Decay constant of $B_c^*$ accurate up to $\mathcal{O}(\alpha_s^3)$

In this paper, we evaluate, up to QCD next-to-next-to-next-to-leading order, the $B_c^*$ 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 $c\bar{b}$, which are functions of the charm quark mass $m_c$, the bottom quark mass $m_b$ and the factorization scale $\mu_\Lambda$, are obtained analytically at $\mathcal{O}(\alpha_s^2)$ and $\mathcal{O}(\alpha_s^3)$. The SDC is calculated up to $\mathcal{O}(\alpha_s^3)$ in perturbative expansion. Explicitly, the $\mathcal{O}(\alpha_s^2)$ correction to the SDC is analytically calculated in terms of logarithmic and polylogarithmic functions of $r\equiv m_b/m_c$, and the $\mathcal{O}(\alpha_s^3)$ 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 $\mathcal{O}(\alpha_s^3)$ can be well estimated by a linear function of $r$. In addition, We find that the $\mathcal{O}(\alpha_s^2)$ and $\mathcal{O}(\alpha_s^3)$ corrections to the decay constant and decay width are considerable and very significant, which indicates a very poor convergence for the perturbative expansion.