Two-body wave functions and compositeness from scattering amplitudes. I. General properties with schematic models

Abstract

For a general two-body bound state in quantum mechanics, both in the stable and decaying cases, we establish a way to extract its two-body wave function in momentum space from the scattering amplitude of the constituent two particles. For this purpose, we first show that the two-body wave function of the bound state corresponds to the residue of the off-shell scattering amplitude at the bound state pole. Then, we examine our scheme to extract the two-body wave function from the scattering amplitude in several schematic models. As a result, the two-body wave functions from the Lippmann--Schwinger equation coincides with that from the Schrödinger equation for an energy-independent interaction. Of special interest is that the two-body wave function from the scattering amplitude is automatically scaled; the norm of the two-body wave function, to which we refer as the compositeness, is unity for an energy-independent interaction, while the compositeness deviates from unity for an energy-dependent interaction, which can be interpreted to implement missing channel contributions. We also discuss general properties of the two-body wave function and compositeness for bound states in the schematic models.

18 pages, 14 eps files; version accepted for publication in PRC; Section IID is added to discuss the model dependence of the compositeness

References in corpus (8)

• Comprehensive analysis of the wave function of a hadronic resonance and its compositeness
• Couplings in coupled channels versus wave functions in the case of resonances: application to the two $\Lambda (1405)$ states
• Remarks on pole trajectories for resonances
• Dynamical coupled-channels study of pi N --> pi pi N reactions
• Generalized weak-binding relations of compositeness in effective field theory
• Hadron mass scaling near the s-wave threshold
• Elementarity of composite systems
• The composite nature of the $\Lambda (1520)$ resonance

Cited by in corpus (2)

• Probing the long-range structure of the $T_{cc}^{+}$ with the strong and electromagnetic decays
• Two-body Wave Functions, Compositeness, And The Internal Structure Of Dynamically Generated Resonances