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

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 spacetime 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 MS scheme requires the calculation of corresponding conversion factors. The latter can only be computed in perturbation theory, by the very nature of MS ; however, the computations are greatly simplified by virtue of the following attributes: (i) In the absence 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 (or in a general gauge, to verify that the result is gauge independent). (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 ψ Γ ψ 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 MS scheme.