Decay constant of [formula omitted] accurate up to [formula omitted

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In this paper, we evaluate, up to QCD next-to-next-to-next-to-leading order, the Bc⁎ decay constant, which, within the nonrelativistic QCD (NRQCD) framework, is factorized as the product of the short-distance coefficient (SDC) and the long-distance matrix element. For the first time, the renormalization constant and the anomalous dimension for the NRQCD vector current composed of cb¯, which are functions of the charm quark mass mc, the bottom quark mass mb and the factorization scale μΛ, are obtained analytically at O(αs2) and O(αs3). The SDC is calculated up to O(αs3) in perturbative expansion. Explicitly, the O(αs2) correction to the SDC is analytically calculated in terms of logarithmic and polylogarithmic functions of r≡mb/mc, and the O(αs3) correction to the SDC is numerically calculated at a series of values of r, ranging from 2.1 to 4.0. Surprisingly, we find that the nontrivial part of the SDC at O(αs3) can be well estimated by a linear function of r. In addition, we find that the O(αs2) and O(αs3) corrections to the decay constant and decay width are considerable and very significant, which indicates a very poor convergence for the perturbative expansion.