Gauge-invariant Renormalization Scheme in QCD: Application to fermion bilinears and the energy-momentum tensor

TL;DR

This paper introduces a gauge-invariant, nonperturbative renormalization scheme for lattice QCD operators, simplifying calculations and reducing operator mixing, and demonstrates its application to fermion bilinears and the energy-momentum tensor.

Contribution

It proposes a novel gauge-invariant, mass-independent renormalization scheme based on coordinate space Green's functions, facilitating nonperturbative renormalization and simplified perturbative conversions.

Findings

Computed lowest-order conversion factors for fermion bilinear operators.

Analyzed mixing of gluon and quark energy-momentum tensors at one loop.

Scheme reduces operator mixing complexity in lattice QCD calculations.

Abstract

We consider a gauge-invariant, mass-independent prescription for renormalizing composite operators, regularized on the lattice, in the spirit of the coordinate space (X-space) renormalization scheme. The prescription involves only Green's functions of products of gauge-invariant operators, situated at distinct space-time points, in a way as to avoid potential contact singularities. Such Green's functions can be computed nonperturbatively in numerical simulations, with no need to fix a gauge: thus, renormalization to this "intermediate" scheme can be carried out in a completely nonperturbative manner. Expressing renormalized operators in the $\overline{\rm MS}$ scheme requires the calculation of corresponding conversion factors. The latter can only be computed in perturbation theory, by the very nature of the $\overline{\rm MS}$; however, the computations are greatly simplified by virtue of the following attributes: i) In the absense of operator mixing, they involve only massless, two-point functions; such quantities are calculable to very high perturbative order. ii) They are gauge invariant; thus, they may be computed in a convenient gauge. iii) Where operator mixing may occur, only gauge-invariant operators will appear in the mixing pattern: Unlike other schemes, involving mixing with gauge-variant operators (which may contain ghost fields), the mixing matrices in the present scheme are greatly reduced. Still, computation of some three-point functions may not be altogether avoidable. We exemplify the procedure by computing, to lowest order, the conversion factors for fermion bilinear operators of the form $\bar\psi\Gamma\psi$ in QCD. We also employ the gauge-invariant scheme in the study of mixing between gluon and quark energy-momentum tensor operators: We compute to one loop the conversion factors relating the nonperturbative mixing matrix to the $\overline{\rm MS}$ scheme.