Dimensional Reduction applied to QCD at three loops

TL;DR

This work applies Dimensional Reduction (.9 DRED) to QCD to compute the DRbar-renormalized $\beta$-function and quark-mass anomalous dimension at three loops, accounting for $\ abla$-scalars (epsilon-scalars) and evanescent couplings $\alpha_e$ and $\eta_r$. The authors perform a direct three-loop calculation of the relevant renormalization constants and provide explicit coefficients for the DRbar $\beta$-function and the mass anomalous dimension, offering a corrected three-loop result for $\beta^{\overline{\rm DR}}_s$ and new three-loop $\gamma_m^{\overline{\rm DR}}$, while validating DR/MS equivalence through both direct computation and MS-based relations. A key finding is that misidentifying the evanescent coupling with the standard gauge coupling leads to inconsistencies in $Z_m$ and erroneous higher-order terms, resolving a discrepancy with previous literature (Bern et al.) and highlighting the importance of treating evanescent couplings distinctly. The results have practical implications for precision calculations in SUSY theories (e.g., MSSM) and for consistency checks in QCD computations that leverage SUSY Ward identities, with concrete implications for running couplings and quark masses across scales.

Abstract

Dimensional Reduction is applied to \qcd{} in order to compute various renormalization constants in the \drbar{} scheme at higher orders in perturbation theory. In particular, the $β$ function and the anomalous dimension of the quark masses are derived to three-loop order. Special emphasis is put on the proper treatment of the so-called $ε$-scalars and the additional couplings which have to be considered.