Comprehensive analysis of the wave function of a hadronic resonance and its compositeness

# Comprehensive analysis of the wave function of a hadronic resonance and its compositeness

Takayasu Sekihara Tetsuo Hyodo, and Daisuke Jido
###### Abstract

We develop a theoretical framework to investigate the two-body composite structure of a resonance as well as a bound state from its wave function. For this purpose, we introduce both one-body bare states and two-body scattering states, and define the compositeness as a fraction of the contribution of the two-body wave function to the normalization of the total wave function. Writing down explicitly the wave function for a resonance state obtained with a general separable interaction, we formulate the compositeness in terms of the position of the resonance pole, the residue of the scattering amplitude at the pole and the derivative of the Green function of the free two-body scattering system. At the same time, our formulation provides the elementariness expressed with the resonance properties and the two-body effective interaction, and confirms the sum rule showing that the summation of the compositeness and elementariness gives unity. In this formulation the Weinberg’s relation for the scattering length and effective range can be derived in the weak binding limit. The extension to the resonance states is performed with the Gamow vector, and a relativistic formulation is also established. As its applications, we study the compositeness of the resonance and the light scalar and vector mesons described with refined amplitudes in coupled-channel models with interactions up to the next to leading order in chiral perturbation theory. We find that and are dominated by the and composite states, respectively, while the vector mesons and are elementary. We also briefly discuss the compositeness of and obtained in a leading-order chiral unitary approach.

## 1 Introduction

In hadron physics, the internal structure of an individual hadron is one of the most important subjects. Traditionally, the excellent successes of constituent quark models lead us to the interpretation that baryons consist of three quarks () and mesons of a quark-antiquark pair (Agashe:2014kda []. At the same time, however, there are experimental indications that some hadrons do not fit into the classification suggested by constituent quark models. One of the classical examples is the hyperon resonance , which has an anomalously light mass among the negative parity baryons. In addition, the lightest scalar mesons [, , , and ] exhibit inverted spectrum from the naïve expectation with the configuration. These observations motivate us to consider more exotic structure of hadrons, such as hadronic molecules and multiquarks Dalitz:1960du [], Dalitz:1967fp [], Jaffe:1976ig [], Jaffe:1976ih [], Weinstein:1982gc [], Weinstein:1983gd [].

It is encouraging that there have been experimental reports on the candidates of manifestly exotic hadrons such as charged quarkonium-like states by Belle collaboration Belle:2011aa []. Moreover, the LEPS collaboration observed the “ signal” Nakano:2003qx [], Nakano:2008ee [], but its interpretation is still controversial MartinezTorres:2010zzb [], Torres:2010jh []. The accumulation of the observations of unconventional states in the heavy quark sector reinforces the existence of hadrons with exotic structure Swanson:2006st [], Brambilla:2010cs []. In fact, recent detailed analyses of in various reactions Niiyama:2008rt [], Moriya:2013eb [], Lu:2013nza [], Agakishiev:2012xk [] and of the - mixing in decay Ablikim:2010aa [] are providing some clues for unusual structure of these hadrons. The exotic structure is also investigated by analyzing the theoretical models; the meson-baryon components of by using the natural renormalization scheme Hyodo:2008xr [], the scaling behaviors of scalar and vector mesons Pelaez:2003dy [], Pelaez:2004xp [] and of  Hyodo:2007np [], Roca:2008kr [], spatial size of  Sekihara:2008qk [], Sekihara:2010uz [], Sekihara:2012xp [], meson Albaladejo:2012te [], and  Sekihara:2012xp [], the nature of the meson from the partial restoration of chiral symmetry Hyodo:2010jp [], and the structure of and mesons studied by their Regge trajectories Londergan:2013dza []. The possibilities to extract the hadron structure from the production yield in relativistic heavy ion collisions Cho:2010db [], Cho:2011ew [] and from the high-energy exclusive productions Kawamura:2013iia [], Kawamura:2013wfa [] are also suggested.

