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arXiv:hep-ph/0008206v118Aug2000QCDINAFINITEVOLUMEaINLO-PUB-08/00PIERREVANBAALInstituut-LorentzforTheoreticalPhysics,UniversityofLeiden,P.O.Box9506,Nl-2300RALeiden,TheNetherlandsWewillreviewourunderstandingofnon-abeliangaugetheoriesinfinitephysicalvolumes.Itallowsoneinareliablewaytotracesomeofthenon-perturbativedynamics.Theroleofgaugefixingambiguitiesrelatedtolargefieldfluctuationsisanimportantlessonthatcanbelearned.Thehamiltonianformalismisthemaintool,partlybecausesemiclassicaltechniquesaresimplyinadequateoncethecouplingbecomesstrong.Usingperiodicboundaryconditions,continuumresultscanbecomparedtothoseonthelattice.Resultsinasphericalfinitevolumewillbediscussedaswell.1IntroductionWehavedecidedtotakethisopportunityofcontributingtoahandbookofQCD,inhonorofBorisLazarevichIoffe,tobringtogetherresultsandmethodsthatweredevelopedinsolvingforthelow-lyingspectruminafinitevolume.Theemphasiswillbeonthedynamicalaspectsoftheclassicallyscaleinvarianttheoryfornon-abeliangaugetheoriesin3+1dimensions.1Thechallengeistounderstandhowthemassgapisgenerated.Duetotheneedforregularization,atthequantumlevelbreakingthescaleinvariance,arunningcouplingappears.Thedifficultyliesinthefactthatthiscouplingincreasesatlow-energiesandlong-distancescales,beyondthepointwhereonehascontroloverthefieldfluctuations,whichbecomesolargethattheyprobetheessentialnon-linearitiesofthetheory.Afinitevolumeexplicitlybreaksthescaleinvariance.However,itdoessoinarathermildway.Classicallyonecanusethescaletransformationtogofromonephysicalvolumetoanother,andonlytherunningofthecouplingconstantpreventsusfromtakingtheinfinitevolumelimit.Asthelongestdistancescale,i.e.thelowestenergyscale,issetbythevolume,onecankeepincheckthegrowthoftherunningcouplingconstant.Wewilldescribetheresultsmainlyinthecontextofahamiltonianpicture2withwavefunctionalsonconfigurationspace.Althoughrathercumbersomefromaperturbativepointofview,wherethecovariantpathintegralapproachofFeynmanisvastlysuperior,itprovidesmoreintuitiononhowtodealwithnon-perturbativecontributionsinsituationswheresemi-classicaltechniquescannolongerbeused,likeforobservablesthatdonotvanishinperturbationtheory.Thehighenergymodescanbewell-approximatedbyaharmonicoscil-latorcontributiontothewavefunctional.Inthedirectionofthesefieldmodesthepotentialenergyrisessteeply.Theircontribution,includingregulatingtheaToappearintheBorisIoffeFestschrift,editedbyM.Shifman(WorldScientific).1ultravioletbehavior,istreatedperturbatively,givinginparticularrisetotherunningofthecoupling.Thefinitevolumeallowsustohaveawell-definedmodeexpansion.Duetotheclassicalscaleinvariance,thehamiltoniancanbeformulatedintermsofdimensionlessfields.Thiscanbeextendedtothequan-tumtheory,asWardidentitiesallowforafielddefinitionwithoutanomalousscaling.Apartfromtheoverallscalingdimensionofthehamiltonian,onlytherunningcouplingintroducesanon-trivialvolumedependence.Duetothenon-abeliannatureofthetheorythephysicalconfigurationspace,formedbythesetofgaugeorbitsA/G(Aisthecollectionofconnec-tions,Gthegroupoflocalgaugetransformations)isnon-trivial.3Mostfre-quently,coordinatesofthisorbitspacearechosenbypickingarepresentativegaugefieldontheorbitinasmoothandpreferablyuniqueway.LineargaugeconditionsliketheLandauorCoulombgaugesufferfromwhatisknownasGribovambiguities.4Theregionoffieldspacethatcontainsnofurthergaugecopiesiscalledafundamentaldomainfornon-abeliangaugetheories.5Havingarrangedthelow-energyfieldmodes(andallthosemodesnotaf-fectedbythecutoff)tobescaleinvariant,thespreadingofthewavefunctionaliscompletelycausedbyanincreasingcoupling.Thisiswhatleadstonon-perturbativeeffects.Asymptoticfreedomontheotherhandguaranteesthatinsmallvolumestherunningcouplingissmallanditthuskeepsthewavefunc-tionallocalizedneartheclassicalvacuummanifold.Inaperiodicgeometry,thisperturbativeanalysiswaspioneeredbyBjorken6andfurtherdevelopedbyL¨uscher7.Theessentialingredientwehaveaddedtoaddressnon-perturbativeeffectsisboundaryconditionsinfieldspace,attheboundaryofthefundamen-taldomain,withgaugeinvarianceimplementedproperlyatallstages.Thenon-perturbativefeaturesduetospreadingofthewavefunctionalcanbefollowedouttoaphysicalvolumeofaboutonecubicfermi(settingthescalebythestringtensionorlowestglueballstate).Thisis,onthebasisofacomparisonwithlatticeMonteCarlodata,thepointwhereinthepuregaugetheorytheconfiningstringisbeingformed,withnosignificantfinitevolumedependencebeyondthisvolume.Whathasbecomeclearisthatthetransitionfromthefinitetotheinfinitevolumeisdrivenbyfieldfluctuationsthatcrossthebarrierwhichisassociatedwithtunnelingbetweendifferentclassicalvacua.Thisisnatural,sincethisbarrier(thefinitevolumesphaleron),willbethedirectionbeyondwhichthewavefunctionalcanfirstspreadmostsignificantly,asitprovidesthelowestmountainpassintheenergylandscape.InSec.2wedescribetheprocessofcompletegaugefixing,andthefactthattheboundaryofthefundamentaldomain,unlikeitsinterior,hasgaugecopiesthatimplementthenon-trivialtopologyofconfigurationspace.ThisisappliedinSec.3toformulatingnon-abeliangaugetheoriesonatorus,both2forSU(2)andinSec.3.1forSU(3).Theinfluenceofmasslessquarksonthesmallvolumevacuum