The standard model and beyond

Physics

Series in High Energy Physics, Cosmology, and Gravitation

Langacker

Series in High Energy Physics, Cosmology, and Gravitation

Series Editors: Brian Foster and Edward W. Kolb

The Standard Model and Beyond
Paul Langacker
“The Standard Model and Beyond is a state-of-the-art description of what we
know about the particles and forces that build up the world we see. Most books
that cover these topics are quantum field theory books that treat the quarks and
leptons, and the electromagnetic, weak, and strong forces as examples. This
is the first treatment with the opposite priorities, focusing on the structure and
applications of the standard model and bringing in the field theory as needed, in
a pedagogically reliable and thorough treatment. Langacker knows well that the
standard model is the platform on which a deeper understanding of the laws of
nature will be constructed, perhaps from clues soon to come from the Large Hadron
Collider, and provides preparation so the reader can participate in that progress.”

The
Standard Model
and Beyond

—Gordon Kane, Victor Weisskopf Collegiate Professor of Physics and Director
of the Michigan Center for Theoretical Physics, University of Michigan,
Ann Arbor, USA
The Standard Model and Beyond presents an advanced introduction to the

physics and formalism of the standard model and other non-abelian gauge
theories. Thoroughly covering gauge field theories, symmetries, and topics beyond
the standard model, this text equips readers with the tools to understand the
structure and phenomenological consequences of the standard model, to construct
extensions, and to perform calculations at tree level. It establishes the necessary
background for readers to carry out more advanced research in particle physics.
Features
• Covers the fundamental interactions
• Describes the construction, experimental tests, and phenomenological
consequences of the standard model
• Presents a self-contained treatment of the complicated technology
needed for tree-level calculations
• Explores applications in astrophysics and cosmology
• Lists many useful reference books, review articles, research papers, and
Web links
• Offers supplementary materials on the author’s Web site

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Paul Langacker


The Standard Model and Beyond

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Series in High Energy Physics, Cosmology, and Gravitation
Series Editors: Brian Foster, Oxford University, UK
Edward W Kolb, Fermi National Accelerator Laboratory, USA
This series of books covers all aspects of theoretical and experimental high energy physics,
cosmology and gravitation and the interface between them. In recent years the fields of particle
physics and astrophysics have become increasingly interdependent and the aim of this series
is to provide a library of books to meet the needs of students and researchers in these fields.
Other recent books in the series:
Particle and Astroparticle Physics
Utpal Sakar
Joint Evolution of Black Holes and Galaxies

M Colpi, V Gorini, F Haardt, and U Moschella (Eds.)
Gravitation: From the Hubble Length to the Planck Length
I Ciufolini, E Coccia, V Gorini, R Peron, and N Vittorio (Eds.)
Neutrino Physics
K Zuber
The Galactic Black Hole: Lectures on General Relativity and Astrophysics
H Falcke, and F Hehl (Eds.)
The Mathematical Theory of Cosmic Strings: Cosmic Strings in the Wire Approximation
M R Anderson
Geometry and Physics of Branes
U Bruzzo, V Gorini and, U Moschella (Eds.)
Modern Cosmology
S Bonometto, V Gorini and, U Moschella (Eds.)
Gravitation and Gauge Symmetries
M Blagojevic
Gravitational Waves
I Ciufolini, V Gorini, U Moschella, and P Fré (Eds.)
Classical and Quantum Black Holes
P Fré, V Gorini, G Magli, and U Moschella (Eds.)
Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics
F Weber

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Series in High Energy Physics, Cosmology, and Gravitation

The Standard Model

and Beyond

Paul Langacker
Institute for Advanced Study
Princeton, New Jersey, USA

Boca Raton London New York

CRC Press is an imprint of the
Taylor & Francis Group, an informa business

A TA Y L O R & F R A N C I S B O O K

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CRC Press
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Contents

Preface

xi

1 Notation and Conventions
1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Review of Perturbative Field Theory
2.1 Creation and Annihilation Operators . . . . . . . . . . .
2.2 Lagrangian Field Theory . . . . . . . . . . . . . . . . . .

2.3 The Hermitian Scalar Field . . . . . . . . . . . . . . . .
2.3.1 The Lagrangian and Equations of Motion . . . .
2.3.2 The Free Hermitian Scalar Field . . . . . . . . .
2.3.3 The Feynman Rules . . . . . . . . . . . . . . . .
2.3.4 Kinematics and the Mandelstam Variables . . . .
2.3.5 The Cross Section and Decay Rate Formulae . .
2.3.6 Loop Effects . . . . . . . . . . . . . . . . . . . . .
2.4 The Complex Scalar Field . . . . . . . . . . . . . . . . .
2.4.1 U (1) Phase Symmetry and the Noether Theorem
2.5 Electromagnetic and Vector Fields . . . . . . . . . . . .
2.5.1 Massive Neutral Vector Field . . . . . . . . . . .
2.6 Electromagnetic Interaction of Charged Pions . . . . . .
2.7 The Dirac Field . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 The Free Dirac Field . . . . . . . . . . . . . . . .
2.7.2 Dirac Matrices and Spinors . . . . . . . . . . . .
2.8 QED for Electrons and Positrons . . . . . . . . . . . . .
2.9 Spin Effects and Spinor Calculations . . . . . . . . . . .
2.10 The Discrete Symmetries P , C, CP , T , and CP T . . . .
2.11 Two-Component Notation and Independent Fields . . .
2.12 Quantum Electrodynamics (QED) . . . . . . . . . . . .
2.12.1 Higher-Order Effects . . . . . . . . . . . . . . . .
2.12.2 The Running Coupling . . . . . . . . . . . . . . .
2.12.3 Tests of QED . . . . . . . . . . . . . . . . . . . .
2.12.4 The Role of the Strong Interactions . . . . . . . .
2.13 Mass and Kinetic Mixing . . . . . . . . . . . . . . . . .
2.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

