Introduction To Quantum Field Theory And The Standard Model

253 Pages·2022·13.147 MB·English

by  Wolfgang Hollik

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Quantum Field Theory
Standard Model
and the
World Scientifi c
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Library of Congress Cataloging-in-Publication Data
Names: Hollik, W. (Wolfgang), 1951– author.
Title: Introduction to quantum field theory and the standard model /
Wolfgang Hollik, Max Planck Institute for Physics, Germany.
Description: New Jersey : World Scientific, [2022] |
Includes bibliographical references and index.
Identifiers: LCCN 2021050559 (print) | LCCN 2021050560 (ebook) |
ISBN 9789811242175 (hardcover) | ISBN 9789811242182 (ebook) |
ISBN 9789811242199 (ebook other)
Subjects: LCSH: Quantum field theory.
Classification: LCC QC174.45 .H647 2022 (print) | LCC QC174.45 (ebook) |
DDC 530.14/3--dc23/eng/20211201
LC record available at https://lccn.loc.gov/2021050559
LC ebook record available at https://lccn.loc.gov/2021050560
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
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Preface
These lecture notes are based on material presented during courses at the
Technical University Munich (TUM) to physics students with interest in
particlephysicsanditstheoreticalfoundation. Thismaterialhasbeenelab-
orated and expanded, to provide both the conceptual ideas regarding the
Standard Model of particle physics and some technical details on the for-
mulation within the context of gauge theories. The scope ranges from rel-
ativistic quantum mechanics to an introduction to quantum field theory
with quantum electrodynamics as an example of a successful theory of a
fundamentalinteraction,andisfurtherextendedtowardsquantumchromo-
dynamicsforthestronginteractionandtotheelectroweakStandardModel
fortheunifiedelectromagneticandweakinteractions,whichcouldcelebrate
anenormoussuccessbythe2012discoveryofanewspin-0particleshowing
the properties of the predicted Higgs boson.
Quantumfieldtheoryis the primarytheoreticaltool for the description
of the microscopic dynamics of the strong and electroweak interactions. It
is thus necessary to provide an introduction to the basic concepts of rela-
tivistic quantumfieldtheory, perturbationtheory, Feynmangraphs, before
advancingtoAbelianandnon-Abeliangaugetheoriesandtheirapplication
to the fundamental forces and the resulting phenomenological implications
with their experimental tests. The success of perturbative QCD at high
energies goes with the basic features of asymptotic freedom and the par-
ton model, displayed in the chapter on QCD. Phenomenology of W and Z
bosons as well as Higgs bosons is part of the electroweak chapter includ-
ing recent experimental results, precision tests and current status of the
Standard Model.
v
December9,2021 12:55 IntroductiontoQuantumFieldTheory...-9inx6in b4460-fm pagevi
vi Introduction toQuantum Field Theory and the Standard Model
The style is elementary and pedagogical. The text is at a level appro-
priateforstudentswhohadalreadyacourseinquantummechanicsandare
familiar with classical electrodynamics and the basics of special relativity.
Special relativity in the first chapter is a compact recapitulation serving
as a summary and for setting the language and notations. The formalism
of quantum field theory is kept at a minimum. Especially the Feynman
rulesarenotderivedinasystematicway;instead,themethodiselucidated
on the basis of examples to convey a more intuitive understanding that is
thought to be more helpful for an early overview rather than proofs and
advanced technical effort. Explicit calculations are performed for selected
examplessothatthereaderbecomesacquaintedwithpracticalcalculations
in particular for scattering amplidudes, cross sections and decay rates at
lowest order. Extra material on specific theoretical issues is included in
various places for the interested student; it can be skipped during a first
reading.
The lecture notes provide a compact introduction, convenient for stu-
dentsdealingwithparticlephysicsandrelatedareastogetfirstinformation
before specializing towards experimental or theoretical topics. They may
also serve as a basis to study elaborate textbooks, like Michael Peskin and
Daniel Schroeder’s An Introduction to Quantum Field Theory (Westview
Press, 1995), Matthew Schwartz’s Quantum Field Theory and the Stan-
dard Model (Cambridge University Press, 2014), or Steven Weinberg’s The
Quantum Theory of Fields (Cambridge University Press, 1995).
