QCD corrections to large-m$_{t}$ electroweak effects in $\Delta$r: an effective field theory point of view

By using effective field theory techniques for the standard model, we discuss the issue of what \mu scale is the appropriate one in the QCD corrections to the large-\mt electroweak contributions. This needs the construction of an effective field theory below the top quark. We argue that, while matching corrections do verify the simple prescription of taking \mu \simeq \mt in \a(\m), logarithmic (i.e. \sim \log \mt) corrections in general do not, and require the use of the running \a(\mu) in the corresponding renormalization group equation. We exemplify this with a calculation of \Delta r_W. We also present an argument by which the "intrinsic" QCD corrections to the non-universal \log \mt piece of the Zb \bar b vertex are shown to vanish; i.e. the final answer is of the form one--loop QCD (m_tM_Z) times one--loop electroweak (m_tM_Z).