\begin{document}$ B_c\to J/\psi $\end{document} helicity form factors (HFFs) up to twist-4 accuracy by using the light-cone sum rules (LCSR) approach. After extrapolating those HFFs to the physically allowable \begin{document}$ q^2 $\end{document} region, we investigate the \begin{document}$ B^+_c $\end{document}-meson two-body decays and semi-leptonic decays \begin{document}$ B_c^+ \to J/\psi+(P, V, \ell^+ \nu_\ell) $\end{document}, where \begin{document}$ P/V $\end{document} stand for light pseudoscalar and vector mesons, respectively. The branching fractions can be derived using the CKM matrix element and the \begin{document}$ B_c $\end{document} lifetime from the Particle Data Group, and we obtain \begin{document}${\cal{B}}(B_c^+ \to J/\psi \pi^+) = (0.136^{+0.002}_{-0.002}){\text{%}}$\end{document}, \begin{document}${\cal{B}}(B_c^+ \to $\end{document}\begin{document}$ J/\psi K^+) = (0.010^{+0.000}_{-0.000}){\text{%}}$\end{document}, \begin{document}$ {\cal{B}}(B_c^+ \to J/\psi \rho^+) = (0.768^{+0.029}_{-0.033}){\text{%}} $\end{document}, \begin{document}$ {\cal{B}}(B_c^+ \to J/\psi K^{\ast +}) = (0.043^{+0.001}_{-0.001}){\text{%}} $\end{document}, \begin{document}${\cal{B}}(B_c^+ \to $\end{document}\begin{document}$ J/\psi \mu^+\nu_\mu) = (2.802^{+0.526}_{-0.675}){\text{%}}$\end{document} and \begin{document}$ {\cal{B}}(B_c^+ \to J/\psi \tau^+\nu_\tau) = (0.559^{+0.131}_{-0.170}){\text{%}} $\end{document}. We then obtain \begin{document}${\cal{R}}_{\pi^+/\mu^+\nu_\mu} = 0.048^{+ 0.009}_{-0.012}$\end{document} and \begin{document}$ {\cal{R}}_{K^+ / \pi^+} = 0.075^{+0.005}_{-0.005} $\end{document}, which agree with the LHCb measured value within \begin{document}$ 1\sigma $\end{document}-error. We also obtain \begin{document}$ {\cal{R}}_{J/\psi} = 0.199^{+ 0.060}_{-0.077} $\end{document}, which like other theoretical predictions, is consistent with the LHCb measured value within \begin{document}$ 2\sigma $\end{document}-error. These imply that the HFFs under the LCSR approach are also applicable to the \begin{document}$ B^+_c $\end{document} meson two-body decays and semi-leptonic decays \begin{document}$ B_c^+ \to J/\psi+(P, V, \ell^+ \nu_\ell) $\end{document}, and the HFFs obtained using LCSR in a new way implies that there may be new physics in the \begin{document}$ B_c\to J/\psi \ell^+ \nu_\ell $\end{document} semi-leptonic decays."> <inline-formula><tex-math id="M1">\begin{document}$ {\boldsymbol B_c \to \boldsymbol J/\boldsymbol\psi }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/da4fc7a5-72cf-4e1d-ba13-404b552ebf75/CPC-2021-0546_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/da4fc7a5-72cf-4e1d-ba13-404b552ebf75/CPC-2021-0546_M1.png"/></alternatives></inline-formula> helicity form factors and the <inline-formula><tex-math id="M2">\begin{document}$ {\boldsymbol B_{\boldsymbol c}^+ \to \boldsymbol J/\boldsymbol\psi+\boldsymbol(P, V, \boldsymbol\ell^+ \boldsymbol\nu_{\boldsymbol\ell}) }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/da4fc7a5-72cf-4e1d-ba13-404b552ebf75/CPC-2021-0546_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/da4fc7a5-72cf-4e1d-ba13-404b552ebf75/CPC-2021-0546_M2.png"/></alternatives></inline-formula> decays -
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