Among various exotic structures, hadronic molecular configurations are of special interest. These states are composed of two (or more) constituent hadrons by strong interaction between them without losing the character of constituent hadrons, in a similar way with the atomic nuclei as bound states of nucleons. The quasi-bound picture for is one of the examples. In contrast to the quark degrees of freedom, the masses and interactions of hadrons are defined independently of the renormalization scheme of QCD, because hadrons are color singlet states. This fact implies that the structure of hadrons may be adequately defined in terms of the hadronic degrees of freedom. This viewpoint originates in the investigations of the elementary or composite nature of particles in terms of the field renormalization constant Salam:1962ap [], Weinberg:1962hj [], Ezawa:1963zz []. Indeed, it is shown in this approach that the deuteron is dominated by the loosely bound proton-neutron component Weinberg:1965zz []. The study of the structure of hadrons from the field renormalization constant have been further developed in Refs. Baru:2003qq [], Hanhart:2007cm [], Hanhart:2011jz [], Hyodo:2011qc [], Aceti:2012dd [], Xiao:2012vv [], Hyodo:2013iga [], Hyodo:2013nka [], Sekihara:2013sma [], Aceti:2014ala [], Aceti:2014wka [], Nagahiro:2014mba [], Sekihara:2014qxa [].

Motivated by these observations, in this study we develop a framework to investigate hadronic two-body components inside a hadron by analyzing comprehensively the wave function of a resonance state. For this purpose, we explicitly introduce one-body bare states in addition to the two-body components so as to form a complete set within them and to measure the elementary and composite contributions. The one-body component has not been taken into account in the preceding studies on wave functions (see Refs. Aceti:2012dd [], Gamermann:2009uq [], YamagataSekihara:2010pj []). For the resonance state we employ the Gamow vector Gamow:1928zz [], which ensures a finite normalization of the resonance wave function. The wave function from a relativistically covariant wave equation is also discussed. Making a good use of a general separable interaction, we analytically solve the wave equations.

In the present formulation, the compositeness and elementariness are respectively defined as the fractions of the contributions from the two-body scattering states and one-body bare states to the normalization of the total wave function. They are further expressed with the quantities in the scattering equation with a general separable interaction. As a consequence, the compositeness can be written in terms of the residue of the scattering amplitude at the pole position, i.e., the coupling constant, and the derivative of the Green function of the free two-body scattering system at the pole. This means that the compositeness can be obtained solely with the pole position of the resonance and the residue at the pole but without knowing the details of the two-body effective interaction. On the other hand, the elementariness is obtained with the residue of the scattering amplitude, the Green function and the derivative of the two-body effective interaction at the pole. It is an interesting finding that with this expression we are allowed to interpret the elementariness as the contributions coming from one-body bare states and implicit two-body channels which do not appear as explicit degrees of freedom but are effectively taken into account for the two-body interaction in the practical model space. Through the discussion on the multiple bare states, we show that our formulation of the compositeness and elementariness can be applied to any separable interactions with arbitrary energy dependence. Based on this foundation, as applications we evaluate the compositeness of hadronic resonances, such as , the light scalar mesons and vector mesons described in the chiral coupled-channel approach with the next-to-leading order interactions so as to discuss their internal structure from the viewpoint of hadronic two-body components.

This paper is organized as follows. In Sec. 2, we formulate the compositeness and elementariness of a physical particle state in terms of its wave function, and show their connection to the physical quantities in scattering equation. We first consider a two-body bound state in the nonrelativistic framework, and later extend the formulation to a resonance state in a relativistic covariant form with the Gamow vector. In Sec. 3 numerical results for the applications to physical resonances are presented. Section 4 is devoted to drawing the conclusion of this study.

## 2 Compositeness and elementariness from wave functions

In this section, we define the compositeness (and simultaneously elementariness) of a particle state, i.e., a stable bound state or a unstable resonance, using its wave function and link the compositeness to the physical quantities in scattering equation. For this purpose, we consider two-body scattering states222We note that the two-body wave functions are given by the asymptotic states of the system. In the application to QCD, the basis should be spanned by the hadronic degrees of freedom. The compositeness in terms of quarks cannot be defined in this approach. coupled with each other and one-body bare states. The one-body bare states have not been introduced in the studies of wave functions before, and the introduction of the one-body bare states makes it clear to implement the elementariness into the formulation. To solve the scattering equation analytically, we make use of the separable type of interaction. We will concentrate on an s-wave scattering system, and thus the two-body wave function and the form factors are assumed to be spherical.