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v


vi

The Standard Model and Beyond

3 Lie Groups, Lie Algebras, and Symmetries
3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Groups and Representations . . . . . . . . . . . . .
3.1.2 Examples of Lie Groups . . . . . . . . . . . . . . .
3.1.3 More on Representations and Groups . . . . . . . .
3.2 Global Symmetries in Field Theory . . . . . . . . . . . . .
3.2.1 Transformation of Fields and States . . . . . . . .
3.2.2 Invariance (Symmetry) and the Noether Theorem .
3.2.3 Isospin and SU (3) Symmetries . . . . . . . . . . .
3.2.4 Chiral Symmetries . . . . . . . . . . . . . . . . . .
3.2.5 Discrete Symmetries . . . . . . . . . . . . . . . . .
3.3 Symmetry Breaking and Realization . . . . . . . . . . . .
3.3.1 A Single Hermitian Scalar . . . . . . . . . . . . . .
3.3.2 A Digression on Topological Defects . . . . . . . .
3.3.3 A Complex Scalar: Explicit and Spontaneous
Symmetry Breaking . . . . . . . . . . . . . . . . .
3.3.4 Spontaneously Broken Chiral Symmetry . . . . . .
3.3.5 Field Redefinition . . . . . . . . . . . . . . . . . . .
3.3.6 The Nambu-Goldstone Theorem . . . . . . . . . .
3.3.7 Boundedness of the Potential . . . . . . . . . . . .
3.3.8 Example: Two Complex Scalars . . . . . . . . . . .
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

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157
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160
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177
179

Strong Interactions and QCD
The QCD Lagrangian . . . . . . . . . . . . . . . . . . . . .
Evidence for QCD . . . . . . . . . . . . . . . . . . . . . . .
Simple QCD Processes . . . . . . . . . . . . . . . . . . . . .
The Running Coupling in Non-Abelian Theories . . . . . .
5.4.1 The RGE Equations for an Arbitrary Gauge Theory

Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . .
5.5.1 Deep Inelastic Kinematics . . . . . . . . . . . . . . .
5.5.2 The Cross Section and Structure Functions . . . . .
5.5.3 The Simple Quark Parton Model (SPM) . . . . . . .
5.5.4 Corrections to the Simple Parton Model . . . . . . .
Other Short Distance Processes . . . . . . . . . . . . . . . .
The Strong Interactions at Long Distances . . . . . . . . . .
The Symmetries of QCD . . . . . . . . . . . . . . . . . . . .

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183
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4 Gauge Theories
4.1 The Abelian Case . . . . . .
4.2 Non-Abelian Gauge Theories
4.3 The Higgs Mechanism . . .
4.4 The Rξ Gauges . . . . . . .
4.5 Anomalies . . . . . . . . . .
4.6 Problems . . . . . . . . . .
5 The
5.1
5.2
5.3
5.4
5.5

5.6
5.7
5.8

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vii

Table of Contents
5.8.1 Continuous Flavor Symmetries
5.8.2 The (3∗ , 3) + (3, 3∗ ) Model . . .
5.8.3 The Axial U (1) Problem . . . .
5.8.4 The Linear σ Model . . . . . .
5.8.5 The Nonlinear σ Model . . . .
5.9 Other Topics . . . . . . . . . . . . . .
5.10 Problems . . . . . . . . . . . . . . . .

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6 The Weak Interactions
239
6.1 Origins of the Weak Interactions . . . . . . . . . . . . . . . . 239
6.2 The Fermi Theory of Charged Current Weak Interactions . . 245
6.2.1 µ Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.2.2 νe e− → νe e− . . . . . . . . . . . . . . . . . . . . . . . 256
6.2.3 π and K Decays . . . . . . . . . . . . . . . . . . . . . 258
6.2.4 Nonrenormalization of Charge and the Ademollo-Gatto

