# $X(3872)$ as a four-quark state in a Dyson-Schwinger/Bethe-Salpeter approach

## TL;DR

This paper extends the Dyson-Schwinger and Bethe-Salpeter framework to study four-quark states with unequal quark masses, applying it to the $X(3872)$ and related states, revealing dominant meson-meson components and predicting their masses.

## Contribution

It introduces a generalized approach for four-quark states with unequal masses and applies it to the $X(3872)$, providing insights into its internal structure and mass predictions.

## Key findings

- Heavy-light meson-meson component is dominant in $X(3872)$
- Predicted mass of $X(3872)$ is 3916(74) MeV
- Predicted mass of $csar{s}ar{c}$ state is 4068(61) MeV

## 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.