Canonical interpretation of the $X(4140)$ state within the $^3P_0$ model

Recently, the LHCb Collaboration has confirmed the state $X(4140)$, with a mass $M=4146.5\pm 4.5^{+4.6}_{-2.8}$~MeV, and a much larger width $\Gamma=83\pm21^{+21}_{-14}$~MeV than the previous experimental measurements, which has confused the understanding of its nature. We will investigate the possibility of the $\chi_{c1}(3P)$ interpretation for the $X(4140)$, considering the mass spectra predicted in the quark model, and the strong decay properties within the $^3P_0$ model. We also predict the strong decay properties of the charmonium states $\chi_{c0}(3P)$ and $\chi_{c2}(3P)$. Our results show that the $X(4140)$ state with the small width given in PDG can be explained as the charmonium state $\chi_{c1}(3P)$ in the $^3P_0$ model, and high precision measurement of the width of the $X(4140)$ is crucial to understand its nature.