean Space Feynman Rules in Covariant Gauge

er

p

rooaoao'

A

P-

B

-----B P-

A

[(r - q)µ8vp + (q - P)p8µv [p+ q+ r = OJ

- igµ 2-w /ABC

-g2µ4-2w[JABE JCDE

(8µp8vu _ 8vp8µu)

+ JCBE JADE (8µp8vu + /DBE JCAE (8up8vµ

319

+ (p -

- 8vµ8pu) - 8vp8µu )]

r)v8pµ]

320

EUCLIDEAN SPACE FEYNMAN RUl:.ES IN COVARIANT GAUGE

,,

p~C

----- ---A

-p

-q

B

fermion line

i a

b

Bibliography

The purpose of this annotated bibliography is to offset the introductory character of the text by providing the reader with a list of references where the material is discussed extensively, both in length and in depth. The author apologizes for any om1ss1on. There are many fine textbooks on Quantum Field Theory. Here we give a few, listed alphabetically by author: V. D. Derestetskii, E. M. Lifshitz and L. P. Pitaevskii, Relativistic Quantum Theory, part I and II (Oxford University Press, Pergamon Press, 1971). J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics and Relativistic Quantum Fields (McGraw-Hill, New York, (1965). N. N. Bogoliubov and D. V. Shirkov An Introduction to the Theory of Quantized Fields (John Wiley & Sons, Inc. - Interscience, New York, 1959). T. P. Cheng and Li L. F. Gauge Theory of Elementary Particle Physics (Oxford University Press, Oxford, 1984). L. D. Faddeev and A. A. Slavnov, Gauge Fields, Introduction to Quantum Theory, Advanced Book Program (Benjamin-Cummings, Reading, MA, 1980). J. 1\1. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison\Vesley, Reading, MA, 1955). C. Itzykson and J .-B. Zuber Quantum Field Theory (McGraw-Hill, New York, 1980). E. l\L Lifshitz and L. P. Pitaevskii, Relativistic Quantum Theory, part II (Pergamon Press, Oxford, UK, 1974). F. Mandi, Introduction to Quantum Field Theory (John \Viley & Son, Inc. Interscience, New York, 1959). C. Nash, Relativistic Quantum Fields (Academic Press, New York, 1978).

321

322

BIBLIOGRAPHY

S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper and Row, New York, 1961). J.C. Taylor, Gauge Theory of Weak Interactions (Cambridge University Press, Cambridge, UK, 1976). See also the extraordinary lectures by S. Coleman, delivered at the International School of Subnuclear Physics "Ettore Majorana"; they are all collected in Aspects of Symmetry (Cambridge University Press, Cambridge, 1985). More advanced readers will enjoy: E. S. Abers and B. W. Lee, "Gauge Theories", Phys. Reports 9C, 1-141. R. Balian and J. Zinn-Justin, Methods in Field Theory (North Holland, Amsterdam, 1975), which contains lectures delivered at the 1975 Les Houches Summer School. We present in the following additional references for each of the chapters in the text: CHAPTER 1. Section 1: An extensive treatment of the relation between invariances of the Lagrangian and conservation laws can be found in E. L. Hill, Rev. Mod. Phys. 23, 253-260 (1951). Sections 2 and 3: Standard references for the Lorentz group are: 1.M. Gel'fand, R.A. Minlos and Z.Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon Press, Oxford, 1963), and 1.M. Gel'fand, M.I. Graev and N. Ya. Vilenkin, Generalized Functions, vol. 5 (Academic Press, New York, 1966). Many of the original papers on the presentation theory of the Lorentz and Poincare groups are reprinted in F.J. Dyson, Symmetry Groups in Nuclear and Particle Physics (Benjamin, New York, 1966). Another beautiful exposition of the representation theory can be found in the lecture notes of F. Giirsey on Field Theory, edited by M. Giinaydin, 1970 (unpublished). Sections 4, 5, 6 and 7: The Transformation properties of fields are discussed by T. W. Kibble, J. Math. Phys. 2, 212-221 (1961). Section 8: Supersymmetry first appears in the context of string models: P. Ramond, Phys. Rev. D3, 2415-1418 (1971), A. Neveu and J. Schwarz, Nucl. Phys. B31, 86-112 (1971); and also as a generalization of the Poincare group, Yu. A. Gol'fand and E. P. Likhtman. JETP Letters 13, 323-326 (1971), D. V. Volkov and V. P. Akulov, Phys. Lett. 46B, 109-110 (1973), J. Wess and B. Zumino, Nucl. Phys. D70, 39-50 (1974). For an extensive review of supersymmetry, see P. Fayet and S. Ferrara, Phys. Reports 32C, 249-334 (1977).

