Approaching conformality in non-Abelian gauge theories

175 Pages·2017·4.29 MB·English

Approaching conformality in non-Abelian gauge theories
Nunes da Silva, Tiago Jose
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2016
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Nunes da Silva, T. J. (2016). Approaching conformality in non-Abelian gauge theories [Groningen]:
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Download date: 09-01-2017
Approaching Conformality in
non-Abelian Gauge Theories
PhD thesis
to obtain the degree of PhD at the
University of Groningen
on the authority of the
Rector Magnificus Prof. E. Sterken
and in accordance with
the decision by the College of Deans.
This thesis will be defended in public on
Friday 26 February 2016 at 12.45 hours
by
Tiago Jose Nunes da Silva
born on 5 December 1986
in Jaboatão dos Guararapes, Brazil
Supervisor
Prof. E. Pallante
Assessment Committee
Prof. R.G.E. Timmermans
Prof. V. Miransky
Prof. B. Lucini
ISBN 978-90-367-8648-5 (printed)
ISBN 978-90-367-8647-8 (digital)
Aosmeuspaiseà minhaavó Salete,comcarinho...
©2015–TiagoJoseNunesdaSilva
allrightsreserved.Nopartofthispublicationmaybereproducedortransmit-
tedinanyformorbyanymeanswithoutpermissionoftheauthorandthepub-
lisherholdingthecopyrightofthepublishedarticles.
ThisworkispartoftheresearchprogramoftheFoundationforFundamental
ResearchonMatter(FOM),whichispartoftheNetherlandsOrganisationfor
ScientificResearch(NWO).
PrintedbyIpskampDrukkers,Enschede.
Contents
0 Introduction 1
0.1 QuantumChromodynamicsinanutshell . . . . . . . . . . . . . . . . . . . . 2
0.2 ThePhaseDiagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
0.3 HigherDimensionalGroupRepresentationsandtheconnectiontophenomenology 22
0.4 LatticeMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
0.5 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1 TheStrongCouplingRegimeoftwelveflavourQCD 43
1.1 TheActions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2 TheObservables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3 Theeffectofimprovement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2 OnTheParticleSpectrumandtheConformalWindow 61
2.1 Thespectrum:Theoreticalpremise . . . . . . . . . . . . . . . . . . . . . . . 62
2.2 Numericalsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3 ApproachingConformality 105
3.1 NumericalSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3 Discussionofthelatticeresults . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.4 Theanomalousdimensionofthescalarglueballoperatorinperturbationtheory
andlarge-N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4 ConcludingRemarks 133
AppendixA Samenvatting 141
AppendixB Acknowledgments 147
References 167
v

