DGLAP evolution at N$^3$LO with the $\texttt{Candia}$ algorithm

TL;DR

This work extends the $x$-space Candia algorithm to $N^3LO$ for DGLAP evolution of unpolarized PDFs, leveraging logarithmic expansions to obtain exact non-singlet power-series solutions and extended truncated recursions for singlet and non-singlet sectors. It provides a detailed framework for scale dependence and heavy-quark matching within VFNS, and validates the approach with $aN^3LO$ benchmarks against Hoppet-v2, demonstrating sub-percent agreement in most regions. The Candia-v2 implementation, now in C++ with performance optimizations, offers a publicly available tool for high-precision PDF evolution at higher perturbative orders, paving the way for consistent N$^3$LO phenomenology at current and future colliders. The results illuminate the analytical structure of $x$-space DGLAP solutions and establish a robust reference for high-precision QCD evolution in both theory and global PDF analyses.

Abstract

We present a generalization of the $x$-space $\texttt{Candia}$ algorithm to next-to-next-to-next-to-leading order (N$^3$LO) accuracy in Quantum Chromodynamics (QCD) for solving the DGLAP evolution equations for unpolarized parton densities in the nucleon. The algorithm is based on logarithmic expansions of the solution and can be extended to all orders in QCD. An expansion equivalent to the exact solution of the DGLAP equation at N$^3$LO is presented in the non-singlet sector. Results for approximate N$^3$LO PDFs, evolved using the most recent approximations to the N$^3$LO DGLAP splitting functions, are provided for benchmarking. The new version of the code, $\texttt{Candia-v2}$, is publicly available at https://github.com/champso1/candia-v2.

Figures (2)

• Figure 1: N$^3$LO/NNLO PDF ratios in the VFNS scheme at $\mu_F=100~\unit{\textrm{GeV}}$ with N$^3$LO matching conditions.
• Figure 2: Same as in Fig. \ref{['fig:ratios1']}, but with NNLO matching conditions (with the $A^{(2)}$ OMEs only).