Factorization of heavy-to-light form factors in soft-colline(13)

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

Figure3:Diagrammaticandoperator/matrixelementrepresentationofthehard-

collinear(leftcolumn),soft(middlecolumn)andcollinearcontributiontothedia-

gramofFigure1.Eachcolumnshows:theoriginaldiagramwiththehard-collinear

subgraphmarkedbybold-facelines(upperrow),andwiththedashedlineindicating

wherethegraphfactorizesintoashort-distanceandlong-distancesubgraph;theop-

eratorvertexinthee ectivetheorycorrespondingtothecontractedhard-collinear

subgraph(middlerow);thecontributiontotheoperatormatrixelement γ|Oi|q¯b correspondingtotheoriginaldiagram(lowerrow).

softandcollinearmodes)thisiswrittenas

C0(E,n l)FT γ|[Aγ(sn+)]c[¯q(tn )hv(0)]s|q¯b (E,n l).(17)

ThesymbolFT ... meansthataFouriertransformofthematrixelementwithrespecttothepositionargumentsofthe eldsistaken,with(E,n l)thevariablesconjugateto(sn+,tn ).Theindexonproductsof eldsindicateswhethertheyaresoftorcollinear,andthenon-localityoftheoperatorisrelatedtothenon-polynomialdependenceofthehard-collinearpropagatoronthemomentumcomponentn lofthelightexternalquark,andthemomentumcomponentn+p′oftheexternalphoton.Thematrixelementfactorizestriviallyinto

q(tn )hv(0)]s|q¯b (n l).(18)FT γ|[Aγ(sn+)]c|0 (E)FT 0|[¯¯meson,Thephotonmatrixelementcanbecalculated.Whentheq¯bstateisreplacedbyaB

thesoftmatrixelementgivestheBmesonlight-conedistributionamplitude.Hence(17)assumestheformofaconvolutionofahard-collinearcoe cientfunctionwiththeBmesonlight-conedistributionfunction,whichreproducesthefactorizationpropertyoftheB→γtransitionatleadingorderin1/mb,andatleadingorderinαs[5].

Thehard-collinearcontributiontothetoyintegralanditsoperatorinterpretationisshownintheleftcolumnofFigure3.Whenthehard-collinearsubgraphiscontractedtoa“point”,thecorrespondingoperatorhasthesame eldcontentasin(17),butwitha