$B_c \to \eta_c$ form factors at large recoil: Interplay of soft-quark and soft-gluon dynamics

TL;DR

This paper analyzes double-logarithmic corrections in heavy-to-light transition form factors at large recoil, revealing a complex interplay of soft-gluon and soft-quark effects that influence Sudakov suppression.

Contribution

It introduces a novel coupled integral equation framework to describe the interplay of soft-gluon and soft-quark effects in form factors, extending understanding of endpoint divergences.

Findings

Double-logarithmic corrections originate from soft-gluon and soft-quark sources.

The structure of these corrections is governed by coupled integral equations.

Sudakov suppression is weakened by the interplay of soft effects.

Abstract

We perform an all-order analysis of double-logarithmic corrections to the so-called soft-overlap contribution to heavy-to-light transition form factors at large hadronic recoil. Specifically, we study $B_c \to \eta_c$ transitions within a perturbative non-relativistic framework, treating both the bottom and charm quarks as heavy with the hierarchy $m_b \gg m_c \gg\Lambda_{\rm QCD}$. Our diagrammatic analysis shows that double-logarithmic corrections arise from two distinct sources: Exponentiated soft-gluon effects described by standard Sudakov factors, and rapidity-ordered soft-quark configurations, leading to implicit integral equations, which so far have only been studied in the context of energetic muon-electron backward scattering. We find that the all-order structure of the double logarithms is governed by a novel type of coupled integral equations, which encode the non-trivial interplay between these two effects. Whereas a closed-form solution to these equations is currently unknown, we present useful iteration formulas, and derive the asymptotic behaviour of the soft-overlap form factor for infinitely large recoil energies, showing that the Sudakov suppression is somewhat weakened by the intertwined soft-quark and soft-gluon corrections. In a broader context, our findings shed light onto the physical origin and mathematical structure of endpoint divergences arising from soft-collinear factorization and the related Feynman mechanism for power-suppressed hard exclusive processes.