Factorization of heavy-to-light form factors in soft-colline(14)

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

di erentcoe cientfunctionC1(E,n l)asaresultoftheloopintegration.WeidentifyC1asa1-loopcorrectiontothehard-scatteringkernel.Theexplicitcalculationshowsthatthereisadoublepolein1/ ,leavingadouble-logarithmicdependenceonthefactorizationscaleµ.

Considernowthesoftcontribution(middlecolumnintheFigure).Thehard-collinearsubgraphhasanadditionalexternalsoftgluonline,sotheoperatorinthee ectivetheoryhasthestructure[¯qAhv]s[Aγ]c(secondlineintheFigure).ThematrixelementinthethirdlineofFigure3takestheform

FT γ|[Aγ(sn+)]c|0 (E) dωC2(E,n l,ω)FT 0|[¯q(t2n )A(t1n )hv(0)]s|q¯b (n l,ω).(19)Thesoftmatrixelementcanbeidenti edwithathree-particlelight-conedistributionamplitudeφq¯paringthisexpressionto(12),weseethatthen k¯bgoftheq

integralin(12)correspondstotheintegrationoverω,whilethetransversemomentumintegralisincludedinthede nitionofthelight-conedistributionamplitude.

Intheconventionalhard-scatteringformalismthescaledependenceofthedistributionamplitudewouldcancelagainstthescaledependenceofahard-scatteringkernel(suchasC1).Thiscannotbecompletelycorrecthere,sincetheω-integralhasanendpointdivergenceasω→0,whichcorrespondstothe1/δsingularityin(12).Theassociatedν-dependenceisnotcancelledbyahard-scatteringkernel,butbythecollinearcontributionasseenfromthetoyexample.Theexistenceofanendpointdivergenceimpliesthatexpression

(19)initsentiretyhasascale-dependencedi erentfromthetwomatrixelementsinthefactorizedexpression.Thispossibilityisnotconsideredintheconventionalhard-scatteringformalism.5

TheoperatorinterpretationofthecollinearintegralisillustratedinthethirdcolumnofFigure3.Thephotonlineisnotdirectlyconnectedtothehard-collinearsubgraphinthiscase.Rathertheoperatorthatresultsaftercontractingthehard-collinearsubgraphhas eldcontent[¯qhv]s[¯qq]c(secondlineintheFigure).Thematrixelement(thirdline)canbewrittenas

q(tn )hv(0)]s|q¯b (n l)FT 0|[¯ 1

0duC3(E,u,n l)FT γ|[¯q(s1n+)q(s2n+)]c|0 (E,u).(20)

Thisseemstorepresenttheconvolutionofahard-scatteringkernelC3withthetwo-particlelight-conedistributionamplitudeoftheq¯bstateφq¯light-conedistribution¯bandtheqq

amplitudeofthephotonφγqq¯.Thisisonlycorrectwiththeunderstandingthattheu-integralisdivergentandmustberegularizedinawaythatisconsistentwiththeregularizationoftheω-integralinthesoftcontribution.Theadditionaldivergence,whichisnotrelatedto