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 the three-body systems, two of which can form a bound states.


Introduction
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 been recently explored in lattice QCD. Using the Nambu-Bethe-Salpeter (NBS) wave function, linked to the S-matrix in QCD [1][2][3][4][5][6][7][8][9], the hadron interactions have been investigated mainly by two methods: the finite volume method [1] and the HAL QCD potential method [5][6][7]. Theoretically the two methods in principle give same results of the scattering phase shift between two hadrons, while in practice they sometimes show different numerical results for two baryon systems, whose origin has been clarified recently in Refs. [10,11].
The first method relies on Lüscher's finite volume formula [1] that relates energies of two hadrons on finite volume to the phase shifts in infinite volume by utilizing the NBS wave function in the asymptotic region. In practice, energies of the two hadron system are extracted from the temporal correlation of the NBS wave function summed over spatial coordinates and are transformed into the corresponding phase shifts via the finite volume formula. The second method utilizes the NBS wave function in non-asymptotic (interacting) region, and extracts the non-local but energy-independent potentials from the space and time dependences of the NBS wave function. Physical observables such as phase shifts and binding energies are then calculated by solving the Schrödinger equation in infinite volume using the obtained potentials, since the asymptotic behavior of the NBS wave function is related to the T -matrix element and thus to the phase shifts [9]. In practice, the non-local potential is given by the form of the derivative expansion, which is truncated by the first few orders [12]. This method has been successfully applied to a wide range of two (or three) hadron systems at heavy pion masses [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] as well as at the nearly physical mass [29][30][31][32][33][34][35].
While a relation of the asymptotic behaviors of the NBS wave functions to the scattering phase shift (or more generally the S-matrix) is important for both methods, in particular, for the HAL QCD potential method, the theoretical arguments for the relation are rather limited: Two-body relativistic systems without bound states have been discussed in several different ways [2][3][4]36], while n-body non-relativistic systems without bound states have been considered in Ref. [9], using the Lippmann-Schwinger equation.
In these systems, its Hilbert space is of course expanded only by scattering states, so that the asymptotic states are composed of also 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. [9] to the two-and three-body systems with a bound state. In the approach, we split the Hamiltonian into a free part which reproduces all energy spectrum and an interacting part. The free part includes not only scattering states but also a bound state, since both appear as asymptotic states and thus there are no reason to exclude the latter. We first consider the Lippman-Schwinger equations for the two-body scalar system with a bound state in Sec.2, and then derive the asymptotic behaviors of the corresponding NBS wave functions in Sec.3. We then generalize our analysis to the threebody scalar system with a bound state: We consider the three-body Lippmann-Schwinger equations in Sec.4, and derive the asymptotic behaviors of the corresponding NBS wave functions in Sec.5.

Lippmann-Schwinger equation for two scalar fields
Let us first consider a Hamiltonian H composed of two complex scalar fields with the same (physical) mass m a = m b = m (denoted by φ a and φ b ) whose interaction leads to an S-wave bound state with the (physical) mass m B (< 2m). In this section, we derive the Lippmann-Schwinger equation in the two-body system, following the definition and notation in Refs. [9] and [37]. The Hamiltonian is divided into two terms, a free Hamiltonian H 0 and an interaction V , where a free eigenstate |α 0 and an asymptotic in-state |α in satisfy for the same energy E α . In order to deal with the bound state in this description, we include the scalar field φ B corresponding to the bound state composed of φ a and φ b in the 2/16 Hamiltonian as with π i (i = a, b) and π B conjugate momenta for φ i (i = a, b) and φ B , respectively. The Heisenberg operator for the scalar fields at t = 0 can be expressed in terms of the creation operator of the free anti-particle and annihilation operator of the free particle as with E ki = m 2 i + (k i ) 2 . Note that this form fixed at t = 0 does not hold generally at t = 0 if V = 0.
In general, eigen-states |α 0,in contain, n a (nā) φ a -particles (anti-particles), n b (nb) φ bparticles (anti-particles), and n B (nB) φ B -bound states (anti-bound states). In this section, to consider two-body scattering, we focus on states with n a = n b = 1 and n B = 1, denoted as whose eigen-energies are given by The Lippmann-Schwinger equation formally relates the in-state |α in to the free-particle state |α 0 as By inserting a complete set of free-particle states into the second term, the equation reduces to where dβ represents both summation over all bound states and integration over all scattering states. Note that unlike the system without a bound state discussed in Ref. [9], the complete set of free particle states includes the bound states. Let us consider the Lippmann-Schwinger equation for the two particle (scattering) state k a , k b in as where the third term appears since the bound state has the same quantum number of the two-particle state with n a = n b = 1. We therefore need to consider the equation for the 3/16 bound state as where the LSZ reduction formula tells us that T q B ,k B vanishes. Let us also remind readers that T q a ,q b ;k a ,k b has a pole corresponding to the bound state at an imaginary momentum. For simplicity we consider the center-of-mass frame, which implies k ≡ k a = −k b for the scattering state and k B = 0 for the bound state. By using the momentum conservation, the equations can be written as with q = q a = −q b . We have removed some of "iǫ" if the corresponding denominator does not lead to any real poles. As mentioned before,

