The Quantum Theory of Fields, Volume 3: Supersymmetry

443 Pages·2005·28.046 MB·English

by  Steven Weinberg

1m.,.
C.\I\IIUtlD(.I· information - www.cambridge.org/9780S21660006
In this third volume of The Quantum Theory of Fields, Nobel Laureate Steven
Weinberg continues his masterly exposition of quantum field theory. This volume
presents a self-contained, up-to-date, and comprehensive introduction to super-
symmetry, a highly active area of theoretical physics that is likely to be at the
center of future progress in the physics of elementary particles and gravitation.
The text introduces and explains a broad range of topics, including supersymmet-
ric algebras, supersymmetric field theories, extended supersymmetry, supergraphs,
non-perturbative results, theories of supersymmetry in higher dimensions, and su-
pergravity. A thorough review is given of the phenomenological implications of
supersymmetry, including theories of both gauge- and gravitationally mediated
supersymmetry breaking. Also provided is an introduction to mathematical tech-
niques, based on holomorphy and duality, that have proved so fruitful in recent
developments. This book contains much material not found in other books on
supersymmetry, including previously unpublished results. Problems are included.
From reviews of Volume I
'. .. an impressively lucid and thorough presentation of the subject ... Weinberg
manages to present difficult topics with richness of meaning and marvellous
clarity. Full of valuable insights, his treatise is sure to become a classic, doing
for quantum field theory what Dirac's Quantum Mechanics did for quantum
mechanics. I eagerly await the publication of the second volume.'
S. S. Schweber, Nature
'For over twenty years there has been no good modern textbook on the subject.
For all that time, Steven Weinberg has been promising to write one. That he has
finally done it is cause for celebration among those who try to teach and try to
learn the subject. Weinberg's book is for serious students of field theory ... it is the
first textbook to treat quantum field theory the way it is used by physicists today.'
Howard Georgi, Science
'Steven Weinberg, who contributed to the development of quantum chromo-
dynamics and shared the Nobel Prize in Physics for his contributions to the
electroweak theory, has written a definitive text on the physical foundations of
quantum field theory. His book differs significantly from the long line of previous
books on quantum field theory ... To summarize, Foundations builds the structure
of quantum field theory on the sure footing of physical insight. It is beautifully
produced and meticulously edited ... and it is a real bargain in price. If you want
to learn quantum field theory, or have already learned it and want to have a
definitive reference at hand, purchase this book.'
O. W. Greenberg, Physics Today
'In addition to a superb treatment of all the conventional topics there are
numerous sections covering areas that are not normally emphasized, such as the
subject of field redefinitions, higher-rank tensor fields and an unusually clear
and thorough treatment of infrared effects ... this latest book reinforces his high
scholarly standards. It provides a unique exposition that will prove invaluable
both to new research students as well as to experienced research workers. Together
with Volume II, this will become a classic text on a subject of central importance
to a wide area of theoretical physics.'
M. B. Green, CERN Courier
'I believe that what readers will find particularly helpful in this volume is the
consistency of the whole approach, and the emphasis on quantities and properties
that are directly useful to particle physicists. This is particularly true for those
who are interested in the more phenomenological aspects. The reader only needs
limited background knowledge, and a clear line is followed throughout the book,
making it easy to follow. The author presents extremely thorough but elementary
discusssions of important physical questions, some of which seem to be an original
way of addressing the subject.'
J. Zinn-Justin, Physics World
'This is a well-written book by one of the masters of the subject ... it is certainly
destined to become a standard text book and should find its way to the shelves
of every physics library.'
J. Madore, Classical and Quantum Gravity
'The book starts out with an excellent historical introduction, not found anywhere
else, giving citations to many by now classic papers ... a valuable reference work
as well as a textbook for graduate students.'
G. Roepstorff, Zentralblatt fur Mathematik
From reviews of Volume II
'It is a majestic exposition. The two volumes are structured in a logical way.
Everything is explained with incisive clarity. Weinberg always goes to the heart
of any argument, and includes many things that cannot be found elsewhere
in the literature. Often I find myself thinking: "Ah! Now I understand that
properly." ... I find it hard to imagine a better treatment of quantum field theory
than Weinberg's. All serious students and researchers will want to have these
volumes on their shelves.'
John C. Taylor, Nature
'Weinberg's Modern Applications goes to the boundaries of our present under-
standing of field theory. It is unmatched by any other book on quantum field
theory for its depth, generality and definitive character, and it will be an essential
reference for serious students and researchers in elementary particle physics.'
