QCD corrections to large-$m_t$ electroweak effects in $Δr$. An effective field theory point of view

Abstract

By using effective field theory techniques for the standard model, we discuss the issue of what $μ$ scale is the appropriate one in the QCD corrections to the large-$\mt$ electroweak contributions to $Δr$. 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 $μ\simeq \mt$ in $\a(\m)$, logarithmic (i.e. $\sim \log \mt$) corrections do not, and require the use of the running $\a(μ)$ in the corresponding renormalization group equation.