We perform a lattice study of charmonium-like mesons with $J^{\mathrm{PC}}=1^{++}$ and three quark contents $\overline{c}c\overline{d}u$, $\overline{c}c(\overline{u}u+\overline{d}d)$ and $\overline{c}c\overline{s}s$, where the later two can mix with $\overline{c}c$. This simulation with $N_f=2$ and $m_{\pi }\simeq 266\mathrm{MeV}$ aims at the possible signatures of four-quark exotic states. We utilize a large basis of $\overline{c}c$, two-meson and diquark-antidiquark interpolating fields, with diquarks in both antitriplet and sextet color representations. A lattice candidate for $X(3872)$ with $I=0$ is observed very close to the experimental state only if both $\overline{c}c$ and $D{\overline{D}}^{*}$ interpolators are included; the candidate is not found if diquark-antidiquark and $D{\overline{D}}^{*}$ are used in the absence of $\overline{c}c$. No candidate for neutral or charged $X(3872)$, or any other exotic candidates are found in the $I=1$ channel. We also do not find signatures of exotic $\overline{c}c\overline{s}s$ candidates below 4.2 GeV, such as $Y(4140)$. Possible physics and methodology related reasons for that are discussed. Along the way, we present the diquark-antidiquark operators as linear combinations of the two-meson operators via the Fierz transformations.

• Received 18 March 2015

DOI:https://doi.org/10.1103/PhysRevD.92.034501