Radially Excited States of $η_c$

In the framework of chiral quark model, the mass spectrum of $η_c(ns) (n=1,...,6)$ is studied with Gaussian expansion method. With the wave functions obtained in the study of mass spectrum, the open flavor two-body strong decay widths are calculated by using $^3P_0$ model. The results show that the masses of $η_c(1S)$ and $η_c(2S)$ are consistent with the experimental data. The explanation of X(3940) as $η_c(3S)$ is disfavored for X(3940) is a narrow state, $Γ=37^{+26}_{-15} \pm 8$ MeV, while the open flavor two-body strong decay width of $η_c(3S)$ is about 200 MeV in our calculation. Although the mass of X(4160) is about 100 MeV less than that of $η_c(4S)$, the assignment of X(4160) as $η_c(4S)$ can not be excluded because the open flavor two-body strong decay width of $η_c(4S)$ is consistent with the experimental value of X(4160) and the branching ratios of $η_c(4S)$ are compatible with that of X(4160), and the mass of $η_c(4S)$ can be shifted downwards by taking into account the coupling effect of the open charm channels. There are still no good candidates to $η_c(5S)$ and $η_c(6S)$.

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