Asymptotic behaviors of the NBS wave functions for two-body scalar system with a bound state
In this section, we derive the asymptotic behavior of the equal-time NBS wave functions for two scalar fields, φ a and φ b , in the center of mass system as with r = x a − x b . By substituting Eq.(12) and Eq.(13) into Eq. (15), we obtain As

4/16
Consequently, in the asymptotic region, x a − x b ≫ 1, the NBS wave functions reduce to The rotational invariance implies that Z(q) = Z q with q = |q|, and we can write the spherical expansions as After performing the integration over Ω q , we obtain If k 2 is below the 4-particle threshold (2 does not have any poles or cuts on the real q axis. Then the integration in Eq. (26) can be performed at large r [3,4,6,36], by picking up the contribution from the pole at where the contributions from the singularities in the upper half-plane become exponentially small in the asymptotically large r region [36]. Using the relation between the T -matrix and the phase of the S-matrix in Appendix A, Ψ lm ab (r, k) is expressed as which shows that the NBS wave function encodes the information of the scattering phase shift in its asymptotic behavior as if it were the wave function of the quantum mechanics [3,4,36], so that the potential defined from the NBS wave functions can reproduce the scattering phases shift of QCD [6,7,9] even if the bound state exists in this channel. The integral in Eq. (27), on the other hand, is evaluated at large r as r + other exponentially damping contributions, (30) with κ B = m 2 − m 2 B /4. In the above, other exponentially damping contributions come from poles or cuts in Z q E q or the half-off shell T -matrix T 2−1 0 (q) in the upper half plane of the complex q, which are expected to of the order of the typical mass scale of the system such as m. Therefore Ψ B,00 ab (r) is dominated by the first term as long as κ B is smaller than the typical mass scale, which is Λ QCD or m π in the case of QCD. Therefore, the potential constructed from the NBS wave functions including Ψ B 00 ab (r) can reproduce the binding energy of the bound state. This result ensures that the time dependent HAL QCD method [38] works to extract the potential correctly from the correlation function, 0|φ a (x a , t)φ b (x b , t)J (0)|0 with a source operator J (0), which couples to the bound state together with scattering states. This is a main result of this paper.

Lippmann-Schwinger equation in the nonrelativistic three-body scalar system
with a bound state in a two-body sub system As an extension of the analysis in the previous sections, we consider the system of three complex scalar fields, φ i (i = a, b, c) with the same mass m, where the interaction between φ a and φ b leads to one bound state denoted by φ B with a mass m B (< 2m), and the other two-body interactions do not lead to any other bound states. We examine how interactions between the fundamental scalar particles and the bound particle are encoded in their NBS wave functions. In this and the following sections, we consider which are eigenstates of the free Hamiltonian H 0 coupled with each other by the interaction, where the corresponding energies are given by As before, the scalar fields in the Heisenberg representation at t = 0 can be expressed in terms of the creation and annihilation operators as 6/16 For the three-body system, it is convenient to introduce the modified Jacobi coordinates [9] as,x where x J i and k J i are the standard Jacobi coordinates, In the center of mass frame, we have where the nonrelativistic approximation is used for the energy. In these coordinates, threebody noninteracting state and in-state in the center of mass frame can be parametrized as The Lippmann-Schwinger equations in the center of mass frame for k a ,k b in and k c , k B = −k c in are given within the nonrelativistic approximation as where T q a ,q b ,q c ;k c ,k B = δ(q a + q b + q c )T 3−2 (q a ,q b ; k c ).
For the latter convenience, by arranging the momenta as Q 2 = q c , K 2 = k c , Q 3 = (q a ,q b ) and K 3 = (k a ,k b ), we rewrite the above coupled channel T -matrix (or S-matrix) compactly as T l−m (Q l ; K m ) (or S l−m (Q l ; K m )) with l, m = 2, 3.