Thereisaremarkablycloseparallelbetweentheproblemsofthephysicistand
thoseofthecryptographer.Thesystemonwhichamessageisencipheredcor-
respondstothelawsoftheuniverse,theinterceptedmessagestotheevidence
available,thekeysforadayoramessagetoimportantconstantswhichhave
tobedetermined.Thecorrespondenceisveryclose,butthesubjectmatterof
cryptographyisveryeasilydealtwithbydiscretemachinery,Physicsnotso
easily.
AlanTuring
0
Introduction
AncientGreeksandIndiansdevelopedtheatomistconcept140,228: thateverythinginnaturecon-
sistedofsmall,immutableandindestructiblebasicparticlesofmatter. Ourunderstandingofthe
fundamentalstructureofmatterhascomealongwaysincetheancientdays. Itissynthesisedin
theStandardModelofParticlePhysics,whichamalgamatesthreeofthefourfundamentalforcesof
nature,describedbygaugetheories: theStrong,ElectromagneticandWeakinteractions. Matteris
constitutedbysixquarks(up,down,strange,charm,bottomandtop),eachinthreecolours,andsix
leptons(electron,muon,tauandtheirrespectiveneutrinos).Vectorbosonsmediatetheinteractions
betweentheparticles: thestronginteractionismediatedbyeightgluons,weakinteractionsbythe
(cid:6)
W andZbosons,andtheElectromagneticinteractionbythephoton.
Theworkpresentedinthisthesisisdedicatedtofurtherunderstandingthestronginteraction.
Morespecifically,itaimsatunderstandingthephasediagramofstronglyinteractingmatterandthe
1
phenomenonoftheemergenceofconformality. Alargepartoftheworkpresentedthroughthis
bookisbasedonLatticeGaugeTheorymethods,whichareanonperturbativewaytonumerically
studygaugetheories. ThischapterprovidesthereaderwithanIntroductiontothesesubjects. It
startswithabriefoverviewofQuantumChromodynamics(QCD),thetheoryofstrongInterac-
tions. Thiswillbefollowedbyatheoreticaloverviewoftheconjecturedpictureforthephasedi-
agramofstronginteractionsandtheassociatedphenomenawhichplayaroleinitsshape. Itends
withanintroductiontothelatticemethodsusedduringtheresearch.
0.1 QuantumChromodynamicsinanutshell
QuantumChromodynamics(QCD)describesthefundamentalstronginteractionbetweenparticles
thatcontainacolourcharge: quarksandgluons. Itisanon-Abeliangaugetheorywithsymmetry
groupSU(3).
QuarksaredescribedinQCDastwelvecomponentfieldstransforminginthefundamentalrep-
resentationofthecolourSU(3)gaugegroup.TheyarerepresentedbyDirac4-spinors
ψ(f)(x)α;c;ψ(f)(x)α;c; (1)
wherexdenotesthespace-timeposition,α=1;2;3;4istheDiracindexandc=1;2;3(or,equiv-
alently,c=red,green,blue)isthecolourindex. Quarksexistinnatureindifferentflavours,withsix
flavoursbeingobservedsofarinnature.Theindexfrepresentstheflavourandingeneralrunsfrom
1tothetotalnumberofflavoursN usedinthecalculations.
f
GluonsaredescribedinQCDbygaugefieldswhichcarrytwocolourindices
A (x) ;
μ cd
whereagainxisthespace-timecoordinate,canddarethecolourindicesandμisaLorentzindex
indicatingthedirectionofdifferentcomponentsinspacetime.*
*WewillworkinEuclideanspacetime.Therefore,theindexμisEuclideanandthemetrictensorreduces
totheidentitymatrix.
2
TheQCDactioncanbewrittenas
∫ ( 1 ) [ ] ∑Nf ( )
S = d4x Tr F (x)F (x) + ψ(f)(x) γ (@ +iA (x))+m(f) ψ(f)(x):
QCD 2g2 μν μν μ μ μ
f=1
(2)
Thefirsttermontheright-handsideofequation2isthegluonicpartoftheaction,describinggluon
propagationandgluoninteraction,andthesecondtermcorrespondstothefermionicpart,includ-
ingquarkfieldsandtheirinteractionwiththegluons†.
ThequantitygisthestrongcouplingconstantofQCDandF isthegluonfield-strengthtensor
μν
[ ] [ ]
F (x)=(cid:0)i D (x);D (x) =@ A (x)(cid:0)@ A (x)+i A (x);A (x) ; (3)
μν μ ν μ ν ν μ μ ν
heredefinedintermsofthecovariantderivative
D (x)=@ +iA (x): (4)
μ μ μ
Theγ-matricesintheactionareEuclideanversionsoftheMinkowskiγ-matricesoftheDiracequa-
tion. Theyare4(cid:2)4matricesinDiracspace,obeyingtheEuclideananti-commutationrelations
{ }
γ ;γ =2δ I; (5)
μ ν μν
andtheymixthedifferentDiraccomponentsofthequarkfields.
TheactioninEquation2isobtainedthroughageneralisationofthegaugeinvarianceofelectro-
dynamics,byrequiringinvarianceunderlocalrotationsamongthequarkscolourindices,i.e.,we
requiretheactiontobeinvariantunderthetransformation
ψ(x)!ψ′(x)=Ω(x)ψ(x); ψ′(x)=ψ(x)γ Ω(x)y: (6)
0
Here,Ω(x)are3(cid:2)3independentcomplexmatriceschosenateachspacetimepointxactingonthe
red,greenandbluecolourcomponentsofψ. Theyarerequiredtosatisfytheunitaritycondition
Ω(x)y = Ω(x)(cid:0)1 andtohavedet[Ω(x)] = 1. TheseΩ(x)matricesconstitutethedefiningrep-
resentationofthespecialunitarygroupSU(3),the3beingrelatedtothematricesdimension. The
†Becausegluonscoupletothemselves,relevantcouplingsarisefromnon-lineartermsinthefieldstrength.
ThispropertyofQCDistheorigintoitsseveralnon-trivialfeatures
3