$O(\alpha_s^3 T_F^2 N_F)$ Contributions to the Heavy Flavor Wilson Coefficients of the Structure Function $F_2(x,Q^2)$ at $Q^2 \gg m^2$

TL;DR

This paper calculates specific 3-loop fermion-loop corrections to heavy flavor Wilson coefficients in the structure function F_2(x,Q^2), providing new results for Q^2 much larger than the quark mass squared, using a direct integration method.

Contribution

It introduces a direct integration approach to compute massive 3-loop corrections to operator matrix elements, avoiding integration-by-parts, and confirms existing anomalous dimension results.

Findings

Computed 3-loop fermion-loop corrections for general N.

Provided contributions to asymptotic heavy flavor Wilson coefficients.

Confirmed existing results for 3-loop anomalous dimensions.

Abstract

The massive 3-loop fermion-loop corrections $\propto C_A N_f T_F^2$ and $C_F N_f T_F^2$ to the massive operator matrix elements $A_{Qg}$, $A_{Qq}^{\rm{PS}}$, $A_{qq,Q}^{\rm{PS}}$, $A_{qq,Q}^{\rm{NS}}$ and $A_{qq,Q}^{\rm{NS,TR}} have been obtained for general values of $N$. Thereby the corresponding contributions to the asymptotic heavy flavor Wilson coefficients of the structure function $F_2(x,Q^2)$ and of transversity in the region $Q^2 \geq 10 \cdot m^2$ are known. Our method is based on direct integration, avoiding the integration-by-parts technique, which is advantageous due to the compactness of the intermediate and final results. We also obtain the corresponding contributions to the 3-loop anomalous dimensions and confirm results in the literature.