New method to extract information of near-threshold resonances: Uniformized Mittag-Leffler expansion of Green's function and T matrix

Wren Yamada,Osamu Morimatsu

doi:10.1103/physrevc.102.055201

Abstract

We propose a new, simple model-independent method to extract information of near-threshold resonances, such as complex energies and residues. The method is based on the observation that the Green's function and the T-matrix can be represented as the sum of all poles, both bound and resonant poles, in the complex plane of a variable in which the Green's function and the T-matrix are single-valued functions. The symmetries of poles, which arise from the unitarity of the S-matrix, naturally impose the sum to obey the proper threshold behaviors. The imaginary part of Green's function and the T-matrix are directly related to observables such as scattering cross sections or invariant or missing mass distributions of hadron resonances. Thus we can determine their pole positions and residues by fitting their imaginary part to observables. We also test the new method by regarding the imaginary part of the $T$-matrix calculated exactly in a model theory as virtual experimental data. As a model theory, we take double-channel meson-baryon scatterings in the chiral unitary model with channels, $\overline{K}N (I=0)$, and $\pi\Sigma (I=0)$. By fitting the imaginary part of the $T$-matrix calculated in the model theory by that of the uniformized pole-sum, we obtain the pole positions and residues. Comparing the obtained results with those of the exact calculation in the model theory, we conclude that our new method works very well.

Highlights

Resonances and threshold behaviors of hadron scatterings are characteristic nonperturbative phenomena in strong interaction physics
To extract information on resonances from experimental data, we must link the experimental observables to analytic functions, such as the T matrix or the Green’s function. We briefly review their relations and explain our notations used in the present paper, having in mind the resonances in the meson-baryon system
The experimental circumstances may not be perfect. Even in such a situation, the use of the uniformized Mittag-Leffler expansion would provide us with a framework which is theoretically more reasonable and practically more useful than the usual methods in the sense that it automatically incorporates the proper threshold behaviors

Summary

Introduction

Resonances and threshold behaviors of hadron scatterings are characteristic nonperturbative phenomena in strong interaction physics. Resonances and hadron scattering processes, threshold behaviors, in particular, correspond to poles and branch points of an analytic function, the S matrix. A resonance is defined by the pole of the scattering amplitude, A, as a Breit-Wigner form [1], √ A( s) ∼ √ R (1) s − MR i. [2]), A(s) ∼ MR MR2 − iMR (2). Where s is the center-of-mass energy squared, and MR and R are the mass and the width of the resonance, respectively

Methods

Results

Conclusion