In Sec. 2.1 we consider a bound state333In general, there can be several bound states in the system. In such a case, we just focus on a bound state out of these bound states. Nothing changes in the following discussion. in two-body scattering. We first introduce a one-body bare state and a single scattering channel, and give the expressions of the compositeness and the elementariness in terms of the wave function of the bound state. In Sec. 2.2 we extend the discussion to a system with multiple bare states and coupled scattering channels, in order to clarify further the meaning of the compositeness and elementariness obtained in Sec. 2.1. Here we also discuss a way to introduce general energy dependent interaction into the formulation. In Sec. 2.3 we consider the weak binding limit to derive the Weinberg’s relation for the scattering length and the effective range Weinberg:1965zz []. Generalization to resonance states is discussed in Sec. 2.4. Finally we give a relativistic covariant formulation in Sec. 2.5.

### 2.1 Bound state in the nonrelativistic scattering

We consider a two-body scattering system in which there exists a discrete energy level below the scattering threshold energy. We call this energy level bound state since it is located below the two-body scattering threshold. We do not assume the origin and structure of the bound state at all. We take the rest frame of the center-of-mass motion, namely two scattering particles have equal and opposite momentum and the bound state is at rest with zero momentum. The system in this frame is described by Hamiltonian which consists of the free part and the interaction term

 ^H=^H0+^V. (1)

We assume that the free Hamiltonian has continuum eigenstates for the scattering state and one discrete state for the one-body bare state. The eigenvalues of the Hamiltonian are set to be

 ^H0|q⟩ =(Mth+q22μ)|q⟩,⟨q|^H0=(Mth+q22μ)⟨q|, (2) ^H0|ψ0⟩ =M0|ψ0⟩,⟨ψ0|^H0=M0⟨ψ0|, (3)

where is the reduced mass of the two-body system, is the mass of the bare state, and . We include the sum of the scattering particle masses, , which is just the scattering threshold energy, into the definition of the eigenenergy for later convenience. These eigenstates are normalized as

 ⟨q′|q⟩=(2π)3δ3(q′−q),⟨ψ0|ψ0⟩=1,⟨ψ0|q⟩=⟨q|ψ0⟩=0. (4)

These states form the complete set of the free Hamiltonian, and thus we can decompose unity in the following way

 1=|ψ0⟩⟨ψ0|+∫d3q(2π)3|q⟩⟨q|. (5)

The bound state is realized as an eigenstate of the full Hamiltonian:

 ^H|ψ⟩=MB|ψ⟩,⟨ψ|^H=MB⟨ψ|, (6)

where is the mass of the bound state. The bound state wave function is normalized as

 ⟨ψ|ψ⟩=1. (7)

We take the matrix element of Eq. (2.1) in terms of the bound state :

 1=⟨ψ|ψ0⟩⟨ψ0|ψ⟩+∫d3q(2π)3⟨ψ|q⟩⟨q|ψ⟩. (8)

The first term of the right-hand side is the probability of finding the bare state in the bound state and also corresponds to the field renormalization constant in the field theory. Thus, we call this quantity elementariness :

 Z≡⟨ψ|ψ0⟩⟨ψ0|ψ⟩. (9)

Because , is always real and nonnegative. The second term, on the other hand, represents the contribution from the two-body state and we call it compositeness :

 X≡∫d3q(2π)3⟨ψ|q⟩⟨q|ψ⟩. (10)

The elementariness and compositeness satisfy the sum rule

 1=⟨ψ|ψ⟩=Z+X. (11)

Introducing the momentum space wave function for the two-body state, ,

 ~ψ(q)=⟨q|ψ⟩,~ψ∗(q)=⟨ψ|q⟩, (12)

the compositeness can be expressed as

 X=∫d3q(2π)3∣∣~ψ(q)∣∣2. (13)

Again, is real and nonnegative.

