We generalize the framework of Dyson-Schwinger and Bethe-Salpeter equations for four-quark states to accommodate the case of unequal quark masses. As a first application, we consider the quantum numbers $I(J^{PC})=0(1^{++})$ of the $X(3872)$ and study the four-quark states with quark contents $cq\overline{q}\overline{c}$ and $cs\overline{s}\overline{c}$. Their Bethe-Salpeter amplitudes are represented by a basis of heavy-light meson-meson, hadrocharmonium and diquark-antidiquark operators, which allows for a dynamical distinction between different internal configurations. In both cases we find the heavy-light meson-meson component to be dominant. For the putative $X(3872)$ we obtain a mass of 3916(74) MeV; the corresponding $cs\overline{s}\overline{c}$ state is predicted at 4068(61) MeV.

• Received 17 May 2019

DOI:https://doi.org/10.1103/PhysRevD.100.014033

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Published by the American Physical Society