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

TL;DR

This paper calculates QCD corrections to $B_c$ meson semi-leptonic decays using NLO corrections and the PMC scale setting, providing refined decay width predictions that align better with experimental data.

Contribution

It introduces the use of the principle of maximum conformality (PMC) for setting renormalization scales in $B_c$ decay calculations, improving the accuracy of theoretical predictions.

Findings

Predicted decay widths with uncertainties for various $B_c$ decay channels.

Optimal PMC scale found to be approximately 0.8 GeV for all TFFs.

Better agreement with experimental measurements of $B_c$ decay ratios.

Abstract

We present a detailed analysis on the $B_c$ meson semi-leptonic decays, $B_c \to \eta_c (J/\psi) \ell \nu$, 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 $B_c\to\eta_c(J/\psi)$ transition form factors (TFFs). Because of the same $\{\beta_0\}$-terms, the optimal PMC scales at the NLO level are the same for all those TFFs, i.e. $\mu_r^{\rm PMC} \approx 0.8{\rm GeV}$. 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 $B_c\to\eta_c(J/\psi)$ TFFs to all their allowable $q^2$ region. Then, we predict $\Gamma_{B_c \to \eta_c \ell \nu}(\ell=e,\mu) =(71.53^{+11.27}_{-8.90})\times 10^{-15} {\rm GeV}$, $\Gamma_{B_c \to \eta_c \tau \nu}=(27.14^{+5.93}_{-4.33})\times 10^{-15} {\rm GeV}$, $\Gamma_{B_c \to J/\psi \ell \nu}(\ell=e,\mu) =(106.31^{+18.59}_{-14.01}) \times 10^{-15} {\rm GeV}$, $\Gamma_{B_c \to J/\psi \tau \nu} =(28.25^{+6.02}_{-4.35})\times 10^{-15} {\rm GeV}$, 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 $B^+_c\to J/\psi \ell^+ v$ relative to that for $B^+ \to J/\psi K^+$, i.e. $\Re(J/\psi \ell^+ \nu)$, is in a better agreement with CDF measurements than the previous predictions.