For the explicit calculation, we assume the separable form of the matrix elements of in the momentum space. The matrix elements are given by

 ⟨q′|^V|q⟩=vf∗(q′2)f(q2),⟨q|^V|ψ0⟩=g0f∗(q2),⟨ψ0|^V|ψ0⟩=0, (14)

where is the interaction strength between the scattering particles, and is the coupling constant of the bare state to the scattering state. As we will see later, the one-body state is the source of the energy dependence of the effective interaction between the scattering particles. The matrix element is taken to be zero since it can be absorbed into without loss of generality, and throughout this study the mass of the bare state, , is not restricted to be smaller than but is allowed to take any value with this condition. The form factor is responsible for the off-shell momentum dependence of the interaction and suppresses the high momentum contribution to tame the ultraviolet divergence. The normalization is chosen to be . The hermiticity of the Hamiltonian ensures that is real and

 ⟨ψ0|^V|q⟩=g∗0f(q2). (15)

In this study we further assume the time-reversal invariance of the scattering process, which constraints the interaction, with an appropriate choice of phases of the states, as

 ⟨q′|^V|q⟩=⟨q|^V|q′⟩=vf(q′2)f(q2),⟨q|^V|ψ0⟩=⟨ψ0|^V|q⟩=g0f(q2),⟨ψ0|^V|ψ0⟩=0. (16)

Thus all of the quantities , , and are now real. We emphasize that the assumptions made in the present framework are just the factorization of the momentum dependence and the time-reversal invariance of the interaction. With the interaction (2.1), we obtain the exact solution of this system without introducing any further assumptions.

For the separable interaction, the wave function can be analytically obtained Yamaguchi:1954mp []. To this end, we multiply and to Eq. (2.1):

 ⟨q|^H|ψ⟩ =(Mth+q22μ)~ψ(q)+vf(q2)∫d3q′(2π)3f(q′2)~ψ(q′)+g0f(q2)⟨ψ0|ψ⟩=MB~ψ(q), (17) ⟨ψ0|^H|ψ⟩ =M0⟨ψ0|ψ⟩+g0∫d3q(2π)3f(q2)~ψ(q)=MB⟨ψ0|ψ⟩, (18)

where we have inserted Eq. (2.1) between and . Eliminating from these equations, we obtain the Schrödinger equation for in an integral form:

 (Mth+q22μ)~ψ(q)+veff(MB)f(q2)∫d3q′(2π)3f(q′2)~ψ(q′)=MB~ψ(q), (19)

where we have defined the energy-dependent interaction as

 veff(E)≡v+(g0)2E−M0. (20)

Equation (2.1) is the single-channel Schrödinger equation for the relative motion of the scattering particles under the presence of the bare state interacting with them by . The effect of the bare state is incorporated into the energy dependent interaction .

The solution of Eq. (2.1) can be obtained as

 ~ψ(q)=−cf(q2)B+q2/(2μ), (21)

where we have defined the binding energy and the normalization constant

 c≡veff(MB)∫d3q′(2π)3f(q′2)~ψ(q′). (22)

In general, Eq. (2.1) is an integral equation to determine the wave function . For the separable interaction, however, the integral in Eq. (2.1) (and hence, the constant ) is independent of . In this way, the wave function is analytically determined by the form factor and the constant , which will be determined through the comparison with the scattering amplitude. Substituting the wave function (2.1) into Eq. (2.1), we obtain

 c=−veff(MB)∫d3q(2π)3[f(q2)]2B+q2/(2μ)c. (23)

For the existence of the bound state at , Eq. (2.1) should be satisfied with nonzero . The nontrivial solution can be obtained by

 1=veff(MB)G(MB), (24)

where we have introduced a function

 G(E)=∫d3q(2π)3[f(q2)]2E−Mth−q2/(2μ), (25)

which plays an important role in the following discussion and is called the loop function. As we will see later, the loop function is equivalent to the Green function of the free two-body Hamiltonian. We note that here and in the following the energy in the denominator of the loop function is considered to have an infinitesimal positive imaginary part : .

The normalization constant is equal to the square root of the residue of the scattering amplitude at the pole position of the bound state. To prove this, we first represent the compositeness and elementariness using . With the explicit form of the wave function (2.1) and the loop function (2.1), the compositeness for the separable interaction can be expressed with the derivative of the loop function as

 X=∫d3q(2π)3∣∣~ψ(q)∣∣2=−|c|2[dGdE]E=MB. (26)

We note that both the wave function and the loop function have the same structure of at . Substituting the wave function into Eq. (2.1), we obtain

 ⟨ψ0|ψ⟩=cg0MB−M0G(MB), (27)

and hence

 Z=⟨ψ|ψ0⟩⟨ψ0|ψ⟩=|c|2G(MB)(g0)2(MB−M0)2G(MB)=−|c|2[GdveffdEG]E=MB, (28)

where we have used the derivative of Eq. (2.1). We note that Eqs. (2.1) and (2.1) provide a sum rule

