The relation between the $\bar{\rm MS}$ and the on-shell quark mass at order $\alpha_s^3$

TL;DR

This paper computes the three-loop relation between the on-shell and bar;MS; quark masses using advanced approximation techniques, reducing theoretical uncertainties in quark mass determination.

Contribution

It provides a novel numerical approximation for the bar;MS; and on-shell quark mass relation at order _s^3, employing conformal mapping and Pade9 methods.

Findings

Achieved an error estimate below 3% in the mass relation.

Significantly reduces theoretical uncertainty in quark mass measurements.

Discussed implications for top and bottom quark production near threshold.

Abstract

The relation between the on-shell and $\bar{\rm MS}$ mass can be expressed through scalar and vector part of the quark propagator. In principle these two-point functions have to be evaluated on-shell which is a non-trivial task at three-loop order. Instead, we evaluate the quark self energy in the limit of large and small external momentum and use conformal mapping in combination with Pad\'e improvement in order to construct a numerical approximation for the relation [1]. The errors of our final result are conservatively estimated to be below 3%. The numerical implications of the results are discussed in particular in view of top and bottom quark production near threshold. We show that the knowledge of new ${\cal O}(\alpha_s^3)$ correction leads to a significant reduction of the theoretical uncertainty in the determination of the quark masses.