$B_c$ Exclusive Decays to Charmonia and Light Mesons in QCD Factorization at Next-to-Leading Order Accuracy

In this paper, the branching ratios of $B_c$ decays to S-wave charmonia along with light mesons including $\pi$, $\rho$, $k$ and $k^*$ up to twist-3 light-cone distribution amplitudes at next-to-leading order (NLO) are performed both in non-relativistic QCD and in light-cone QCD framework. Renormalization scale dependence for the decay widths is obviously depressed at NLO correction, hence the theoretic uncertainty is correspondingly reduced. Numerical results show that NLO QCD corrections markedly enhanced the branching ratios with a K factor of 1.75 for $B_{c}^{\pm}\to \eta_{c} \pi^{\pm}$ and 1.31 for $B_{c}^{\pm}\to J/\Psi \pi^{\pm}$, which can now be checked in the LHCb experiment. Besides, in the heavy quark limit $m_b \to \infty$, the factorization is shown to hold at two scenarios: non-relativistic (NR) factorization scheme and light-cone (LC) factorization scheme. We note that the annihilation diagrams contribute trivia in this limit, and the leading order contribution in $z= m_c/m_b$ expansion comes from form factors and hard spectator interactions. Analytic investigations ensure that the comparison for the processes within7 between $\mathrm{NRQCD}\otimes \mathrm{NRQCD} \otimes \mathrm{SCET}$ and $\mathrm{SCET}\otimes \mathrm{SCET}\otimes \mathrm{SCET}$ is more clear. At last, some related phenomenologies are also discussed.