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 266
6.2.5 β Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.2.6 Hyperon Decays . . . . . . . . . . . . . . . . . . . . . 273
6.2.7 Heavy Quark and Lepton Decays . . . . . . . . . . . . 274
6.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
7 The Standard Electroweak Theory
281
7.1 The Standard Model Lagrangian . . . . . . . . . . . . . . . . 281
7.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . 287
7.2.1 The Higgs Mechanism . . . . . . . . . . . . . . . . . . 287
7.2.2 The Lagrangian in Unitary Gauge after SSB . . . . . . 289
7.2.3 Effective Theories . . . . . . . . . . . . . . . . . . . . . 304
7.2.4 The Rξ Gauges . . . . . . . . . . . . . . . . . . . . . . 306
7.3 The Z, the W , and the Weak Neutral Current . . . . . . . . . 308
7.3.1 Purely Weak Processes . . . . . . . . . . . . . . . . . . 309
7.3.2 Weak-Electromagnetic Interference . . . . . . . . . . . 321
7.3.3 Implications of the WNC Experiments . . . . . . . . . 328
7.3.4 Precision Tests of the Standard Model . . . . . . . . . 330
7.3.5 The Z-Pole and Above . . . . . . . . . . . . . . . . . . 340
7.3.6 Implications of the Precision Program . . . . . . . . . 350
7.4 Gauge Self-Interactions . . . . . . . . . . . . . . . . . . . . . . 358
7.5 The Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
7.5.1 Theoretical Constraints . . . . . . . . . . . . . . . . . 362
7.5.2 Experimental Constraints and Prospects . . . . . . . . 368
7.6 The CKM Matrix and CP Violation . . . . . . . . . . . . . . 371
7.6.1 The CKM Matrix . . . . . . . . . . . . . . . . . . . . . 372
7.6.2 CP Violation and the Unitarity Triangle . . . . . . . . 376
7.6.3 The Neutral Kaon System . . . . . . . . . . . . . . . . 378
7.6.4 Mixing and CP Violation in the B System . . . . . . 391
7.6.5 Time Reversal Violation and Electric Dipole Moments 397

viii

The Standard Model and Beyond
7.7

7.8

7.6.6 Flavor Changing Neutral Currents (FCNC)
Neutrino Mass and Mixing . . . . . . . . . . . . . .
7.7.1 Basic Concepts for Neutrino Mass . . . . .
7.7.2 The Propagators for Majorana Fermions . .
7.7.3 Experiments and Observations . . . . . . .
7.7.4 Neutrino Oscillations . . . . . . . . . . . . .
7.7.5 The Spectrum . . . . . . . . . . . . . . . . .
7.7.6 Models of Neutrino Mass . . . . . . . . . .
7.7.7 Implications of Neutrino Mass . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .

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401
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8 Beyond the Standard Model
453
8.1 Problems with the Standard Model . . . . . . . . . . . . . . . 453
8.2 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 463
8.2.1 Implications of Supersymmetry . . . . . . . . . . . . . 463
8.2.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 469
8.2.3 Supersymmetric Interactions . . . . . . . . . . . . . . 482
8.2.4 Supersymmetry Breaking and Mediation . . . . . . . . 490
8.2.5 The Minimal Supersymmetric Standard Model (MSSM)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
8.2.6 Further Aspects of Supersymmetry . . . . . . . . . . . 505
8.3 Extended Gauge Groups . . . . . . . . . . . . . . . . . . . . . 508
8.3.1 SU (2) × U (1) × U (1) Models . . . . . . . . . . . . . . 510
8.3.2 SU (2)L × SU (2)R × U (1) Models . . . . . . . . . . . . 519
8.4 Grand Unified Theories (GUTs) . . . . . . . . . . . . . . . . . 525
8.4.1 The SU (5) Model . . . . . . . . . . . . . . . . . . . . . 527
8.4.2 Beyond the Minimal SU (5) Model . . . . . . . . . . . 534
8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
A Canonical Commutation Rules

541

B Derivation of a Simple Feynman Diagram

545

C Unitarity, the Partial Wave Expansion and the Optical
Theorem

547

D Two, Three, and n-Body Phase Space

549

E Calculation of the Anomalous Magnetic Moment of the
Electron

555

F Breit-Wigner Resonances

559

G Implications of P , C, T , and G-parity for Nucleon
Matrix Elements

563


ix

Table of Contents

H Collider Kinematics

567

I

573

Quantum Mechanical Analogs of Symmetry Breaking

References
I.1
Field Theory . . . . . . . . . . . . . . . . . . . . .
I.2
The Standard Model and Particle Physics . . . . .
I.3
The Strong Interactions, QCD, and Collider Physics
I.4
The Electroweak Interactions . . . . . . . . . . . .
I.5
CP Violation . . . . . . . . . . . . . . . . . . . . .
I.6
Neutrinos . . . . . . . . . . . . . . . . . . . . . . .
I.7
Supersymmetry, Strings, and Grand Unification
.
I.8
Astrophysics and Cosmology . . . . . . . . . . . .
I.9
Groups and Symmetries . . . . . . . . . . . . . . .

I.10 Articles . . . . . . . . . . . . . . . . . . . . . . . .

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577
577
578
579
580
581
581
582
583
584
584

Web Sites

631

Index

635



Preface

In the last few decades there has been a tremendous advance in our understanding of the elementary particles and their interactions. We now have a
mathematically consistent theory of the strong, electromagnetic, and weak
interactions—the standard model—most aspects of which have been successfully tested in detail at colliders, accelerators, and non-accelerator experiments. It also provides a successful framework and has been strongly constrained by many observations in cosmology and astrophysics. The standard
model is almost certainly an approximately correct description of Nature down
to a distance scale 1/1000th the size of the atomic nucleus.
However, nobody believes that the standard model is the ultimate theory:
it is too complicated and arbitrary, does not provide an understanding of the
patterns of fermion masses and mixings, does not incorporate quantum gravity, and it involves several severe fine-tunings. Furthermore, the origins of
electroweak symmetry breaking, whether by the Higgs mechanism or something else, are uncertain. The recent discovery of non-zero neutrino mass can
be incorporated, but in more than one way, with different implications for
physics at very short distance scales. Finally, the observations of dark matter
and energy suggest new particle physics beyond the standard model.
Most current activity is directed towards discovering the new physics which
must underlie the standard model. Much of the theoretical effort involves
constructing models of possible new physics at the TeV scale, such as supersymmetry or alternative models of spontaneous symmetry breaking. Others
are examining the extremely promising ideas of superstring theory, which
offer the hope of an ultimate unification of all interactions including gravity. There is a lively debate about the implications of a landscape of possible
string vacua, and serious efforts are being made to explore the consequences
of string theory for the TeV scale. It is likely that a combination of such
bottom-up and top-down ideas will be necessary for progress. In any case,

