The strong decays of

The new mesons X(3940) and X(4160) have been found by Belle Collaboration in the processes $$e^+e^-\rightarrow J/\psi D^{(*)}{\bar{D}}^{(*)}$$. Considering X(3940) and X(4160) as $$\eta _c(3S)$$ and $$\eta _c(4S)$$ states, the two-body open charm OZI-allowed strong decay of $$\eta _c(3S)$$ and $$\eta _c(4S)$$ are studied by the improved Bethe–Salpeter method combined with the $$^3P_0$$ model. The strong decay width of $$\eta _c(3S)$$ is $$\Gamma _{\eta _c(3S)}=(33.5^{+18.4}_{-15.3})$$ MeV, which is close to the result of X(3940); therefore, $$\eta _c(3S)$$ is a good candidate of X(3940). The strong decay width of $$\eta _c(4S)$$ is $$\Gamma _{\eta _c(4S)}=(69.9^{+22.4}_{-21.1})$$ MeV, considering the errors of the results, it is close to the lower limit of X(4160). But the ratio of the decay width $$\frac{\Gamma (D{\bar{D}}^*)}{\Gamma (D^*{\bar{D}}^*)}$$ of $$\eta _c(4S)$$ is larger than the experimental data of X(4160). According to the above analysis, $$\eta _c(4S)$$ is not the candidate of X(4160), and more investigations of X(4160) is needed