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

TL;DR

This paper computes the quark-mass conversion factor between the $\ overline{\mathrm{MS}}$ scheme and the RI/mSMOM scheme up to three loops in QCD for nonzero quark masses, enabling precise matching relevant to Lattice QCD determinations of $\overline{m}_R$. The authors perform a massive off-shell three-loop calculation using differential equations for 1115 master integrals, validating the result against the chiral (SMOM) limit and RG constraints. They demonstrate a $m_s$-dependent window where $C_m$ receives smaller perturbative corrections than the SMOM case, which can reduce systematic uncertainties in $\overline{m}_c$ extractions. Additionally, they explore a weaker interpretation of DR-based mSMOM conditions that yields the same finite $C_m$ with simpler renormalization constants, potentially easing higher-loop computations and singlet-diagram handling, with broad implications for lattice-continuum matching and future extensions.

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.

Figures (2)

• Figure 1: Representative Feynman diagrams at three loops
• Figure 2: Numerical results for $\mathrm{C}_m$ as functions in $m_s \equiv m^2_R/\mu^2_s$ up to 3-loop order evaluated in QCD with $n_l=3$ at $\bar{\mu}=\mu_s=2$ GeV in Landau gauge