Abstract
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 the three-body systems, two of which can form a bound states.
Highlights
Lattice quantum chromodynamics (QCD) is a successful non-perturbative method to study hadron physics from the underlying degrees of freedom, i.e. quarks and gluons
Since the integrand in the last line does not have any poles on the real axis of |k a | as in the case of Sect. 3, this contribution vanishes exponentially in large xa − xb
Note that complex poles from the denominator appear at k 2 = −m2 or on the unphysical sheet of k 2 which satisfies 2Ek + mB = 0
Summary
Lattice quantum chromodynamics (QCD) is a successful non-perturbative method to study hadron physics from the underlying degrees of freedom, i.e. quarks and gluons. The aim of this paper is to relate the asymptotic behaviors of NBS wave functions in scalar systems to their phase shifts and binding energy in the presence of one bound state. We first consider the Lippman–Schwinger equations for the two-body scalar system with a bound state, and derive the asymptotic behaviors of the corresponding NBS wave functions in Sect. 3. We generalize our analysis to the three-body scalar system with a bound state: We consider the three-body Lippmann–Schwinger equations, and derive the asymptotic behaviors of the corresponding NBS wave functions in Sect. 3. Asymptotic behaviors of the NBS wave functions for a two-body scalar system with a bound state. 5. Asymptotic behaviors of NBS wave functions for three-body non-relativistic scalar system with a bound state. The final result agrees with the one obtained in the absence of bound states [9]
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