Searching for $X(3872)$ using lattice QCD

For decades, many excited charmonium states have been discovered that cannot be explained within the conventional quark model. Among the those mesons, the narrow charmonium-like state $X(3872)$ has been examined using various phenomenological models, however, the question for its constituent still remains open. One of the strong candidates is a $D\bar D^*$ molecular state because its mass is within $1~$MeV of the $D\bar D^*$ threshold, however, such a molecular state can’t be directly studied by perturbative QCD in such a low energy regime where the interaction of the colored quarks and gluons is very strong. Numerical simulation with lattice QCD provides a nonperturbative, ab initio method for studying this mysterious meson state. In this talk, I present preliminary simulation results for this charmonium-like states with quantum numbers $J^{\rm PC} = 1^{++}$ in both the isospin $0$ and $1$ channels. We use interpolating operators including both the conventional excited P-wave charmonium state ($\chi_{c1}$) and the $D\bar D^*$ open charm state for the isospin $0$ channel, but only $D\bar D^*$ for the isospin $1$ channel. We extract large negative S-wave scattering length and find an $X(3872)$ candidate $13\pm6$~MeV below the $D\bar D^*$ threshold in the isospin $0$ channel.