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Relativistic corrections to the axial vector and vector currents in the bar{b}c meson system at order alpha_s

We compute the short distance coefficients for the NRQCD factorization formulas of the meson-to-vacuum matrix elements for the axial vector and vector parts of the charged weak current in the $S$-wave spin-singlet and -triplet bar{b}c mesons, respectively. The computation is carried out to order alpha_s including relativistic corrections of all orders in vec{q}^{2n}, where vec{q} is the relative momentum of the bar{b} and c in the meson rest frame. The relativistic corrections at order alpha_s are new. The results reveal that the relativistic corrections to the leptonic decay rate of the B_c meson at order alpha_s or less converge rapidly, which shows a strong contrast to the uncomfortably large corrections of order alpha_s^2|vec{q}|^0. The short distance coefficients listed in this paper can be employed to compute the resummation of relativistic corrections to the phenomenological measurables that involve B_c and B_c^* production and decay.