3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering with Two Heavy Quark Lines

TL;DR

This paper calculates 3-loop gluonic heavy flavor corrections in deep-inelastic scattering with two heavy quark lines, providing exact results for equal and unequal masses using advanced mathematical functions.

Contribution

It presents the complete 3-loop results for heavy flavor Wilson coefficients with two heavy quark lines, including cases of equal and unequal masses, using nested sums and integrals.

Findings

Exact 3-loop results for equal mass case in $A_{gg,Q}^{(3)}$

Results expressed via nested binomial sums and iterated integrals

Calculations performed without further approximation for unequal masses

Abstract

We consider gluonic contributions to the heavy flavor Wilson coefficients at 3-loop order in QCD with two heavy quark lines in the asymptotic region $Q^2 \gg m_{1(2)}^2$. Here we report on the complete result in the case of two equal masses $m_1 = m_2$ for the massive operator matrix element $A_{gg,Q}^{(3)}$, which contributes to the corresponding heavy flavor transition matrix element in the variable flavor number scheme. Nested finite binomial sums and iterated integrals over square-root valued alphabets emerge in the result for this quantity in $N$ and $x$-space, respectively. We also present results for the case of two unequal masses for the flavor non-singlet OMEs and on the scalar integrals ic case of $A_{gg,Q}^{(3)}$, which were calculated without a further approximation. The graphs can be expressed by finite nested binomial sums over generalized harmonic sums, the alphabet of which contains rational letters in the ratio $\eta = m_1^2/m_2^2$.