QCD corrections to large-$m_t$ electroweak effects in $Δr$. 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 to $\Delta r$. 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 do not, and require the use of the running $\a(\mu)$ in the corresponding renormalization group equation.