Large N gauge theories -- Numerical results

Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

February1,200823:52WSPC-ProceedingsTrimSize:9inx6intalk

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002 luJ 02 1v941706/0ht-pe:hviXra1

LargeNgaugetheories–Numericalresults

R.Narayanan

DepartmentofPhysics,FloridaInternationalUniversity,

Miami,FL33199,USA

E-mail:rajamani.narayanan@ u.edu

H.Neuberger

RutgersUniversity,DepartmentofPhysicsandAstronomy,

Piscataway,NJ08855,USA

E-mail:neuberg@physics.rutgers.edu

SomephysicalresultsinfourdimensionallargeNgaugetheoriesonaperiodictorusaresummarized.

Keywords:LargeNQCD,decon nement,phasetransitions.

1.Introduction

tticeQCDisausefultechniqueforextractingfundamentalresultsinthelargeNlimitofQCD.Fermionsarenaturallyquenchedinthe’tHooftlimitoflargeNQCDandthissigni cantlyreducesthecomputationalcostinalatticecalculation.Inaddition,thereisaconceptofcontinuumreduction3,namely,physicsdoesnotdependonthesizeofboxl4forl>lcandlcisaphysicalcriticalsize.ThesetwoobservationshavebeenusedtoextractphysicalresultsinthelargeNlimitofQCDusingnumericaltechniquesonthelattice.

2.PhasesoflargeNQCD

LargeNQCDonacontinuumtorusl4hasseveralphases4dependinguponthesizeofthetorusasshowninFig.1.ThecontinuumactionhasU4(1)symmetriesassociatedwiththePolyakovloopsinthefourdirectionsandthevariousphasescorrespondtothenumberofdirectionsinwhichthis

Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

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symmetryisbroken.Thecontinuumlimitisobtainedbygoingtothetop-rightcornerofFig.1anddi erentapproachestothiscornerwillresultinoneofthe vecontinuumphases.

The0h-phasepresentforb<0.36forallLisanunphysicalphasethatdoesnotsurvivethecontinuumlimit.The0hto0ctransitionisassoci-atedwiththesingleplaquetteoperatoropeningupagaparoundπinitseigenvaluedistribution.Gauge eldscomeindisconnectedpiecesinalltheXc-phasesduetothepresenceofthegapinthesingleplaquetteoperator5.

The0c-phaseisthecon nedphaseoflargeNQCDandthe1c-phaseisthedecon nedphaseandlc=1/tc.Animmediateconsequenceofcon-tinuumreductionisthatlargeNQCDdoesnotfeeltemperaturesbelowtc6.Numericalanalysishasshownthatlatticespacinge ectsaresmallinthe0c-phaseanditissu cienttoworkatL～9andN～30toextractcontinuumresults.Therefore,numericalcomputationscanbeperformedonaserialcomputerandaclusterofcomputerscanbee cientlyusedtogeneratestatisticsinaMonte-Carlocalculation.

Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

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tticecoupling.

3.Chiralsymmetrybreakingin nitevolume

Sincephysicsdoesnotdependontheboxsizein0c-phase,oneshouldshowthatchiralsymmetryisspontaneouslybrokenin nitevolumeinordertoproperlyreproducephysicsinthisphase7.Theorderoflimitsareimpor-tantandonehastotakethelargeNlimitbeforetakingthequarkmasstozeroat nitephsyicalvolume.ThelowlyingspectrumofthemasslessDiracoperatorshowsevidenceforspontaneouschiralbreakingsincetheeigenval-ues,iλ,scaleslikez=λΣNl4withzobeyingauniversaldistribution.Thechiralcondesate,Σ,isindependentofl.ResultsofacalculationofthechiralcondensateonthelatticeisshowninFig.2.Resultsfromchiralrandommatrixtheory8wereusedtoextractthechiralcondensateata xedN,Landlatticecoupling.Twolowestnon-zeroeigenvaluesintheQ=0andQ=1topologicalsectorswereusedtoshowconsistency.TheplotshowsthatthereisalimitasN→∞andthislimitisindependentofL.Results

obtainedatdi erentlatticecouplingsyieldl3

c

Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

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4.PionsinlargeNQCD

SincechiralsymmetryisbrokeninlargeNQCDevenin nitevolume,oneshouldbeabletoobservemasslesspionsin nitevolume.Thisresultemergesinthefollowingmanner9.Propertiesofasinglequarkinaback-groundgauge eldAµ(x)cannotdependonashiftofAµ(x)→Aµ(x)+p

µ

otherseesAµ(x)→Aµ(x)

pµ

2l

andthe

Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

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curveasshownin g.3andfπlc=

12Λπ

≈0.269.

