Contributions of ${\psi }_2(3823)$ and $\psi (4040)$ charmonium in $B^{+}\to J\mathrm{/}\psi \eta K^{+}$ decay

Recently, a study on the $J\mathrm{/}\psi \eta$ mass spectrum from $B^{+}\to J\mathrm{/}\psi \eta K^{+}$ decays was reported by the LHCb detector. The results of this study are reported as a ratio of branching fractions as $F_X\equiv \frac{\mathcal{B}r(B^{+}\to X{K}^{+})\times \mathcal{B}r(X\to J\mathrm{/}\psi \eta )}{\mathcal{B}r(B^{+}\to \psi (2S)K^{+})\times \mathcal{B}r(\psi (2S)\to J\mathrm{/}\psi \eta )}$ for $X={\psi }_2(3823),\psi (4040)$, which are $(5.95_{-2.55}^{+3.38})\times {10}^{-2}$ and $(40.60\pm 11.20)\times {10}^{-2}$, respectively. Also, the products related to $B_X\equiv \mathcal{B}r(B^{+}\to X{K}^{+})\times \mathcal{B}r(X\to J\mathrm{/}\psi \eta )$ branching fractions are $B_{{\psi }_2(3823)}=(1.25_{-0.53}^{+0.71}\pm 0.04)\times {10}^{-6}$ and $B_{\psi (4040)}=(8.53\pm 2.35\pm 0.30)\times {10}^{-6}$. For the first time, we calculated this branching fraction using factorization. According to our calculations, $F_X$ to be $F_{{\psi }_2(3823)}=(6.55\pm 1.88)\times {10}^{-2}$ and $F_{\psi (4040)}=(14.33\pm 4.15)\times {10}^{-2}$ at $\mu =m_b\mathrm{/}2$. We have estimated $B_{{\psi }_2(3823)}=(0.26\pm 0.05)\times {10}^{-6}$ at $\mu =m_b\mathrm{/}2$ and $B_{\psi (4040)}=(2.88\pm 0.64)\times {10}^{-6}$ at $\mu =2m_b$.