The Pinch Technique and its Applications to Non-Abelian Gauge Theories

306 Pages·2011·1.86 MB·English

by  Cornwall J.M.

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THE PINCH TECHNIQUE AND ITS APPLICATIONS
TO NON-ABELIAN GAUGE THEORIES
Non-Abelian gauge theories, such as quantum chromodynamics (QCD) or
electroweak theory, are best studied with the aid of Green’s functions that are
gauge invariant off-shell, but unlike for the photon in quantum electrodynamics,
conventional graphical constructions fail. The pinch technique provides a sys-
tematic framework for constructing such Green’s functions and has many useful
applications.
Beginning with elementary one-loop examples, this book goes on to extend the
methodtoallorders,showingthatthepinchtechniqueisequivalenttocalculations
in the background-field Feynman gauge. The pinch technique Schwinger-Dyson
equations are derived and used to show how a dynamical gluon mass arises in
QCD. Applications are given to the center vortex picture of confinement, the
gauge-invariant treatment of resonant amplitudes, the definition of non-Abelian
effectivecharges,high-temperatureeffects,andevensupersymmetry.Thisbookis
idealforelementaryparticletheoristsandgraduatestudents.
john m. cornwall is Distinguished Professor of Physics Emeritus in the
Department of Physics and Astronomy, University of California, Los Angeles.
Inventor of the pinch technique, he has made many other contributions to the
formalism and applications of quantum field theory, as well as to space plasma
physics.Hehascontributedtothetechnicalanalysisofmanypublicpolicyissues,
rangingfromballisticmissiledefensetothehumangenome.
joannis papavassiliou is a researcher in the Department of Theoretical
Physics and IFIC, the University of Valencia–CSIC. A large part of his work has
beendevotedtothedevelopmentofthepinchtechnique,bothitsformalfoundation
and its many applications, and he has published articles on quantum field theory
andparticlephenomenology.
daniele binosi is a researcherat the EuropeanCentre for Theoretical Studies
∗
in Nuclear Physics and Related Areas (ECT ) and Fondazione Bruno Kessler. In
addition to his work on extending the pinch technique and its applications, he
leads several policy-related European projects on the development of the vision
andsustainabilityofquantuminformationfoundationsandtechnologies.
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31. J.M.Cornwall,J.Papavassiliou,andD.Binosi:ThePinchTechniqueandItsApplicationstoNon-Abelian
GaugeTheories
THE PINCH TECHNIQUE AND
ITS APPLICATIONS TO NON-ABELIAN
GAUGE THEORIES
JOHN M. CORNWALL
UniversityofCaliforniaatLosAngeles,USA
JOANNIS PAPAVASSILIOU
UniversityofValencia–CSIC,Spain
DANIELE BINOSI
EuropeanCentreforTheoreticalStudiesinNuclearPhysicsand
RelatedAreas,Italy
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(cid:2)C J.Cornwall,J.Papavassiliou,D.Binosi2011
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Cornwall,JohnM.
Thepinchtechniqueanditsapplicationstonon-Abeliangaugetheories/JohnM.Cornwall,Joannis
Papavassiliou,DanieleBinosi.
p. cm.–(Cambridgemonographsonparticlephysics,nuclearphysics,andcosmology;31)
Includesbibliographicalreferencesandindex.
isbn978-0-521-43752-3
1.Quantumchromodynamics–Mathematics. 2.Gaugefields(Physics)–Mathematics.
3.Green’sfunctions. 4.Gaugeinvariance. I.Papavassiliou,Joannis. II.Binosi,Daniele. III.Title.
QC793.3.Q35C675 2011
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Contents
Introduction:Whythepinchtechnique? pagexi
1 Thepinchtechniqueatoneloop 1
1.1 Abriefhistory 1
1.2 Notationandconventions 3
1.3 Thebasicone-looppinchtechnique 6
1.4 Anotherwaytothepinchtechnique 17
1.5 Pinchtechniquevertices 20
1.6 Thepinchtechniqueinthelight-conegauge 31
1.7 Theabsorptivepinchtechniqueconstruction 34
1.8 Positivityandthepinchtechniquegluonpropagator 42
References 43
2 Advancedpinchtechnique:Stilloneloop 45
2.1 Thepinchtechniqueandtheoperatorproductexpansion:Running
massandcondensates 45
2.2 Thepinchtechniqueandgauge-bosonmassgeneration 47
2.3 Thepinchtechniquetoday:Background-fieldFeynmangauge 62
2.4 Whattoexpectbeyondoneloop 72
References 73
3 Pinchtechniquetoallorders 75
3.1 Thes-t cancellationtoallorders 75
3.2 Quark-gluonvertexandgluonpropagatortoallorders 79
References 85
vii
viii Contents
4 ThepinchtechniqueintheBatalin–Vilkoviskyframework 86
4.1 AnoverviewoftheBatalin–Vilkoviskyformalism 88
4.2 Examples 93
4.3 PinchingintheBatalin–Vilkoviskyframework 100
References 102
5 Thegaugetechnique 104
5.1 TheoriginalgaugetechniqueforQED 105
5.2 Masslesslongitudinalpoles 108
5.3 ThegaugetechniqueforNAGTs 109
References 113
6 Schwinger–Dysonequationsinthepinchtechnique
framework 114
6.1 Latticestudiesofgluonmassgeneration 115
6.2 Theneedforagauge-invarianttruncationschemeforthe
Schwinger–DysonequationsofNAGTs 117
6.3 ThepinchtechniquealgorithmforSchwinger–Dysonequations 119
6.4 PinchtechniqueGreen’sfunctionsfromSchwinger–Dysonequations 120
6.5 SolutionsofthepinchtechniqueSchwinger–Dysonequationsand
comparisonwithlatticedata 131
6.6 TheQCDeffectivecharge 134
References 141
7 Nonperturbativegluonmassandquantumsolitons 144
7.1 Notation 144
7.2 Introduction 144
7.3 Thequantumsolitons 150
7.4 Thecentervortexsoliton 152
References 165
8 Nexuses,sphalerons,andfractionaltopologicalcharge 167
8.1 Introductiontonexusesandjunctions 167
8.2 NexusesinSU(N) 170
8.3 TheQCDsphaleron 181
8.4 Chiralsymmetrybreakdown,nexuses,andfractionaltopologicalcharge 186
References 188
9 Abriefsummaryofd = 3NAGTs 190
9.1 Introduction 190