用微扰QCD方法研究B (s)→φρ衰变

131ò16Ï2007c6pUÔn†ØÔnHIGHENERGYPHYSICSANDNUCLEARPHYSICSVol.31,No.6Jun.,2007PerturbativeQCDStudyofB(s)!DecaysLIJing-Wu1)YOUFu-Yi2)(DepartmentofPhysics,XuzhouNormalUniversity,Xuzhou221116,China)AbstractWestudyB(s)!decaysinaperturbativeQCDapproachbasedonkTfactorization.Inthisapproach,wecalculatefactorizableandnon-factorizablecontributions,therearenoannihilationcontributionsduetoquarkcontent.WegetthebranchingratiosandpolarizationfractionsforB(s)!decays.Ourpredictionsareconsistentwiththecurrentexperimentaldata.KeywordsPQCD,Bmesondecay,branchingratio,polarizationfraction1IntroductionExclusiveBmesondecays,especiallyB!VVmodes,havearousedmoreandmoreinterestboththeoreticallyandexperimentally.Sincetherstob-servedcharmlessB!VVmode,theB!Kdecay[1],manyB!VVdecaychannelshavebeenstudiedinPQCDapproach,suchasB!K,K![2],B!KK[3],Bs!(!)K[4],B!(!)(!)[5],andB0![6].ItoersanexcellentplacetostudytheCPviolationandsearchfornewphysicshints[7].Be-causethehadronizationprocessisnon-perturbativeinnature,theessentialprobleminhandlingthedecayprocessesistheseparationofdierentenergyscales,i.e.,thefactorizationassumption.Manyapproachesbasedonthefactorizationassumptionhavebeende-veloped,suchasthenaivefactorization[8],thegen-eralizedfactorization[9,10],theQCDfactorization[11],andtheperturbativeQCDapproachwhichisbasedonkTfactorization[12,13].Recently,B!Kdatarevealalargetransversepolarizationfraction,whichhasbeenconsideredasapuzzle,manytheoreticaleortshavebeenputtoclarifyit[14|21].ThissuggeststhatB!VVmodesmustbemorecomplicatedthanalltheothermodesandneedtobestudieddeeply.Motivatedbythis,westudyanotherB!VVmodeintheperturba-tiveapproach(PQCD)withintheStandardModel.ForB!decay,onlypenguinoperatorscontributeandwendthatthebranchingratioisattheor-derof109.ForB0s!0decay,current-currentoperatorsandpenguinoperatorscancontributeandwendthatthebranchingratioisattheorderof107.Thelongitudinalpolarizationpredominatesovertransversepolarizationanditsfractionisfoundtogobeyond70%.Ourpredictionsareconsistentwiththecurrentexperimentalvalues.Wehopethatourstudywillhelptoresolvetheabove-mentionedpuzzleabit.Theremainingpartofthispaperisorganizedasfollows.InSec./,wecalculateanalyticallytherelatedFeynmandiagramsandpresentthevariousdecayamplitudesforthedecaymodesstudied.InSec.III,wegivethenumericalanalysisforthebranch-ingratiosandpolarizationfractionoftherelatedde-caymodesandcomparethemwiththemeasuredval-ues.Thesummaryandsomediscussionareincludedinthenalsection.Received24July2006,Revised22January20071)E-mail:[e-mail]2)E-mail:[e-mail]523|531524pUÔn†ØÔn(HEP&NP)131ò2Theoreticalframeworkandpertur-bativecalculationInPQCDapproach,thedecayamplitudeisex-pressedastheconvolutionofthemesons'light-conewavefunctions,thehardscatteringkernelsandtheWilsoncoecients,whichstandforthesoft,hardandharderdynamics,respectively.Theformalismcanbewrittenas:MZdx1dx2dx3b1db1b2db2b3db3TrC(t)B(x1;b1)(x2;b2)(x3;b3)H(xi;bi;t)St(xi)eS(t);(1)whereTrdenotesthetraceoverDiracandcolorin-dices.C(t)isWilsoncoecientofthefour-quarkop-eratorwhichresultsfromtheradiativecorrectionsatshortdistance.ThewavefunctionMabsorbsnon-perturbativedynamicsoftheprocess,whichispro-cessindependent.ThehardpartHisratherprocess-independentandcanbecalculatedinperturbativeapproach.Thebiistheconjugatespacecoordinateofthetransversemomentum,whichrepresentsthetransverseintervalofthemeson.tisthelargesten-ergyscaleinhardfunctionH,whilethejetfunc-tionSt(xi)comesfromtheresummationofthedoublelogarithmsln2xi,calledthresholdresummation[22,23],whichbecomeslargerneartheendpoint.TheSu-dakovformfactorS(t)isfromtheresummationofdoublelogarithmsln2Qb[24].Inthispaper,weusethelight-conecoordi-natestodescribethefour-dimensionalmomentumas(p+;p;PT).WeworkintheframewiththeBmesonatrest,sothemesonmomentumcanbewrittenasP1=MBp2(1;1;0T);P2=MBp2(1r2;r2;0T);P3=MBp2(r2;1r2;0T);(2)inwhichr;risdenedbyr=M=MBandr=M=MB.P1;P2;P3refertoB;;respectively.Toextractthehelicityamplitudes,weparameterizethefollowingpolarizationvectors.Thelongitudinalpo-larizationmustsatisfytheorthogonalityandnormal-ization:2LˆP2=0,3LˆP3=0,and22L=23L=1.Thenwecangivethemanifestformasfollows:2L=1p2r(1r2;r2;0T);3L=1p2r(r2;1r2;0T):(3)Astothetransversepolarizationvectors,wecanchoosethesimpleform:2T=1p2(0;0;1T);3T=1p3(0;0;1T):(4)Thedecaywidthforthesechannelis:=G2FjPcj16M2BX=L;TMyM;(5)wherejPcjisthethree-dimensionalmomentumofthenalstatemeson,andjPcj=MB2(1r2r2).Thesubscriptdenotesthehelicitystatesofthetwovec-tormesonswithL(T)standingforthelongitudinal(transverse)component.AsdiscussedinRef.[1],theamplitudeMisdecomposedintoM=M2BML+M2BMN2(=T)ˆ3(=T)+iMT23P2P3:(6)WecandenethelongitudinalH0,transverseHhelicityamplitudesH0=M2BML;H=M2BMNMMpr21MT:(7)wherer=P2ˆP3=(MM).AndwecandeducethattheysatisfytherelationX=L;TMyM=jH0j2+jH+j2+jHj2:(8)Thereisanothersetofdenitionofhelicityam-plitudesA0=M2BML;Ak=p2M2BMN;A?=MMp2(r21)MT;(9)withthenormalizationfactor=pG2FPc=(16M2B).Thesehelicityamp