 1=−|c|2[dGdE+GdveffdEG]E=MB. (29)

Next the scattering amplitude is obtained by taking the matrix element of the -operator for the scattering state with the on-shell condition as for the separable interaction. The -operator satisfies the Lippmann-Schwinger equation

 ^T=^V+^V1E−^H0^T. (30)

Inserting the complete set (2.1) between the operators and eliminating the bare state component from the equation, we obtain the Lippmann-Schwinger equation for the scattering state as

 ^T=^Veff(E)+^Veff(E)1E−^H0^T, (31)

where we have introduced the operator of the effective interaction for the scattering state as

 ^Veff(E)≡^V+^V|ψ0⟩1E−M0⟨ψ0|^V. (32)

This operator acts only on the two-body state and its matrix element leads to . Taking matrix element of the two-body state in Eq. (2.1), we obtain the amplitude algebraically as

 t(E)=veff(E)+veff(E)G(E)t(E)=veff(E)1−veff(E)G(E), (33)

where is the same form as Eq. (2.1), i.e., the Green function of the free two-body Hamiltonian. The bound state condition (2.1) ensures that the amplitude has a pole at . The residue of the amplitude at the pole reflects the properties of the bound state. The residue turns out to be real and positive, so we represent the residue as :

 |g|2≡limE→MB(E−MB)t(E)=−1[dGdE+1(veff)2dveffdE]E=MB. (34)

We can interpret as the coupling constant of the bound state to the two-body state. Using the bound state condition (2.1), we obtain the relation

 1=−|g|2[dGdE+GdveffdEG]E=MB. (35)

Comparing this with Eq. (2.1), we find with an appropriate choice of the phase.

The equality is also confirmed by the following form of the -operator:

 ^T=^Veff(E)+^Veff(E)1E−^H0−^Veff(E)^Veff(E). (36)

As we have seen before, the operator corresponds to the full Hamiltonian for the two-body system with the implicit bare state. Near the bound state pole, the amplitude is dominated by the pole term in the expansion by the eigenstates of the full Hamiltonian as

 limE→MB^T(E) ∼^Veff(MB)|ψ⟩1E−MB⟨ψ|^Veff(MB), (37)

and hence, taking the matrix element of the scattering states, we have

 limE→MBt(E) ∼∫d3q(2π)3∫d3p(2π)3veff(MB)f(q2)⟨q|ψ⟩⟨ψ|p⟩E−MBf(p2)veff(MB)→|c|2E−MB, (38)

where we have used Eq. (2.1). From the definition of the residue (2.1), this verifies .

Here we emphasize that, as seen in Eq. (2.1), the compositeness is expressed with the residue of the scattering amplitude at the pole position and the energy derivative of the loop function , and hence the compositeness does not explicitly depend on the effective interaction .444Since the bound state properties are determined by the interaction, the compositeness depends implicitly on the effective interaction . Therefore, the compositeness can be obtained solely with the bound state properties without knowing the details of the effective interaction, once we fix the loop function, which coincides with fixing the model space to measure the compositeness via the Green function of the free two-body Hamiltonian.

We also note that, as seen in Eq. (2.1), the elementariness is proportional to the energy derivative of the interaction at the bound state energy. This is instructive to interpret the origin of the elementariness . In quantum mechanics, the two-body interaction should not depend on the energy to have an appropriate normalization. In the present case, the energy dependence of stems from the bare state channel . Strong energy dependence of the interaction at the bound-state pole position emerges when the involved bare state lies close to the physical bound state, and provides . This means that the effect from the bare state is responsible for the formation of the bound state. Weak energy dependence, which corresponds to , can be understood that the bare state exists far away from the pole position of the physical bound state, and is insensitive to form the bound state. In this case, the bound state is composed dominantly of the scattering channels considered. This shares viewpoints with Ref. Hyodo:2008xr [], where it was discussed that the energy-dependent Weinberg-Tomozawa term can provide the effect of the CDD pole Castillejo:1955ed [].

