Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence

TL;DR

This paper analytically computes all relevant two-loop planar master integrals for Higgs decay into three partons, incorporating full heavy-quark mass dependence, essential for precise NNLO and NLO Higgs production predictions.

Contribution

It provides the first fully analytic calculation of four-point multiscale Feynman integrals involving elliptic functions with full mass dependence.

Findings

Analytic expressions for all planar master integrals with heavy-quark mass.

Introduction of a parametrization simplifying elliptic integral solutions.

Expressions suitable for numerical evaluation of complex integrals.

Abstract

We present the analytic computation of all the planar master integrals which contribute to the two-loop scattering amplitudes for Higgs$\to 3$ partons, with full heavy-quark mass dependence. These are relevant for the NNLO corrections to fully inclusive Higgs production and to the NLO corrections to Higgs production in association with a jet, in the full theory. The computation is performed using the differential equations method. Whenever possible, a basis of master integrals that are pure functions of uniform weight is used. The result is expressed in terms of one-fold integrals of polylogarithms and elementary functions up to transcendental weight four. Two integral sectors are expressed in terms of elliptic functions. We show that by introducing a one-dimensional parametrization of the integrals the relevant second order differential equation can be readily solved, and the solution can be expressed to all orders of the dimensional regularization parameter in terms of iterated integrals over elliptic kernels. We express the result for the elliptic sectors in terms of two and three-fold iterated integrals, which we find suitable for numerical evaluations. This is the first time that four-point multiscale Feynman integrals have been computed in a fully analytic way in terms of elliptic functions.