Heavy-to-light form factors to three loops

We compute three-loop corrections of $𝒪⁡(𝛼^3_𝑠)$ 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 semianalytic results for the form factors. We implement our results in a Mathematica and a 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 lightlike momentum transfer are used to extract the hard function in $¯B → X_sγ$ to three loops.