On evolution kernels of twist-two operators

TL;DR

This paper constructs a transformation to restore conformal symmetry in evolution kernels of twist-two operators beyond one loop, enabling the derivation of invariant kernels from anomalous dimensions with applications in QCD and N=4 SYM.

Contribution

It introduces a method to restore conformal invariance in evolution kernels at higher loops, linking eigenvalues to anomalous dimensions and providing explicit NNLO kernels.

Findings

Derived invariant kernels for twist-two operators in QCD at NNLO.

Established a method to reconstruct kernels from anomalous dimensions.

Applied the method to N=4 SYM for planar anomalous dimensions.

Abstract

The evolution kernels that govern the scale dependence of the generalized parton distributions are invariant under transformations of the $\mathrm{SL}(2,\mathrm R)$ collinear subgroup of the conformal group. Beyond one loop the symmetry generators, due to quantum effects, differ from the canonical ones. We construct the transformation which brings the {\it full} symmetry generators back to their canonical form and show that the eigenvalues (anomalous dimensions) of the new, canonically invariant, evolution kernel coincide with the so-called parity respecting anomalous dimensions. We develop an efficient method that allows one to restore an invariant kernel from the corresponding anomalous dimensions. As an example, the explicit expressions for NNLO invariant kernels for the twist two flavor-nonsinglet operators in QCD and for the planar part of the universal anomalous dimension in $ N=4$ SYM are presented.