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 $\pi \pi$ and $K\overline{K}$ but something else.

• Received 14 March 2012

DOI:https://doi.org/10.1103/PhysRevD.86.014012