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

TL;DR

This work delivers a complete four-loop calculation of the $\overline{\rm MS}$–on-shell heavy-quark mass relation for a general $SU(N_c)$ gauge group, including a full colour-structure decomposition and careful treatment of master integrals via Mellin-Barnes techniques and sector decomposition with $\text{FIESTA}$. It provides both analytic inputs and highly precise numerical results, including explicit transformations for $t$, $b$, and $c$ masses and their connections to threshold masses PS, RS, and RS$'$, along with the dependence on renormalization and factorization scales. The authors demonstrate substantial cancellations and improved perturbative stability in threshold-mass relations, yielding MeV-scale four-loop corrections for realistic quark masses, and present detailed $z_m^{(4)}$ and $c_m^{(4)}$ coefficients that form a general framework for precision heavy-quark mass determinations. Overall, the paper advances the precision frontier in heavy-quark mass theory and provides a versatile methodology applicable to arbitrary $SU(N_c)$ gauge groups.

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.

Figures (10)

• Figure 1: Sample Feynman diagrams contributing to $\Sigma_S$ and $\Sigma_V$ at one-, two-, three- and four-loop order. The solid lines represent quarks and the curly lines gluons.
• Figure 2: Four-loop prototype families needed to generate the four-loop on-shell integral families shown in Appendix \ref{['app::intfam']}.
• Figure 3: FIESTA results for three typical integrals for various choices of $N$. The corresponding master integrals are shown to the left of the plots (see caption of Fig. \ref{['fig::MB_BB']} for the meaning of the lines). In this plot the FIESTA uncertainties have been multiplied by a factor ten. For each $\epsilon$ coefficient on the $x$ axis results for different numbers of sampling points, $N$, are shown. For all plots we show results for $N=5\times 10^k$ with $k=5,6,7,8$. The bottom plot also contains results for $N=2\times 10^9$. In each case we normalize the results to the most precise one and then subtract 1.
• Figure 5: Sample master integrals which are treated with the Mellin-Barnes method. The dimension of the Mellin-Barnes integration is specified below the diagrams.
• Figure 6: $n_l$-dependence of $z_m^{(4)}(M)$.