### 2.2 Coupled scattering channels with multiple bare states

The framework in the last subsection is straightforwardly generalized to the coupled-channel scattering with multiple one-body bare states. The eigenstates of the free Hamiltonian now include several bare states labeled by and two-body scattering states of several channels labeled by . We assume that the bound state of which the components we want to examine is located below the lowest threshold of the two-body channels to make the state stable. The normalization and the completeness relation are given by

 ⟨q′j|qk⟩=(2π)3δjkδ3(q′−q),⟨ψa|ψb⟩=δab,⟨ψa|qj⟩=⟨qj|ψa⟩=0, (39)
 1=∑a|ψa⟩⟨ψa|+∑j∫d3q(2π)3|qj⟩⟨qj|. (40)

The matrix elements of the interaction are

 ⟨q′j|^V|qk⟩=vjkfj(q′2)fk(q2),⟨qj|^V|ψa⟩=⟨ψa|^V|qj⟩=ga,j0fj(q2),⟨ψa|^V|ψb⟩=0, (41)

where, due to the time-reversal invariance, is a real symmetric matrix and and are real with an appropriate choice of phases of states. The total normalization of the bound state wave function now leads to

 1=∑aZa+∑jXj, (42)

with the elementariness

 Za≡⟨ψ|ψa⟩⟨ψa|ψ⟩, (43)

and the compositeness given by the wave function for each channel

 Xj≡∫d3q(2π)3∣∣~ψj(q)∣∣2, (44)

where

 ~ψj(q)=⟨qj|ψ⟩,~ψ∗j(q)=⟨ψ|qj⟩. (45)

We follow the same procedure as the single channel case; incorporating the one-body bare states to the effective interaction for the two-body states, we obtain the coupled Schrödinger equation as

 (Mthj+q22μj)~ψj(q)+∑kveffjk(MB)fj(q2)∫d3q′(2π)3fk(q′2)~ψk(q′)=MB~ψj(q), (46)

where and are the threshold and the reduced mass in channel , respectively, and we have defined the energy-dependent effective interaction as

 veffjk(E)≡vjk+∑aga,j0ga,k0E−Ma, (47)

which is a real symmetric matrix for a real energy and is the mass of the bare state. The Schrödinger equation can be solved algebraically again for the separable interaction:

 ~ψj(q)=−cjfj(q2)Bj+q2/(2μj), (48)

where is the binding energy measured from the -channel threshold. The normalization constant is given by

 cj≡∑kveffjk(MB)∫d3q(2π)3fk(q2)~ψk(q). (49)

With substitution of Eq. (2.2) to Eq. (2.2), the bound state condition for nonzero can be summarized as

 det[1−veff(MB)G(MB)]=0, (50)

with the loop function

 Gj(E)=∫d3q(2π)3[fj(q2)]2E−Mthj−q2/(2μj), (51)

which is diagonal with respect to the channel index.

The coupled-channel scattering equation is in the matrix form

 t(E)=[1−veff(E)G(E)]−1veff(E), (52)

where the channel index runs through only the scattering channels, since the one-body bare states are incorporated into the effective interaction . Equation (2.2) ensures the existence of the bound state pole at . The residue of the amplitude at the pole, which is real for the bound state, is interpreted as the product of the coupling constants555Since an interaction of a symmetric matrix leads to a symmetric -matrix, , the residue of the -matrix is also symmetric and can be factorized as .

 gjgk=limE→MB(E−MB)tjk(E). (53)

On the other hand, using the coupled-channel version of Eq. (2.1), the amplitude near the bound state pole is given by

 limE→MBtjk(E) ∼∑l,m∫d3q(2π)3∫d3p(2π)3veffjl(MB)fl(q2)⟨ql|ψ⟩⟨ψ|pm⟩E−MBfm(p2)veffmk(MB) →cjc∗kE−MB, (54)

which shows that with an appropriate choice of the phase.

