Three loop anomalous dimension of non-singlet quark currents in the RI' scheme

TL;DR

The paper performs a full three-loop renormalization of QCD in the RI$^ extprime$ scheme in arbitrary covariant gauge and demonstrates that the RI$^ extprime$ $eta$-function matches the MSbar result to the relevant order, implying equivalence at four loops. It then determines RI$^ extprime$ anomalous dimensions for the scalar, vector, and tensor quark currents via operator insertions in quark two-point functions, and provides explicit conversion functions to MSbar. The authors reproduce known MSbar results in appropriate gauges, verify gauge-parameter consistency, and extend the framework to the tensor current, offering a path for lattice-QCD matching and future exploration of additional operators. These results supply the necessary renormalization and conversion machinery to relate lattice RI$^ extprime$ data to continuum MSbar predictions with high precision. The work therefore strengthens the bridge between lattice calculations and continuum perturbation theory for important quark-current observables, including the quark mass and transversity-related operators.

Abstract

We renormalize QCD at three loops in the modified regularization invariant, RI', scheme in arbitrary covariant gauge and deduce that the four loop beta-function is equivalent to the MSbar result. The anomalous dimensions of the scalar, vector and tensor currents are then determined in the RI' scheme at three loops by considering the insertion of the operator in a quark two-point function. The expression for the scalar current agrees with the quark mass anomalous dimension and we deduce an expression for the four loop RI' mass anomalous dimension in arbitrary covariant gauge and for any Lie group.