new experimental data are urgently needed. At the time of this writing the
particle physics community is eagerly awaiting the results of the Large Hadron
Collider (LHC) and is optimistic about a possible future International Linear
Collider. Future experiments to elucidate the properties of neutrinos and to
explore aspects of flavor, and more detailed probes of the dark energy and
dark matter, are also anticipated.
The purpose of this volume is to provide an advanced introduction to the
physics and formalism of the standard model and other non-abelian gauge
theories, and thus to provide a thorough background for topics such as superxi


xii

The Standard Model and Beyond

symmetry, string theory, extra dimensions, dynamical symmetry breaking,
and cosmology. It is intended to provide the tools for a researcher to understand the structure and phenomenological consequences of the standard
model, construct extensions, and to carry out calculations at tree level. Some
“old-fashioned” topics which may still be useful are included. This is not a
text on field theory, and does not substitute for the excellent texts that already exist. Ideally, the reader will have completed a standard field theory
course. Nevertheless, Chapter 2 of this book presents a largely self-contained
treatment of the complicated technology needed for tree-level calculations involving spin-0, spin- 12 , and spin-1 particles, and should be useful for those
who have not studied field theory recently, or whose exposure has been more
formal than calculational∗ . It does not attempt to deal systematically with
the subtleties of renormalization, gauge issues, or higher-order corrections.
An introductory-level background in the ideas of particle physics is assumed,
with occasional reference to topics such as gluons or supersymmetry before
they are formally introduced. Similarly, occasional reference is made to applications to and constraints from astrophysics and cosmology. The necessary
background material may be found in the sources listed in the bibliography.
Chapter 1 is a short summary of notations and conventions and of some

basic mathematical machinery. Chapter 2 contains a review of calculational
techniques in field theory and the status of quantum electrodynamics. Chapters 3 and 4 are concerned with global and local symmetries and the construction of non-abelian gauge theories. Chapter 5 examines the strong interactions and the structure and tests of Quantum Chromodynamics (QCD).
Chapters 6 and 7 examine the electroweak interactions and theory, including neutrino masses. Chapter 8 considers the motivations for extending the
standard model, and examines supersymmetry, extended gauge groups, and
grand unification. There are short appendices on additional topics. The bibliographies list many useful reference books, review articles, research papers,
and Web sites. No attempt has been made to list all relevant original articles,
with preference given instead to later articles and books that can be used to
track down the original ones. Supplementary materials and corrections are
available at Comments, corrections,
and typographical errors can also be sent through that site.
I would like to thank Mirjam Cvetiˇc, Jens Erler, Hye-Sung Lee, Gil Paz,
Liantao Wang, and Itay Yavin for reading and commenting on parts of the
manuscript, Lisa Fleischer for help in the preparation of the manuscript, and
my wife Irmgard for her extreme patience during the writing.
Paul Langacker
July 4, 2009
∗ Most

calculations, especially at the tree-level, are now carried out by specialized computer
programs, many of which are included in the list of Web sites, but it is still important to
understand the techniques that go into them.


1
Notation and Conventions

In this chapter we briefly survey our notation and conventions.
Conventions
We generally follow the conventions used in (Langacker, 1981). In particular, (µ, ν, ρ, σ) are Lorentz indices; (i, j, k = 1 · · · 3) are three-vector indices;
(i, j, k = 1 · · · N ) are also used to label group generators or elements of the adjoint representation; (a, b, c) run over the elements of a representation, while

(α, β, γ) and (r, s, t) refer to the special cases of color and flavor, respectively. (α, β) are also occasionally used for Dirac indices. (m, n) are used
as horizontal (family) indices, labeling repeated fermions, scalars, and representations. The summation convention applies to all repeated indices except
where indicated. Operators are represented by capital letters (T i , Q, Y ), their
eigenvalues by lower case letters (ti , q, y), and their matrix representations
by (Li , LQ , LY ). In Feynman diagrams, ordinary fermions are represented by
solid lines, spin-0 particles by dashed lines, gluons by curly lines, other gauge
bosons by wavy lines, and gluinos, neutralinos, and charginos by single or
double lines. Experimental errors are usually quoted as a single number, with
statistical, systematic, and theoretical uncertainties combined in quadrature
and asymmetric errors symmetrized.
Units and Physical Constants
We take = c = 1, implying that E, p, m, x1 , 1t have “energy units.” One
can restore units at the end of a calculation using the values of , c, and c
listed in Table 1.1.
Operators and Matrices
The commutator and anti-commutator of two operators or matrices are
[A, B] = AB − BA,

{A, B} = AB + BA.

