Abstract
We generalise 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 $cq\bar{q}\bar{c}$ and $cs\bar{s}\bar{c}$. Their Bethe-Salpeter amplitudes are represented by a basis of heavy-light meson-meson, hadro-charmonium 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 $cs\bar{s}\bar{c}$ state is predicted at $4068(61)$ MeV.
Highlights
With the spectacular success of Belle, BABAR, BES III and the LHC experiments and their discovery of an ever increasing and largely unexplained number of potentially exotic states, hadron spectroscopy in the heavy-quark region has become a fascinating topic in recent years; see e.g. [1,2,3,4,5,6] for recent reviews
We solve the four-body equation depicted in Fig. 1 with the rainbow-ladder kernel including all permutations, where the dressed light and charm-quark propagators are obtained from their Dyson-Schwinger equations (DSEs)
We approximate the structure of the tetraquark amplitude by its three dominant components in Eq (14), which has the advantage of reducing the complexity of the four-body equation while the system still dynamically decides which of the three configurations— heavy-light meson, hadrocharmonium or dq-dq—is most important
Summary
With the spectacular success of Belle, BABAR, BES III and the LHC experiments and their discovery of an ever increasing and largely unexplained number of potentially exotic states, hadron spectroscopy in the heavy-quark region has become a fascinating topic in recent years; see e.g. [1,2,3,4,5,6] for recent reviews. [25] employed the functional approach of Dyson-Schwinger equations (DSEs) and a four-body Bethe-Salpeter equation (BSE) to describe the lowest scalar meson octet and successfully reproduced the mass hierarchy of the f0ð500Þ, the κ and the f0=a0ð980Þ To this end, a special role of internal meson-meson configurations in the pseudoscalar meson channels has been identified: the strong binding in these channels due to dynamical chiral symmetry breaking induces a drastic reduction of the mass of the fourbody states from the natural scale of 1300–1500 MeV (four valence quarks) down to a mass of roughly 400–500 MeV for the f0ð500Þ. We consider the quantum numbers 1þþ of the Xð3872Þ channel We study this state in a spin-flavor basis which includes all three internal configurations: pairs of heavy-light mesons, hadrocharmonium and dq-dq, letting the dynamics decide which of these structures is favored. Ð4Þ avoids overcounting and ensures the separability of the four-body correlation function obtained from one channel
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