ItismyspecialconcerntothankmycolleaguesfromthePhysicsDepart-
mentatTUM,PeterFierlinger,LotharOberauer,StephanPaul,andStefan
Scho¨nert,forthepleasantcooperationduringvariouslecturecoursescover-
ingbothexperimentalandtheoreticalaspects,overmanyyearsofteaching.
Wolfgang Hollik
December9,2021 12:55 IntroductiontoQuantumFieldTheory...-9inx6in b4460-fm pagevii
Contents
Preface v
1. Special Relativity 1
1.1 Notations and Conventions . . . . . . . . . . . . . . . . . . 1
1.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . 3
1.2.1 Examples of Lorentz tranformations . . . . . . . . 3
1.2.2 General 4-vector . . . . . . . . . . . . . . . . . . . 5
1.2.3 Tensor . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Covariant equation of motion . . . . . . . . . . . . 8
1.4 LagrangianFormulation . . . . . . . . . . . . . . . . . . . 9
1.4.1 Free particle . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 Particle in an electromagnetic field . . . . . . . . . 11
1.5 Fields and Derivatives . . . . . . . . . . . . . . . . . . . . 12
1.5.1 Scalar field . . . . . . . . . . . . . . . . . . . . . . 12
1.5.2 Vector field . . . . . . . . . . . . . . . . . . . . . . 13
1.5.3 Tensor field . . . . . . . . . . . . . . . . . . . . . . 13
1.5.4 Partial derivatives . . . . . . . . . . . . . . . . . . 13
1.6 Electrodynamics. . . . . . . . . . . . . . . . . . . . . . . . 14
1.6.1 Potentials . . . . . . . . . . . . . . . . . . . . . . . 14
1.6.2 Field strengths . . . . . . . . . . . . . . . . . . . . 14
1.6.3 Gauge transformations . . . . . . . . . . . . . . . 15
1.6.4 Electromagnetic current . . . . . . . . . . . . . . . 15
1.6.5 Maxwell’s equations . . . . . . . . . . . . . . . . . 15
1.6.6 Energy-momentum tensor . . . . . . . . . . . . . . 16
vii
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viii Introduction to Quantum Field Theory and the Standard Model
2. Elements of Relativistic Quantum Field Theory 19
2.1 Klein–Gordon Equation . . . . . . . . . . . . . . . . . . . 20
2.1.1 Relativistic quantum mechanics of
spin-0 particles . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Quantum field theoretical formulation . . . . . . . 22
2.1.3 Current and charge . . . . . . . . . . . . . . . . . 24
2.1.4 Mechanical observables . . . . . . . . . . . . . . . 25
2.2 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Relativistic quantum mechanics of
spin-1/2 particles . . . . . . . . . . . . . . . . . . 26
2.2.2 Solutions of the Dirac equation . . . . . . . . . . . 31
2.2.3 Quantum field theoretical formulation . . . . . . . 36
2.3 Lorentz Symmetry and Dirac Equation . . . . . . . . . . . 41
2.3.1 Lorentz transformations and spinors . . . . . . . . 41
2.3.2 Covariance of the Dirac equation . . . . . . . . . . 46
2.3.3 Lorentz covariants . . . . . . . . . . . . . . . . . . 49
2.4 Dirac Particle in an External Electromagnetic Field . . . . 49
2.4.1 Electrostatic Coulomb field . . . . . . . . . . . . . 50
2.4.2 Static magnetic field . . . . . . . . . . . . . . . . . 51
3. Quantum Electrodynamics 55
3.1 Free Electromagnetic Field . . . . . . . . . . . . . . . . . . 55
3.1.1 Quantized electromagnetic field . . . . . . . . . . 56
3.1.2 Mechanical observables . . . . . . . . . . . . . . . 57
3.2 Interacting Electromagnetic Field . . . . . . . . . . . . . . 59
