A semi-numerical method for one-scale problems applied to the $\overline{\rm MS}$-on-shell relation

TL;DR

This paper introduces a semi-numerical method for calculating master integrals in one-scale problems, demonstrated through four-loop QCD corrections relating different heavy quark mass schemes.

Contribution

The paper presents a novel semi-numerical approach for computing master integrals in one-scale problems, specifically applied to complex four-loop QCD corrections.

Findings

Effective computation of four-loop QCD corrections

Improved understanding of heavy quark mass relations

Validation of semi-numerical method in high-order calculations

Abstract

We discuss a practical approach to compute master integrals entering physical quantities which depend on one parameter. As an example we consider four-loop QCD corrections to the relation between a heavy quark mass defined in the $\overline{\rm MS}$ and on-shell scheme in the presence of a second heavy quark.