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, 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

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 (

It is encouraging that there have been experimental reports on candidates for manifestly exotic hadrons, such as charged quarkonium-like states by the Belle collaboration [

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 the constituent hadrons, in a similar way to atomic nuclei as bound states of nucleons. The

Motivated by these observations, in this study we develop a framework to investigate hadronic two-body components inside a hadron by comprehensively analyzing 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. [

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

This paper is organized as follows. In Sect. 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 the 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 Sect. 3, numerical results for the applications to physical resonances are presented. Section 4 is devoted to drawing the conclusions of this study.

In this section, we define the compositeness (and simultaneously elementariness) of a particle state, i.e., a stable bound state or an unstable resonance, using its wave function, and link the compositeness to the physical quantities in the scattering equation. For this purpose, we consider two-body scattering states^{1} coupled with each other and one-body bare states. The one-body bare states have not been introduced in studies of wave functions before, and the introduction of the one-body bare states makes it possible to establish the meaning of the elementariness in 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 Sect. 2.1 we consider a bound state^{2} 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 Sect. 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 Sect. 2.1. Here we also discuss a way to introduce a general energy-dependent interaction into the formulation. In Sect. 2.3 we consider the weak binding limit to derive Weinberg's relation for the scattering length and the effective range [

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 a 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

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

We take the matrix element of Eq. (

For the explicit calculation, we assume the separable form of the matrix elements of

For the separable interaction, the wave function

The solution of Eq. (

The normalization constant

Next, the scattering amplitude

The equality

Here, we emphasize that, as seen in Eq. (^{3} 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. (

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

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

The coupled-channel scattering equation is, in matrix form,
^{4}

Now the compositeness in channel

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 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

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

In this subsection, we consider the weak binding limit to derive Weinberg's compositeness condition [

In the single-channel problem, the elastic scattering amplitude

The first term in the parenthesis in Eq. (

As a consequence, we obtain the expression of the scattering length in terms of the compositeness

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 by^{5}

In this way, the structure of the bound state can be determined from

Now we generalize our argument to a resonance state. We first introduce the Gamow state [^{6}

To determine the wave function, we solve the Schrödinger equation

Also, for the resonance pole the residue of the scattering amplitude is interpreted as the product of the coupling constants

From Eq. (^{7}

By definition, the compositeness for the resonance state becomes complex. Therefore, strictly speaking, it cannot be interpreted as a probability of finding the two-body component. Nevertheless, because it represents the contribution of the channel wave function to the total normalization, the compositeness

Finally, we consider the coupled-channel two-body scattering in a relativistic form. Here we do not consider the intermediate states with more than two particles but simply solve the two-body wave equation.^{8} To describe the wave function of the resonances, we extract the relative motion of the two-body system from a relativistic scattering equation with a three-dimensional reduction [

According to Appendix B, we introduce the state

The scattering state

The dynamics of the system are determined by the interaction operator ^{9} In order to make a three-dimensional reduction of the scattering equation, we assume that the form factor

As in Sect. 2.3, the wave function is determined as

In Appendix B we confirm that the wave equation (

By comparing the residue of the resonance pole as in Eq. (

Diagrammatic interpretation of the compositeness

We note that, although both the compositeness

Having established the compositeness and elementariness in Eqs. (

The compositeness and elementariness have been evaluated in the chiral model with the simple leading order chiral interaction for

As we will show below, the scattering amplitudes in Refs. [

In this study, we utilize the set of the one- and two-body states introduced in Sect. 2 as the basis for interpreting the structure of the hadronic resonances in the coupled-channel chiral model. On the assumption that the energy dependence of the interaction originates from channels which do not appear as explicit degrees of freedom, it has been shown that the final expression of the compositeness is given by Eq. (

Let us summarize the interpretation of the compositeness and elementariness for resonances. As shown in Sect. 2.3,

In Refs. [

With the formulae in Sect. 3.1 we calculate the pole positions, compositeness

Compositeness

On the other hand, for the lower pole, there is a certain amount of cancellation (

The compositeness and elementariness of

Before closing this subsection, we mention that the structure of

The lowest-lying scalar and vector mesons in the meson–meson scattering have been studied in Ref. [

To evaluate the compositeness and elementariness, we rewrite the amplitude (

Before evaluating the compositeness, let us focus on the structure of the interaction kernel (

Now let us evaluate the compositeness and elementariness of the lightest scalar and vector mesons described by the coupled-channel IAM developed in Ref. [

Compositeness

— | |||

— | |||

— | |||

— | — | ||

— | — | ||

Compositeness

— | ||

— | ||

— | ||

— | ||

In the vector channels,^{10} we find that, for both the

The compositeness of scalar mesons [

In the preceding subsections we have evaluated the compositeness and elementariness of

Using

Compositeness

Here we emphasize that both

In this study we have developed a framework to investigate the internal structure of bound and resonance states with their compositeness and elementariness by using their wave functions. For this purpose we have explicitly taken into account both one-body bare states and two-body scattering states as the basis to interpret the structure of bound and resonance states. Compositeness and elementariness are respectively defined as the contributions from the two-body scattering states and the one-body bare states to the normalization of the total wave function. After reviewing the formulation for the bound state, we have discussed the extension to the resonance state.

