Renormalization of gluonic leading-twist operators in covariant gauges

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.

Renormalization of twist-two operators in QCD and its application to singlet splitting functions

“…Most recently, Falcioni and Herzog [39,40] revisited the conditions formulated by Joglekar and Lee [38] and established a framework based on BRST symmetry, that allows inferring the gaugevariant operators and their associated renormalization factors at fixed 𝑛 order-by-order in the loop expansion and the strong coupling constant. They demonstrated their method with the derivation of the three-loop anomalous dimensions for 𝑛 = 6 and of the four-loop anomalous dimensions for 𝑛 up to 4.…”

Renormalisation of singlet operators to four loops

“…The coefficients ; 1 ... 22)-( 24) are further expanded into free coupling constants 1 ... and colour factors. The latter correspond to the colour structures of the + 1-point gluonic subdiagrams that appear in the OMEs [42]. We get…”

“…Recent results in this direction were discussed in detail during this conference [40,41]. Here we present the work [42], which provides the general construction of aliens in pure Yang Mills theory. This covers to the most complicated pattern of operator mixing and generalises to the construction of [33], extending it to all order.…”

“…The main point of ref. [42] is that we can extend the Yang Mills Lagrangian by including the gauge invariant operator defined in eq. ( 6) and the alien operators that are required to maintain invariance under a generalised version of the BRST transformations.…”

“…Renormalisation of singlet operators to four loops Giulio Falcioni where ; 1 ... 1 ... are coefficients constrained only by colour conservation [42]. In turn, O ( ) EOM reads…”