Basis States for Hamiltonian QCD with Dynamical Qu

arXiv:hep-lat/9210006v12Oct1992BUHEP-92-33October,1992BasisStatesforHamiltonianQCDwithDynamicalQuarksTimothyE.VaughanDepartmentofPhysicsBostonUniversity590CommonwealthAve.Boston,MA02215email:[e-mail]onstructionofbasisstatesforHamiltonianQCDonthelattice,inparticularstateswithdynamicalquarkpairs.Wecal-culatethematrixelementsoftheoperatorsintheQCDHamiltonianbetweenthesestates.Alongwiththe“harmonicoscillator”statesin-troducedinpreviouspureSU(3)work,thesestatesformaworkingbasisforcalculationsinfullQCD.1IntroductionAsdetailedinpaperswithBronzan[2,3],wehavebeenstudyinglatticeQCDusinganoperatorandstatesapproachratherthantheusualMonteCarlosimulationoftheFeynmanpathintegral.Aswehaveemphasized,thisrequirestheuseofasuitablesetofsingledegree-of-freedom(DOF)statesontheSU(3)manifold.Here“suitable”meansthatmatrixelementsoftheQCDHamiltonianmustbecalculableinclosedformusingthesestates,andthatthestateshavetunableparameterssotheycanmodeltheQCDwavefunctionsatallvaluesofthecouplingconstant.Wehaveintroduceda“harmonicoscillator”basisofstatesasonewhichsatisfiestheabovecriteriaandisthereforeusefulinmakingcalculationsinHamiltonianQCD.Sincethesestatesdescribeonlygaugedegreesoffreedom,theyareactuallydesignedforstudyingonlyapureSU(3)gaugetheory.Inthispaperwewillintroduceasimilarlysuitablesetofdynamical-quarkbasisstatestocomplementtheharmonicoscillatorstates.Wewilldiscusshowweconstructaquarkgroundstateatarbritraryvaluesofthecouplingconstant,createquarkexcitations,andcalculatematrixelementsoftheQCDHamiltonianbetweenthesestates.Togetherwiththeharmonicoscillatorstates,then,wehaveacompletebasisforthestudyoffullQCD.Thelayoutofthispaperisasfollows.InSection2webrieflyreviewourworkontheconstructionofharmonicoscillatorstates.InSection3wedescribetheconstructionofthequarkvacuumaswellasthestateswhichincludevirtualquarkpairs.InSection4wediscussthecombinationoftheharmonicoscillatorstateswiththequarkstatestoformabasisforsimulatingfullQCD.InSection5wederivethematrixelementsoftheoperatorsoftheQCDHamiltonianbetweenthesestates.WeconcludeinSection6withadiscussionoftheusefulnessofthesestates.2ReviewofpuregaugestatesEachgaugedegreeoffreedom(linkvariableonthelattice)canbedescribedbyawavefunctionwhichisa“harmonicoscillator”stateontheSU(3)man-ifold.Theharmonicoscillatorstatesarederivedfroma“Gaussian”stateon1themanifold.ThisGaussiancanbewrittenψα1(α,t)=Xp,qd(p,q)e−tλ(p,q)χ(p,q),(2.1)whered(p,q)=(p+1)(q+1)(p+q+2)/2isthedimensionofthe(p,q)representationofSU(3),λ(p,q)=(p+q)2/4+(p−q)2/12+(p+q)isthequadraticCasimireigenvalue,andχ(p,q)isthecharacter.α≡(α1,α2,...,α8)isaparameterizationoftheadjointrepresentationofSU(3).TheparametertcontrolsthewidthoftheGaussian,whichiscenteredatα1.Tomakeconnectionwithamorefamiliarobject,notethatthecorre-spondingwavefunctionontheflat,three-dimensionalmanifoldR3wouldbeψx1(x,t)=e−(x−x1)2/4t(4πt)4,(2.2)wherexandx1areordinaryvectors.Onthisflatmanifold,harmonicos-cillatorstatescanbegeneratedbyapplyingapolynomialintheoperators−i∂∂x1atothewavefunction,thensettingx1to0.OntheSU(3)manifold,thecorrespondingoperatorsareJLa(α1),thegeneratorsofSU(3)indifferentialform.1TheharmonicoscillatorstatesthatwehaveformedonthemanifoldcanbelabeledbythenumberofSU(3)indicesonthestate.Thus,thezero-,one-,andtwo-colorstatesareφ=φ(α,t)=ψα1(α,t)|α1=0,φa=φa(α,t)=JLa(α1)ψα1(α,t)|α1=0,φab=φab(α,t)={JLa(α1),JLb(α1)}ψα1(α,t)|α1=0.(2.3)Forcomparison,theanalogousstatesonaflatmanifoldareφ(x,t)=e−x2/4t,φa(x,t)=xae−x2/4t,φab(x,t)=xaxbe−x2/4t.(2.4)1WhenappliedtoarepresentationofSU(3),JLa(α1)hastheeffectofleft-multiplyingtherepresentationbytheathgenerator.TherearealsooperatorsJRa(α1)whichright-multiplytherepresentationandcouldbeusedtogenerateaslightlydifferentsetofhar-monicoscillatorstates.2InthepaperswehavecitedwedescribedtheorthogonalizationofthesestatesaswellasthecalculationofmatrixelementsoftheQCDHamiltonianbetweenthem.Wewillnotneedthosedetailshere,however.Thesesingle-DOFwavefunctionsmustnowbepiecedtogethertoformmulti-DOFstatessuitableforcomputationsinHamiltonianQCD.Eachmulti-DOFstatewillbeaproductofsingle-DOFstates.Themostbasicstateisonewithnoexcitations.Specifically,eachdegreeoffreedomisazero-colorstate:|Ψ=|φφφφ···.(2.5)Thenextsimpleststatesarethosewithjustoneexcitation:|Ψ=|φaφφφ···.(2.6)However,thesestatesarenotSU(3)singlets,asevidencedbythecolorindexwhichislefthanging.InordertomaintainglobalSU(3)invariance,wemustsumoverallindices.Therefore,thenextpossiblestatesare|Ψ=δab|φaφbφφ···,(2.7)whereδabisaKroneckerδ-function.Thereareofcoursesimilarstateswheretheexcitationslieondifferentdegreesoffreedom.Wecalltheseone-pairstates.Inanalogousfashion,wecanformtwo-pairstates,three-pairstates,andsoon.Arbitrarilyintricatestatescanbeconstructedbyusinghighlycolored,single-DOFstatesalongwithcomplicatedcouplingschemes.Forinstance,thefollowingisavalid(thatis,gauge-invariant)state:|Ψ=δabTr(λcλdλe)|φacφbdφφe···,(2.8)wheretheλi’sareGell-Mannmatrices.ThestatesdescribedaboveprovidethebasisforapureSU(3)gaugetheorycalculation,andtheywilllikewiseprovidethebasisforthegauge