$\overline{\rm MS}$-on-shell quark mass relation up to four loops in QCD and a general SU$(N)$ gauge group

TL;DR

This paper calculates the four-loop relation between the on-shell and MS-bar heavy quark masses in QCD for a general SU(N) gauge group, providing precise numerical results and updated mass relations.

Contribution

It presents the first four-loop computation of the heavy quark mass relation in QCD for a general SU(N) gauge group, including detailed numerical evaluations of master integrals.

Findings

Four-loop on-shell to MS-bar quark mass relation computed.

Updated mass relations for charm, bottom, and top quarks provided.

Numerical methods for master integrals discussed in detail.

Abstract

In this paper we compute the relation between heavy quark masses defined in the modified minimal subtraction and on-shell scheme. Detailed results are presented for all coefficients of the SU$(N_c)$ colour factors. The reduction of the four-loop on-shell integrals is performed for a general QCD gauge parameter. Some of the about 380 master integrals are computed analytically, others with high numerical precision based on Mellin-Barnes representations, and the rest numerically with the help of {\tt FIESTA}. We discuss in detail the precise numerical evaluation of the four-loop master integrals. Updated relations between various short-distance masses and the $\overline{\rm MS}$ quark mass to next-to-next-to-next-to-leading order accuracy are provided for the charm, bottom and top quark. We discuss the dependence on the renormalization and factorization scale.