$B_c\to J\mathrm{/}\psi$ helicity form factors and the $B_c^{+}\to J\mathrm{/}\psi +(P,V,{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }})$ decays

In this paper, we calculate the $B_c\to J\mathrm{/}\psi$ 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${\cal B}(B_c^+ \to J/\psi \tau^+\nu\tau)=(0.559^{+0.131}{-0.170})%$\mathrm{.}Wethenobtain${\cal R}{\pi^+/\mu^+\nu_\mu} = 0.048^{+ 0.009}{-0.012}$and${\cal R}{K^+ / \pi^+} = 0.075^{+0.005}{-0.005}$,whichagreewiththeLHCbmeasuredvaluewithin$1\sigma$-error\mathrm{.}Wealsoobtain${\cal R}{J/\psi}=0.199^{+ 0.060}{-0.077}$, 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 $B_c^{+}\to J\mathrm{/}\psi +(P,V,{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }})$, 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.