The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$

TL;DR

This paper computes the two-mass contributions to a three-loop gluonic operator matrix element in QCD, providing analytic results that depend on the heavy quark mass ratio, crucial for flavor scheme matching.

Contribution

It presents the first analytic calculation of two-mass three-loop gluonic operator matrix elements with full mass ratio dependence in Mellin and z-space.

Findings

Analytic expressions in Mellin N- and z-space with harmonic sums and iterated integrals.

Results enable precise flavor scheme matching with two heavy quarks.

Numerical evaluations demonstrate the impact of two-mass effects.

Abstract

We calculate the two-mass QCD contributions to the massive operator matrix element $A_{gg,Q}$ at $\mathcal{O} (\alpha_s^3)$ in analytic form in Mellin $N$- and $z$-space, maintaining the complete dependence on the heavy quark mass ratio. These terms are important ingredients for the matching relations of the variable flavor number scheme in the presence of two heavy quark flavors, such as charm and bottom. In Mellin $N$-space the result is given in the form of nested harmonic, generalized harmonic, cyclotomic and binomial sums, with arguments depending on the mass ratio. The Mellin inversion of these quantities to $z$-space gives rise to generalized iterated integrals with square root valued letters in the alphabet, depending on the mass ratio as well. Numerical results are presented.