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

TL;DR

The paper delivers an independent, massive-operator-method calculation of the three-loop QCD splitting functions $P_{qg}^{(2)}(x)$ and $P_{gg}^{(2,\mathrm{N_F})}(x)$, enabling NNLO accuracy with heavy-quark effects. It employs large Mellin moments and recurrences to obtain $P_{qg}^{(2)}$ and directly computes $P_{gg}^{(2,\mathrm{N_F})}$, cross-validating with established massless results. The anomalous dimensions $\\gamma_{qg}^{(2)}(N)$ and $\\gamma_{gg}^{(2),\mathrm{N_F}}(N)$ are extracted via pole structures of massive OMEs and various analytic techniques (differential equations, hypergeometric representations, and harmonic sums). The results are in agreement with prior literature (e.g., Vogt et al.), and the work showcases a robust framework for incorporating heavy-flavor contributions into NNLO QCD predictions and heavy-quark Wilson coefficients.

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.

Figures (2)

• Figure 1: Diagrams for the tadpole master integrals. Dashed lines represent massless propagators and massive propagators are depicted as solid lines.
• Figure 2: Diagrams for the master integrals with propagators of two different masses. Thin solid lines denote propagators with mass $m_1$ and thick solid lines correspond to propagators with mass $m_2$.