# Two-loop evolution equations for flavor-singlet light-ray operators

## TL;DR

This paper derives two-loop evolution equations for flavor-singlet light-ray operators in QCD, confirming previous results and enhancing understanding of operator mixing and conformal symmetry in quantum field theory.

## Contribution

It introduces a novel method leveraging conformal invariance in non-integer dimensions to compute two-loop evolution kernels for flavor-singlet operators.

## Key findings

- Calculated two-loop evolution kernels for flavor-singlet operators.
- Confirmed previous evolution equations for generalized parton distributions.
- Demonstrated the utility of conformal symmetry in higher-order QCD calculations.

## Abstract

QCD in non-integer $d=4-2\epsilon$ space-time dimensions enjoys conformal invariance at the special fine-tuned value of the coupling. Counterterms for composite operators in minimal subtraction schemes do not depend on $\epsilon$ by construction, and therefore the renormalization group equations for composite operators in physical (integer) dimensions inherit conformal symmetry. This observation can be used to restore the complete evolution kernels that take into account mixing with the operators containing total derivatives from their eigenvalues (anomalous dimensions). Using this approach we calculate the two-loop (NLO) evolution kernels for the leading twist flavor-singlet operators in the position space (light-ray operator) representation. As the main result of phenomenological relevance, in this way we are able to confirm the evolution equations of flavor-singlet generalized hadron parton distributions derived earlier by Belitsky and M\"uller using a different approach.