On the relation between the $\overline{\mathrm{MS}}$ and the kinetic mass of heavy quarks

TL;DR

This paper calculates the precise three-loop relation between the pole, $ m MS$, and kinetic masses of heavy quarks, improving accuracy for charm and bottom quarks, which is vital for CKM matrix element extraction.

Contribution

It provides the first three-loop conversion formulas between the $ m MS$ and kinetic masses of heavy quarks, including finite charm mass effects and a four-loop large-$eta_0$ estimate.

Findings

Achieved three-loop accuracy in mass conversion relations.

Estimated a 15 MeV uncertainty in bottom quark mass conversion.

Enhanced precision aids CKM matrix element determination at Belle II.

Abstract

We compute the relation between the pole mass and the kinetic mass of a heavy quark to three loops. Using the known relation between the pole and the $\overline{\rm MS}$ mass we obtain precise conversion relations between the $\overline{\rm MS}$ and kinetic masses. The kinetic mass is defined via the moments of the spectral function for the scattering involving a heavy quark close to threshold. This requires the computation of the imaginary part of a forward scattering amplitude up to three-loop order. We discuss in detail the expansion procedure and the reduction to master integrals. For the latter analytic results are provided. We apply our result both to charm and bottom quark masses. In the latter case we compute and include finite charm quark mass effects. Furthermore, we determine the large-$\beta_0$ result for the conversion formula at four-loop order. For the bottom quark we estimate the uncertainty in the conversion between the $\overline{\rm MS}$ and kinetic masses to about 15 MeV which is an improvement by a factor two to three as compared to the two-loop formula. The improved precision is crucial for the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ at Belle II.