Properties Of Heavy-Light Four-Quark States From Functional Methods

Abstract

In this work we study the properties – specifically mass spectra and internal structure – of exotic mesons in the charmonium and bottomonium energy regions. Most of these mesons are deemed candidates for four-quark states featuring two heavy and two light quarks. For our calculations, we employ the functional framework of Dyson-Schwinger and Bethe-Salpeter equations (DSEs and BSEs) using a four-quark formulation. We begin by solving the quark DSEs for different quark flavours and calculating mass spectra from several two-body meson and diquark BSEs. These results serve as input for the four-quark BSE, the central object of this work. Using a physically motivated Ansatz, we describe the heavy-light four-quark states in terms of all possible internal two-quark groupings and the attractive and repulsive forces between them. We compare the mass spectra obtained using only the attractive and the attractive plus repulsive components to experimental measurements or theoretical predictions. In this way, we are able to assess the importance of repulsive forces for our description. Furthermore, we are able to investigate the favoured internal structure of the four-quark states by calculating the contribution of each internal component to the total normalization. We observe that the inclusion of the repulsive forces leads to a much better agreement of our calculated mass spectra with the observed experimental spectra for the hidden-charm and hidden-bottom four-quark states. For the open-flavour four-quark states with total spin 𝐽 = 1, the effect of including the repulsive forces is found to be especially significant, rendering our masses for the $T_{bb}^-$, $T_{bb\bar{s}}^-$, $T_{bc}$, $T_{cc}(3875)^+$ in quantitative agreement with predictions from the literature. Regarding the internal structure, we see that most of the investigated hidden-flavour four-quark states are purely dominated by the respective lightest internal heavy-light meson-meson components. The $𝐽^{𝑃𝐶} = 1^{+−}$ , including the 𝑇𝑐𝑐1̄ (3900) and 𝑇𝑏𝑏1̄ (10610), and the 0−+ channel show a very different picture, however, with a strong tendency towards a dominant hadro-quarkonium component. In all cases, the contribution coming from the diquark-antidiquark pairings is almost negligible. For the open-flavour states, we observe an interesting trend when considering the binding energy of the states with respect to the lowest heavy-light meson-meson threshold. For very shallow bound states, e.g., $T_{cc}^+$, $T_{bc}$, the corresponding heavy-light meson-meson component is found to be dominating. However, the deeper the state is bound, e.g., $T_{bb\bar{s}}^-$ and $T_{bb}$ , the stronger the contribution coming from the diquark-antidiquark pairings becomes.