Relation between the pole and the minimally subtracted mass in dimensional regularization and dimensional reduction to three-loop order

TL;DR

This work derives the three-loop relation between the pole quark mass and the minimally subtracted DR-bar mass in QCD, carefully incorporating dimensional reduction with evanescent couplings and epsilon-scalar mass renormalization. It provides a detailed analytic construction, including the on-shell counterterms in DREG for cross-checks ($Z_m^{\rm OS}$ and $Z_2^{\rm OS}$) and the finite ratio $z_m^{\rm OS,\overline{\rm DR}}$ up to three loops, with extensive color-structure decompositions and master-integral reductions. The main result furnishes the explicit three-loop DR-bar mass relation, validates it against known limits (e.g., $\alpha_e=\alpha_s$ and $n_h=0$), and illustrates the numerical size of the three-loop corrections for bottom and top quarks. The findings are especially pertinent for supersymmetric contexts, where DR-based regularization is favored, and provide a groundwork for precise mass conversions in beyond-Standard-Model theories.

Abstract

We compute the relation between the pole quark mass and the minimally subtracted quark mass in the framework of QCD applying dimensional reduction as a regularization scheme. Special emphasis is put on the evanescent couplings and the renormalization of the epsilon-scalar mass. As a by-product we obtain the three-loop on-shell renormalization constants Zm(OS) and Z2(OS) in dimensional regularization and thus provide the first independent check of the analytical results computed several years ago.

Figures (4)

• Figure 1: Sample three-loop diagrams. Solid lines denote massive quarks with mass $m_q$ and curly lines denote gluons. In the closed fermion loops all quark flavours have to be considered.
• Figure 2: Three-loop master integrals. Solid lines denote massive and dashed lines are massless scalar propagators.
• Figure 3: One- and two-loop Feynman diagrams contributing to the $\varepsilon$-scalar propagator. Dashed lines denote $\varepsilon$ scalars, curly lines denote gluons and solid lines denote massive quarks with mass $m_q$.
• Figure 4: Sample three-loop diagrams contributing to the quark propagator which have to be considered additionally in case dimensional reduction is used for the regularization. Solid lines denote massive quarks with mass $m_q$, curly lines denote gluons and the $\varepsilon$ scalars are represented by dashed lines. In the closed fermion loops all quark flavours have to be considered.