323 CHAPTER 2. Section 1: For more details on canonical transformations see H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1950) and N. Mukunda and E. C. G. Sudarshan, Classical .Dynamics: A Modern Perspective (John Wiley & Son. Inc. - Interscience, New York, 1974). V.1. Arnold Mathematical Methods of Classical Physics (Springer-Verlag, New York, 1978). Sections 2 and 3: Original papers on the Path Integrals are to be found in the reprint volume Selected Papers on Quantum Electrodynamics, J. Schwinger, ed. (Dover, New York, 1958). The uses of the Path Integral are detailed in R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw Hill, New York, 1965). CHAPTER 3. Sections 1, 2, 3 and 4: The treatment follows closely that of E. Abers and B. Lee, op. cit.; D. Lurie Particles and Fields (John Wiley & Son. Inc. Interscience, New York, 1968). Sections 5 and 6: For the (-function evaluation of determinants see D. B. Ray and I. M. Singer, Adv. Math. 7, 145 (1971); J. S. Dowker and R. Critchley, Phys. Rev. D13, 3224-3232 (1976); E. Corrigan, P. Goddard, H. Osborn and S. Templeton, Nucl. Phys. Bl59, 469---496 (1979). Section 7: For a lucid account of finite temperature field theory and some of its applications, see A.D. Linde, Rep. Prog. Phys. 42, 25(1979). CHAPTER 4. Section 3: Standard references on dimensional regularization (besides the one quoted in the text) are C. G. Bellini and J. J. Giambiagi, Nuovo Cimento 12B, 20-26 (1972); J. F. Ashmore, Nuovo Cimento Lett. 4, 289-290 (1972); for a review see G. Leibbrandt, Rev. Mod. Phys. 47, 849-876 (1975). Section 5: A lucid explanation of renormalization is to be found in Coleman's Erice lectures op. cit.. See also J. C. Collins, Phys. Rev. D10, 1213-1218 (1974); for proofs of renormalizability see W. Zimmerman in Lectures on Elementary Particles and Quantum Field Theory, S. Deser, M. Grisaru and H. Pendleton, eds. (MIT Press, Cambridge, MA, 1970), and E. Abers and B. Lee, op. cit. On the renormalization group: E. C. G. Stuckelberg and A. Petermann, Helv. Pysica Acta 26, 499-520 (1953); M. Gell-Mann and F. Low, op. cit.; C. C. Callan in Methods in Field Theory op. cit. (1975). For the difference between different renormalization group equations, see S. Weinberg, Phys. ll,ev. D8, 3497-3509 (1973). Section 8 and 9: R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The Analytic S-Matn·x (Cambridge University Press, Cambridge, UK, 1966), and E. M. Lifshitz and L. P. Pitaevskii, op. cit.