Because the wave function is analytically obtained for a separable interaction, we have explicitly written down the wave function for a bound state in a general separable interaction and obtained the expressions for the compositeness and elementariness. We have demonstrated that the compositeness is determined by the residue of the scattering amplitude and the energy dependence of the loop function at the pole position. Therefore, once one has the loop function, which is the Green function of the free two-body Hamiltonian, one can obtain the compositeness only from the bound state properties. On the other hand, we have found that the elementariness is obtained with the energy dependence of the effective two-body interaction. It is an interesting finding that the energy dependence of the two-body effective interaction arises from implicit 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. These implicit channels contain the two-body scattering states as well as the one-body bare states. We have also shown the sum rule of the compositeness and elementariness. We have proved that, with multiple bare states, the formulae of the compositeness and elementariness can be applied to interactions with an arbitrary energy dependence. Of particular value is the derivation of Weinberg's relation for the scattering length and effective range in the weak binding limit. In the present formulation, thanks to the separable interaction, the scattering amplitude is analytically obtained. With this fact we have explicitly performed the expansion of the amplitude around the threshold to derive Weinberg's relation. In this derivation, the higher-order corrections come from the explicit expression of the form factor as well as higher-order derivatives in the expansion. The limitation of the formula due to the existence of the CDD pole is clearly linked to the breakdown of the effective range expansion.

Our discussion on the wave function has been extended to resonance states with Gamow vectors. The use of the Gamow vector enables us to have finite normalization of the resonance wave function. For a resonance state, by definition both the compositeness and elementariness become complex, which are difficult to interpret. Nevertheless, utilizing the fact that the compositeness and element-ariness are defined by the wave functions, we have proposed the interpretation of the structure of a certain class of resonance states, on the basis of the similarity of the wave function of the bound state. Namely, if the compositeness in a channel (elementariness) is close to unity with small imaginary part and all the other components have small absolute values, this resonance state can be considered to be a composite state in the channel (an elementary state). Finally, we have given the expressions for the compositeness and elementariness with a general separable interaction in a relativistic covariant form by considering a relativistic scattering with a three-dimensional reduction.

As applications, the expression for the compositeness in a relativistic form has been used to investigate the internal structure of hadronic resonances, on the assumption that the energy dependence of the interaction originates from the implicit channels. By employing chiral coupled-channel scattering models with interactions up to the next-to-leading order, we have observed that the higher pole of

Finally, we emphasize that the fact that constituent hadrons are observable as asymptotic states in QCD is essential to constructing the two-body wave functions and to determining the compositeness for hadronic resonances.

Open Access funding: SCOAP^{3}.

The authors greatly acknowledge T. Myo and A. Doté for discussions on the Gamow vectors for resonance states, J. Nebreda on the theoretical description of scalar and vector mesons, and H. Nagahiro and A. Hosaka on the physical interpretation of compositeness. The authors also thank E. Oset for his careful reading of the manuscript and stimulating discussions during the stay of T. S. in Valencia supported by JSPS Open Partnership Bilateral Joint Research Projects. This work is partly supported by Grants-in-Aid for Scientific Research from MEXT and JSPS (24740152 and 25400254) and by the Yukawa International Program for Quark-Hadron Sciences (YIPQS).

First, we consider an on-shell one-body state of a scalar field of mass

Next, we construct a two-body state, in which both particles are on the mass shell and the relative momentum is denoted as

With the definition of the two-body state

Now we would like to confirm that the wave equation (

In general, the wave equation can be composed of the free two-body Green's operator ^{11}
^{12} Here we also assume that the form factor

Let us now derive the scattering equation with the above normalizations. To this end, we define the

At last we emphasize that the normalization (

We 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.

In general, there can be several bound states in the system. In such a case, we just focus on one of these bound states. Nothing changes in the following discussion.

Since the bound state properties are determined by the interaction, the compositeness depends implicitly on the effective interaction

Since an interaction of a symmetric matrix

The relation (

The eigenvectors

In Sect. 3.2 we will compare the structure of

In general relativistic field theory, there are infinitely many diagrams which contribute to the scattering amplitude. The present formulation picks up the summation of the

In relativistic field theory, the coupling

In the framework of IAM, the loop function in

In the nonrelativistic framework the two-body Green's operator is

By using the notations in Sect. 2.4, the two-body interaction operator