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

In this paper, we calculate the $B_c\to J/ψ$ 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 $B_c^+ \to J/ψ+(P, V, \ell^+ ν_\ell)$ 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 ${\cal B}(B_c^+ \to J/ψπ^+)=(0.136^{+0.002}_{-0.002})\%$, ${\cal B}(B_c^+ \to J/ψK^+)=(0.010^{+0.000}_{-0.000})\%$, ${\cal B}(B_c^+ \to J/ψρ^+) =(0.768^{+0.029}_{-0.033})\%$, ${\cal B}(B_c^+ \to J/ψK^{\ast +})=(0.043^{+0.001}_{-0.001})\%$, ${\cal B}(B_c^+ \to J/ψμ^+ν_μ)=(2.802^{+0.526}_{-0.675})\%$ and ${\cal B}(B_c^+ \to J/ψτ^+ν_τ)=(0.559^{+0.131}_{-0.170})\%$. We then obtain ${\cal R}_{π^+/μ^+ν_μ} = 0.048^{+ 0.009}_{-0.012}$ and ${\cal R}_{K^+ / π^+} = 0.075^{+0.005}_{-0.005}$, which agree with the LHCb measured value within $1σ$-error. We also obtain ${\cal R}_{J/ψ}=0.199^{+ 0.060}_{-0.077}$, which like other theoretical predictions, is consistent with the LHCb measured value within $2σ$-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 $B_c^+ \to J/ψ+(P, V, \ell^+ ν_\ell)$, and the HFFs obtained by using LCSR in a new way implies that there may be new physics in the $B_c\to J/ψ\ell^+ ν_\ell$ semi-leptonic decays.