Now the compositeness in channel can be expressed as

 Xj=∫d3q(2π)3∣∣~ψj(q)∣∣2=−|cj|2[dGjdE]E=MB=−|gj|2[dGjdE]E=MB. (55)

The overlap of the bound state wave function with the bare state is given by

 ⟨ψa|ψ⟩=1MB−Ma∑jcjga,j0Gj(MB), (56)

and thus we obtain

 Za=⟨ψ|ψa⟩⟨ψa|ψ⟩=∑j,kckc∗jGj(MB)Gk(MB)ga,j0ga,k0(MB−Ma)2. (57)

The total elementariness , which contains all contributions from the implicit channels, is

 Z≡∑aZa=∑j,kckc∗jGj(MB)Gk(MB)∑aga,j0ga,k0(MB−Ma)2=−∑j,kgkgj⎡⎢⎣GjdveffjkdEGk⎤⎥⎦E=MB. (58)

From the normalization (2.2), we obtain the sum rule

 −∑j,kgkgj⎡⎢⎣δjkdGjdE+GjdveffjkdEGk⎤⎥⎦E=MB=1. (59)

This corresponds to the nonrelativistic counterpart of the generalized Ward identity derived in Ref. Sekihara:2010uz []. We note that the sum rule (59) as the normalization of the wave function can be obtained by the explicit treatment of both the two-body states and the one-body bare states, which complement the discussion of the bound-state wave function with an energy-independent separable interaction done in Ref. Gamermann:2009uq [].

So far, we have regarded the components coming from the one-body bare states as the elementariness. On the other hand, sometimes it happens that some of the two-body channel thresholds are so high enough that these channels may play a minor role. In such a case, these channels can be also included into implicit channels of the effective interaction by, e.g., the Feshbach method Feshbach:1958nx [], Feshbach:1962ut []. These implicit channels also provide energy dependence of the effective interaction which acts on the reduced model space (see also Ref. Aceti:2014ala []), and accordingly we are allowed to interpret the contributions coming from these channels as the elementariness. For instance, the implementation of a scattering channel into the effective interaction can be done by replacing as:

 wjk(E)=veffjk+veffjNGN(E)1−veffNNGN(E)veffNk,j,k≠N, (60)

where the -th channel has been absorbed in the effective interaction in the same manner as in Hyodo:2007jq []. In this case the elementariness may be able to be calculated by the derivative of the effective interaction as

 Zw=−∑j,k≠Ngkgj[GjdwjkdEGk]E=MB. (61)

Interestingly, the elementariness can be expressed as the sum of the elementariness with the full two-body channels, , and the -th channel compositeness , namely,

 Zw=Z+XN, (62)

with

 (63)

The proof is shown in Appendix A. In this way, the elementariness can be redefined by Eq. (61). With this expression the elementariness measures contributions coming from both one-body bare states and two-body channels which are implemented into the effective interaction and do not appear as explicit degrees of freedom.

At the end of this subsection, we mention that our formulation of the compositeness and elementariness can be applied to any separable interactions with arbitrary energy dependence by interpreting that the energy dependence on the effective interaction comes from the implicit channels. Actually, when the compositeness and elementariness are formulated with multiple one-body bare states, all of these bare states are included in the effective two-body interaction and the total elementariness is calculated as the sum of each bare-state contribution, which is essentially the derivative of the effective two-body interaction as in Eq. (58). It is important that in this case we can produce any energy dependent interactions with suitable bare states. In order to see this, for instance, we assume that the mass of a bare state is large enough, and by expanding the bare-state term in the effective interaction as

 1E−M0=−1M0(1+EM0+⋯), (64)

we have polynomial energy dependence in the effective interaction. This fact enables us to apply the formulae of the compositeness and elementariness to interactions with an arbitrary energy dependence. This is the foundation of the analysis of physical hadronic resonances in Sec. 3.

### 2.3 Weak binding limit and threshold parameters

In this subsection, we consider the weak binding limit to derive the Weinberg’s compositeness condition Weinberg:1965zz [] on the scattering length and the effective range . This ensures that the expression for the compositeness in this paper correctly reproduces the model-independent result of Ref. Weinberg:1965zz [] in the weak binding limit. For simplicity we consider a system with one scattering channel like in Sec. 2.1.