(1.1)

The transpose, adjoint, and trace of an n × n matrix M are
transpose: M T

T
(Mab
= Mba ),

adjoint: M † = M T ∗

(1.2)
1


2

The Standard Model and Beyond

TABLE 1.1

Conversions and physical constants. For more precise values, see (Amsler
et al., 2008). The fine structure constant is α = e2 /4π, where e > 0 is the
−1/2
charge of the positron. The Planck constant is MP = GN .
∼ 6.6 × 10−22 MeV-s
α ∼ 137.04
αg (MZ2 ) ∼ 0.034
GF ∼ 1.17 × 10−5 GeV−2
me ∼ 0.511 MeV
mp ∼ 938 MeV
MP ∼ 1.22 × 1019 GeV
−1

c ∼ 3.0 × 1010 cm/s
α−1 (MZ2 ) ∼ 128.9
αg (MZ2 ) ∼ 0.010
MW ∼ 80.40 GeV
mµ ∼ 105.7 MeV
mπ± ∼ 140 MeV

c ∼ 197 MeV-fm
sin2 θˆW (MZ2 ) ∼ 0.2312
αs (MZ2 ) ∼ 0.118
MZ ∼ 91.19 GeV
mτ ∼ 1.78 GeV
mK ± ∼ 494 MeV
k ∼ 1.16 × 104 ◦ K/eV

n

trace : Tr M =

Maa ,

Tr (M1 M2 ) = Tr (M2 M1 ),

a=1

Tr M = Tr M T .
(1.3)

Vectors, Metric, and Relativity
Three-vectors and unit vectors are denoted by x and x
ˆ = x/|x|, respectively.
We do not distinguish between upper and lower indices for three-vectors; e.g.,
the inner (dot) product x · y may be written as xi y i , xi yi , or xi yi . The LeviCivita tensor is ijk , where i, j, k = 1 · · · 3, is totally antisymmetric, with
123 = 1. Its contractions are
ijk ijk
ijk imn

= 6,

ijk ijm

= 2δkm

= δjm δkn − δjn δkm ,

(1.4)

where the Kronecker delta function is
δij =
ijk

1, i = j
.
0, i = j

(1.5)

is useful for vector cross products and their identities. For example,
(A × B)i =

(A× B)·(C × D) =

ijk ilm Aj Bk Cl Dm

ijk Aj Bk

(1.6)

= (A· C)(B · D)−(A· D)(C · D). (1.7)

Notations for four-vectors and the metric are given in Table 1.2.
The four-momentum of a particle with mass m is pµ = (E, p ) with p2 =
2
E − p 2 = m2 . (The symbol p is occasionally used to represent |p | rather
than a four-vector, but the meaning should always be clear from the context.)
The velocity β and energy are given by
β=

p
,
E

γ≡

E
=
m

1
1 − β2

.

(1.8)

3

Notation and Conventions

Under a Lorentz boost by velocity βL (the relativistic addition of βL to β,
which is equivalent to going to a new Lorentz frame moving with −βL )
pµ → p µ = (E , p ),
where
with

E = γL (E + βL · p ),
p = βˆL βˆL · p,

(1.9)

p = p⊥ + γL (p + βL E),

p⊥ = p − p ,

γL =

1
1 − βL2

(1.10)

.

(1.11)

TABLE 1.2

Notations and conventions for four-vectors and the metric
contravariant four-vector Aµ = (A0 , A ),

xµ = (t, x)

covariant four-vector

Aµ = gµν Aν = (A0 , −A ),

xµ = (t, −x)

metric

gµν = g µν = diag(1, −1, −1, −1)
1, µ = ν

gµν ≡ g νσ gµσ = δµν =
Lorentz invariant

0, µ = ν
A · B ≡ Aµ B µ = gµν Aµ B ν = A0 B 0 − A · B

derivatives

∂µ ≡

∂
∂xµ

=

✷ ≡ ∂µ ∂ µ =

∂
∂t , ∇
∂2
∂t2

,

∂µ ≡

∂
∂xµ

=

∂
∂t , −∇

− ∇2
0

∂ · A = ∂µ Aµ = ∂A
∂t + ∇ · A
←
→µ
a ∂ b = a∂ µ b − (∂ µ a)b

antisymmetric tensor

µνρσ

contractions

µνρσ

, with

0123

µνρσ

= −24

µντ ω

= −2 (gτρ gωσ − gωρ gτσ )

µνρσ

= +1 and
µνρσ

0123

= −1

µνρτ

= −6gτσ

Translation Invariance
Let P µ be the momentum operator, |i and |f momentum eigenstates,
P µ |i = pµi |i ,

P µ |f = pµf |f ,

(1.12)

and let O(x) be an operator defined at spacetime point x, so that
O(x) = eiP ·x O(0)e−iP ·x .

(1.13)


4

The Standard Model and Beyond

Then the x dependence of the matrix element f |O(x)|i is given by
f |O(x)|i = ei(pf −pi )·x f |O(0)|i .

(1.14)

The combination of Lorentz and translation invariance is Poincar´e invariance.
The Pauli Matrices
The 2 × 2 Pauli matrices σ = (σ1 , σ2 , σ3 ) (also denoted by τ , especially for
internal symmetries) are Hermitian, σi = σi† , and defined by

[σi , σj ] = 2i

ijk σk .

(1.15)

A convenient representation is
0
1

σ1 =

1
,
0

σ2 =

0 −i
,
i 0

σ3 =

1 0
.
0 −1

(1.16)

There is no distinction between σi and σ i . Some useful identities include
2

Tr σi =

σiaa = 0,

Tr (σi σj ) = 2δij

(1.17)

a=1

σi2

{σi , σj } = 2δij I ⇒

= I,

σi σj = δij I + i

ijk σk .

