Factorization of heavy-to-light form factors in soft-colline(16)

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

Figure 4: Kinematics of an exclusive heavy-to-light transition in SCET(c,s). The heavy quark and the soft pa

rtons in the B meson must be converted into a cluster of collinear partons.[16, 17]. We will be mainly concerned with the second matching step, in which the hardcollinear modes are integrated out and the transition current is nally represented in terms of operators constructed only from soft and collinear elds. We refer to the theory of soft and collinear elds as SCET(c,s) (also called SCETII ). The kinematics of a heavy-to-light transition is illustrated in Figure 4. In contrast to[17] the invariant mass of the nal state is restricted to orderλ4～Λ2 as appropriate to an exclusive decay. This implies that the nal state must now consist only of collinear lines, since the addition of a soft line would increase the virtuality toλ2 . The initial state is described by a heavy quark and soft lines with total invariant mass near m2 . The SCET(c,s) transition current has to turn a cluster b¯ of soft modes with the quantum numbers of the B meson into a cluster of collinear lines with the quantum numbers of the nal-state meson. We begin with a brief description of the SCET(c,s) elds, gauge symmetries and Lagrangian. This theory is in many ways simpler than SCET(hc,c,s), because the dynamics of the soft-collinear transition resides only in the e ective current. We then discuss in detail the representation of the heavy-to-light current for an exclusive decay. In this section we restrict ourselves to tree-level matching. The general case will be considered in Section 4 to the extent that is necessary to prove the factorization of form factors. However, we brie y sketch the structure of transition operators and their coe cient functions beyond tree level at the end of this section.

3.1

Elements of SCET(c,s)

Fields. The SCET(c,s) Lagrangian and operators are built from a collinear light quark eldξc, a collinear gluon eld Ac, and soft light quark, heavy quark and gluon elds, denoted by qs, hv, As, respectively. As in[17] we assume that collinear elds describe particles with large momentum in the direction of the light-like vector n . n+ is another light-like vector, satisfying n n+= 2, and v will be the velocity vector labelling soft heavy quark elds. We will present our results in a general frame subject to the conditions n+ v～ 1, n v～ 1, v⊥～λ2 . The scaling of quark and gluon elds can be read o from the corresponding propagators

15

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

inmomentumspace.Forthequark eldsone nds

(22)2

The“opposite”projectionsofthefullcollinearquark eldψcandthefullsoftheavyquark eldsQv(de nedastheheavyquark eldwiththerapidvariationse imbvxremoved)areλ2suppressed,andintegratedout.Gluon eldsscaleasthecorrespondingderivatives,ξc=

n+Ac～1,A⊥c～λ2,n Ac～λ4,As～λ2.(23)n/ n/+Qv～λ3,qs～λ3.

Inderivingthis,weusedthattheintegrationmeasured4x～1/λ8,whentheintegralisoverproductsofonlycollinear eldsorproductsofonlysoft elds.Thisfollowsfromthefactthatanx-componentscalesinverselytothecorrespondingmomentumcomponent.Multipoleexpansion.Sincesoftandcollinear eldshavesigni cantvariationsoverdif-ferentlengthscalesinthen andn+directions,theyhavetobemultipole-expandedinproductsofsoftandcollinear elds.ThemultipoleexpansioninSCET(c,s)isdi erentfromthemultipoleexpansionde nedin[17],whichappliestoatheorywithhard-collinearandsoft elds,andnocollinear elds.Hereweneed

n xφs(x)=φs(x )+

2

where

x ≡n+xn

2+x⊥.(25)[n φc](x+)+...,(24)

Thecorrectiontermsinthetwoexpansionsof(24)arebothλ2suppressedrelativetotheleadingterms.

Gaugesymmetry.Thee ectivetheoryshouldbeinvariantundercollinearandsoftgaugetransformations,de nedastherestrictionofgaugefunctionsU(x)tothecorrespondingspatialvariations.Theimplementationofgaugetransformationsinthee ectivetheoryisnotunique,since eldrede nitionsorapplicationsofthe eldequationscanbeusedtoalterthegauge-transformationproperties[17,19,27].

InSCET(c,s)collinearandsoft eldsdecoupleatleadingpowerintheλexpansionaswillbeseenbelow.Furthermore,theproductofacollinearandasoft eldhashard-collinearmomentummodes,thereforegeneralsoftgaugetransformationsactingoncollinear elds(andviceversa)arenotallowed.Anaturalchoiceisthentode ne[20]

ξc→Ucξc, Ac→UcAcUc+i

Us ,Us g

.(26)