Introduction to Strong Interactions: Theory and Applications

318 Pages·2022·13.042 MB·English

by  Andrey Grabovsky

Introduction to
Strong Interactions
This is a problem-oriented introduction to the main ideas, methods, and problems
needed to form a basic understanding of the theory of strong interactions. Each
section contains solid but concise technical foundations to key concepts of the
theory, and the level of rigor is appropriate for readers with a background in
physics (rather than mathematics). It begins with a foundational introduction
to topics including SU(N) group, hadrons, and effective SU(3) symmetric flavor
lagrangians, constituent quarks in hadrons, quarks, and gluons as fundamental
fields. It then discusses quantum chromodynamics as a gauge field theory,
functional integration, and Wilson lines and loops, before moving on to discuss
gauge–fixing and Faddeev–Popov ghosts, Becchi-Rouet-Stora-Tyutin symmetry,
and lattice methods. It concludes with a discussion on the anomalies and the
strong CP problem, effective action, chiral perturbation theory, deep inelastic
scattering, and derivation and solution of the Dokshitzer–Gribov–Lipatov–
Altarelli–Parisi equations.
Constructed as a one-term course on strong interactions for advanced students,
it will be a useful self-study guide for graduate and PhD students of high energy
physics, quantum chromodynamics, and the Standard Model.

Introduction to
Strong Interactions
Theory and Applications
Andrey Grabovsky
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Andrey Grabovsky
Reasonable efforts have been made to publish reliable data and information, but the author and
publisher cannot assume responsibility for the validity of all materials or the consequences of their use.
The authors and publishers have attempted to trace the copyright holders of all material reproduced
in this publication and apologize to copyright holders if permission to publish in this form has not
been obtained. If any copyright material has not been acknowledged, please write and let us know so
we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information
storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.com
or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. For works that are not available on CCC, please contact mpkbookspermissions@tandf.
co.uk
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are
used only for identification and explanation without intent to infringe.
ISBN: 978-1-032-20675-2 (hbk)
ISBN: 978-1-032-22393-3 (pbk)
ISBN: 978-1-003-27240-3 (ebk)
DOI: 10.1201/9781003272403
Typeset in Latin Modern font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface vii
Chapter 1■ SU(N) Group 1
Chapter 2■ SU(3) Color Gauge Invariance 39
Chapter 3■ Functional Integration 71
Chapter 4■ Gauge Fixing and Calculation Rules 99
Chapter 5■ Strong Coupling 141
Chapter 6■ Effective Action 161
Chapter 7■ Renormalization 189
Chapter 8■ Scale Anomaly 209
Chapter 9■ Adler-Bell-Jakiw Anomaly 227
v
vi ■ Contents
Chapter 10■ Continuous Non-Color Symmetries 241
Chapter 11■ Collinear Factorization: Deep Inelastic
Scattering 269
Index 307
Preface
This material was given to the fifth year High Energy Physics students of
the Novosibirsk State University in 2017–2022. These students studied quan-
tum electrodynamics in the preceding semester and Quantum and Analytical
Mechanics, Mathematical Physics and Introduction into Field Theory in pre-
viousyears.Therefore,theyhadknowledgeofGroupTheory,Lagrangianand
Hamiltonian approaches, Canonical Quantization and Feynman diagrams, in-
cluding one-loop integration and renormalization in Quantum Electrodynam-
ics. However, no knowledge of Functional Integration was assumed. This is
an introductory compulsory course spanning one semester of 18 weeks, 18
lectures, and 18 problem solving sessions. The material in each chapter usu-
ally took from one to three weeks. More advanced topics are covered in the
follow-up optional course on QCD.
The course persues the following aims:
1. Givesthemainphysicalideasofthetheoryofstronginteractions:asymp-
totic freedom, confinement, chiral symmetry breaking, anomalies, con-
densates;
2. Introducestheprincipalconceptsandmethods:SU(N)groupsymmetry,
gauge invariance, renormalisation, functional integration, lattice, chiral
perturbation theory, parton distribution functions, evolution equations;
3. Links the theory and experiment discussing high energy scattering and
low energy effective Lagrangians.
This course tries to give solid but as short as possible technical foundation.
Thematerialineachsectionismostlytakenfromthesourcesgivenattheend
of the section. The level of rigor is physical rather than mathematical.
I am grateful to V. S. Fadin and R. N. Lee who taught strong interactions
tomewhenIwasastudent,D.Yu.Ivanov,M.G.Kozlov,A.V.Reznichenko,
whotaughtthiscoursewithme.IthankthesepeopleandI.I.Balitsky,A.V.
Bogdan, R. Boussarie, A. L. Feldman, L. V. Kardapoltsev, A. I. Milstein, L.
Szymanowski,S.WallonandmanyotherpeopleImetatdifferentschoolsand
conferences for discussions improving my understanding of the subject. I also
thank the students who took this course and helped to make it better. The
project is being implemented by the winner of the master’s program faculty
grant competition 2020/2021 of the Vladimir Potanin fellowship program.
vii

1
CHAPTER
SU(N) Group
AKEY mathematical object we deal with in the theory of strong interac-
tions is the SU(3) group. Therefore, we have to take a close look at it
and at the SU(N) group in general. In this chapter we will cover the prop-
erties of these groups necessary for further discussion of the theory. We need
the following facts.
1. SU(N) is a non-Abelian group, i.e. g g ≠ g g for g ∈ SU(N),
1 2 2 1 1,2
N ≥2.
2. In the fundamental (defining) representation, Uj ∈ SU(N) is a
i
 ψ1 
N ×N matrix acting in the space of vertical vectors ψi =  ... ,
ψN
ψj′ =Ujψi. These functions ψi are often called the contravariant com-
i
ponents of spinors.
3. The Hermitian conjugate vectors ϕ = (ψ†) = (ψ∗,...,ψ∗ ) are trans-
j j 1 N
formed by the Hermitian conjugate matrices U†j, ϕ′ = ϕ (U†)i, which
i j i j
wewillcallthematricesintheantifundamentalrepresentation.The
functions ψ are often called the covariant components of spinors.
j
4. One can form tensors with an arbitrary number of the covariant and
contravariant indices with the transformation law
ψn1...nk(2) =Un1...Unkψi1...ik(1)U†j1...U†jl. (1.1)
p1...pl i1 ik j1...jl p1 pl
5. SU(N) is a unitary group, i.e. U ∈ SU(N) =⇒ U† = U−1, U = eiθ,
with a Hermitian matrix θ :
θ =θ†. (1.2)
Hence, δi is an invariant tensor
j
δk =UkδiU†j. (1.3)
l i j l
DOI:10.1201/9781003272403-1 1