The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg,Q}$ and $\Delta A_{gg,Q}$

TL;DR

This paper presents the calculation of three-loop gluonic operator matrix elements for unpolarized and polarized cases, crucial for precise gluon distribution matching in the variable flavor number scheme.

Contribution

It provides the first complete three-loop results for gluon transition matrix elements in both unpolarized and polarized scenarios, including analytic expressions and numerical evaluations.

Findings

Calculated three-loop gluonic operator matrix elements in unpolarized and polarized cases.

Derived analytic expressions and performed numerical evaluations.

Complete the set of gluon transition matrix elements at three-loop order.

Abstract

We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,\mu^2)$ and $\Delta A_{gg,Q}(x,\mu^2)$, at three-loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$-space and iterated integrals over square root-valued alphabets in momentum fraction $x$-space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction $x$-space and derive numerical results. The present results complete the gluon transition matrix elements both of the single- and double-mass variable flavor number scheme to three-loop order.