Wave functions of composite hadron states and relationship to couplings of scattering amplitudes for general partial waves

TL;DR

This paper generalizes the relationship between scattering amplitudes and wave functions for all partial waves, extending Weinberg's compositeness condition to coupled channels and resonances, with an application to the $ ho$ meson.

Contribution

It introduces a generalized sum rule linking scattering amplitude residues to wave functions for any partial wave and coupled channels, broadening the scope of Weinberg's compositeness condition.

Findings

The $ ho$ meson is not a simple $ ext{pi} ext{pi}$ composite.

The sum rule applies to bound states and resonances.

The method uses only experimental data.

Abstract

In this paper we present the connection between scattering amplitudes in momentum space and wave functions in coordinate space, generalizing previous work done for s-waves to any partial wave. The relationship to the wave function of the residues of the scattering amplitudes at the pole of bound states or resonances is investigated in detail. A sum rule obtained for the couplings provides a generalization to coupled channels, any partial wave and bound or resonance states, of Weinberg's compositeness condition, which was only valid for weakly bound states in one channel and s-wave. An example, requiring only experimental data, is shown for the $\rho$ meson indicating that it is not a composite particle of $\pi \pi$ but something else.