$B_c \to J/\psi$ helicity form factors and the $B_c^+ \to J/\psi+(P, V, \ell^+ \nu_\ell)$ decays

In this paper, we calculate the $B_c\to J/\psi$ helicity form factors (HFFs) up to twist-4 accuracy by using light-cone sum rules (LCSR) approach. After extrapolating those HFFs physically allowable $q^2$ region, investigate $B^+_c$-meson two-body decays and semi-leptonic $B_c^+ \to J/\psi+(P, V, \ell^+ \nu_\ell)$ with $P/V$ stands for light pseudoscalar/vector meson, respectively. The branching fractions can be derived CKM matrix element $B_c$ lifetime from Particle Data Group, obtain ${\cal B}(B_c^+ J/\psi \pi^+)=(0.136^{+0.002}_{-0.002})\%$, K^+)=(0.010^{+0.000}_{-0.000})\%$, \rho^+) =(0.768^{+0.029}_{-0.033})\%$, K^{\ast +})=(0.043^{+0.001}_{-0.001})\%$, \mu^+\nu_\mu)=(2.802^{+0.526}_{-0.675})\%$ \tau^+\nu_\tau)=(0.559^{+0.131}_{-0.170})\%$. We then R}_{\pi^+/\mu^+\nu_\mu} = 0.048^{+ 0.009}_{-0.012}$ R}_{K^+ / \pi^+} 0.075^{+0.005}_{-0.005}$, which agree LHCb measured value within $1\sigma$-error. also R}_{J/\psi}=0.199^{+ 0.060}_{-0.077}$, like other theoretical predictions, is consistent $2\sigma$-error. Those imply that under LCSR approach are applicable $B^+_c$ meson \nu_\ell)$, obtained in a new way implies there may physics \nu_\ell$ decays.