Searching for $X(3872)$ and $Z_c(3900)$ using lattice QCD

In the past decade, many excited charmonium states have been discovered that cannot be explained within the conventional quark model. Among those, the narrow charmonium-like state $X(3872)$ and the charged charmonium-like state $Z_c(3900)$ have been examined using various phenomenological models. Since $X(3872)$ mass is within 1~MeV of the $D\bar D^*$ threshold and $Z_c(3900)$ must contain at least four quarks, one strong candidate of these states is a $D\bar D^*$ molecular state. However, such molecular state cannot be directly studied by perturbative QCD in a low energy regime. Numerical simulation with lattice QCD can provide a nonperturbative, {\it ab initio} method for studying these mysterious meson states. In this talk, I present simulation results for $X(3872)$ and $Z_c(3900)$ with quantum numbers $J^{\rm PC} = 1^{++}$ in the isospin 0 and $J^{\rm PC} = 1^{-+}$ in the isospin 1 using lattice QCD, respectively. We use interpolating operators including both the conventional excited P-wave charmonium state and the $D^0\bar D^{0*}$ open charm state for $X(3872)$ and $J/\psi~\pi^{\pm}$ and the $D^{\pm} \bar D^{*0}$ open charm state for $Z_c(3900)$. We find an $X(3872)$ candidate close to and below the $D\bar D^*$ threshold, however, only scattering states for $Z_c(3900)$.