Quark Mass Effects in Fermionic Decays of the Higgs Boson in $O(\alpha_s^2)$ Perturbative QCD

The results of analytical evaluation of $O(\alpha_s^2)$ QCD contributions due to the nonvanishing quark masses to $\Gamma_{H\rightarrow q_f\overline{q}_f}$ are presented. The ``triangle anomaly'' type contributions are included. As a byproduct the $O(\alpha_s^3)$ logarithmic contributions are evaluated. The results are presented both in terms of running and pole quark masses. The partial decay modes $H \rightarrow b\overline{b}$ and $H \rightarrow c\overline{c}$ are considered. The calculated corrections decrease the absolute value of large and negative $O(\alpha_s^2)$ massless limit coefficient by $\leq 1\%$ in the intermediate mass region and by 1\%$ - $20\% in the low mass region which, however, is experimentally ruled out. The results are relevant for $H \rightarrow t\overline{t}$ decay mode for the higher Higgs mass region where the mass effects are large and important. The high order corrections remove a very large discrepancy between the results for $\Gamma_{H\rightarrow q_f\overline{q}_f}$ in terms of running and pole quark masses almost completely and reduce the scale dependence from about 40\% to nearly 5\%.