Heavy-to-light form factors to three loops

TL;DR

This paper presents the calculation of three-loop corrections to form factors involving one massive and one massless quark for various currents, providing analytic and semi-analytic results useful for precision QCD predictions.

Contribution

It provides the first complete three-loop corrections to these form factors, including analytic results for color-planar and specific loop contributions, with implementations for numerical evaluation.

Findings

Analytic results for color-planar contributions

Semi-analytic form factors using expand and match

Applications to soft-collinear effective theory and B decay processes

Abstract

We compute three-loop corrections of $\mathcal{O}(\alpha_{s}^3)$ to form factors with one massive and one massless quark coupling to an external vector, axialvector, scalar, pseudoscalar, or tensor current. We obtain analytic results for the color-planar contributions, for the contributions of light-quark loops, and the contributions with two heavy-quark loops. For the computation of the remaining master integrals we use the "expand and match" approach which leads to semi-analytic results for the form factors. We implement our results in a {\tt Mathematica} and a {\tt Fortran} code which allows for fast and precise numerical evaluations in the physically relevant phase space. The form factors are used to compute the hard matching coefficients in Soft-Collinear Effective Theory for all currents. The tensor coefficients at light-like momentum transfer are used to extract the hard function in $\bar B \to X_s \gamma$ to three loops.