Renormalization of gluonic leading-twist Operators in covariant Gauges

TL;DR

This paper develops an all-loop framework for renormalizing gauge-variant twist-two operators in Yang-Mills theory, utilizing a generalized gauge symmetry and BRST formalism, and applies it to compute high-loop Mellin moments of gluonic splitting functions.

Contribution

It introduces a comprehensive all-loop structure for gauge-variant operators in covariant gauges, employing a generalized BRST symmetry and background field formalism for high-order calculations.

Findings

Explicit basis for operators up to 4-loop order

Derived relations among anomalous dimension matrices

Computed Mellin moments of gluonic splitting functions up to 4 loops

Abstract

We provide the all-loop structure of gauge-variant operators required for the renormalisation of Green's functions with insertions of twist-two operators in Yang-Mills theory. Using this structure we work out an explicit basis valid up to 4-loop order for an arbitrary compact simple gauge group. To achieve this we employ a generalised gauge symmetry, originally proposed by Dixon and Taylor, which arises after adding to the Yang-Mills Lagrangian also operators proportional to its equation of motion. Promoting this symmetry to a generalised BRST symmetry allows to generate the ghost operator from a single exact operator in the BRST-generalised sense. We show that our construction complies with the theorems by Joglekar and Lee. We further establish the existence of a generalised anti-BRST symmetry which we employ to derive non-trivial relations among the anomalous dimension matrices of ghost and equation-of-motion operators. For the purpose of demonstration we employ the formalism to compute the N=2,4 Mellin moments of the gluonic splitting function up to 4 loops and its N=6 Mellin moment up to 3 loops, where we also take advantage of additional simplifications of the background field formalism.