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Two-loop QCD corrections to $b\to c$ transitions 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 $η_{A,V}$ are calculated analytically in the limit of equal beauty and charm masses, and a series appoximation is obtained for the general mass case. $η_A$ is crucial for the determination of the absolute value of the Cabibbo-Kobayashi-Maskawa matrix element $V_{cb}$. The two-loop effects enhance the one-loop corrections by 21\%, removing a major theoretical uncertainty in the value of $|V_{cb}|$. Including two-loop QCD effects and previously neglected electroweak and Coulomb corrections we find $|V_{cb}| = 0.0379 \pm 0.0020({\rm stat}) \pm 0.0025({\rm syst}) \pm 0.0011({\rm theory})$.