We perform a lattice study of charmonium-like mesons with $J^{PC}=1^{++}$ and three quark contents $bar cc bar du$, $bar cc(bar uu+bar dd)$ and $bar cc bar ss$, where the later two can mix with $bar cc$. This simulation with $N_f=2$ and $m_pi=266$ MeV aims at the possible signatures of four-quark exotic states. We utilize a large basis of $bar cc$, two-meson and diquark-antidiquark interpolating fields, with diquarks in both anti-triplet and sextet color representations. A lattice candidate for X(3872) with I=0 is observed very close to the experimental state only if both $bar cc$ and $Dbar D^*$ interpolators are included; the candidate is not found if diquark-antidiquark and $Dbar D^*$ are used in the absence of $bar cc$. 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 $bar ccbar ss$ candidates below 4.3 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.
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