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