Exploring improved holographic theories for QCD Pa

arXiv:0707.1349v3[hep-th]5Mar2008PreprinttypesetinJHEPstyle-HYPERVERSIONArXiv:0707.1349CPHT-RR028.0507ExploringimprovedholographictheoriesforQCD:PartIIU.G¨ursoy1,2,E.Kiritsis1,3,F.Nitti11CPHT,EcolePolytechnique,CNRS,91128,Palaiseau,France(UMRduCNRS7644).2LaboratoiredePhysiqueTh´eorique,EcoleNormaleSup´erieure,24,RueLhomond,Paris75005,France.3DepartmentofPhysics,UniversityofCrete71003Heraklion,GreeceAbstract:ThispaperisacontinuationofArXiv:0707.1324whereimprovedholo-graphictheoriesforQCDweresetupandexplored.Here,theIRconfininggeometriesareclassifiedandanalyzed.Theyallendina“good”(repulsive)singularityintheIR.Theglueballspectraaregappedanddiscrete,andtheyfavorablycomparetothelatticedata.Quitegenerally,confinementanddiscretespectraimplyeachother.Asymptoticallylinearglueballmassescanalsobeachieved.Asymptoticmassratiosofvariousglueballswithdifferentspinalsoturnouttobeuniversal.MesondynamicsisimplementedviaspacefillingD4−¯D4branepairs.Theassociatedtachyondy-namicsisanalyzedandchiralsymmetrybreakingisshown.ThedynamicsoftheRRaxionisanalyzed,andthenon-perturbativerunningoftheQCDθ-angleisobtained.ItisshowntoalwaysvanishintheIR.Keywords:AdS/CFT,holography,gaugetheory,QCD,Large-Nlimit.Contents1.Introductionandsummary32.BuildingblocksofholographicQCD52.1The5Dbackgrounds52.2Holographicdictionary82.3InfraredpropertiesofasymptoticallyAdS5backgrounds93.Confiningbackgrounds113.1TheWilsonlooptest113.2ConfiningIRasymptotics133.3Generalconfinementcriteria153.4Magneticchargescreening173.4.1Absenceofscreeninginhard-wallmodels204.TheQCDvacuumenergy205.Theaxionbackground225.1ScreeningofCPviolationintheIR266.Theparticlespectrum286.1Generalpropertiesofthespectra296.2Existenceofamassgap306.3Largenmassasymptoticsandlinearconfinement316.4Universalasymptoticmassratios326.5Glueballspectra336.5.1Scalarglueballs336.5.2Tensorglueballs346.5.3Pseudo-scalarglueballs346.6Singularityatfiniter0366.6.1Thepathologiesfor0δ1376.7Addingflavor376.7.1Tachyondynamics416.7.2Vectormesons427.Theparametersofthecorrespondence447.1Parametersinthegravitationalaction447.2Reparametrizationsymmetryandintegrationconstants45–1–8.Concretebackgrounds488.1BackgroundI:unboundedconformalcoordinate498.1.1TheglueballspectrainbackgroundI508.1.2LatticeData538.1.3FitforReferenceI548.1.4FittoReferenceII588.1.5Dependenceofthespectrumonthespectralparameterα608.2BackgroundII:singularityatfiniter618.2.1TheglueballspectrainbackgroundII618.3EstimatingtheeffectoftheUVrunning628.4Discussion63Acknowledgments65Appendices66A.Characterizationofconfiningbackgrounds66A.1Unboundedconformalcoordinate67A.1.1Logarithmicdivergence67A.1.2Power-lawdivergence68A.2Finiterangeoftheconformalcoordinate69A.2.1FiniteA(r0)69A.2.2Power-lawdivergence70A.2.3Logarithmicdivergence71B.Magneticchargescreening:thefiniterange72B.1A(r0)finite72B.2A(r0)→−∞72C.Fundamentalstringworld-sheetembeddingsinthepresenceofanon-trivialdilaton73D.Singularitiesofthetachyon75E.Thesuperpotentialversusthepotential76F.StandardAdS/QCDGlueballspectrum78G.AsimpleanalyticsolutionwithAdSandconfiningasymptotics79References80–2–1.IntroductionandsummaryThispaperisadirectsequelof[1],andthereaderisguidedthereforacomprehensiveintroductionandsummaryofresultsofbothpapers.Inthefirstpartofthiswork,twooftheauthorsestablishandmotivateageneral5Dholographicsetuptodescribe4Dgaugetheorieswithalargenumberofcolors(largeNc).Thesetupdescribedthereconstitutesabottom-upapproach,motivatedinpartfromknownfeaturesof5Dnon-criticalstringtheoryandinpartbywhatweexpectfromQCD.Thepuregaugedynamicsisencodedholographicallyinthesolutionofatwo-derivativeactionfortheexpected5Dfields:the5Dmetric(dualtotheYMstresstensor),ascalar(thedilaton,dualtoTr[F2]),andapseudoscalar(theaxion,dualtoTr[F∧F]).Thedilatonpotentialisexpectedtobenon-trivialandisexpectedtoobtainnon-trivialcontributionsfromthenon-propagatingfour-form.Inpractice,thepotential(andtheassociatedsuperpotential)isinone-to-onecorrespondencewiththeQCDβ-functionandischoseninsuchawayastoreproduceknownfeatures(e.g.UVasymptoticfreedomandIRconfinement)ofthegaugetheory.Thisiswhatmakesourapproachphenomenological.Theresultingbackgroundspresentanimprovementoverpre-existingmodelsof“phenomenologicalholography”,e.g.[2,3]:amongotheradvantages,theback-groundswepresentincorporatetherunningofthecouplingandasymptoticfreedom;establishaone-to-onecorrespondencebetweenthe5Dgeometryandthegaugethe-oryparameters,namelytheβ-functionβ(λ),andthedynamicallygeneratedIRscaleΛQCDanddonotrequirespecifyingtheboundaryconditionsintheIR.Moreover,theyprovideanaturalenvironmenttostudynon-perturbativedynamicalphenomenasuchasconfinement,generationofthemassgapandchiral-symmetrybreaking.PartIisdevotedtoestablishandmotivatethesetup,andtoanalyzetheper-turbativeUVregimeofthecorrespondence.Thepresentwork,ontheotherhand,focusesontheanalysisofthenon-perturbativeregime.Asoneofourmainresults,weestablisharelationbetweencolorconfinement(i.e.anarealawfortheWilsonloop)andthepropertiesofthegeometryintheIR,andweshowthatconfiningbackgroundsalwaysexhibitamassgapandgenericallyadiscretespectrum.Thisisano