TTP93-08 Complete QCD Corrections of Order $\alpha_s^2$ to the Z Decay Rate

The axial part of the Z decay rate into hadrons ($\G^A_Z$) gets extra singlet contributions due to the huge splitting between top and bottom quark masses. By treating the axial neutral current in the large top-mass limit we partially re-derive in a systematic manner the known results on the contribution of the $\as^2$ order anomalous “double triangle” diagrams to $\G^A_Z$ and resolve an ambiguity concerning the scale at which the strong coupling constant $\alpha_s$ is to be taken in these diagrams. We evaluate and sum all leading log and next to leading log QCD corrections of orders $\as^{1+n}(\ln(M_Z/4m^2_t))^n$ and $\as^{2+n}(\ln(M_Z/4m^2_t))^n$ respectively.

We examine the sensitivity of our results to a change of the renormalization scales to estimate the size of yet unknown corrections of order $\alpha_s^3$ to $\G^A_Z$. The analysis implies that the corrections might change the magnitude of the singlet term by up to 30\% while they hardly might be of noticeable numerical significance in comparison with the total contribution from non-singlet terms.