Revisiting evolution equations for generalised parton distributions

TL;DR

This paper revisits and reformulates the evolution equations for generalized parton distributions (GPDs) in momentum space, providing a new open-source numerical implementation suitable for comprehensive kinematic analysis and future high-precision studies.

Contribution

It introduces a novel formulation of GPD evolution kernels at one-loop in pQCD, enabling accurate numerical evolution across the entire kinematic region, including heavy-quark thresholds, with open-source tools.

Findings

The implementation reproduces DGLAP and ERBL limits accurately.

It ensures GPD continuity and polynomiality.

Benchmarking shows good agreement with analytical solutions.

Abstract

We revisit the evolution of generalised parton distributions (GPDs) in momentum space. We formulate the evolution kernels at one-loop in perturbative QCD (pQCD) in a form suitable for numerical implementation and that allows for an accurate study of their properties. This leads to the first open-source implementation of GPD evolution equations able to cover the entire kinematic region and allowing for heavy-quark-threshold crossings. The numerical implementation of the GPD evolution equations is publicly accessible through the APFEL++ evolution library and is available within the PARTONS framework. Our formulation makes use of the operator definition of GPDs in light-cone gauge renormalised in the $\overline{\mbox{MS}}$ scheme. For the sake of clarity, we recompute the evolution kernels at one-loop in pQCD, confirming previous calculations. We obtain general conditions on the evolution kernels deriving from the GPD sum rules and show that our formulation obeys these conditions. We analytically show that our calculation reproduces the DGLAP and the ERBL equations in the appropriate limits and that it guarantees the continuity of GPDs. We numerically check that the evolved GPDs fulfil DGLAP and ERBL limits, continuity, and polynomiality. We benchmark our numerical implementation against analytical evolution in conformal space. Finally, we perform a numerical comparison to an existing implementation of GPD evolution finding a general good agreement on the kinematic region accessible to the latter. This work provides a pedagogical description of GPD evolution equations which benefits from a renewed interest as future colliders, such as the electron-ion colliders in the US and in China, are being designed. It also paves the way to the extension of GPD evolution codes to higher accuracies in pQCD desirable for precision phenomenology at these facilities.