A semi-analytical $x$-space solution for parton evolution -- Application to non-singlet and singlet DGLAP equation

TL;DR

This paper introduces a semi-analytical method for solving the DGLAP equations in quantum chromodynamics, transforming integro-differential equations into a manageable system of ordinary differential equations with applications to non-singlet and singlet cases.

Contribution

The authors develop a novel semi-analytical approach that constructs a basis of functions for parton evolution, enabling direct x-space solutions and providing analytic insight, which is validated against traditional methods.

Findings

Good agreement with Mellin-space methods.

Provides analytic expressions for evolved distributions.

Implemented in Mathematica and Python.

Abstract

We present a novel semi-analytical method for parton evolution. It is based on constructing a family of analytic functions spanning $x$-space which is closed under the considered evolution equation. Using these functions as a basis, the original integro-differential evolution equation transforms into a system of coupled ordinary differential equations, which can be solved numerically by restriction to a suitably chosen finite subsystem. The evolved distributions are obtained as analytic functions in $x$ with numerically obtained coefficients, providing insight into the analytic behavior of the evolved parton distributions. As a proof-of-principle, we apply our method to the leading order non-singlet and singlet DGLAP equation. Comparing our results to traditional Mellin-space methods, we find good agreement. The method is implemented in the code $\texttt{POMPOM}$ in $\texttt{Mathematica}$ as well as in $\texttt{Python}$.