In the single-channel problem, the elastic scattering amplitude is written with the -matrix given in Eq. (2.1) as

 F(E)=−1(2π)3(2π)2μt(E)[f(k2)]2, (65)

with . The scattering length is defined as the value of the scattering amplitude at the threshold:

 a≡−F(Mth)=μ2πt(Mth)=μ2π1v−1(Mth)−G(Mth), (66)

where we have abbreviated as for simplicity. Now we perform the expansion in terms of the energy around by considering to be small. To expand the denominator, we write

 (67)
 (68)

where we have defined

 Δv−1≡∞∑n=2Bnn![dnv−1dEn]E=MB,ΔG≡∞∑n=2Bnn![dnGdEn]E=MB. (69)

Here we allow arbitrary energy dependence for , as stated in the end of the last subsection, but assume that the effective range expansion is valid up to the energy of the bound state, which is a precondition for the formula in Ref. Weinberg:1965zz []. In this case there should exist no singularity of between and and expansion (67) is safely performed up to the threshold, and hence . Otherwise the singularity of around the threshold spoils the effective range expansion, as the divergence of leads to the existence of the CDD pole. As a result, with the bound state condition (2.1), the scattering length is now given by

 a=μ2π(B[dv−1dE−dGdE]E=MB−ΔG+O(B2))−1. (70)

The first term in the parenthesis in Eq. (70) is calculated as

 B[dv−1dE−dGdE]E=MB =−B[G2dvdE+dGdE]E=MB =B|g|2 =−BX[dGdE]E=MB =BX∫d3q(2π)3[f(0)]2+O(q2)[B+q2/(2μ)]2 =μ4πX1R+O(B), (71)

where we have used Eqs. (2.1), (2.1), (2.1), and the normalization , and we have defined in the last line. To evaluate , we first note that

 [dnGdEn]E=MB =∫d3q(2π)3(−1)nn![f(q2)]2[MB−Mth−q2/(2μ)]n+1 =−n!∫d3q(2π)3[f(q2)]2[B+q2/(2μ)]n+1 =−n!Bn2μR∫d3q′(2π)3[f(2μBq′2)]2(q′2+1)n+1, (72)

where . Thus summing up all contributions we have

 ΔG =−∞∑n=22μR∫d3q′(2π)3[f(0)]2(q′2+1)n+1+O(B) =−μπ2R∫∞0dxx2∞∑n=21(x2+1)n+1+O(B) =−μ4π1R+O(B), (73)

where we have used the summation relation

 ∞∑n=21(x2+1)n+1=1x2(x2+1)2(x≠0). (74)

As a consequence, we obtain the expression of the scattering length in terms of the compositeness from Eqs. (70), (71) and (73):

 a=μ2π(μ4πX1R+μ4π1R+O(B))−1=R2X1+X+O(B0), (75)

which agrees with the result in Ref. Weinberg:1965zz [] with . It is important that in the weak binding limit the details of the form factor are irrelevant to the determination of the compositeness of the bound state from the scattering length of two constituents. In contrast, the correction terms of depend on the explicit form of the function .

Because we have assumed that the bound state pole lies within the valid region of the effective range approximation, the relation between the scattering length and the effective range is given by666The relation (76) can be obtained from the condition at .

 re=2R(1−Ra) (76)

Comparing it with Eq. (75), we find

 re=RX−1X+O(B0). (77)

This again corresponds to the expression in Ref. Weinberg:1965zz [].

In this way, the structure of the bound state can be determined from and unambiguously in the weak binding limit. This means that, in principle, tuning and could lead to arbitrary structure of the bound state. It is however shown in Ref. Hanhart:2014ssa [] that the bound state with naturally appears when the state exists near the threshold, and a significant fine tuning is required to realize in this small binding region. This behavior can be understood by considering the value of in the exact limit. Actually, the value of is shown to vanish in the limit, as far as the bound state pole exist in the scattering amplitude Hyodo:2014bda []. It is therefore natural to expect that the bound state should be in the small binding region.

### 2.4 Generalization to resonances

Now we generalize our argument to a resonance state. We first introduce the Gamow state Gamow:1928zz [] denoted as to express the resonance state. The eigenvalue of the Hamiltonian is allowed to be complex for the Gamow state:

 ^H|ψ)=(MR−iΓR2)|ψ). (78)

Here and are the mass and width of the resonance state, respectively. The state with a complex eigenvalue cannot be normalized in the ordinary sense. To establish the normalization, we define the corresponding bra-state as the complex conjugate of the Dirac bra-state:

 (ψ|≡⟨ψ∗|, (79)

which was firstly introduced to describe unstable nuclei Hokkyo:1965zz [], Berggren:1968zz [], Romo:1968zz []. As a consequence, the eigenvalue of the Hamiltonian is the same with the ket vector:777The eigenvectors and have the eigenvalue