Factorization of heavy-to-light form factors in soft-colline(7)

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

n,theresultisthatonlythesoftmomentumregion,n=2,contributes.Thecorrespondingintegralwillbecalculatedbelow.

Sincethereareexternallineswithlargemomentumandsmallvirtuality,weshouldalsoconsiderloopmomentumcon gurations,wheren+kisthelargestcomponent.Thatis,wetaken+k～λnandk2～λ2mwithm<n,expandtheintegrand,anddeterminetheintegralsthatdonotvanish.2Theresultistwonon-vanishingcontributions,onefromn+k～1andk2～λ2,whichweidentifyashard-collinear,andtheotherfromn+k～1andk2～λ4,whichwecallcollinear,seeTable1.Regionswithk2<λ4donotappearduetotheinternalmassesm2～λ4.Wewillnowverify(atleadingorder)thatthesumofthethreeregionsconstructstheexpansionofourintegralI.

Thehard-collinearregion.Expandingthepropagatorssystematicallytheleadinghard-collinearintegralis

Ihc= [dk]

1

1

2p·l′2p′·lln2ln22p·l′

12 .(7)

Theexpansionhasrenderedtheintegralinfrareddivergent.Ifweperformthen kinte-grationbycontourmethods,thek⊥integralisdivergentfork⊥→0(physically,k⊥ λ)foranyn+k,butthen+kintegralconvergesat xedk⊥.Thedoublepoleoriginatesfromn±k→0,k⊥→0simultaneously.

Thecollinearregion.Inthisregionthe“lightquark”propagatorswithmomentap′ kand karecollinearandhavevirtualityoforderλ4.The“gluon”propagatorishard-collinearwithvirtualityλ2.One ndsthatthecollinearandsoftintegralsarenotwell-de nedseparatelyindimensionalregularization.Thisalsooccurredinpreviousapplicationsofthemethodofexpansionbyregionstocollinearintegrals[24],andisrelatedtothefactthedimensionalregulatorisattachedtothetransversemomentumcomponents.Ifadditionaldivergencesarisefromthen+korn kintegrations,theymaynotberegularized.Asin

[24]weintroduceanadditional“analytic”regularizationbysubstituting

1

[(k l)2]1+δ,(8)

whereνisaparameterwithmassdimensionone.Theleadingcollinearintegralis

Ic=

2 [dk][ ν2]δSomeintegralsvanishindependentofanyregularization,becauseallpoleslieinoneofthecomplexhalf-planes.Otherintegralsvanishonly,becauseweassumearegularizationthatdoesnotintroduceanadditionalscaleintotheintegral.