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

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.