Asymptotic behaviors of NBS wave functions for three-body nonrelativistic scalar system with a bound state
The NBS wave functions in the coupled channel are compactly written as where By using the Lippmann-Schwinger equation given in Eq. (38) and Eq.(39), the NBS wave functions within the nonrelativistic approximation can be written as where [dP] n ≡ 3 (6−3n)/2 d (3n−3) P n = 3 (6−3n)/2 P 3n−4 n dP n dΩ Pn , P n = |P n |, As shown in Appendix B, we have in the asymptotic region, |x i − x j | ≫ 1 for i, j = a, b, c or |x c − x B | ≫ 1, where Thus the asymptotic behaviors of NBS wave functions reduce to Note that the third and forth terms appear due to the transition from vacuum to the state with anti-bound particle and the a-and b-particles. They do not appear in the coupled channel three-body system in the absence of bound states, as was studied in Ref. [9]. We 8/16 introduce the hyperspherical expansion as where X l = |X l |, and j D L is the hyperspherical Bessel function of the first kind, defined by with L D = L + D−2 2 and the Bessel function of the first kind J LD (x). Explicitly, they are expressed as and Y [L]2 corresponds to the spherical harmonic function Y lm . Within the non-relativistic approximation, we can write D l (K l ) and D 23 ( where In the nonrelativistic expansion, The integration over P l for the large X l in the second term can be performed by picking up the poles inside the closed contour in a complex plane as has been done in Ref. [9], where K l and K m satisfy E Kl = E Km = E for a given total energy E, and N L (x) is the Bessel function of the second kind. Explicitly, K 2 and K 3 are written as Using the parametrization of the T -matrix derived in Appendix A as and the asymptotic behaviors of J L (x) and N L (x) for large |x| as we obtain the asymptotic forms related to the phase shifts as for l, m = 2 ( B, c ) and l, m = 3 (a, b, c). The final result agrees with the one obtained in the absence of bound states [9].

Summary and concluding remarks
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 S-matrix via its unitary, while the asymptotic form for the NBS function of the bound state decays exponentially with its binding energy as long as the binding energy is smaller than other mass scales such as the mass of the scattering particle. This result establishes that the potential extracted via the time-dependent HAL QCD method [38] from the four-point correlation functions, which couple to the bound state as well as scattering states, can correctly reproduce both binding energy and scattering phase shift. This is the main result of this paper. 10/16 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 threeparticles, three particles are scattered into one fundamental particle plus one bound state and vice versa. Although 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. [9].

A. On-shell T-matrix and its parametrization
In this appendix, we parametrize the T -matrix, where S = 1 − iT , using the unitarity of the S-matrix, S † S = 1. We first discuss a simple two-body system and then consider a three-body system with a bound state.

A.1. Two-body case
In the center-of-mass frame that k a = −k b = k, the S-matrix is denoted as where q a = −q b = q, E q = 2 q 2 + m 2 , and E k = 2 √ k 2 + m 2 with q = |q| and k = |k|. The unitarity of the S-matrix reads Introducing expansions in terms of the spherical harmonic functions as and we obtain which is solved as 11/16 with the phase shift δ 2−2 l (q). Thus, the T -matrix, defined from the S-matrix as δ(E q − E k )S 2−2 (q; k) = δ(q − k) − iδ(E q − E k )T 2−2 (q; k) , is expressed as where

A.2. Three-body case
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 S-matrix as S l−m (Q l ; K m ), as for the T -matrix in the main text, the unitarity condition is expressed as where l, n, m = 2, 3 represent a number of particles involved in the state, dP n = 3 (3n−6)/2 d (3n−3) P n = 3 (3n−6)/2 (P n ) 3n−4 dP n dΩ Pn , P n = |P n |.
Introducing the hyperspherical expansion as [9] S(Q l ; K m ) = and using where m 2 = m red , m 3 = m, and Q l (E) is a solution of E Ql = E, the unitarity condition for a total energy E reads where C l (E) = 3 (3l−6)/2 m l {Q l (E)} 3l−5 .
(A17) 12/16 Thus we can parametrize the above S-matrix as we evaluate the vacuum contribution to the two-body and three-body NBS wave functions.
Extensions to the general n-body NBS wave functions are straightforward.

B.1. Two-body case
As seen in Eq. (16) and Eq. (17),  (4). Nonzero contribution in Eq.( B2), which comes from is evaluated as where Since the integrand in the last line does not have any poles on the real axis of |k a | as in the case of Sec. 3, this contribution vanishes exponentially in large x a − x b . Note that complex poles from the denominator appear at k 2 = −m 2 or on the unphysical sheet of k 2 which satisfies 2E k + m B = 0. 13/16 (ii) In the same manner, we obtain As shown in Appendix A of Ref. [9], this reduces to in the center of mass system, where Z(q) corresponds to the renormalization factor for the vacuum, whose explicit form can be found in Ref. [9].

B.2. Three-body case
As seen in Eq.(46), we need to evaluate in 0|Φ l (x l )|K m 0 . Using the Lippmann-Schwinger equation for the vacuum state, we have We first consider the case with l = m, which has already been analyzed in Appendix A of Ref. [9] for n = 2 and n = 3, and the result is given by for the asymptotic region, where Here m a,b,c = m, k c + k B = 0 (k b + k b + k c = 0) for D 2 (D 3 ), and T 0;iī is the off-shell Tmatrix from vacuum to a pair of particle-antiparticle with a flavor i. We consider other cases with l = m.
whose second term vanishes as |x B − x c | → ∞. Thus we have . (B16)