Two-loop QCD corrections to

Complete two-loop QCD corrections to b ! c transitions are presented in the limit of zero recoil. Vector and axial-vector coefficients j 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. j 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 jV cb j. Including two-loop QCD effects and previously neglected electroweak and Coulomb corrections we find jV cb j = 0:0379 \Sigma 0:0020(stat) \Sigma 0:0025(syst) \Sigma 0:0011(theory). Elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix are fundamental input parameters of the Standard Model. Their precise measurements have been the subject of vast experimental efforts and will remain prominent issues in forthcoming projects, most notably the B-factories. The values of CKM ma..