The last identity implies
(A · σ) (B · σ) = A · B I + i(A × B) · σ,

(1.18)

where A and B are any three-vectors (including operators) and A · σ is a 2 × 2
matrix. Thus, (A · σ)2 = A2 I for an ordinary real vector A with A ≡ |A |, and

eiA·σ = (cos A)I + i(sin A)Aˆ · σ.

(1.19)

Any 2 × 2 matrix M can be expressed in terms of σ and the identity by
M=

1
1
Tr (M )I + Tr (M σ) · σ.
2
2

(1.20)

The SU (2) Fierz identity is given in Problem 1.1.
The Delta and Step Functions
The Dirac delta function δ(x) is defined (for our purposes) by
+∞

δ(x − a)g(x)dx = g(a)

(1.21)

−∞

for sufficiently well-behaved g(x). Useful representations of δ(x) include
δ(x − a) =

1

2π

+∞

eik(x−a) dk =
−∞

1
γ
lim
.
γ→0
π
(x − a)2 + γ 2

(1.22)


5

Notation and Conventions
The derivative of δ(x) is defined by integration by parts,
+∞

+∞

δ (x − a)g(x)dx ≡
−∞

−∞

dδ(x − a)
dg
g(x)dx = −
dx
dx

.

(1.23)

x=a

Suppose a well-behaved function f (x) has zeroes at x0i . Then
δ(f (x)) =
i

δ(x − x0i )
.
|df /dx|x0i

(1.24)

1,
0,

(1.25)

The step function, Θ(x), is defined by
Θ(x − x ) =

x>x
,
x
from which δ(x) = dΘ/dx.

1.1

Problems

1.1 Let χn , n = 1 · · · 4 be arbitrary Pauli spinors (i.e., two-component complex column vectors). Then the bilinear form χ†m σi χn is an ordinary number.
Prove the Fierz identity
(χ†4 σχ3 ) · (χ†2 σχ1 ) = 2ηF (χ†4 χ1 )(χ†2 χ3 ) − (χ†4 χ3 )(χ†2 χ1 ),
where ηF = +1. (The identity also holds for anticommuting two-component
fields if one sets ηF = −1.) Hint: expand the 2×2 matrix χ1 χ†2 in (χ†4 χ1 )(χ†2 χ3 )
using (1.20).
1.2 Justify the result (1.24) for δ(f (x)).
1.3 Calculate the surface area dΩn of a unit sphere in n-dimensional Eu∞
clidean space, so that dn k = dΩn 0 k n−1 dk. Show that the general
formula yields
dΩ1 = 2,

dΩ2 = 2π,

dΩ3 = 4π,

dΩ4 = 2π 2 .

Hint: Use the Gaussian integral formula

+∞

2

e−αx dx =
−∞

to integrate

2

π
for
α

eα > 0

dn k e−αk in both Euclidean and spherical coordinates.


6

The Standard Model and Beyond

1.4 Show that the Lorentz boost in (1.10) can be written as
E
p

=

cosh yL sinh yL
sinh yL cosh yL

where
yL =
is the rapidity of the boost.

1 1 + βL
ln
2 1 − βL

E
p

,


2
Review of Perturbative Field Theory

Field Theory is the basic language of particle physics (i.e., of point particles).
It combines quantum mechanics, relativistic kinematics, and the notion of
particle creation and annihilation. The basic framework is remarkably successful and well-tested. In this book we will work mainly with perturbative
field theory, characterized by weak coupling.
The Lagrangian of a field theory contains the interaction vertices. Combined with the propagators for virtual or unstable particles one can compute
scattering and decay amplitudes using Feynman diagrams. Here we will review
the rules (but not the derivations) for carrying out field theory calculations
of amplitudes and the associated kinematics for processes involving spin-0,
spin- 12 , and spin-1 particles in four dimensions of space and time, and give
examples, mainly at tree level. Much more detail may be found in such field

theory texts as (Bjorken and Drell, 1964, 1965; Weinberg, 1995; Peskin and
Schroeder, 1995).

2.1

Creation and Annihilation Operators

Let |0 represent the ground state or vacuum, which we define as the no particle state (we are ignoring for now the complications of spontaneous symmetry
breaking). The vacuum is normalized 0|0 = 1. We will use a covariant
normalization convention∗ . For a spin-0 particle, define the creation and annihilation operators for a state of momentum p as a† (p ) and a(p ), respectively,
i.e.,
a(p )|0 = 0,

a† (p )|0 = |p ,

(2.1)

∗ The

alternative non-covariant convention, in which some formulaspare simpler, is
[an (p), a†n (p )] = δ 3 (p − p ), where an (p) is related to a(p ) by a(p ) = (2π)3 2Ep an (p).
The corresponding non-covariantly normalized single-particle
state is |p n = a†n (p) |0 , and
R
the integration over physical states is simply d3 p. Yet another possibility is box normalization in a volume V = L3 , leading to discrete three-momenta with ith component
pi = ni 2π
, ni = 1, 2, · · · , and commutators [aB (p ), a†B (p )] = δp p .
L

7

8

The Standard Model and Beyond

where |p describes a single-particle state with three-momentum p, energy
Ep = p 2 + m2 , and velocity β = p/Ep . We assume the commutation rules
(for Bose-Einstein statistics)
[a(p ), a† (p )] = (2π)3 2Ep δ 3 (p − p ),

[a(p ), a(p )] = [a† (p ), a† (p )] = 0,
(2.2)
which correspond to the state normalization
p |p

= (2π)3 2Ep δ 3 (p − p ).

