Factorization of heavy-to-light form factors in soft-colline(17)

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

Lagrangian.TheSCET(c,s)Lagrangiandescribesinteractionsofsoftandcollinear elds.Ingeneral,therecanbescatteringprocessesofthetypes+c→s+c.Ithasbeenshownin[20]forquarkscatteringthattheseinteractionsarepower-suppressed.Belowweextendthistogluonsandderivetheexplicitformoftheleadingpower-suppressedinteractions.Becauseofthedecouplingofsoftandcollinearmodesatleadingpower,theLagrangianissimply

L(0)=Ls+Lc(27)

atleadingpower,with

Lc= 1

in+Dc

2tr(FµνsFsµν)+q¯s(iD/s m)qs+LHQET (iD/⊥c+m)n/+(28)

andm～λ2thelightquarkmass.Theheavyquarkinteractionswithsoft eldsaredescribedbythestandardheavyquarke ectivetheory(HQET)LagrangianLHQET=¯viv·Dshv+...h

ForthekinematicsituationshowninFigure4scatteringprocessess+c→s+ccannotoccur.Ontheotherhand,thecrossedprocesss+s→c+cisnotpossiblebymomentumconservationinthen directions,sincen+p>0foracollinearmomentumandn l>0forasoftmomentum.Itfollowsthatinsertionsofthesoft-collinearinteractiontermsfrom¯ matrixelements,sothatwecansimplythesub-leadingLagrangianhavezero π|...|B

workwiththeLagrangianwithoutsoft-collinearinteractionstoanyaccuracy.(See[25]forarelateddiscussion.)BeyondleadingpowerthereareadditionaltermsinthecollinearLagrangianLc,whichariseuponintegratingoutheavy-quarkloopswithexternalcollinearlines.Heavy-quarkloopsalsogenerateadditionalsoftinteractionterms,whichcorrectLs,whichalsoincludesthe1/mbsuppressedtermsfromtheHQETLagrangian.However,thesoft-collinearinteractionsallresideinthee ectivecurrent.

Statesandhadronicmatrixelements.TheBmesonstatesinthee ectivetheorymaybede nedastheeigenstatesoftheleading-ordersoftHamiltonian.ThesestatesareidenticaltothoseusedinHQET.Alternatively,ifweregardthestatesastheeigenstatesoftheexactsoftHamiltonian,thesestatescoincidewiththeBmesonstatesoffullQCD,sincethesoftLagrangiantoallordersisequivalenttothefullQCDLagrangian.Similarly,thelightmesonstateinSCET(c,s)maybede nedastheeigenstateoftheleading-ordercollinearHamiltonian.ThisHamiltonianisequivalenttoQCDwithoutheavyquarks,sothepionstateinthee ectivetheoryisthesameasinQCD(withoutheavyquarks).De ningthepionwithrespecttotheexactcollinearHamiltonianimpliesthatthepionstateisthesameasinfullQCDwithheavyquarks.Inthefollowingweadopttheconventionthatthestatesarede nedwithrespecttotheexactsoftorcollinearHamiltonians,sothatwedonotdistinguishthee ectivetheorystatesfromthoseinQCD.Itwouldbeasimplemattertomaketheλdependenceofthestatesexplicit.Animplicitassumptionhereis