Four-quark states with charm quarks in a two-body Bethe-Salpeter approach

We study the internal structure of a range of four-quark states with charm quark contributions using a two-body Bethe-Salpeter equation. Thereby, we examine charmonium-like states with hidden charm and quark content $c\bar{c}q\bar{q}$, open-charm states with quark content $cc\bar{q}\bar{q}$ and all-charm states with $cc\bar{c}\bar{c}$. In particular we study the internal competition between meson-meson components and diquark-antidiquark components in the wave functions of these states. Our results indicate that the $\chi_{c1}(3872)$ and the $Z_c(3900)$ are predominantly $D\bar D^*$ states and that the recently discovered open-charm state $T_{cc}^+$ is dominated by an internal $DD^*$ component. In both cases the diquark components are negligible. For the all-charm state $X(6900)$ with as yet unknown quantum numbers we identify candidates in the excitation spectra of $0^+$ and $1^+$ states. Furthermore, our framework serves to provide predictions for further, yet undiscovered open and hidden charm four-quark states.Comment: 9 pages, 4 figures; v2: minor changes, version accepted for EPJ

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

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 χ1(3872) and Z (3900) are both dominated by a mesonic ̄ ∗ 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 0(500) and 0/0(980)are dominated by meson-meson correlations( and̄)as well, whereas the diquark-antidiquark and even the ̄ 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

σ\sigma-meson: Four-quark versus two-quark components and decay width in a Bethe-Salpeter approach

We study the dynamical generation of resonances in isospin singlet channels with mixing between two- and four-quark states. To this end we generalise a Bethe-Salpeter approach to four-quark states employed previously \cite{Heupel:2012ua} to accommodate for mixing diagrams. The $q\bar{q}q\bar{q}$ and $q\bar{q}$ components of the Bethe-Salpeter wave function (with light quarks $q\in\{u,d\}$) are determined consistently in a symmetry-preserving truncation of the underlying Dyson-Schwinger equations. As a prominent example we deal with the isospin-singlet $0^{++}$ meson with light quark content. We find that the $\pi\pi$ contribution of the four-quark component is mainly responsible for the low (real part of the) mass of the resulting state. We also study the analytic structure in the complex momentum plane and find a branch cut at the two-pion threshold and a singularity in the second Riemann sheet indicating a considerable decay width. Our findings are in excellent qualitative agreement with the general picture for the $\sigma/f_0(500)$ that emerged in the past two decades from dispersive approaches \cite{Pelaez:2015qba}.Comment: 10 pages, 7 figures, v2: minor changes, version accepted by PR