(2.3)

This is Lorentz invariant because of the Ep factor. This can be seen from the
fact that the integration over physical momenta is
d3 p
d4 p
=
δ(p2 − m2 ) Θ(p0 ),
(2π)3 2Ep
(2π)3

(2.4)

where Θ(x) is the step function and we have used (1.24). The right-hand side
of (2.4) is manifestly invariant. The additional 2(2π)3 factor is for convenience.
The interpretation of (2.2) is that each momentum p of a non-interacting
particle can be described by a simple harmonic oscillator. The number operator N (p ), which counts the number of particles with momentum p in a state,
and the total number operator N , which counts the total number of particles,
are given by
N (p ) ≡ a† (p )a(p ),

N≡

d3 p
N (p ).
(2π)3 2Ep

(2.5)

(2.1)–(2.5) actually hold for any real or complex bosons, provided one adds
appropriate labels for particle type and (in the case of spin-1, 2, · · · ) for spin.
For real (i.e., describing particles that are their own antiparticles) spin-0 fields
of particle types a and b, for example,
[aa (p ), a†b (p )] = δab (2π)3 2Ep δ 3 (p−p ),

[aa (p ), ab (p )] = [a†a (p ), a†b (p )] = 0.
(2.6)
Similarly, for a complex scalar, describing a spin-0 particle with a distinct
antiparticle, it is conventional to use the symbols a† and b† for the particle
and antiparticle creation operators, respectively. (It is a convention which
state is called the particle and which the antiparticle.) E.g., for the π + state,
a† (p )|0 = |π + (p ) ,

b† (p )|0 = |π − (p ) ,

(2.7)

with
[b(p ), b† (p )] = (2π)3 2Ep δ 3 (p − p ),

[a(p ), b† (p )] = [a(p ), b(p )] = 0.
(2.8)
The creation and annihilation operators for fermions are similar, except that
they obey the anti-commutation rules appropriate to Fermi-Dirac statistics.


Review of Perturbative Field Theory

9

The creation operator for a spin- 12 particle is a† (p, s), where s refers to the
particle’s spin orientation, which may be taken with respect to a fixed z axis
or with respect to pˆ (helicity). Then,
{a(p, s), a† (p , s )} ≡ a(p, s)a† (p , s ) + a† (p , s )a(p, s)
= (2π)3 2Ep δ 3 (p − p )δss .

(2.9)

Similarly, for the antiparticle,

while

{b(p, s), b† (p , s )} = (2π)3 2Ep δ 3 (p − p )δss ,

(2.10)

{a, a} = {b, b} = {a, b} = {a, b† } = 0

(2.11)

for all values of p, p , s, and s . Fermion and boson operators commute with
each other, e.g., [aboson , afermion ] = 0.
Non-interacting multi-particle states are constructed similarly. For example, the state for two identical bosons is
|p1 p2 = |p2 p1 = a† (p1 )a† (p2 )|0

(2.12)

with
p1 p2 |p3 p4 = (2π)3 2E1 (2π)3 2E2 δ 3 (p1 − p3 )δ 3 (p2 − p4 )
+ δ 3 (p1 − p4 )δ 3 (p2 − p3 ) .

(2.13)

Similarly, for two identical fermions,
|p1 s1 ; p2 s2 = −|p2 s2 ; p1 s1 = a† (p1 s1 )a† (p2 s2 )|0 ,

(2.14)

with
p1 s1 ; p2 s2 |p3 s3 ; p4 s4 = (2π)3 2E1 (2π)3 2E2 δ 3 (p1 − p3 )δs1 s3 δ 3 (p2 − p4 )δs2 s4
− δ 3 (p1 − p4 )δs1 s4 δ 3 (p2 − p3 )δs2 s3 .

2.2

(2.15)

Lagrangian Field Theory

Consider a real or complex field φ(x), where x ≡ (t, x ). The (Hermitian)
Lagrangian density
L(φ(x), ∂µ φ(x), φ† (x), ∂µ φ† (x))

(2.16)


10

The Standard Model and Beyond

contains information about the kinetic energy, mass, and interactions of φ.
We will generally use the simpler notation L(φ, ∂µ φ), or just L(x), with the
understanding that for a complex field L can depend on both φ and its Hermitian conjugate φ† . (2.16) is trivially generalized to the case in which there
are more than one field. It is useful to also introduce the Lagrangian L(t) and
the action S by integrating L over space and over space-time, respectively,
+∞

L(t) =

d3 x L(φ, ∂µ φ),

S=

dt L(t) =

d4 x L(φ, ∂µ φ). (2.17)

−∞

The Euler-Lagrange equationss of motion for φ are obtained by minimizing
the action with respect to φ(x) and φ† (x),
δL
δL
− ∂µ
= 0,
δφ
δ∂µ φ

(2.18)

and similarly for φ† . The fields φ are interpreted as operators in the Heisenberg
picture, i.e., they are time-dependent while the states are time independent.
Other quantities, such as the conjugate momentum, the Hamiltonian, and the
canonical commutation rules, are summarized in Appendix A.

