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We compute the short distance coefficients for the NRQCD factorizationformulas of the meson-to-vacuum matrix elements for the axial vector and vectorparts of the charged weak current in the $S$-wave spin-singlet and -tripletbar{b}c mesons, respectively. The computation is carried out to order alpha_sincluding 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. Therelativistic corrections at order alpha_s are new. The results reveal that therelativistic corrections to the leptonic decay rate of the B_c meson at orderalpha_s or less converge rapidly, which shows a strong contrast to theuncomfortably large corrections of order alpha_s^2|vec{q}|^0. The shortdistance coefficients listed in this paper can be employed to compute theresummation of relativistic corrections to the phenomenological measurablesthat involve B_c and B_c^* production and decay.

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