Soft-overlap contribution to $B_c \to \eta_c$ form factors: diagrammatic resummation of double logarithms

TL;DR

This paper employs diagrammatic resummation to analyze the soft-overlap contribution to $B_c o a_c$ form factors, addressing endpoint singularities and deriving all-order double logarithm resummation equations in a simplified non-relativistic framework.

Contribution

It introduces a novel diagrammatic resummation method for double logarithms in $B_c o a_c$ form factors, elucidating the all-order structure and addressing endpoint singularities.

Findings

Double logarithms originate from soft-quark endpoint and soft-gluon corrections.

Resummation involves new integral equations for systematic all-order analysis.

Current results include all double logs in the Abelian limit.

Abstract

Using diagrammatic resummation techniques, we investigate the double-logarithmic series of the "soft-overlap" contribution to $B_c \to \eta_c$ transition form factors at large hadronic recoil, assuming the scale hierarchy $m_b \gg m_c \gg \Lambda_{\rm QCD}$. In this case, the hadronic bound states can be treated in the non-relativistic approximation and the relevant hadronic matrix elements can be computed perturbatively. This setup defines one of the simplest examples to study the problem of endpoint singularities appearing in the factorization of exclusive $B$-decay amplitudes. We find that the leading double logarithms arise from a peculiar interplay of soft-quark "endpoint logarithms" from ladder diagrams with energy-ordered spectator-quark propagators, as well as standard Sudakov-type soft-gluon corrections. We elucidate the all-order systematics, and show that their resummation proceeds via a novel type of integral equations. The current status of the calculation, which includes all double logarithms in the Abelian limit, is reported.