2.3

The Hermitian Scalar Field

A real (or more accurately, Hermitian) spin-0 (scalar) field, which satisfies
φ(x) = φ† (x), is suitable for describing a particle such as the π 0 which has no
internal quantum numbers and is therefore the same as its antiparticle.

2.3.1

The Lagrangian and Equations of Motion

The Lagrangian density for a Hermitian scalar is
1
(∂µ φ)(∂ µ φ) − m2 φ2 − VI (φ)
2
1
≡
(∂µ φ)2 − m2 φ2 − VI (φ).
2

L(φ, ∂µ φ) =

(2.19)

The first two terms correspond respectively to canonical kinetic energy and
mass (the 12 is special to Hermitian fields), while the last describes interactions.
(∂µ φ)2 is a shorthand for (∂µ φ)(∂ µ φ). The interaction potential is
VI (φ) = κ

φ3
φ4
φ5
+λ
+ c + d1 φ + d5
+ · · · + non − perturbative, (2.20)
3!
4!

5!


11

Review of Perturbative Field Theory

where the k! factors are for later convenience in cancelling combinatoric factors† , and “non-perturbative” allows for the possibility of non-polynomial
interactions. The constant c is irrelevant unless gravity is included. A nonzero d1 will necessarily induce a non-zero vacuum expectation value (VEV),
0|φ|0 = 0, suggesting that one is working in the wrong vacuum. The d1
term can be eliminated by a redefinition of φ → φ = constant + φ, as will
be described in Chapter 3. L and φ have dimensions of 4 and 1, respectively,
in mass units, so the coefficient of φk has the mass dimension 4 − k. The
dk terms with k ≥ 5 are known as non-renormalizable or higher-dimensional
operators. They lead to new divergences in each order of perturbation theory,
with dk typically of the form dk = ck /Mk−4 , where ck is dimensionless and
M is a large scale with dimensions of mass. Such terms would be absent in
a renormalizable theory, but may occur in an effective theory at low energy,
where they describe the effects of the exchange of heavy particles (or other
degrees of freedom) of mass M that are not explicitly taken into account in
the field theory. (In Chapters 6 and 7 we will see that an example of this is
the four-fermi operators that are relevant to describing the weak interactions
at low energy.) Keeping just the renormalizable terms (and c = d1 = 0), one
has
φ4
φ3
+λ ,
(2.21)
VI (φ) = κ
3!

4!
where κ (dimensions of mass) and λ (dimensionless) describe three- and fourpoint interactions, respectively, as illustrated in Figure 2.1. From the Euler-

−iλ

−iκ

FIGURE 2.1
Three- and four-point interactions of a Hermitian scalar field φ. The factor is
the coefficient of φn /n! in iL, as described in Appendix B.
Langrange equation (2.18), one obtains the field equation
✷ + m2 φ +

∂VI
φ2
φ3
= ✷ + m2 φ + κ
+λ
= 0,
∂φ
2
6

(2.22)

where ✷ + m2 = ∂µ ∂ µ + m2 is the Klein-Gordon operator. The expression for
the Hamiltonian density is given in Appendix A.
† Conventions

for such factors may change, depending on the context.

– Typeset by FoilTEX –


12

The Standard Model and Beyond

2.3.2

The Free Hermitian Scalar Field

Let φ0 = φ†0 be the solution of (2.22) in the free (or non-interacting) limit
κ = λ = 0, i.e.,
✷ + m2 φ0 (x) = 0.
(2.23)
(2.23) can be solved exactly, and then small values of the interaction parameters κ and λ can be treated perturbatively (as Feynman diagrams). The
general solution is
φ0 (x) = φ†0 (x) =

d3 p
a(p )e−ip·x + a† (p )e+ip·x ,
(2π)3 2Ep

(2.24)

where x = (t, x ) and p = (Ep , p ) with Ep ≡
p 2 + m2 , i.e., the fourmomentum p in the Fourier transform is an on-shell momentum for a particle
of mass m. The canonical commutation rules for φ0 and its conjugate momentum given in Appendix A will be satisfied if the Fourier coefficients a(p )
satisfy the creation-annihilation operator rules in (2.2).

It is useful to define the Feynman propagator for φ0 ,
i∆F (x − x ) ≡ 0|T [φ0 (x), φ0 (x )]|0 ,

(2.25)

where
T [φ0 (x), φ0 (x )] ≡ Θ(t − t )φ0 (x)φ0 (x ) + Θ(t − t)φ0 (x )φ0 (x)

(2.26)

represents the time-ordered product of φ0 (x) and φ0 (x ). In (2.26) Θ(t − t ) is
the step function, defined in (1.25). ∆F (x − x ) is just the Green’s function
of the Klein-Gordon operator, i.e.,
✷x + m2 ∆F (x − x ) = −δ 4 (x − x ),

(2.27)

where ✷x refers to derivatives w.r.t. x. The momentum space propagator
∆F (k) is
∆F (k) =
=

d4 xe+ik·x ∆F (x)
1
1
=
.
2
2
2

k −m +i
k0 − k 2 − m2 + i

(2.28)

k is an arbitrary four-momentum, i.e., it need not be on shell. The on-shell
limit is correctly handled by the i factor in the denominator, where is a
small positive quantity that can be taken to 0 at the end of the calculation.

2.3.3

The Feynman Rules

The Feynman rules allow a systematic diagrammatic representation of the
terms in the perturbative expansion (in κ and λ) of the transition amplitude