Master integrals for the two-loop penguin contribution in non-leptonic B-decays

TL;DR

This paper provides a complete analytic evaluation of the master integrals needed for the two-loop penguin contributions to non-leptonic $B$-decays by employing differential equations in a canonical basis. It introduces and leverages generalized harmonic polylogarithms with rational weights to express the results, including the first application to a two-mass, multi-scale setting. The authors present explicit analytic results for 29 master integrals across multiple topologies, supported by boundary conditions, cross-checks, and alternative representations, enabling precise predictions for the penguin kernels. The work thus enables fully analytic expressions for the hard-scattering kernels $T_i^I$ and has potential applications to inclusive and exclusive $B$-decay observables, where charm-threshold effects can be treated analytically. The methods and results pave the way for broader multi-scale multi-loop calculations in heavy-flavor phenomenology.

Abstract

We compute the master integrals that arise in the calculation of the leading penguin amplitudes in non-leptonic B-decays at two-loop order. The application of differential equations in a canonical basis enables us to give analytic results for all master integrals in terms of iterated integrals with rational weight functions. It is the first application of this method to the case of two different internal masses.