Gauge-invariant renormalization scheme in QCD: Application to fermion bilinears and the energy-momentum tensor

• Published in: Physical review. D 2021-05, Vol.103 (9), p.1, Article 094509
• Main Authors: Costa, M., Karpasitis, I., Pafitis, T., Panagopoulos, G., Panagopoulos, H., Skouroupathis, A., Spanoudes, G.
• Format: Article
• Language: English
• Subjects: Computation; Contact potentials; Conversion; Fermions; Gluons; Green's functions; Invariants; Mathematical analysis; Momentum; Operators (mathematics); Perturbation theory; Quantum chromodynamics; Tensors
• ISSN: 2470-0010; 2470-0029
• DOI: 10.1103/PhysRevD.103.094509

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 ¯ψΓψ 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.