A recent study from the LHCb detector analyzed the \(J/\psi \eta \) mass spectrum from \(B^+\rightarrow J/\psi \eta K^+\) decays, reporting the branching fraction ratios as \(F_{X}\equiv \frac{\mathcal {B}r(B^+\rightarrow \psi ' K^+)\times \mathcal {B}r(\psi '\rightarrow J/\psi \eta )}{\mathcal {B}r(B^+\rightarrow \psi (2S) K^+)\times \mathcal {B}r(\psi (2S)\rightarrow J/\psi \eta )}\) for \(\psi '=\psi _2(3823),\psi (4040)\) , which are \((5.95^{+3.38}_{-2.55})\times 10^{-2}\) and \((40.60\pm 11.20)\times 10^{-2}\) , respectively. Also, the products related to \(B_{X}\equiv \mathcal {B}r(B^+\rightarrow \psi 'K^+)\times \mathcal {B}r(\psi '\rightarrow J/\psi \eta )\) branching fractions are \(B_{\psi _2(3823)}=(1.25^{+0.71}_{-0.53}\pm 0.04)\times 10^{-6}\) and \(B_{\psi (4040)}=(8.53\pm 2.35\pm 0.30)\times 10^{-6}\) . This represents the first calculation of this branching fraction using factorization. Our calculations indicate that \(F_X\) is \(F_{\psi _{2}(3823)}=(6.55\pm 1.88)\times 10^{-2}\) and \(F_{\psi (4040)}=(14.33\pm 4.15)\times 10^{-2}\) at \(\mu =m_b/2\) . We have estimated \(B_{\psi _{2}(3823)}=(0.26\pm 0.05)\times 10^{-6}\) at \(\mu =m_b/2\) and \(B_{\psi (4040)}=(2.88\pm 0.64)\times 10^{-6}\) at \(\mu =2m_b\) .
