Measurement of branching fractions for $\it{B \to J/\psi \eta K}$ decays and search for a narrow resonance in the $\it{J/\psi \eta}$ final state

We report an observation of the $B^{\pm} \to J/\psi \eta K^{\pm}$ and $B^0 \to J/\psi \eta K^0_S$ decays using 772$\times 10^{6}$ $B\bar{B}$ pairs collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. We obtain the branching fractions ${\cal B}(B^{\pm}\rightarrow J/\psi\eta K^{\pm})=(1.27\pm 0.11{\rm (stat.)\pm 0.11{\rm (syst.)})}\times10^{-4}$ and ${\cal B}(B^0\to J/\psi \eta K^0_S)=(5.22 \pm 0.78 {\rm(stat.)} \pm 0.49{\rm(syst.)})\times10^{-5}$. We search for a new narrow charmonium(-like) state $X$ in the $J/\psi \eta$ mass spectrum and find no significant excess. We set upper limits on the product of branching fractions, ${\cal B}(B^\pm \to XK^\pm){\cal B}(X \to J/\psi \eta)$, at 3872 MeV$/c^2$ where a $C$-odd partner of X(3872) may exist, at $\psi(4040)$ and $\psi(4160)$ assuming their known mass and width, and over a range from 3.8 to 4.8 GeV$/c^2$. % at a 5 MeV$/c^2$ step. The obtained upper limits at 90% confidence level for $X^{C{\rm -odd}}(3872)$, $\psi(4040)$ and $\psi(4160)$ are 3.8$\times 10^{-6}$, 15.5$\times 10^{-6}$ and 8.1$\times 10^{-6}$, respectively