A Matrix Approach to Numerical Solution of the DGLAP Evolution Equations

TL;DR

This work introduces a matrix-based discretization of the DGLAP evolution equations in x-space, turning the integro-differential problem into a lower-triangular, banded matrix equation in the evolution variable t. By discretizing x into bins and choosing binning so that the kernel depends only on k−m, the evolution operator becomes a commuting matrix, allowing an exact exponential solution and, when appropriate, a finite polynomial expansion via nilpotent components. The framework extends to singlet densities and higher-order QCD corrections, maintaining computational efficiency, stability, and linear parameter dependence that enables fast, analytic data fitting. Numerical results demonstrate high-precision evolution with modest binning, and the method offers significant speedups and reduced cross-bin correlations compared to traditional functional-parametrization approaches, making it well-suited for large-scale parton-density fits. A note discusses compatibility and contrasts with prior approaches, highlighting the practical advantages and potential extensions of the matrix method.

Abstract

A matrix-based approach to numerical integration of the DGLAP evolution equations is presented. The method arises naturally on discretisation of the Bjorken x variable, a necessary procedure for numerical integration. Owing to peculiar properties of the matrices involved, the resulting equations take on a particularly simple form and may be solved in closed analytical form in the variable t=ln(alpha_0/alpha). Such an approach affords parametrisation via data x bins, rather than fixed functional forms. Thus, with the aid of the full correlation matrix, appraisal of the behaviour in different x regions is rendered more transparent and free of pollution from unphysical cross-correlations inherent to functional parametrisations. Computationally, the entire programme results in greater speed and stability; the matrix representation developed is extremely compact. Moreover, since the parameter dependence is linear, fitting is very stable and may be performed analytically in a single pass over the data values.