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

Abstract

For a general two-body bound state in quantum mechanics, both in the stable and decaying cases, I 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, I 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, I examine my scheme to extract the two-body wave function from the scattering amplitude in several schematic models. As a result, the two-body wave function 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 I 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. I also discuss general properties of the two-body wave function and compositeness for bound states with the schematic models.

Construction of a local K¯N−πΣ−πΛ potential and composition of the ΛMiyahara1, Hyodo2, Weise3 2018Phys. Rev. C403361View full textAdd to dashboardCite

“…At the same time, this formulation substantiates the "elementarity" interpretation of the energy-derivative term, previously discussed in Ref. [27]. We comment briefly on the relation between the energy dependence of the potential and positivity aspects in Q space.…”

Hadronic molecules

“…(Baru et al, 2004), where the spectral density was employed to supplement the parameter λ 2 introduced above for bound states. The subject was later elaborated in various papers Guo and Oller, 2016a;Hyodo, 2013a,b;Kang et al, 2016;Sekihara, 2017;Sekihara et al, 2015;Xiao andZhou, 2016a,b, 2017); however, what is common to all of them is that a quantitative, probabilistic extraction of the level of compositeness is not possible rigorously as soon as one moves to resonances. The reason is that states that belong to poles on the second sheet are not normalizable and as such one looses the condition of Eq.…”

NΩ interaction: Meson exchanges, inelastic channels, and quasibound state

“…In order to investigate the properties of the N Ω( 5 S 2 ) quasibound state, we calculate its wave function from the residue of the T -matrix at the pole position, according to the approach in Ref [47]. The off-shell T -matrix contains the pole in the following expression…”