A New Renormalization Group for Hamiltonian Field

arXiv:nucl-th/9901079v126Jan1999ANEWRENORMALIZATIONGROUPFORHAMILTONIANFIELDTHEORYROBERTJ.PERRYANDS´ERGIOSZPIGELPhysicsDepartment,TheOhioStateUniversity,Columbus,OH43210,USAE-mail:[e-mail],szpigel@pacific.mps.ohio-state.eduTheSchr¨odingerequationwithatwo-dimensionalδ-functionpotentialisasimpleexampleofanasymptoticallyfreetheorythatundergoesdimensionaltransmuta-tion.Renormalizationrequirestheintroductionofamassscale,whichcanbeloweredperturbativelyuntilaninfraredcutoffproducedbynon-perturbativeef-fectssuchasboundstateformationisencountered.Weoutlinetheeffectivefieldtheoryandsimilarityrenormalizationgrouptechniquesforproducingrenormalizedcutoffhamiltonians,andillustratethecontroloflogarithmicandinverse-power-lawerrorsbothtechniquesprovide.1MotivationandOutlineThisarticlefollowsamorecompletediscussioninalongerarticlebeingpre-paredforpublication.1ElsewhereintheseProceedingsthepossibilityofde-rivingavalidconstituentapproximationinlight-frontQCDisdiscussed.2Asimpleprocedurethatproducesarenormalizedlight-frontQCDhamilto-nianstartswiththecomputationofacutoffhamiltonianusingasimilarityrenormalizationgroup(SRG)3,4,5,6andcouplingcoherence.7,8,9Inthisarti-cleweillustratethisstepusingthetwo-dimensionalδ-functioninaone-bodySchr¨odingerequation.10Wealsodiscussarelatedapproach,effectivefieldtheory(EFT).11,12Localityisthesourceofperturbativedivergencesinfieldtheory,anditproducessimilareffectsinsimpleone-bodyquantummechanicsproblems.Lo-calityleadsustoconsiderpotentialsconsistingofδ-functionsandtheirderiva-tives.Thesepotentialsproducedivergencesthatrequiretheintroductionofamassscale.Changingthismassscaleshouldbeequivalenttochangingadimensionlesscouplingconstant(i.e.,dimensionaltransmutation)forascale-invariantlocalhamiltonian.Thetwo-dimensionalδ-functionproblemissolvedinthesecondsection,providingthedataforourerroranalysisofEFTandSRGcalculations.Problemsrequiringrenormalizationresultfromanarbitrarilylargenum-berofdegreesoffreedombeingcoupled.Thisisclearinmomentumspace,wheretheoff-diagonalmatrixelementsofaδ-functionareconstants.Diver-gencesresultfromaninfinitenumberofscalesbeingdirectlycoupled,andrenormalizationrequiresacutoffλtoregulatethedivergences.Physicalre-japanrg:submittedtoWorldScientificonFebruary9,20081sultssuchasbindingenergiesshouldbeindependentofλ,sothereshouldexistalineofcutoffhamiltoniansHλthatallproducethesamephysicalre-sults.Arenormalizationgroup(RG)isbuiltfromtransformations,T(λ,λ′),thatchangethecutoff.TconnectshamiltoniansatdifferentscalesandintheSRGTistypicallyaunitarytransformation.InEFTTexistsinprinciple,butHλisdeterminedateachscalebycon-strainingthecouplingsinfrontofafinitenumberoflocaloperatorstofitdata.Asimplescaletransformationleadstotheclassificationoftheseoper-atorsasrelevant(powersofthecutoffappear),marginal(thecutoffappearsonlyinlogarithmsthatareabsorbedintherunningmarginalcoupling),andirrelevant(inversepowersofthecutoffappear).ThesameoperatorsariseintheSRG,wheretheirstrengthsareapproximatedbyexpansionsinpowersofarunningmarginalcoupling.Forthesimpletwo-dimensionalδ-functiontherearenorelevantoperatorsandonlyonemarginaloperator.Therearealwaysaninfinitenumberofirrelevantoperators,andthesebecomeincreas-inglyimportantasthecutoffapproachesanon-perturbativeenergyscaleintheproblem.Theeasiestwaytoclarifythesepointsisbyexample.Weshowacompletesolutionofthetwo-dimensionalδ-functionproblem,introducingtheK-matrixtocomputescatteringphaseshifts.Thereisoneboundstate,anditsbindingenergysetsthescalefornon-perturbativephysics.TheEFThamiltonianisproducedbytruncatingtheseriesofirrelevantoperators,andthenfittingafinitenumberofremainingcouplingstoanequalnumberoflow-energyscatteringconstraints.Thisleadstopower-lawsuppressionoferrorsinthepredictedbindingenergy,withtheexpansionclearlybreakingdownasthecutoffapproachesthebindingenergy.TheSRGhamiltonianisproducedbysolvingadifferentialequationforHλsubjecttotheconstraintthatallirrelevantcouplingsareanalyticfunctionsofthesinglemarginalcoupling.2Thetwo-dimensionalδ-functionpotentialConsidertheSchr¨odingerequationintwodimensionswithanattractiveDiracδ-functionpotential:10−∇2rΨ(r)−α0δ(2)(r)Ψ(r)=EΨ(r).(1)Thecouplingα0isdimensionless,sothehamiltonianisscaleinvariant(i.e.,thereisnointrinsiclengthorenergyscale).Itisrelativelystraightforwardtosolvethisequationinpositionspacebyintroducingaconvenientdistributionjapanrg:submittedtoWorldScientificonFebruary9,20082functiontoregulatetheδ-function,butwewillworkinmomentumspacetostayclosetotheEFTandSRGcalculations.TheSchr¨odingerequationinmomentumspaceis,p2Φ(p)−α0(2π)2Zd2qΦ(q)=EΦ(p).(2)Asaconsequenceofscaleinvariance,ifthereisaboundstatesolutiontoEq.(2)thenitwilladmitsolutionsforanyE0.Thiscorrespondstoacontinuumofboundstateswithenergiesextendingdownto−∞,sothesystemisnotboundedfrombelow.ByrearrangingthetermsintheSchr¨odingerequationweobtainΦ(p)=α02πΨ(0)(p2+E0),(3)whereΨ(0)isthepositionspacewave-functionattheoriginandE0=−E0isthebindingenergy.Toobtaintheeigenvalueconditionforthebindingenergy,wecanintegratebothsidesofEq.(3):1=α02πZ∞0dpp1(p2+E0).(4)Theintegralonther.h.s.diver