Two-Loop QCD Corrections tob→cTransitions at Zero Recoil

Complete two-loop QCD corrections to $b\to c$ transitions are presented in the limit of zero recoil. Vector and axial-vector coefficients ${\eta }_{A,V}$ are calculated analytically in the limit of equal beauty and charm masses, and a series appoximation is obtained for the general mass case. ${\eta }_A$ is crucial for the determination of the absolute value of the Cabibbo-Kobayashi-Maskawa matrix element $V_{\mathrm{cb}}$. The two-loop effects enhance the one-loop corrections by 22%, removing a major theoretical uncertainty in the value of $|V_{\mathrm{cb}}|$. Including two-loop QCD effects and previously neglected electroweak corrections we find $|V_{\mathrm{cb}}|=0.0383\mathrm{}\pm 0.0021\left(\mathrm{stat}\right)\pm 0.0025\left(\mathrm{syst}\right)\pm 0.0011\left(\mathrm{theory}\right)$.