The hidden charm decay of Y(4140) by the rescattering mechanism

Assuming that Y(4140) is the second radial excitation of the P-wave charmonium $\chi_{cJ}^{\prime\prime}$ ($J=0, 1$), the hidden charm decay mode of Y(4140) is calculated in terms of the rescattering mechanism. Our numerical results show that the upper limit of the branching ratio of the hidden charm decay $Y(4140)\to J/\psi\phi$ is on the order of $10^{-4}\sim 10^{-3}$ for both of the charmonium assumptions for Y(4140), which disfavors the large hidden charm decay pattern indicated by the CDF experiment. It seems to reveal that the pure second radial excitation of the P-wave charmonium $\chi_{cJ}^{\prime\prime}$ ($J=0, 1$) is problematic.