Factorization of heavy-to-light form factors in soft-colline(18)

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

thattheseparationofcollinearandsoftmodesisdonewithoutanexplicitcut-o (dimen-sionaloranalyticregularization).WitharegularizationthatbreaksboostinvariancethecollinearLagrangianisnotequivalenttothefullQCDLagrangian.Fromtheconventionalnormalizationofhadronicstatesitfollowsthat

|B(p) ～λ 3,|π(p′) ～λ 2,(29)

wherethelightmesonisassumedtobeenergetic.

ThatapionismadeonlyofcollinearpartonscanbeunderstoodbynotingthatacollinearpionstatecanbeobtainedfromapionatrestbyalargeLorentzboost.InapionatrestallpartonshavemomentaoforderΛ,sotheboostedsystemcontainsonlycollinearmodes.Addingasoftpartontothecollinearmodesproducesacon gurationofinvariantmassmbΛ,whichcannotcontributetothepionpole.Ontheotherhand,withthesamelineofreasoning,aBmesonconsistsonlyofsoftpartons(andaheavyquark),sinceaddingacollinearmodeproducesacon gurationfarawayfromtheBmesonpole.Anapparentconsequenceoftheabsenceofsoft-collinearinteractionsandthenatureofthestatesisthatanexpression

¯ C π|f(φc)g(φs)|B(30)

wheref(φc)(g(φs))isanon-localproductofcollinear(soft) eldsandthestardenotesconvolutions,factorizesinto

¯ . π|f(φc)|0 C 0|g(φs)|B(31)

AsshowninSection2thisshouldbeconsideredasformal,sincethecollinearandsoftconvolutionintegralscanbedivergent.

AnotherconsequenceisthattheQCDcurrentmatrixelement π(p′)|u¯Γb|B(p) simplymatchesto π(p′)|Je |B(p) ,sotheproblemreducestoobtainingthee ectivecurrent.Alreadyatthispointwemaynotethattheapparentlyleadingtermvanishes,

¯cΓhv|B¯(p) =0, π(p′)|ξ(32)

¯c)dobecausethequantumnumbersoftheproductofcollinear elds(herethesingle eldξ

notmatchthoseofapion,andthequantumnumbersofthesoft elds(hereonlyhv)donot¯meson.Thiscanbeformalizedbysayingthatthee ectiveLagrangianmatchthoseoftheB

isinvariantunderseparatephasetransformationsofthecollinear,softandheavyquark elds,sowecanassign“collinearquarknumber”toproductsofoperators,withξ elds¯havingcharge 1andallotherfundamental eldscarryingcollinearquarkcharge+1,ξ

charge0.Weshallseelaterthatthe rstnon-zeromatrixelementissuppressedbythree3/2powersofλ.Thisistheoriginofthewell-known1/mbsuppressionofheavy-to-lightformfactorsatlargerecoil[11].

Thisleadstotheimportantobservation[15]thatpower-suppressedcurrentsinthee ectivetheorybecomerelevanttotheB→πformfactoratleadingpower.ThederivationofthesecurrentsinSCET(c,s)willbeworkedoutattreelevelbelow,andinmoregenerality