Combining the recent lattice calculation of $a_0(1450)$ and $\sigma(600)$ mesons with the overlap fermion in the chiral regime with the pion mass less than $300 {\rm MeV}$, the quenched lattice calculation of the scalar glueball, and the phenomenological s

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a1PatternofLightScalarMesonsKeh-FeiLiu )Dept.ofPhysicsandAstronomy,UniversityofKentucky,Lexington,KY40506USACombiningtherecentlatticecalculationofa0(1450)andσ(600)mesonswiththeoverlapfermioninthechiralregimewiththepionmasslessthan300MeV,thequenchedlatticecalculationofthescalarglueball,andthephenomenologicalstudyofthemixingofisoscalarscalarmesonsf0(1710),f0(1500),f0(1370)throughtheirdecays,asimplepatternforthelightscalarmesonsbeginstoemerge.Below1GeV,thescalarmesonsformanonetoftetraquarkmesoniums.Above1GeV,thenonentqq¯mesonsaremadeofanoctetwithlargelyunbrokenSU(3)symmetryandafairlygoodsingletwhichisf0(1370).f0(1710)isidenti edasanalmostpurescalarglueballwitha～10%mixtureofqq¯.§1.IntroductionInlightmesonspectroscopy,thepseudoscalar,vector,axial,andtensorsectorsarereasonablywellknownintermsoftheirSU(3)classi cationandquarkcontents.Thescalarsector,ontheotherhand,ispoorlyunderstoodinthisregard.Firstofall,therearetoomanyofthem.Thereare19stateswhicharemorethantwicetheusualqq¯nonetasinothersectors.WeshowinFig.1theknownscalarmesonswhichincludeσ(600),κ(800),andf0(1710)whicharebetterestablishedexperimentallynowadays.1),2)Thereareseveralpuzzlingcharacteristicswhichhavebeenobservedovertheyears.The rstquestiononemightraiseisthewhereaboutoftheqq¯athe3P0partnerofa1(1260)(3P1)anda2(1320)(30,P2)accordingtothequarkmodelclassi cation.Fromtheorderofspin-orbitsplittingoftheP-waveqq¯spectrum,itseemsnaturaltoidentifyitwitha0(980).However,thereareahostofdi cultiesinsuchanassignment: In (e.g.su¯withonestrangequark)isexpectedthiscase,toliethe～member100MeVofabove,theoctetwhichK0wouldplaceitthereisnostatethere,itwouldbe～300MeVbelowK around1100MeV.But(1430)and～300MeV

aboveκ(800).0

Thewidthsoff0(1370).a0(980)andf0(980)aresubstantiallysmallerthanthoseofa0(1450)andInparticular,theyaremuchsmallerthanthatofκ(800)whichshouldbeanonetpartnerofa0(980)andf0(980).

Thestate.γγ3)widthsofa0(980)andf0(980)aremuchsmallerthanexpectedofaqq¯ ItTheisexperimentalhardtounderstanddataonwhyD+sa0(980)andf0(980)arepracticallydegenerate.→f0(980)π+andφ→f0(980)γimplycopiousf0(980)productionviaitsss¯component.Yet,therecannotbess¯ina0(980)sinceitisanI=1state.