Perturbative strong interaction corrections to the heavy quark semileptonic decay rate

We calculate the part of the order αs2 correction to the semileptonic heavy quark decay rate proportional to the number of light quark flavors, and use our result to set the scale for evaluating the strong coupling in the order αs term according to the scheme of Brodsky, Lepage and Mackenzie. Expressing the decay rate in terms of the heavy quark pole mass mQ, we find the scale for the MS strong coupling to be 0.07 mQ. If the decay rate is expressed in terms of the MS heavy quark mass mQ (mQ) then the scale is 0.12 mQ. We use these results along with the existing calculations for hadronic τ decay to calculate the BLM scale for the nonleptonic decay width and the semileptonic branching ratio. The implications for the value of |Vbc| extracted from the inclusive semileptonic B meson decay rate are discussed.