Factorization of heavy-to-light form factors in soft-colline(19)

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

butlessexplicitlyinSection4.NotethatintheintermediateSCET(hc,c,s),whereonlyhardmodesareintegratedout,thepower-countingforhadronicmatrixelementsisnotexplicit.Inparticularin[17]thepionstateincludedsoft-collinearinteractionsandthe¯hcΓhv(withξhcdenotingthehard-collinearquarksuppressionofthematrixelementofξ

eld)withthesepionstateswasnotdetermined(seethediscussionattheendofSection

5.3of[17]andin[20]).

Light-conegaugeandWilsonlines.Itwillsometimesbeconvenient–especiallyforthefollowingtree-levelmatchingofSCET(hc,c,s)toSCET(c,s)–tochoosethegauge

n+Ahc=n+Ac=n As=0.(33)

Theusefulnessofn+Ahc=n+Ac=0gaugefollowsfromthefactthatSCET(hc,c,s)isnon-localonlyduetothepresenceofWilsonlinesinthedirectionofn+,andduetotheappearanceof(in+ ) 1.Withn+Ahc=n+Ac=0allWilsonlinesreduceto1,andthereareno eldsoforder1.Theusefulnessofn As=0gaugeisrelatedtothefactthatinSCET(hc,c,s)softgluonsdecouplefromcollinearandhard-collineargluonsatleadingorderinλinthisgauge.

Onceaparticularresulthasbeenderivedinthisgauge,thecollinearandsoftgaugeinvarianceisrecoveredbytransformingthe eldsbacktoageneralgaugeusingthegaugetransformations(26).Asexplainedin[19],thetransformationmatricesUcandUsthataccomplishthisarethelight-likeWilsonlines

Uc(x)=Wc(x)=Pexpig

Us(x)=Ys (x)=Pexpig 0

∞

∞dsn+Ac(x+sn+),

0dtn As(x+tn ),(34)

andthecorrespondinggaugetransformationofthe eldscanbewrittenas

ξc→Wc ξc,

hv→Ys hv,gAc→Wc [iDcWc]≡Ac,qs→Ys qs,gAs→Ys [iDsYs]≡As.(35)

(Hereandinthefollowingderivativesinsquarebracketsactonlyontheexpressiontotheirrightinsidethebracket.)BecausetheWilsonlinestransformas

cYs,Ys→UsUsYs,Ys→UcUcWc,Wc→UsWc,Wc→U(36)

theexpressionsontheright-handsideof(35)aregauge-singlets.The eldsAc,Ashavebeenintroducedin[20]asbuildingblocksformanifestlygauge-invariantoperators.

InageneralgaugetheWilsonlinesemergeautomaticallyfrommatchinganin nitesetofunsuppressedtree-levelFeynmandiagramswithattachmentsofn+Actosoft elds,andn Asgluonstocollinear eldsassketchedinFigure5.Indeed,atleadingpowerthese