Factorization of heavy-to-light form factors in soft-collinear effective theory

TL;DR

The paper addresses the challenge of factorizing heavy-to-light form factors at large recoil in exclusive B decays. By formulating and matching two soft-collinear effective theories (SCET(hc,c,s) and SCET(c,s)), it derives a factorization theorem F_i = C_i ξ_π + φ_B ⊗ T_i ⊗ φ_π, where a universal soft form factor ξ_π absorbs hard-collinear, collinear, and soft effects, and the remainder is a convergent hard-scattering contribution. The authors show that naive soft-collinear factorization encounters endpoint divergences that can be controlled by the ξ_π definition and by careful handling of convolutions with light-cone distribution amplitudes, enabling all-orders perturbative factorization at leading power in 1/m_b. This work clarifies the structure of endpoint divergences, contrasts with previous treatments, and lays groundwork for resummation and reliable phenomenology of B → π and related heavy-to-light processes.

Abstract

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. In this paper we extend our previous investigation of heavy-to-light currents in soft-collinear effective theory to final states with invariant mass Lambda^2 as is appropriate to exclusive B meson decays. The effective theory contains soft modes and two collinear modes with virtualities of order m_b*Lambda (`hard-collinear') and Lambda^2. Integrating out the hard-collinear modes results in the hard spectator-scattering contributions to exclusive B decays. We discuss the representation of heavy-to-light currents in the effective theory after integrating out the hard-collinear scale, and show that the previously conjectured factorization formula is valid to all orders in perturbation theory. The naive factorization of matrix elements in the effective theory into collinear and soft matrix elements may be invalidated by divergences in convolution integrals. In the factorization proof we circumvent the explicit regularization of endpoint divergences by a definition of the universal form factors that includes hard-collinear, collinear and soft effects.

Figures (6)

• Figure 1: Photon vertex correction to $\bar{B}\to\gamma l\nu$. $\Gamma$ denotes the weak $b\to u$ decay vertex. We consider the corresponding vertex integral with all lines simplified to scalar propagators and all vertex factors set to 1.
• Figure 2: Divergence structure of the integral (\ref{['defI']}) and its off-shell, massless version when expanded by regions. The arrows indicate the divergences in different regions that are related and cancel each other. The dashed lines mark the various factorization steps.
• Figure 3: Diagrammatic and operator/matrix element representation of the hard-collinear (left column), soft (middle column) and collinear contribution to the diagram of Figure \ref{['fig:vertex']}. Each column shows: the original diagram with the hard-collinear subgraph marked by bold-face lines (upper row), and with the dashed line indicating where the graph factorizes into a short-distance and long-distance subgraph; the operator vertex in the effective theory corresponding to the contracted hard-collinear subgraph (middle row); the contribution to the operator matrix element $\langle \gamma|O_i|\bar{q} b\rangle$ corresponding to the original diagram (lower row).
• Figure 4: Kinematics of an exclusive heavy-to-light transition in $\hbox{SCET(c,s)}$. The heavy quark and the soft partons in the $B$ meson must be converted into a cluster of collinear partons.
• Figure 5: Infinite sets of Feynman graphs with attachments of soft gluons to collinear quarks and vice-versa. Integrating out the intermediate hard-collinear propagators leads to the Wilson lines $Y_s^\dagger$ and $W_c$, respectively, see (\ref{['eq:Wilson']}).