QCD corrections to the $B_c$ to charmonia semi-leptonic decays

We present a detailed analysis on the $B_c$ meson semi-leptonic decays, $B_c \to η_c (J/ψ) \ell ν$, up to next-to-leading order (NLO) QCD correction. We adopt the principle of maximum conformality (PMC) to set the renormalization scales for those decays. After applying the PMC scale setting, we determine the optimal renormalization scale for the $B_c\toη_c(J/ψ)$ transition form factors (TFFs). Because of the same $\{β_0\}$-terms, the optimal PMC scales at the NLO level are the same for all those TFFs, i.e. $μ_r^{\rm PMC} \approx 0.8{\rm GeV}$. We adopt a strong coupling model from the massive perturbation theory (MPT) to achieve a reliable pQCD estimation in this low energy region. Furthermore, we adopt a monopole form as an extrapolation for the $B_c\toη_c(J/ψ)$ TFFs to all their allowable $q^2$ region. Then, we predict $Γ_{B_c \to η_c \ell ν}(\ell=e,μ) =(71.53^{+11.27}_{-8.90})\times 10^{-15} {\rm GeV}$, $Γ_{B_c \to η_c τν}=(27.14^{+5.93}_{-4.33})\times 10^{-15} {\rm GeV}$, $Γ_{B_c \to J/ψ\ell ν}(\ell=e,μ) =(106.31^{+18.59}_{-14.01}) \times 10^{-15} {\rm GeV}$, $Γ_{B_c \to J/ψτν} =(28.25^{+6.02}_{-4.35})\times 10^{-15} {\rm GeV}$, where the uncertainties are squared averages of all the mentioned error sources. We show that the present prediction of the production cross section times branching ratio for $B^+_c\to J/ψ\ell^+ v$ relative to that for $B^+ \to J/ψK^+$, i.e. $\Re(J/ψ\ell^+ ν)$, is in a better agreement with CDF measurements than the previous predictions.

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