O ( α$^s_2$ ) polarized heavy flavor corrections to deep-inelastic scattering at Q$^2$ ≫ m$^2$

We calculate the quarkonic O(α$^2_s$) massive operator matrix elements $\Delta$A$_{Qg}$ (N),$\Delta$A$^{PS}_{Qq}$(N) and $\Delta$A$^{NS}_{qq}$,$_Q$(N) for the twist–2 operators and the associated heavy flavor Wilson coefficients in polarized deeply inelastic scattering in the region Q$^2$ ≫ m$^2$ to O(ε) 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$^2$ ≫ m$^2$ derived previously in [1], 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(α$^2_s$ ) for all but the power suppressed terms ∝ (m$^2$/Q$^2$)$^k$ , k ≥ 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(ε).