Factorization of heavy-to-light form factors in soft-colline(19)
时间:2025-04-03
时间:2025-04-03
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
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