Factorization of heavy-to-light form factors in soft-colline(12)

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

occursatweakcoupling.Ifitdoesnot,factorizationofsoftandcollinearmodescannotbeimplementedperturbativelyinQCD.Ifendpointdivergencescanbeshowntobeabsenttoallordersinperturbationtheory,sothatnoregularizationisrequired,thisindicatesthatnaivesoft-collinearfactorizationisvalid.

2.3Interpretationoftheresult

Hard-collinearmodesinSCET.Thetoyintegralclari esthatSCET,de nedasthee ec-tivetheoryafterintegratingouthardmodes,containstwocollinearmodeswithdi erentvirtuality.IfSCETisformulatedwithasinglecollinearquarkandgluon eld,thecollinear eldscannotbeassignedanunambiguousscalinglaw,andpowercountingisnolongermanifestintheverticesofthee ectivetheory,4unlessoneofthetwocollinearmodesisirrelevantforaspeci cprocess.Analternativeistointroduceseparatehard-collinearandcollinear eldsinSCET.ThecorrespondingLagrangiancanbetakenasastartingpointforthesecondmatchingstep,inwhichhard-collinearmodesareintegratedoutandSCETisreducedtoane ectiveLagrangianforsoftandcollinearmodesonly.Thisformulationwillbeusedinsomeofthetechnicalstepsinthefollowingtwosections.

Acommentisnecessaryonthetransversemomentumscalingofhard-collinearmodes.Whenasoftandcollinearmomentumcombinetoahard-collinear uctuationofvirtualityλ2,thehard-collineartransversemomentummustbeoforderλ2bymomentumconser-vation.Thisisthecaseforthehard-collinearpropagatorsinthesoftandcollinearcon-tributionstothetoyintegral(seetheupperrowofFigure3).However,inhard-collinearloopsthetransversemomentumisoforderλ,asonecaneasilyverifyfromthelocationofpolesofthehard-collinearintegrand.Assumingk⊥～λ2wouldmakeallhard-collinear

2loopintegralsvanish,sincetheintegralswouldhavetobeexpandedink⊥.Thiswould

obviouslyfailtoreproducetheexpansionoftheexactintegral.Wethereforeassignthegenericscaling(1,λ,λ2)tohard-collinearmodes,asgiveninTable1.Non-genericscalingintreesubgraphsisnotparticulartothepresentcaseofhard-collinearmodes.Whenoneintegratesouthardheavyquark uctuationsgeneratedbytheinteractionofnearon-shellheavyquarkswithhard-collinearorcollineargluons[16],theo -shellmodeshavemomentum(1,λ,1)or(1,λ2,1),unlikethegenerichardmomentum(1,1,1).

Operatorinterpretationofthetoyintegral.Weproceedtodiscussthethreecontributionstothetoyintegralintermsofoperatorsandmatrixelementsofane ectivetheoryforsoftandcollinearmodes.Thisdiscussionwillbeheuristic,sinceweabstractfromthescalarintegralanduseQCDterminology,butwithoutmakingthenotationcompletelyexplicit.WeimaginethatFigure1representsacorrectiontothematrixelement γ|J|q¯b oftheb→utransitioncurrentbetweenaq¯bstatewith xedlightquarkmomentumn landaphotonwithlargeenergyE=n+p′/2.Thecorrespondingtreediagramhasonehard-collinearlinejoiningtheweakvertextothephotonvertex.Inthee ectivetheory(of