On normalization of QCD effects in $O(m_t^2)$ electroweak corrections

Abstract

We point out that, contrary to some recent claims, there is no intrinsic long-distance uncertainty in perturbative calculation of the QCD effects in the $t\text{tb}$ and $t\text{bb}$ loops giving the electroweak corrections proportional to $m_t^2$. If these corrections are expressed in terms of the ``on-shell" mass $m_t$, the only ambiguity arising is that associated with the definition of the ``on-shell" mass of a quark. The latter is entirely eliminated if the result is expressed in terms of $m_t$ defined at short distances. Applying the Brodsky-Lepage-Mackenzie criterion for determining the natural scale for normalization of $\text{as}$, we find that using the ``on-shell" mass makes this scale numerically small in units of $m_t$. Specifically, we find that by this criterion the first QCD correction to the $O(m_t^2)$ terms is determined by ${\text{as}}^{\text{msb}}(0.15m_t)$. Naturally, a full calculation of three-loop graphs is needed to completely quantify the scale.

Comments: LaTeX, 10 pages + 1 figure (appended as a PostScript file)