Three-loop QCD Mass Relation between the $\overline{\mathrm{MS}}$ and Symmetric-momentum Subtraction Scheme Away from the Chiral Limit

TL;DR

This paper derives the three-loop perturbative relation for quark mass conversion between the $ar{MS}$ scheme and RI/SMOM scheme away from the chiral limit, revealing new insights into renormalization conditions and potential improvements in lattice QCD mass determinations.

Contribution

It extends the three-loop quark-mass conversion calculation to non-zero masses and introduces a novel interpretation of RI/SMOM conditions in dimensional regularization.

Findings

High-order conversion factors show reduced corrections in certain scale windows.

The reinterpretation simplifies high-order computations.

Results can improve the precision of lattice QCD quark-mass determinations.

Abstract

The perturbative result for the quark-mass conversion factor between the $\overline{\mathrm{MS}}$ and regularization-independent symmetric-momentum subtraction scheme (RI/SMOM) away from the chiral limit, i.e. at non-zero quark masses (RI/mSMOM), is derived up to three loops in QCD, extending the existing result by two additional orders. We further explore an illuminating possibility that in Dimensional Regularization, the original RI/(m)SMOM renormalization conditions may be interpreted merely in a weaker sense, namely as equations holding just in the 4-dimensional limit rather than exactly in $d$ dimensions: they result in different, albeit simpler, renormalization constants but still the same finite conversion factor. This novel observation has the added benefit of reducing computational effort, particularly at high orders. Our high-order results for the conversion factor exhibit rich behaviors, and in particular a window is observed in the subtraction scale and mass where it receives less perturbative corrections than the RI/SMOM counterpart up to three loops; this finding may help to further improve the accuracy of $\overline{\mathrm{MS}}$ quark-mass determinations with Lattice QCD.