Asymptotic behavior of Nambu-Bethe-Salpeter wave functions for scalar systems with a bound state

We study the asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave functions, which are important for the HAL QCD potential method to extract hadron interactions, in the case that a bound state exists in the system. We consider the complex scalar particles, two of which lead to the formation of a bound state. In the case of the two-body system, we show that the NBS wave functions for the bound state, as well as scattering states in the asymptotic region, behave like the wave functions in quantum mechanics, which carry the information of the binding energy as well as the scattering phase shift. This analysis theoretically establishes under some conditions that the HAL QCD potential can correctly reproduce not only the scattering phase shift but also the binding energy. As an extension of the analysis, we also study the asymptotic behaviors of all possible NBS wave functions in the case of three-body systems, two of which can form a bound states.