The O(\alpha_s^3 T_F^2) Contributions to the Gluonic Operator Matrix Element

TL;DR

This paper calculates third-order quantum chromodynamics contributions to gluonic operator matrix elements involving two massive fermion lines, revealing complex mathematical structures beyond standard harmonic sums.

Contribution

It presents the first calculation of specific three-loop contributions involving two equal-mass fermion lines, introducing inverse binomial sums and root-valued letters in iterated integrals.

Findings

Computed the $O( ext{alpha}_s^3 T_F^2)$ contributions to $A_{gg,Q}$.

Identified new mathematical structures in the integrals, such as inverse binomial sums.

Provided technical methods applicable to diagrams with two different massive lines.

Abstract

The $O(\alpha_s^3 T_F^2 C_F (C_A))$ contributions to the transition matrix element $A_{gg,Q}$ relevant for the variable flavor number scheme at 3--loop order are calculated. The corresponding graphs contain two massive fermion lines of equal mass leading to terms given by inverse binomially weighted sums beyond the usual harmonic sums. In $x$-space two root-valued letters contribute in the iterated integrals in addition to those forming the harmonic polylogarithms. We outline technical details needed in the calculation of graphs of this type, which are as well of importance in the case of two different internal massive lines.