Recently, a study on the $J/\psi\eta$ mass spectrum from $B^+\rightarrow J/\psi\eta K^+$ decays was reported by the LHCb detector. The results of this study are reported as a ratio of branching fractions as $F_{ X}\equiv\frac{ \mathcal{ B}r(B^+\rightarrow XK^+)\times\mathcal{ B}r(X\rightarrow J/\psi\eta)}{ \mathcal{ B}r(B^+\rightarrow \psi(2S) K^+)\times\mathcal{ B}r(\psi(2S)\rightarrow J/\psi\eta)}$ for $X=\psi_2(3823),\psi(4040)$, which are $(5.95^{ +3.38}_{ -2.55})\times10^{ -2}$ and $(40.60\pm11.20)\times10^{ -2}$, respectively. Also, the products related to $B_{ X}\equiv\mathcal{ B}r(B^+\rightarrow XK^+)\times\mathcal{ B}r(X\rightarrow J/\psi\eta)$ branching fractions are $B_{ \psi_2(3823)}=(1.25^{ +0.71}_{ -0.53}\pm0.04)\times10^{ -6}$ and $B_{ \psi(4040)}=(8.53\pm2.35\pm0.30)\times10^{ -6}$. For the first time, we calculated this branching fraction using factorization. According to our calculations, $F_X$ to be $F_{ \psi_{ 2}(3823)}=(6.55\pm1.88)\times10^{ -2}$ and $F_{ \psi(4040)}=(14.33\pm4.15)\times10^{ -2}$ at $\mu=m_b/2$. We have estimated $B_{ \psi_{ 2}(3823)}=(0.26\pm0.05)\times10^{ -6}$ at $\mu=m_b/2$ and $B_{ \psi(4040)}=(2.88\pm0.64)\times10^{ -6}$ at $\mu=2m_b$.
Categories: hep-ph hep-ex