Factorization of heavy-to-light form factors in soft-colline(4)

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

beenthecauseofsomeconfusion.Sinceinthispaperwewillconstructane ectivetheoryformodeswithvirtualityΛ2only,wewillusethesecondterminology.Thee ectivetheorythencontainssoftandcollinearmodesinagreementwiththestandardQCDterminology.Forpowercountingwede nethescalingparameterλtobeoforder(Λ/mb)1/2.Thisdi ersfromtheconventionin[20]whereλisoforderΛ/mb.

TheexistenceofthevariousmodesfollowsfromtheassumptionthattheexternalmomentaofscatteringamplitudesforexclusiveBdecaysatlargemomentumtransfer

2aresoftorcollinear.1One ndsthethreecharacteristicvirtualitiesm2b,mbΛandΛby

combiningexternalmomenta.Forinstance,m2bisobtainedbyaddingandsquaringaheavyquarkandacollinearmomentum,orbysquaringtheheavyquarkmomentum.Theintermediatevirtualityistypicalforinteractionsofcollineargluonsorlightquarkswithsoftgluonsorlightquarks,whileΛ2arisesintheself-interactionsofcollinearorsoftmodes.Soft-collineare ectivetheoryasde nedin[16,17]isthee ectivetheoryobtainedafterintegratingouthardmodesofvirtualitym2b.Thistheorystillcontainstwotypesofsoftmodes,called“semi-hard”(virtualityofordermbΛ)and“soft”.Thesemi-hardmodescanbeintegratedoutperturbatively,butitappearsthatsemi-hardloopintegralsalwaysvanishindimensionalregularization[17],sosemi-hardmodescanbeignoredinpractice.Thetheoryalsocontainstwotypesofcollinearmodes,called“hard-collinear”and“collinear”accordingtotheirvirtuality.AlthougheachoneofthesetwohasbeendiscussedinpreviousapplicationsofSCET,thesimultaneouspresenceoftwodistinctcollinearmodeshasnotbeenconsideredinmuchdetailuptonow.ThereasonforthisisthatpreviousapplicationsofSCETtosemi-inclusiveBdecay,suchasB→Xsγnearmaximalphotonenergy[18],andtoB→γlν(atleadingorderin1/mb)aresensitiveonlytohard-collinearmodes[7,8].OnecanthereforematchSCETdirectlytothestandardheavyquarke ectivetheory,whichcontainsonlysoftmodes.Ontheotherhand,intheexclusivedecayB→Dπ[3,21]orhardexclusivescatteringoflighthadrons[22]thee ectsofhard-collinearandsoftmodescancelalmosttriviallyatleadingpowerinthepowerexpansion.Thee ectivetheoryatleadingpowercanthenbeformulatedentirelyintermsofcollinearmodes.

Theoutlineofthispaperisasfollows:inSection2westudyascalarintegral,whichwouldberelevanttoB→γlνdecayatsub-leadingorderin1/mb,usingthemethodofexpandingbyregions[23].Wedemonstratewiththisexamplethatseparatecontributionsfromhard-collinearandcollinearloopmomentummustbeincludedtoreproducethein-tegralinfull“QCD”.Weshallalso ndthatthecollinearandsoftcontributionsarenotwell-de nedindividuallyindimensionalregularization.Theinterpretationofthesead-ditionaldivergencesprovidesanimportantcluetotheproblemofendpointdivergences.Inthecontextofe ective eldtheorytheadditionaldivergencesshowthatthematrixelementsinthee ectivetheoryofsoftandcollinear eldsdonotfactorize(naively)intoaproductofsoftandcollinearmatrixelementsasonemighthaveconcludedfromthedecouplingofsoftandcollinear eldsintheLagrangian.