$\mathcal{O}(α_s^3)$ corrections to semileptonic $b \to c$ decays in the heavy daughter approximation

Abstract

We present our recent calculation of the third order corrections to the semileptonic b \to cb→c and the muon decays. The calculation has been performed in an expansion around the limit m_c \sim m_bmc∼mb, but shows decent convergence even for m_c=0mc=0 from which the contribution to the muon decay can be extracted. For the semileptonic b \to cb→c decay we find large perturbative corrections in the on-shell scheme which can be significantly reduced by changing to the kinetic scheme for the bottom quark mass. These results are important input for the inclusive determination of |V_{cb}||Vcb| and the Fermi coupling constant G_FGF.