O. W. Greenberg, Physics Today
'. .. Steven Weinberg is one of our most gifted makers of theoretical tools as well as
a virtuoso in their use. His new book conveys both the satisfaction of understand-
ing nature and the feel of the atelier, for the "modern applications" of its subtitle
include both the derivation of physical consequences and the development of new
tools for understanding and applying field theory itself. .. Modern Applications is
a splendid book, with abundant useful references to the original literature. It is a
very interesting read from cover to cover, for the wholeness Weinberg's personal
perspective gives to quantum field theory and particle physics.'
Chris Quigg, Science
'Experienced researchers and beginning graduate students will delight in the
gems of wisdom to be found in these pages. This book combines exposition
of technical detail with physical insight in a unique manner that confirms the
promise of Volume I and I have no doubt that these two volumes will rapidly
constitute the classic treatment of this important subject.'
M. B. Green, CERN Courier
' ... a valued reference and a mine of useful information for professional field
theorists.'
Tom Kibble, New Scientist
' ... a clear presentation of the subject, explaining the underlying concepts in much
depth and in an accessible style. I expect that these volumes will become the
first source we turn to when trying to answer the challenging questions asked by
bright postgraduates when they first encounter quantum field theory ... I have no
doubt that The Quantum Theory of Fields will soon be found on the bookshelves
of most particle theorists, and that it will be one of the main sources used in the
preparation of lectures on the subject for postgraduate students.'
C. T. C. Sachrajda, The Times Higher Education Supplement
' ... Weinberg has produced a masterpiece that will be a standard reference on the
field for a long time to come.'
B. E. Y. Svensson, Elementa
From reviews of Volume III
' ... has produced a treatise that many of us had long awaited, perhaps without
fully realizing it ... with the publication of The Quantum Theory of Fields, Vol. III,
has performed an analogous service for supersymmetry ... Although this volume
is the third in a trilogy, it is quite different from its two predecessors, and it stands
on its own ... Maya new generation of students imbibe its content and spirit.'
Physics Today
'The third volume of The Quantum Theory of Fields is a self-contained introduction
to the world of supersymmetry and supergravity. It will be useful both for
experienced researchers in the field and for students who want to take the first
steps towards learning about supersymmetry. Unlike other books in this field, it
covers the wide spectrum of possible applications of supersymmetry in physics.'
Hans Peter Nilles, Nature
'Weinberg is of course one of the creators of modern quantum field theory, as
well as of its physical culmination, the standard model of all (nongravitational)
interactions. It is ... very timely that this latest part of his monograph, devoted
to supersymmetry and supergravity, has just appeared. As a text, it has been
pretested by Weinberg for a freestanding one-year graduate course; as a clear
organizing reference to this extremely vast field, it will help the experts as
well ... Weinberg's style of presentation is as clear and meticulous as in his
previous works.'
Stanley Deser, Journal of General Relativity and Gravitation
'Weinberg tries to be as elementary and clear as possible and steers clear of
more sophisticated mathematical tools. Together with the previous volumes, this
volume will serve as an invaluable reference to researchers and a textbook for
graduate students.'
G. Roepstorff, Zentralblatt MATH
The Quantum Theory of Fields
Volume III
Supersymmetry
Steven Weinberg
University of Texas at Austin
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Contents
Sections marked with an asterisk are somewhat out of the book's main line of
development and may be omitted in a first reading.
PREFACE TO VOLUME III XVI
NOTATION xx
24 HISTORICAL INTRODUCTION 1
24.1 Unconventional Symmetries and 'No-Go' Theorems 1
S U( 6) symmetry 0 Elementary no-go theorem for unconventional semi-simple
compact Lie algebras 0 Role of relativity
24.2 The Birth of Supersymmetry 4
Bosonic string theory 0 Fermionic coordinates 0 Worldsheet supersymmetry 0
Wess-Zumino model 0 Precursors
Appendix A SU(6) Symmetry of Non-Relativistic Quark Models 8
Appendix B The Coleman-Mandula Theorem 12
Problems 22
References 22
25 SUPERSYMMETRY ALGEBRAS 25
25.1 Graded Lie Algebras and Graded Parameters 25
Fermionic and bosonic generators 0 Super-Jacobi identity 0 Grassmann par-