5.Chiralsymmetryrestoration

The0cto1cphasetransitionisthecon nement-decon nementphasetransitionsincetheU(1)symmetryassociatedwithoneofthePolyakovloopsisbroken.Thistransitionis rstordersincethereisalatentheatas-sociatedwiththesingleplaquette11.Thefermiondeterminantdoesmatterinthe1c-phaseanditpicksthecorrectboundaryconditionsforfermionsinthebrokendirection,namely,anti-periodicwithrespecttothePolyakovloop12.ThelowesteigenvalueoftheDiracoperatorwiththecorrectbound-aryconditionscanbeusedtostudythegapinthe1c-phaseandone ndsthatchiralsymmetryisrestoredinthe1c-phase12.Furthermore,thechiraltransitionis rstorderasshowninFig.4.Ifoneweretosuper-coolthe1c-phaseintothe0c-phase,asecondordertransitionwithasquarerootsingularitywouldbeobservedat0.93/lc.

Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

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6.PhasetransitionintheWilsonloopoperator

Non-abeliangaugetheoriesinthecon nedphasearestronglyinteractingatlargedistancesandweaklyinteractingatshortdistances.Ifthisisseenasaphasetransitioninsomeobservable,onecouldusetheuniversalbehaviorofthistransitiontoconnectthelowenergyphysicsofQCDtothehighenergyphysicsofQCD.TheWilsonloopoperatorasafunctionofitssizeisthemostlikelycandidatetostudythistransitionwithinthe0c-phaseoflargeNQCD13.SuchatransitionexistsintwodimensionallargeNQCD14anditisclaimedthatthetransitioninfourdimensionalQCDisinthesameuniversalityclass.ThistransitionisreferredtoastheDurhuus-OlesenphasetransitioninFig1.

asafunctionofitssizeinlargeNQCD.ThedistributionoftheeigenvaluesofthesmearedWilsonloopoperatorarecomparedtotheDurhuus-Olesendistributions.Re-sultsareshownforloopsofsizesl/lc=0.740,0.660,0.560,0.503.Theymatchwithk=4.03,2.30,1.41,1.15wherekisthedimensionlessareaintwodimensionlargeNQCD.k=2isthecriticalarea.

Wilsonloopoperatorssu erfromaperimeterdivergenceinfourdimen-sionsandonehastoeliminateitwhende ningthisoperatorifonewereto

Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

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studyitseigenvaluedistribution.Onepossiblewaytosuppresstheperime-terdivergenceistousesmearedoperatorsonthelattice.NumericalstudiesoftheeigenvaluedistributionofthesmearedWilsonloopoperatoronthelatticeshowsclearevidenceforaphasetransitionasafunctionofthesizeoftheWilsonloop13.TheuniversalityclassisthesameastheonefoundintwodimensionallargeNQCDasshowninFig.5andthecriticalloopsizeisroughly0.6lc.Acknowledgements

R.N.wouldliketothanktheorganizersofCAQCD-06forastimulat-ingatmosphere.R.N.acknowledgespartialsupportbytheNSFundergrantnumberPHY-0300065andpartialsupportfromJe ersonLab.TheThomasJe ersonNationalAcceleratorFacility(Je ersonLab)isoperatedbytheSoutheasternUniversitiesResearchAssociation(SURA)underDOEcontractDE-AC05-84ER40150.H.N.acknowledgespartialsupportbytheDOEundergrantnumberDE-FG02-01ER41165atRutgersUniversity.References

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