Practical parametrization of two-pole structure

TL;DR

This paper introduces a practical parametrization method based on the extended Lee-Friedrichs model for analyzing resonance structures, effectively capturing two-pole phenomena and respecting physical constraints, aiding future exotic hadron studies.

Contribution

The paper presents a novel, practical parametrization approach using the extended Lee-Friedrichs model that incorporates physical constraints and effectively identifies two-pole structures in resonance analysis.

Findings

Effectively fits resonance data with comparable quality to Breit-Wigner models.

Successfully extracts poles near the physical region, including two-pole structures.

Demonstrates applicability to various hadron states like $ ho(770)$ and $K_0^*(700)$.

Abstract

We suggest that the extended Lee-Friedrichs model could be directly used as a practical parametrization method for the experimental analysis of resonance structures. This parametrization incorporates the constraints of relativistic phase space and the threshold behavior, and respects both unitarity and analyticity constraints of the scattering amplitude. As such, the poles on unphysical Riemann sheets could be easily extracted. This parametrization method offers a comparable fit quality to the improved Breit-Wigner parametrization with an energy-dependent width function when the coupling strength is moderate. It is found that the parametrization could be used to correctly extract the poles near the physical region correctly. In particular, it can naturally incorporate the two-pole structure in which one pole is shifted from the discrete state and the other is dynamically generated. Moreover, the coupled-channel formulation of the extended Lee-Friedrichs parameterization is straightforward and its relationship with the Flatt\'{e} parametrization form is discussed. Using $\rho(770)$, $\Delta(1232)$, $K^*_0$ and $f_0$ states as illustrative examples, we demonstrate the effectiveness of this parametrization in capturing fit qualities and identifying relevant poles. It is illustrated that the $K_0^*(700)$ and $K_0^*(1430)$ could be perfectly parameterized in the Lee-Friedrichs form as a two-pole structure. A tentative investigation of $f_0$s in coupled-channel parametrization form are discussed, and a possible lineshape contributed by $\chi_{c1}(3872)$ and $\chi_{c1}(4012)$ is presented. The proposed parametrization scheme holds promise for future studies involving exotic hadron states near thresholds, offering a valuable tool for analyzing resonance structures in upcoming experimental investigations.