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

4K.F.Liu

to250MeV.Theloweststateisabout100(30)MeVbelowtheππthresholdwhichisindicatedbythesolidline.ThisisthelowestinteractingstateoftwopionswhichisattractiveintheI=0channelandisreasonablywelldescribedbythequenchedchiralperturbationtheoryofππscattering.13)Thethirdstateisabovethenon-interactingππscatteringstatewitheachpionhavingoneunitoflatticemomentum(i.e.p1=2π/La)andissupposedtoincludethehigherexcitedstateswhicharenot tted.Theinterestingthingisthatthereisanextrastateat～550MeVwhichfallsinbetweenthetwostates.Todiscernthenatureofthisstate,westudiedthevolume dependenceofthespectralweightWifromthe ttingfunctioniWieEitofthetetraquarkcorrelator.ThedetailsaregiveninRef.12)Tosummarizetheresults,wefoundthat,byexaminingthecharacteristic3-volumedependenceofthespectralweight,thestateat～550MeVisaone-particlestate,whiletheloweststateisatwo-particlestatewhichisconsistentwiththequenchedchiralperturbationpredictionoftheinteractingI=0ππscatteringstate.13)

2

Four quark spectrum (GeV) 1.8 1.6 1.4 1.2 1 0.8 0.6

0.4 0.2 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07

22mπ (GeV)

Fig.2.Thelowestthreestatesfromthescalartetraquarkcorrelatorasafunctionofm2πformπ

3from182MeVto250MeVonthe12×28lattice.Thesolidlinesindicatetheenergiesofthe

twolowestnon-interactingpionsinS-wavewithlatticemomentap0=0andp1=2π/La.Thisveri esthatthetetraquarkmesoniumexistsandthelatticecalculation,whichgivesamassof540±170MeVatthechirallimit,suggeststhatσ(600)issuchastate.However,oneimportantquestionremains.Experimentally,σisaverybroadresonancewithawidthof544MeV.8)Howdoesone nditswidthonthelattice?After ndingσ(600)whichisseparatedfromtheππscatteringstatesonthepresentlattice,onecanincreasetheboxwhichwilllowertheenergiesofthescatteringstateaboveit.Whenitisloweredtowithintherangeofthe”width”,itmixeswiththeboundstateandavoidslevelcrossing.FromtheenergyofthemixedstateonecandeducethescatteringphaseshiftfromL¨uscher’sformula.14)Thisisvalidforelasticscatteringirrespectivehowbroadtheresonanceis.Thisisstudiedindetailinaspin