Resummation of rapidity logarithms in $B$ meson wave functions

TL;DR

This paper develops a resummation method for rapidity logarithms in $B$ meson wave functions within the $k_T$ factorization framework, improving the wave functions and affecting $B\to\pi$ transition form factors.

Contribution

It introduces a novel evolution equation that sums rapidity logarithms for $B$ meson wave functions, incorporating heavy-quark effects and renormalization-group evolution.

Findings

Resummation maintains wave function normalization.

Resummation enhances convergence at small spectator momentum.

Leads to ~25% reduction in $B\to\pi$ form factors at large recoil.

Abstract

We construct an evolution equation for the $B$ meson wave functions in the $k_T$ factorization theorem, whose solutions sum the double logarithms associated with the light-cone singularities, namely, the rapidity logarithms. The derivation is subtler than that of the Sudakov resummation for an energetic light hadron, due to the involvement of the effective heavy-quark field. The renormalization-group evolution in the factorization scale needs to be included in order to derive an ultraviolet-finite and scale-invariant kernel for resumming the rapidity logarithms. It is observed that this kernel is similar to that of the joint resummation for QCD processes in extreme kinematic regions, which combines the threshold and $k_T$ resummations. We show that the resummation effect maintains the normalization of the $B$ meson wave functions, and strengths their convergent behavior at small spectator momentum. The resummation improved $B$ meson wave functions are then employed in the leading-order analysis of the $B\to\pi$ transition form factors, which lead to approximately 25% deduction in the large recoil region.