3.3 Interacting Dirac-Field . . . . . . . . . . . . . . . . . . . . 61
3.4 Interaction and Time Evolution . . . . . . . . . . . . . . . 62
3.5 S-Matrix Elements and Feynman Graphs . . . . . . . . . . 64
3.5.1 Modus operandi for the calculation of
matrix elements . . . . . . . . . . . . . . . . . . . 68
3.5.2 Extension to all fermions . . . . . . . . . . . . . . 70
3.6 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6.1 Special case: 2-particle processes . . . . . . . . . . 74
3.6.2 Unpolarized cross section . . . . . . . . . . . . . . 76
3.6.3 Decay width . . . . . . . . . . . . . . . . . . . . . 77
3.7 On the Calculation of Cross Sections . . . . . . . . . . . . 78
3.7.1 Spinors in matrix elements . . . . . . . . . . . . . 78
3.7.2 Traces over Dirac matrices . . . . . . . . . . . . . 79
3.7.3 Polarizationsum for photons . . . . . . . . . . . . 79
December9,2021 12:55 IntroductiontoQuantumFieldTheory...-9inx6in b4460-fm pageix
Contents ix
3.8 Precision Tests of QED . . . . . . . . . . . . . . . . . . . . 82
3.8.1 Precision tests at low energies . . . . . . . . . . . 83
3.9 Addendum: Coulomb Gauge and Feynman Graphs . . . . 86
3.9.1 Classical solution for the radiation field . . . . . . 89
4. Lagrangians and Symmetries 91
4.1 LagrangianFormalism for Fields. . . . . . . . . . . . . . . 91
4.1.1 Scalar field . . . . . . . . . . . . . . . . . . . . . . 93
4.1.2 Dirac field . . . . . . . . . . . . . . . . . . . . . . 94
4.1.3 Vector field . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Space–Time Symmetries . . . . . . . . . . . . . . . . . . . 95
4.2.1 Translational invariance and 4-momentum . . . . 96
4.2.2 Rotational invariance and angular
momentum . . . . . . . . . . . . . . . . . . . . . . 101
4.3 Gauge Symmetry and QED . . . . . . . . . . . . . . . . . 105
4.3.1 Local gauge transformations . . . . . . . . . . . . 107
4.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Non-Abelian Gauge Symmetries . . . . . . . . . . . . . . . 110
4.4.1 Global gauge symmetry . . . . . . . . . . . . . . . 111
4.4.2 Local gauge symmetry. . . . . . . . . . . . . . . . 115
4.4.3 Dynamics of the gauge fields . . . . . . . . . . . . 117
4.4.4 Gauge fixing and ghost fields . . . . . . . . . . . . 122
4.4.5 Perturbation theory and Feynman graphs . . . . . 125
5. Quantum Chromodynamics 127
5.1 Formulation of QCD . . . . . . . . . . . . . . . . . . . . . 127
5.1.1 Feynman rules . . . . . . . . . . . . . . . . . . . . 129
5.2 QCD Processes at High Energies . . . . . . . . . . . . . . 131
5.2.1 Processes at hadron colliders . . . . . . . . . . . . 137
5.3 Running Coupling Constant . . . . . . . . . . . . . . . . . 138
5.3.1 Fine structure constant of QED . . . . . . . . . . 139
5.3.2 Fine structure constant of QCD . . . . . . . . . . 141
5.4 Parton Distributions . . . . . . . . . . . . . . . . . . . . . 146
5.4.1 Parton model. . . . . . . . . . . . . . . . . . . . . 146
5.4.2 Deep-inelastic scattering . . . . . . . . . . . . . . 147
5.4.3 Parton distributions and QCD . . . . . . . . . . . 150
5.5 Hadronic Bound States . . . . . . . . . . . . . . . . . . . . 157
5.5.1 Representations of SU(3) . . . . . . . . . . . . . . 157
5.5.2 Hadron states . . . . . . . . . . . . . . . . . . . . 162