The Complete $O(\alpha_s^2)$ Non-Singlet Heavy Flavor Corrections to the   Structure Functions $g_{1,2}^{ep}(x,Q^2)$, $F_{1,2,L}^{ep}(x,Q^2)$,   $F_{1,2,3}^{\nu(\bar{\nu})}(x,Q^2)$ and the Associated Sum Rules
  We calculate analytically the flavor non-singlet $O(\alpha_s^2)$ massive Wilson coefficients for the inclusive neutral current non-singlet structure functions $F_{1,2,L}^{ep}(x,Q^2)$ and $g_{1,2}^{ep}(x,Q^2)$ and charged current non-singlet structure functions $F_{1,2,3}^{\nu(\bar{\nu})p}(x,Q^2)$, at general virtualities $Q^2$ in the deep-inelastic region. Numerical results are presented. We illustrate the transition from low to large virtualities for these observables, which may be contrasted to basic assumptions made in the so-called variable flavor number scheme. We also derive the corresponding results for the Adler sum rule, the unpolarized and polarized Bjorken sum rules and the Gross-Llewellyn Smith sum rule. There are no logarithmic corrections at large scales $Q^2$ and the effects of the power corrections due to the heavy quark mass are of the size of the known $O(\alpha_s^4)$ corrections in the case of the sum rules. The complete charm and bottom corrections are compared to the approach using asymptotic representations in the region $Q^2 \gg m_{c,b}^2$. We also study the target mass corrections to the above sum rules. 
Johannes Bl\"umlein, Giulio Falcioni, and Abilio De Freitas
hep-ph hep-ex