The three loop relation between the \bar {MS} and the pole quark masses
Kirill Melnikov, Timo van Ritbergen
TL;DR
The paper delivers an analytic three-loop relation between the MSbar mass and the pole mass, obtained by computing the three-loop on-shell mass renormalization factor $Z^{OS}_m$ with integration-by-parts and master integrals, and by combining it with the known MSbar mass renormalization. The resulting exact expression for the ratio $m_{MS}(M)/M$ is decomposed into color and flavor structures and validated against gauge invariance and established results, showing consistency with the large-$beta_0$ picture while exposing nontrivial subleading contributions. The work then applies this relation to semileptonic $B$ decays, analyzing the ${\cal O}(\alpha_s^3)$ corrections to $B \to X_u e \nu_e$ under various scenarios and demonstrating that expressing decay widths in terms of a low-scale short-distance mass (e.g., the $1S$ mass) can improve perturbative convergence for a suitable parameter window. Together, these results provide a crucial analytic ingredient for precision QCD phenomenology and motivate further exploration of alternative approximations (e.g., large-$N_c$) or a full three-loop decay calculation to resolve remaining uncertainties.
Abstract
The analytic relation between the \bar {MS} and the pole quark masses is computed to O(α_s^3) in QCD. Using this exact result, the accuracy of the large β_0 approximation is critically examined and the implications of the obtained relation for semileptonic B decays are discussed.