The Role of Four-Quark States in the Nature of Exotic Hadrons from Bethe-Salpeter Equations

Abstract

In this work we use the framework of Dyson-Schwinger and Bethe-Salpeter equations (DSEs and BSEs) to describe candidates of exotic hadrons in a four-quark (tetraquark) picture. Specifically, we calculate mass spectra on the energy levels of light scalar mesons, ordinary charmonia and fully-charmed tetraquark states. For that, we solve the quark DSE and several two-quark meson and diquark BSEs for different quark masses and quantum numbers in order to use the corresponding propagators and bound state amplitudes for a description of tetraquarks in a reduced two-body approximation of the full four-body BSE that is able to distinguish between different internal structures. Beyond that, we introduce a novel method to couple the two-body tetraquark BSE with the two-quark meson BSE in order to be able to describe mixing effects of tetraquark components with ordinary quarkonia. In the energy region of ordinary charmonia, we observe that the candidates of the χ𝑐1(3872) and Z𝑐 (3900) are both dominated by a mesonic 𝐷𝐷̄ ∗ component, whereas the diquark- antidiquark and the hadro-charmonium component are negligible for the description of those states. The same mostly holds for other hidden and open charm heavy-light ground states. A mixing with ordinary quarkonia was not considered in those channels for technical reasons. Moreover, we observe that the light scalar mesons 𝑓0(500) and 𝑎0/𝑓0(980)are dominated by meson-meson correlations(𝜋𝜋 and𝐾𝐾̄)as well, whereas the diquark-antidiquark and even the 𝑞𝑞̄ components appear to be irrelevant for a description of the ground states. We further show that this is an effect of chiral symmetry breaking as this four-quark dominance is only present for light quark masses. In course of all-charm calculations we are able to extract a whole spectrum for quantum numbers 0+ and 1+, where we find possible candidates for the recently discovered X(6900) in the excitation spectra for both quantum numbers. The 1+ candidates are pure mesonic composite states, whereas the 0+ candidates also have a non-negligible diquark-antidiquark component.