# Understanding $X(3862)$, $X(3872)$, and $X(3930)$ in a Friedrichs-model-like scheme

## TL;DR

This paper introduces a Friedrichs-model-like approach to study hadron resonances, successfully reproducing certain states and predicting new ones by coupling quark model states with continuum states, offering insights into quark model discrepancies.

## Contribution

It presents a novel scheme combining quark potential and pair creation models to analyze hadron resonances as Gamow states, improving understanding of observed and predicted states.

## Key findings

- Successfully reproduces $X(3862)$, $X(3872)$, and $X(3930)$ with a one-parameter model
- Predicts a $h_c(2P)$ state at about 3902 MeV with a 54 MeV width
- Finds large compositeness for the $X(3872)$ state

## Abstract

We developed a Friedrichs-model-like scheme in studying the hadron resonance phenomenology and present that the hadron resonances might be regarded as the Gamow states produced by a Hamiltonian in which the bare discrete state is described by the result of usual quark potential model and the interaction part is described by the quark pair creation model. In a one-parameter calculation, the $X(3862)$, $X(3872)$, and $X(3930)$ state could be simultaneously produced with a quite good accuracy by coupling the three P-wave states, $\chi_{c2}(2P)$, $\chi_{c1}(2P)$, $\chi_{c0}(2P)$ predicted in the Godfrey-Isgur model to the $D\bar D$, $D\bar D^{*}$, $D^*\bar D^*$ continuum states. At the same time, we predict that the $h_c(2P)$ state is at about 3902 MeV with a pole width of about 54 MeV. In this calculation, the $X(3872)$ state has a large compositeness. This scheme may shed more light on the long-standing problem about the general discrepancy between the prediction of the quark model and the observed values, and it may also provide reference for future search for the hadron resonance state.