Virtual states and generalized completeness relation in the Friedrichs Model

Abstract

We study the well-known Friedrichs model, in which a discrete state is coupled to a continuum state. By examining the pole behaviors of the Friedrichs model in a specific form factor thoroughly, we find that, in general, when the bare discrete state is below the threshold of the continuum state, there should also be a virtual-state pole accompanying the bound-state pole originating from the bare discrete state as the coupling is turned on. There are also other second-sheet poles originating from the singularities of the form factor. We give a general argument for the existence of these two kinds of states. As the coupling is increased to a certain value, the second-sheet poles may merge and become higher-order poles. We then discuss the completeness relations incorporating bound states, virtual states, and resonant states corresponding to higher-order poles.

15 pages, 8 figures; Minor modifications, imporved figures, version to appear in Phys. Rev. D

References in corpus (6)

• The $\kappa$ resonance in s wave $\pi K$ scatterings
• Left Hand Singularities, Hadron Form Factors and the Properties of the Sigma Meson
• Hadron loops effect on mass shifts of the charmed and charmed-strange spectra
• Gamow-Jordan Vectors and Non-Reducible Density Operators from Higher Order S-Matrix Poles
• The origin of light $0^{+}$ scalar resonances
• Comprehending heavy charmonia and their decays by hadron loop effects

Cited by in corpus (15)

• Hadronic molecules
• An updated review of the new hadron states
• On the nature of near-threshold bound and virtual states
• A Review on Partial-wave Dynamics with Chiral Effective Field Theory and Dispersion Relation
• Generalization of Weinberg's Compositeness Relations
• Discussions on the Line-shape of $X(4660)$ Resonance
• Comprehending Isospin breaking effects of $X(3872)$ in a Friedrichs-model-like scheme
• Bound state and non-Markovian dynamics of a quantum emitter around a surface plasmonic nanostructure
• Two-pole structures in a relativistic Friedrichs-Lee-QPC scheme
• Partial Wave Decomposition in Friedrichs Model With Self-interacting Continua
• Does the bottomonium counterpart of $X(3872)$ exist?
• Relativistic Friedrichs-Lee model and quark-pair creation model
• Inverse scattering problem with a bare state
• Understanding $X(3872)$ and its decays in the extended Friedrichs scheme
• Understanding the 1P- and 2S-wave nucleon resonances within the extended Lee-Friedrichs Model