^{3}

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.

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. Masses of the single stable hadrons obtained from lattice QCD show good agreement with the experimental results, and even hadron interactions have recently been explored in lattice QCD. Using the Nambu–Bethe–Salpeter (NBS) wave function, linked to the S-matrix in QCD [

The first method relies on Lüscher’s finite volume formula [

While a relation of the asymptotic behaviors of the NBS wave functions to the scattering phase shift (or more generally the

In these systems, its Hilbert space is of course expanded only by scattering states, so that the asymptotic states are also composed of only the scattering states. If the system contains bound states, on the other hand, the Hilbert space is expanded by bound states as well as the scattering states. This situation has never been considered in previous works and will be discussed in this paper.

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 apply the Lippmann–Schwinger approach in Ref. [

Let us first consider a Hamiltonian

In general, eigenstates

The Lippmann–Schwinger equation formally relates the in-state

By inserting a complete set of free-particle states into the second term, the equation reduces to

Let us consider the Lippmann–Schwinger equation for the two particle (scattering) state

For simplicity we consider the center-of-mass frame, which implies

In this section we derive the asymptotic behavior of the equal-time NBS wave functions for two scalar fields,

As shown in

Consequently, in the asymptotic region,

The rotational invariance implies that

After performing the integration over

If ^{1}

The integral in Eq. (

As an extension of the analysis in the previous sections, we consider the system of three complex scalar fields

In this and the following sections, we consider

As before, the scalar fields in the Heisenberg representation at

For the three-body system, it is convenient to introduce the modified Jacobi coordinates [

In the center-of-mass frame, we have

The Lippmann–Schwinger equations in the center-of-mass frame for

For later convenience, by arranging the momenta as

The NBS wave functions in the coupled channel are compactly written as

By using the Lippmann–Schwinger equation given in Eqs. (

As shown in

Thus the asymptotic behaviors of NBS wave functions reduce to

Note that the third and fourth terms appear due to the transition from vacuum to the state with anti-bound particle and the

The integration over

In the nonrelativistic expansion,

The integration over

Using the parametrization of the

In this paper we have investigated the asymptotic behaviors of the NBS wave functions in complex scalar systems in the presence of a bound state. We include a bound state in the asymptotic states of the theory and consider a coupled system of two scattering particles and one bound state. We have shown that the asymptotic form for the NBS wave function of the scattering state is related to the phase shifts of the

As an extension of our analysis, we have considered the three complex scalar systems, two of which can form a bound state. In addition to the the elastic scattering of the three-particles, three particles are scattered into one fundamental particle plus one bound state and vice versa. Although the analysis becomes rather involved in this case, the final result in the non-relativistic limit becomes almost identical to the one for the coupled two- and three-particle systems in Ref. [

The authors thank T. Doi, T. Hyodo, and Y. Ikeda for fruitful discussions and useful comments. S. G. is supported by the Special Postdoctoral Researchers Program of RIKEN. S. A. is supported in part by a Grant-in-Aid of the Japanese Ministry of Education, Sciences and Technology, Sports and Culture (MEXT) for Scientific Research (No. JP16H03978), by a priority issue (Elucidation of the fundamental laws and evolution of the universe) to be tackled by using Post “K” Computer, and by the Joint Institute for Computational Fundamental Science (JICFuS).

Open Access funding: SCOAP

In this appendix, we parametrize the

In the center-of-mass frame such that

The unitarity of the

Introducing expansions in terms of the spherical harmonic functions as

We next consider the scattering of the three-body system, where not only the elastic scattering but also the bound particle production and its inverse process occur:

Denoting the corresponding

Introducing the hyperspherical expansion as [

Thus we can parametrize the above

The corresponding

Using the Lippmann–Schwinger equation for the vacuum in-state,

As seen in Eqs. (

(i) Using the Lippmann–Schwinger equation for the vacuum in-state in Eq. (

(ii) In the same manner, we obtain

As shown in

As seen in Eq. (

We first consider the case with

Here,

We consider other cases with

(i)

Equation (

(ii)

Equation (

^{1} The condition for