O ( α${}_2^s$ ) polarized heavy flavor corrections to deep-inelastic scattering at Q${}^2$ ≫ m${}^2$

We calculate the quarkonic O(α${}_s^2$) massive operator matrix elements $\mathrm{\Delta }$A${}_{Qg}$ (N),$\mathrm{\Delta }$A${}_{Qq}^{PS}$(N) and $\mathrm{\Delta }$A${}_{qq}^{NS}$,${}_Q$(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${}_s^2$ ) 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).