Looking at the QCD corrections for large M$_{t}$ : an effective lagrangian point of view

We discuss the QCD corrections to the large-m_t electroweak contributions to \Delta r and to the process Z\to b \bar b as two of the most representative examples. This needs the construction of an effetive field theory below the top quark. We discuss the issue of what \mu scale is the appropriate one at every stage and argue that while matching corrections do verify the simple prescription of taking \mu \simeq m_t in \alpha_s(\mu), logarithmic (i.e. \sim \log m_t) corrections do not, and require the use of the running \alpha_s(\mu) in the corresponding renormalization group equations. In particular we obtain the \alpha_s correction to the non-universal \log m_t contribution to the Zb\bar b vertex.