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

TL;DR

This paper computes the three-loop (${\cal O}(\alpha_s^3)$) relation between the MSbar quark mass and the on-shell (pole) mass by analyzing the quark self-energy in both small- and large-momentum limits. A conformal mapping and Padé approximation are used to construct a numerical interpolation for the mass relation, with rigorous error estimates. The results, validated against known lower-order coefficients, reduce theoretical uncertainties in quark mass determinations and have direct implications for top- and bottom-quark threshold phenomena and the use of short-distance mass schemes. The methodology also yields detailed information about the perturbative quark propagator in QCD, potentially applicable to broader momentum regimes and gauge choices.

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 Padé 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 ${\cal O}(α_s^3)$ correction leads to a significant reduction of the theoretical uncertainty in the determination of the quark masses.

Figures (6)

• Figure 1: Sample diagrams contributing to the fermion propagator. The fermions in the closed loops may either also carry mass $M$ or considered to be massless.
• Figure 2: $z$ dependence of $-g(z)$ for the structures $F$, $FF$ and $FA$ for different values of $\xi$. The solid curves belong to $\xi=-2$. The dashed curves correspond to $\xi=-5,0,+2,+5$ (from short to long dashes).
• Figure 3: Generic three-loop bubble diagram. In general each line may be massless or carry mass $M$.
• Figure 4: Master diagrams corresponding to the expression in Eq. (\ref{['eq:master1']}). The full lines carry mass $M$ whereas the dashed ones are massless.
• Figure 5: Two- and three-loop master diagrams corresponding to the expressions in Eqs. (\ref{['eq:master2']}) and (\ref{['eq:master2l']}). The full lines carry mass $M$ whereas the dashed ones are massless.