ameters 0 Structure constants from supergroup multiplication rules 0 Complex
conjugates
25.2 Supersymmetry Algebras 29
Haag-Lopuszanski-Sohnius theorem 0 Lorentz transformation of fermionic gen-
erators 0 Central charges 0 Other bosonic symmetries 0 R-symmetry 0 Simple
Vll
Vlll Contents
and extended supersymmetry 0 Four-component notation 0 Superconformal
algebra
25.3 Space Inversion Properties of Supersymmetry Generators 40
Parity phases in simple supersymmetry 0 Fermions have imaginary parity 0
Parity matrices in extended supersymmetry 0 Dirac notation
25.4 Massless Particle Supermultiplets 43
Known particles are massless for unbroken supersymmetry 0 Helicity raising and
lowering operators 0 Simple supersymmetry doublets 0 Squarks, sleptons, and
gauginos 0 Gravitino 0 Extended supersymmetry multiplets 0 Chirality problem
for extended supersymmetry
25.5 Massive Particle Supermultiplets 48
Raising and lowering operators for spin 3-component 0 General massive multi-
plets for simple supersymmetry 0 Collapsed supermultiplet 0 Mass bounds in
extended supersymmetry 0 BPS states and short supermultiplets
Problems 53
References 54
26 SUPERS YMMETRIC FIELD THEORIES 55
26.1 Direct Construction of Field Supermultiplets 55
Construction of simplest N = 1 field multiplet 0 Auxiliary field 0 Infinitesi-
mal supersymmetry transformation rules 0 Four-component notation 0 Wess-
Zumino supermultiplets regained
26.2 General Superfields 59
Superspace spinor coordinates 0 Supersymmetry generators as superspace dif-
ferential operators 0 Supersymmetry transformations in superspace 0 General
superfields 0 Multiplication rules 0 Supersymmetric differential operators in su-
perspace 0 Supersymmetric actions for general superfields 0 Parity of component
fields 0 Counting fermionic and bosonic components
26.3 Chiral and Linear Superfields 68
Chirality conditions on a general superfield 0 Left- and right-chiral superfields 0
Coordinates x~ 0 Differential constraints 0 Product rules 0 Supersymmetric ~­
terms 0 ~ -terms equivalent to D-terms 0 Superpotentials 0 Kahler potentials
o Partial integration in superspace 0 Space inversion of chiral superfields 0
R-symmetry again 0 Linear superfields
26.4 Renormalizable Theories of Chiral Superfields 75
Counting powers 0 Kinematic Lagrangian 0 ~ -term of the superpotential 0
Complete Lagrangian 0 Elimination of auxiliary fields 0 On-shell superalgebra 0
Vacuum solutions 0 Masses and couplings 0 Wess-Zumino Lagrangian regained
Contents IX
26.5 Spontaneous Supersymmetry Breaking in the Tree Approximation 83
O'Raifeartaigh mechanism 0 R-symmetry constraints 0 Flat directions 0 Gold-
stino
26.6 Superspace Integrals, Field Equations, and the Current Superfield 86
Berezin integration 0 D- and .?-terms as superspace integrals 0 Potential su-
perfields 0 Superspace field equations 0 Conserved currents as components of
linear superfields 0 Conservation conditions in superspace
26.7 The Supercurrent 90
Supersymmetry current 0 Superspace transformations generated by the super-
symmetry current 0 Local supersymmetry transformations 0 Construction of
the supercurrent 0 Conservation of the supercurrent 0 Energy-momentum ten-
sor and R-current 0 Scale invariance and R conservation 0 Non-uniqueness of
supercurrent
26.8 General Kahler Potentials' 102
Non-renormalizable non-derivative actions 0 D-term of Kahler potential 0
Kahler metric 0 Lagrangian density 0 Non-linear a-models from spontaneous
internal symmetry breaking 0 Kahler manifolds 0 Complexified coset spaces
Appendix Majorana Spinors 107
Problems 111
References 112
27 SUPERS YMMETRIC GAUGE THEORIES 113
27.1 Gauge-Invariant Actions for Chiral Superfields 113
Gauge transformation of chiral superfields 0 Gauge superfield V 0 Extended
gauge invariance 0 Wess-Zumino gauge 0 Supersymmetric gauge-invariant kine-
matic terms for chiral superfields
27.2 Gauge-Invariant Action for Abelian Gauge Superfields 122
Field strength supermultiplet 0 Kinematic Lagrangian density for Abelian gauge
supermultiplet 0 Fayet-Iliopoulos terms 0 Abelian field-strength spinor super-
field W~ 0 Left- and right-chiral parts of Wx 0 Wa as a superspace derivative of
V 0 Gauge invariance of W~ 0 'Bianchi' identities in superspace
27.3 Gauge-Invariant Action for General Gauge Superfields 127
Kinematic Lagrangian density for non-Abelian gauge supermultiplet 0 Non-
Abelian field-strength spinor superfield WA~ 0 Left- and right-chiral parts of
WAa 0 8-term 0 Complex coupling parameter r
27.4 Renormalizable Gauge Theories with Chiral Superfields 132
Supersymmetric Lagrangian density 0 Elimination of auxiliary fields 0 Condi-
tions for unbroken supersymmetry 0 Counting independent conditions and field