arXiv:hep-ph/0111158v1 13 Nov 2001DCPT/01/108
IPPP/01/54
QCDFiniteEnergySumRules
andtheIsoscalarScalarMesons
S.N.CherryaandM.R.Penningtonb
aGroupedePhysiqueTh´eorique,IPNUniversit´edeParis-Sud,F-91406OrsayC´edex,France.
bInstituteforParticlePhysicsPhenomenology,UniversityofDurham,Durham,DH13LE,U.K.
Abstract
WeapplyQCDFiniteEnergySumRulestothescalar-isoscalarcurrenttodeter-minethelightestudmesoninthischannel.Weuse‘pinch-weights’toimprovethereliabilityoftheQCDpredictionsandreducethesensitivitytothecut-offs0.Adecayingexponentialisincludedintheweightfunctiontoallowustofocusonthecontributionfromlowmassstatestothephenomenologicalintegral.OnthetheoreticalsideweincludeOPEcontributionsuptodimensionsixandacontribu-tionduetoinstantonstakenfromtheInstantonLiquidModel.Phenomenologically,weincorporateexperimentaldatabyusingacouplingschemeforthescalarcurrentwhichlinksthevacuumpolarisationtotheππscatteringamplitudeviathescalarformfactor.Wefindthatthesumrulesarewellsaturatedforcertaininstantonparameters.Weconcludethatthef0(400−1200)definitelycontainsalargeudcomponent,whereasthef0(980)mostlikelydoesnot.Weareabletoestimatetheaveragelightquarkmassandfindmq(1GeV2)=5.2±0.6MeV.
11Introduction
Currently,experimentsuggeststhattherearemorescalarmesonsthancanbeaccommo-
datedinasinglequark-modelnonet.ManyQCDmotivatedmodelspredicttheexistenceofnon-qqq[3]andK
qstates,whicharetheextrastates?Intheworkthatfollowswehopetoshedsomelightonthisquestionbyattemptingtodeterminewhichof
the5scalar-isoscalarmesonscurrentlylistedintheParticleDataGrouptables[5]isthe
lightestudmesoninthischannel.ToaddressthisquestionwewillusetheQCD
sumruletechnique.TheanswerhasimportanceforchiralsymmetrybreakinginQCD,seeforinstance[6,7].
QCDsumrulesareintegralexpressionsthatrelatethehadronicandpartonicregimes,i.e.thelowenergyworldofresonanceswiththehighenergyworldof(tractable)QCD.
Sincetheirconceptionovertwentyyearsago[8],QCDsumruleshavebecomeanestab-
lishedtechniquebothforcalculatinghadronicpropertiesinchannels,wheretheQCDexpressionsareundercontrol,and,conversely,estimatingQCDparameters(suchasthe
massesofthequarks)inchannelswherethereisgoodexperimentalinformation.
UsingQCDsumrulesinthescalarchannelisnotanewidea.Theywerefirstapplied
tothescalarmesonsin[9]whereLaplacesumruleswereused(forarecentreviewof
LaplaceSumRulessee[10]).Herethephenomenologicalsidewasrepresentedusing
the,nowstandard,resonance+continuumansatz,withresonancesbeingrepresentedasδ-functionsandthecontinuumbeingcalculatedentirelywithinperturbativeQCD.As
onlytheOperatorProductExpansion(OPE)terms,discussedinSect.3,weretaken
intoaccountonthetheoryside,exactmassdegeneracybetweenthelightestisoscalarandisovectorwasfound,withmf0=ma0=1.00±0.03GeV.Thes
qinterpretationofthesetwomesonscouldnotberuledoutwiththe
datathenavailable.
TheeffectsofgoingbeyondtheOPEwerestudiedin[12],whereinstantons(see
Sect.4)wereincludedonthetheoreticalside,butnotintheQCDcontinuumonthe
phenomenologicalside.Onceagaintheresonancesweretreatedasδ-functionsandonlytheisospin-1channelwasconsidered.ItwasconcludedthatthelightestqForthelightscalarmesonsthezero-widthapproximationisnotagoodoneandthefirst
attempttogobeyonditinasumruleinvestigationwasbyEliasetal[13].HereresonanceswererepresentedbyBreit-Wignerformulae,which,whilstbetterthanaδ-function,stilldonotadequatelydescribethecomplexstructureinthescalarsector.Afurtherabstraction
wasintroducedbyreplacingtheseBreit-WignershapeswithaRiemannsumofrectangular
pulses.Laplacesumruleswereused,butinanintegralformwhichrequiresthecalculation
ofperturbativeexpressionsatratherlowenergyscales.Againinstantonswereincluded,butnotinthecontinuum.Intheisovectorchannel,thea0(980)wasfoundtodecouple
fromthesumruleandthemassofthelightestquarkoniumisovectorwasconsistent
withthea0(1450).Intheisoscalarchanneltheconclusionswerenotsoclearcut,itwas
predictedthatthelightestquarkoniumhereshouldhaveawidthlessthanhalfofitsmass,andalthoughthef0(980)couldnotberuledoutasthemaincontributoralighterstate
waspreferred.Similarconclusionswerereachedin[14].ThisstudyusedbothLaplace
andFiniteEnergysumrules,againrequiringperturbativeexpressionstobeevaluatedatlowenergies.Theauthorsnoted,thatforaconsistenttreatmentoftheBorelsum
rules,theinstantoneffectsshouldalsobeincludedintheQCDcontinuumcontribution.
Thesumrulesweredominatedbyanisoscalarwithmassaround1GeVandanisovector
withmassaround1.5GeV,thussuggestingthef0(980)anda0(1450)asthelightestq
q,K∗0(1430).InstantonswereincludedandallQCDintegralswere
carriedoutinthecomplexplane,whichimprovestheconvergenceofperturbativeexpres-
sions.Theisovectordecayconstantswerefoundtobeofcomparablesize,suggestingasimilarstructureforboth.TheauthorconcludedthatthisfavouredaUnitarisedQuark
Model[16]scenario,wheretwophysicalhadronscanarisefromone‘seed’state.
IntheworkthatfollowswewillapplyQCDFiniteEnergySumRulesintroducedin
Sect.2tothescalar-isoscalarchannel.OnthetheoreticalsidewewillincludeinSects.3
and4instantoneffectsaswellastheOPEtermsuptodimensionsixandusethecontourimprovementprescriptiontoincreasetheconvergenceofperturbativeexpressions.Onthe
thephenomenologicalsideinSect.5wewillincorporateexperimentaldatadirectlyby
relatingthehadronicvacuumpolarisationtotheππscatteringamplitudeviathepion’s
scalarform-factor.ThesumrulesareevaluatedinSect.6andwepresentourconclusionsinSect.7.
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