324

BIBLIOGRAPHY

CHAPTER 5. The fundamental reference for this chapter is F. A. Berez in, The Method of Second Quantization (Academic Press, New York, 1966), and inevitably S. Coleman, The Uses of Instantons, op. cit . . CHAPTER 6. Sections 1 and 2: In addition to the Yang-Mills paper op. cit., see R. Utiyama, 1597-1607 (1956); T. W. Kibble, op. cit., (1961) and M. Gell-Mann and S. L. Glashow, Ann. Phys. (New York) 15, 437-460 )1961). Section 3: For instantons: S. Coleman, The Uses of Instantons, in op. cit.; J. C. Taylor, op. cit. (1976). Path-ordered exponentials were first introduced by S. Mandelstam, Ann. Phys. (New York) 19, 1-24 (1962). Section 4: For the vierbein approach see R. Utiyama, op. cit., and T. W. Kibble, op. cit. See also the beautiful text by S. ·weinberg Gravity and Cosmology (John Wiley & Sons, New York 1972). CHAPTER 7. Section 1: A beautiful exposition of systems with constrained Hamiltonians is to be found in P. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva University Press, New York, 1964, and Lectures on Quantum Field Theory (Yeshiva University Press, New York, 1965). Section 2: See the text by L. D. Faddeev and A. A. Slavnov, op. cit. (1980); L. D. Faddeev in Methods of Field Theory, op. cit. (1975). CHAPTERS. Section 1: The necessity for ghost fields in covariant gauges was first recognized by R. R. Feynman, Acta Physics Polonica 24, 697-722 (1963); B. S. De Witt, Phys. Rev. 162, 1195-1239 (1967). Sections 2 and 4: Quantum Electrodynamics has been extensively discussed in many textbooks. Some additional ones are A. I. Akhiezer, V. B. Berestetskii, Quantum Electrodynamics (John Wiley & Son, Inc. -Interscience, New York, 1965); R. P. Feynman, Quantum Electrodynamics (Benjamin, New York, 1961); G. Kallen, Quantum Electrodynamics (Springer-Verlag, Berlin, (1972); for dimensionally regularized QED, see for instance J. C. Collins and A. J. McFarlane, Phys. Rev. D10, 1201-1212 (1973). Section 3: J. C. Ward, Phys. Rev. 78, 182-183 (1950); Y. Takal1aski, Nuovo Cimento 6, 370-375 {1957). Section 4: For use of dimensional regularization to tag infrared divergences, see R. Gastmans and R. Mendermans, Nucl. Phys. B63, 277-284 (1973); ,v. J. Marciano, Phys. Rev. D12, 3861-3871 (1975). Section 5: Yang-Mills theories are proved renormalizable by G. 't Hooft, Nucl. Phys. D33, 173-199 (1971); see also E. Abers and B. ,v. Lee, op. cit..

325 Section 7: A. A. Slavnov, Theor. & Math. Phys. 10, 99-104 (1972); J.C. Taylor, Nucl. Phys. B33, 436-444 (1971). Section 8: The phenomenon of asymptotic freedom was first noted by G. 't Hooft (1972) at the Conference on Lagrangian Field Theory, Marseille; for more details and references see H. D. Politzer, "Asymptotic Freedom: An Approach to Strong Interactions," Phys. Reports 14C, 129-180 (1974). For applications of QCD: R. D. Field, Applications of Quantum Chromodynamics, lectures given at the La Jolla Institute Summer School (1978). Section 9: See Current Algebra and Anomalies by Treiman, Jackiw, Zumino, and Witten, (Princeton University Press, Princeton, 1985).

Index

Analytical continuation, 59, 136, 148 Anomaly, 185, 192, 294 Anomalous dimension, 143 Anomalous magnetic moment, 268 Asymptotic freedom, 141, 290 Auxiliary fields, 42 Axial gauge, 228 Bare fields, 132, 27 4 BRS transformation, 260, 286 Belinfante tensor, 33, 38 ,8-function, 87, 135, 138, 296 Bianchi identities, 195, 215 Boosts, 6 Callan-Symanzik equation, 133 Canonical transformation, 26, 45, 46 Casimir operator, 8, 11 Charge conjugation, 20 Chiral symmetry, 37 Chiral transformation, 34, 36, 170 Christoffel symbols, 219 Classical limit, 52 Compton Amplitude, 271 Confinement, 292 Conformal Invariance, 29 Connected Green's functions, 76, 101, 103 Consistency condition, 310 Coulomb gauge, 227

