# The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2, N_F)}$

## TL;DR

This paper presents the calculation of three-loop unpolarized splitting functions in quantum chromodynamics, providing precise results for parton evolution with advanced computational techniques.

## Contribution

It introduces a novel calculation of the three-loop splitting functions $P_{qg}^{(2)}$ and $P_{gg}^{(2,
m N_F)}$ using massive operator matrix elements and recurrence relations.

## Key findings

- Confirmed previous results in the literature.
- Calculated $P_{gg}^{(2,
m N_F)}(x)$ directly.
- Derived $P_{qg}^{(2)}(x)$ from 1200 moments.

## Abstract

We calculate the unpolarized twist-2 three-loop splitting functions $P_{qg}^{(2)}(x)$ and $P_{gg}^{(2,\rm N_F)}(x)$ and the associated anomalous dimensions using massive three-loop operator matrix elements. While we calculate $P_{gg}^{(2,\rm N_F)}(x)$ directly, $P_{qg}^{(2)}(x)$ is computed from 1200 even moments, without any structural prejudice, using a hierarchy of recurrences obtained for the corresponding operator matrix element. The largest recurrence to be solved is of order 12 and degree 191. We confirm results in the foregoing literature.