Reference results for time-like evolution up to $\mathcal{O}(\alpha_s^3)$

TL;DR

This paper provides high-precision numerical results for time-like evolution in QCD up to NNLO, including validation of splitting functions, implementation in two codes, and benchmark tables for future fragmentation function studies.

Contribution

It offers the first NNLO time-like evolution results, validated analytical splitting functions, and introduces two independent codes with benchmark tables for the community.

Findings

Validated analytical splitting functions in x and N space.

Implemented time-like evolution in two independent codes.

Provided benchmark tables for future fragmentation function analyses.

Abstract

We present high-precision numerical results for time-like Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution in the $\bar{\rm MS}$ factorisation scheme, for the first time up to next-to-next-to-leading order accuracy in quantum chromodynamics. First, we scrutinise the analytical expressions of the splitting functions available in the literature, in both x and N space, and check their mutual consistency. Second, we implement time-like evolution in two publicly available, entirely independent and conceptually different numerical codes, in x and N space respectively: the already existing APFEL code, which has been updated with time-like evolution, and the new MELA code, which has been specifically developed to perform the study in this work. Third, by means of a model for fragmentation functions, we provide results for the evolution in different factorisation schemes, for different ratios between renormalisation and factorisation scales and at different final scales. Our results are collected in the format of benchmark tables, which could be used as a reference for global determinations of fragmentation functions in the future.