Single Heavy Flavour Baryons using Coulomb plus Po

arXiv:0805.3439v2[hep-ph]27Aug2008SingleHeavyFlavourBaryonsusingCoulombplusPowerlawinterquarkPotentialAjayMajethiya,BhavinPatelandP.C.VinodkumarDepartmentofPhysics,SardarPatelUniversity,VallabhVidyanagar-388120,Gujarat,INDIAAbstractPropertiesofsingleheavyflavorbaryonsinanonrelativisticpotentialmodelwithcolourcoulombpluspowerlawconfinementpotentialhavebeenstudied.Thegroundstatemassesofsingleheavybaryonsandthemassdifferencebetweenthe(JP=32+andJP=12+)statesarecomputedusingaspindependenttwobodypotential.Usingthespin-flavourstructureoftheconstitutingquarksandbydefininganeffectivecon-finedmassoftheconstituentquarkswithinthebaryons,themagneticmomentsarecomputed.Themassesandmagneticmomentsofthesingleheavybaryonsarefoundtobeinaccordancewiththeexistingexperimentalvaluesandwithothertheoreticalpredictions.Itisfoundthatanadditionalattractiveinteractionoftheorderof−200MeVisrequiredfortheantisymmetricstatesofΛQ(Q∈c,b).Itisalsofoundthatthespinhyperfineinteractionparametersplaydecisiveroleinhadronspectroscopy.1IntroductionConfirmationoftheexistenceofthecharmedbaryonsatFermilab[1]aroseanincreas-inginterestonheavy-baryonspectroscopy.Itisstrikingthatbaryonscontainingoneortwoheavycharmorbeautyflavourcouldplayanimportantroleinourunderstand-ingofQCDatthehadronicscale[2].ThecopiousproductionofheavyquarksatLEP,FermilabTevatron,CERNandBfactories,openeduprichspectroscopicstudyofheavyhadrons.Manytheoreticalmodels[3,4,5,6,7,8,9,10,11,12]havealsopredictedtheheavybaryonmassspectrum.Thenon-relativisticquarkmodel(NRQM)[13]hasbeenabletoexplainverynicelythemassspectrumoflightbaryons.Thoughtheexperimentalandtheoreticaldataonthepropertiesofheavyflavourmesonsareavailableplentyinliterature,themassesofmostoftheheavybaryonshavenotbeenmeasuredyetexperimentally[14].ThustherecentpredictionsabouttheheavybaryonmassspectrumhavebecomeasubjectofrenewedinterestduetotheexperimentalfacilitiesatBelle,BABAR,DELPHI,CLEO,CDFetc[15,16,17,18,19,20].Theseexperimentalgroupshavebeensuccessfulindiscoveringheavybaryonicstatesalongwithotherheavyflavourmesonicstatesanditisexpectedthatmoreheavyflavourbaryonstateswillbedetectedinnearfuture.Mostofthenewstatesarewithinthe1heavyflavoursectorwithoneormoreheavyflavourcontentandsomeofthemarefarfrommostofthetheoreticalpredictions.Thoughthereareconsensusamongthetheoreticalpredictionsonthegroundstatemasses[6,7],therearelittleagreementamongthemodelpredictionsofthepropertieslikespin-hyperfinespilttingamongtheJP=12+andJP=32+baryons,theformfactors[6],magneticmomentsetc[9].Allthesereasonsmakethestudyoftheheavyflavourspectroscopyextremelyrichandinteresting.Thestudyofheavybaryonsfurtherprovideexcellentlaboratorytounder-standthedynamicsoflightquarksinthevicinityofheavyflavourquarksinboundstates.Forthepresentstudyofthepropertiesofthelow-lyingbaryonicstateswithaheavyquark,weconsidercoulombpluspowerformastheinterquarkinteractionpotential[9,21,22].Thespinhyperfineinteractionssimilartotheoneemployedin[2]havebeenusedwithmassdependanceoftheconstitutingquarksinthepresentstudy.Themagneticmomentsofthebaryonsarecomputedbasedonthenon-relativisticquarkmodelusingspin-flavourwavefunctionsoftheconstitutingquarks[9].Thepaperisorganizedasfollows.Insect.IIofthispaper,thebasicmethodologyadoptedforthepresentstudyofcomputingthebindingenergyoftheconstitutingquarkswithinthebaryoncontainingsingleheavyquarksisdescribed.Insect.III,wepresentthecalculationofmagneticmomentsofthebaryons.Insect.IV,wepresentourresultsanddiscusstheimportantfeaturesandconclusionsofthepresentstudy.2MethodologyandBindingenergyoftheSin-gleHeavyFlavourBaryonsWestartwiththecolorsingletHamiltonianofthesystemasH=−3Xi=1∇2i2mi+XijVij(1)Where,theinterquarkpotentialVij=−2αs31xij+βxνij+Vspin(ij);xij=|−→xi−−→xj|(2)Followingref[8],thesingleparticlepositionco-ordinatesarereplacedbytheCMco-ordinatesplustheinterquarkdistancesofq1,q2andq3=Q,as−→X=1Pmi3Xi=1mi−→xi;(3)2Table1:TheModelParameterswiththeVariationalParameterλSystemνλinunitsofβinunitsofBohrradius(BohrEnergy)ν+1CharmBaryons01.311.01813.071.26223.641.830BeautyBaryons01.314.78215.5710.00026.7226.530mu=md=338,ms=420,mc=1380,mb=4275(inMeV)Table2:HyperfineParametersforthesymmetricspin-flavourcombinationsofsingleheavybaryonsAwith(r0=10−6MeV−1)Systemν=0ν=1ν=2cqq1310.625cqs3931.875bqq7810.625bqs46863.750−→r1=−→x1−−→x3;−→r2=−→x2−−→x3(4)andr12=|−→r2−−→r1|=|−→x2−−→x1|;(5)mi3=mim3mi+m3;m3=mQ(6)Accordingly,theHamiltoniancanbeseparatedintothetranslationallyinvariantCMpartandthepartgoverningtherelativemotiongivenby,hash=−∇2r12m13−∇2r22m23−∇r1·∇r2m3+V(r1,r2)(7)Here,m1,m2,m3aretheconstituentquarkmassesandalltheindependentorientationsofco-ordinates−→r1and−→r2areassumed.Underthesphericallysymmetricapproxima-tion,thepotentialisthenwrittenintermsofthemagnitudesoftheco-ordinates−→r13and−→r2asV(r1,r2)=−2αs31r1+1r2+1r12+β(rν1+rν2+rν12)+Vspin(8)Here,αsistherunningstrongcouplingconstantandVspinisthespindependentpartofthethreebodysystem.Wechoosethespatialpartofthetrialwavefunctionas[8]Ψ(r1,r2)=f(r1)f(r2)(9)withanormalizationconditiongivenbyR|Ψ(r1,r2)|2d3r1d3r2=1.Here,f(r)=λ3/2√πe−λrandλisacommon