Counterterms, 127 Counterterm Lagrangian, 132 Covariant derivative, 193 Curvature, 213 Cut diagrams, 158 Dilatation, 32, 38, 87 Dilogarithm, 126 Dimensional regularization, 115 Dinosaur diagram, 111 Dirac spinor, 19 Double scoop diagram, 122, 130 Dynkin index, 279 Energy momentum tensor, 30 Euclidean space, 65 Effective action, 71 Effective potential, 73 Entropy, 92 Euler-Mascheroni Constant, 97 Euler-Mascheroni constant, 118 Equivalence Principle, 210, 211 Exceptional groups, 191 Faddeev-Popov ghosts, 242 Feynman diagrams, 59 Feynman gauge, 244 Feynman parameters, 121 Feynman rules for scalar fields, 101

327

328 Feynman rules for spinors, 171 Feynman rules for gauge fields, 319 Fierz matrix, 41 Finite part of counterterms, 137 Fish diagram, 109, 120 Fixed points, 141 Functional determinant, 75 Furry's theorem, 250 -y matrices, 19 Gauge invariance, 184 Gauge transformations, 194 Gell-Mann matrices, 188 Gell-Mann-Low equation, 134 Generating functional, 65 Ghost fields, 242 Global anomalies, 295 Global transformations, 37 Gluons, 292 Grassmann numbers, 17 Graviton, 23 Green's function, 59, 65 Gribov ambiguity, 234

Haar measure, 238 Ilamilton-J acobi equation, 49 Harmonic oscillator, 56, 90 Heat Function, 81 Homotopy, 202

Index

Legendre transformation, 64, 70 Lie algebra, 186, 198 Lie groups, 191 Local invariance, 184 Lorentz gauge, 242 Majorana flip, 40 Majorana mass, 35, 38 Majorana spinor, 20 Mandelstam variables, 128 Mass-independent renormalization, 137 Neutrino, 35 Noether's theorem, 3, 27, 31 Noether Current, 295 One-loop diagrams, 120 One-particle irreducible graphs, 80 Optical theorem, 156 Ordering, 50, 54 Orthogonal group, 187 Overlapping divergences, 131, 135

Infrared divergence, 109, 263, 265 Infrared stable, 141 Instantons, 201

Palatini formalism, 215 Parity, 19 Partition function, 88 Pauli-Lubanski vector, 11 Pontryagin index, 203, 236 Poisson brackets, 224 Primitive divergences, 113 Proper vertices, 73

Klein-Gordon equation, 68 Killing equation, 222 Killing vector, 222

QCD, 139, 292 Quarks, 139, 248 QED, 185

Lagrange multiplier, 232 Lamb shift, 269 Landau gauge, 244 Landau point, 139 Landau singularity, 258 Lcvi-Civita tensor, 8

Rarita- Schwinger field, 21 Renormalization group equation, 133 Renormalization prescription, 136 Renormalized Lagrangian, 132 Ricci tensor, 214 Rotation group, 8

329

Index

S-matrix, 153 Scale dependent parameters, 84 Setting sun diagram, 123, 150 u-model, 38 Sine-Gordon Lagrangian, 30 Slavnov-Taylor identities, 275 Spin matrices, 188 Steepest descent, 85 Superficial degree of divergence, 109 Supersymmetry, 39 Symplectic group, 190

Thermodynamic potential, 91 Tadpole diagram, 120, 127 Thomson limit, 136 Thresholds, 150 Transition amplitude, 154 Tree diagrams, 79 Torsion, 213 Triangle diagram, 304 Two-loop diagrams, 123

Ultraviolet divergences, 112 Unitarity, 153 Unitary group, 188

Vacuum polarization, 174, 252, 280 Vertex correction, 253 Vierbein, 209

Ward identities, 28, 262, 275 Weinberg's theorem, 113 Wess-Zumino consistency condition, 310 Weyl group, 29 Weyl spinors, 9 Weyl tensor, 215 Wick rotation, 59

Yang-Mills equation, 198 Yukawa coupling, 173

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