$O(\alpha_s^2$) Polarized Heavy Flavor Corrections}to Deep-Inelastic Scattering at $Q^2 \gg m^2$

TL;DR

This paper computes the $O(eta_s^2)$ polarized heavy flavor corrections to deep-inelastic scattering at high energy, providing compact operator matrix elements and Wilson coefficients in Mellin space, with implications for small $x$ behavior.

Contribution

It presents the first calculation of polarized heavy flavor Wilson coefficients at $O(eta_s^2)$ in the region $Q^2 \\gg m^2$, improving previous results and correcting earlier approximations.

Findings

Operator matrix elements expressed in harmonic sums.

Wilson coefficients for $g_1(x,Q^2)$ derived to $O(eta_s^2)$.

Analysis of small $x$ effects in polarized scattering.

Abstract

We calculate the quarkonic $O(\alpha_s^2)$ massive operator matrix elements $\Delta A_{Qg}(N), \Delta A_{Qq}^{\rm PS}(N)$ and $\Delta A_{qq,Q}^{\rm NS}(N)$ for the twist--2 operators and the associated heavy flavor Wilson coefficients in polarized deeply inelastic scattering in the region $Q^2 \gg m^2$ to $O(\varepsilon)$ in the case of the inclusive heavy flavor contributions. The evaluation is performed in Mellin space, without applying the integration-by-parts method. The result is given in terms of harmonic sums. This leads to a significant compactification of the operator matrix elements and massive Wilson coefficients in the region $Q^2 \gg m^2$ derived previously in \cite{BUZA2}, which we partly confirm, and also partly correct. The results allow to determine the heavy flavor Wilson coefficients for $g_1(x,Q^2)$ to $O(\alpha_s^2)$ for all but the power suppressed terms $\propto (m^2/Q^2)^k, k \geq 1$. The results in momentum fraction $z$-space are also presented. We also discuss the small $x$ effects in the polarized case. Numerical results are presented. We also compute the gluonic matching coefficients in the two--mass variable flavor number scheme to $O(\varepsilon)$.