Factorization and the Soft Overlap Contribution to Heavy-to-Light Form Factors

TL;DR

This work delivers a complete SCET-based separation of soft-overlap and hard-scattering contributions to heavy-to-light form factors at large recoil. It shows that the soft-overlap piece, encoded in $\zeta_M(E)$, cannot be fully factorized into simple convolutions with meson LCDAs due to endpoint configurations, which are accounted for by soft-collinear messenger fields in SCET$_{\rm II}$. The authors derive a rigorous operator basis and perform RG analysis, finding that Sudakov suppression is modest and that the intermediate hard-collinear scale $\sqrt{E\Lambda}$ is not physically relevant for the soft overlap. Consequently, the soft overlap exists as a long-distance component that resists full perturbative factorization, reinforcing the role of nonperturbative inputs in heavy-to-light decays and challenging some pQCD expectations.

Abstract

Using the formalism of soft-collinear effective theory, a complete separation of short- and long-distance contributions to heavy-to-light transition form factors at large recoil is performed. The universal functions $ζ_M(E)$ parameterizing the ``soft overlap'' contribution to the form factors are defined in terms of matrix elements in the effective theory. Endpoint configurations corresponding to kinematic situations where one of the valence partons in the external mesons carries very small momentum are accounted for in terms of operators involving soft-collinear messenger fields. They contribute at leading order in $Λ_{\rm QCD}/E$ and spoil factorization. An analysis of operator mixing and renormalization-group evolution in the effective theory reveals that the intermediate scale $\sqrt{EΛ}$ is without significance to the soft functions $ζ_M(E)$, and that the soft overlap contribution does not receive a significant perturbative (Sudakov) suppression.

Figures (6)

• Figure 1: Gluon exchange contributions to heavy-to-light form factors. The flavor-changing current is denoted by a dashed line. The lines to the left belong to the $B$ meson, and those to the right belong to the light meson $M$.
• Figure 2: Diagrams relevant to the matching calculation for operators containing an extra collinear gluon.
• Figure 3: Triangle subgraph whose spectral function can be used to study endpoint singularities on the collinear side.
• Figure 4: Collinear and soft-collinear contributions to the scalar triangle graph in SCET$_{\rm II}$. Dashed (dotted) lines represent collinear (soft-collinear) propagators.
• Figure 5: An artist's view of the soft-collinear messenger contribution to the form factors. The shaded region contains soft-collinear interactions. The arrows indicate the flow of the components $n\cdot p_s$ (left) and $\bar{n}\cdot p_c$ (right) of soft and collinear momenta, which do not enter the soft-collinear block.