^{*}

^{†}

^{3}.

We discuss the relation between the analytic structure of the scattering amplitude and the origin of an eigenstate represented by a pole of the amplitude. If the eigenstate is not dynamically generated by the interaction in the channel of interest, the residue of the pole vanishes in the zero coupling limit. Based on the topological nature of the phase of the scattering amplitude, we show that the pole must encounter with the Castillejo-Dalitz-Dyson (CDD) zero in this limit. It is concluded that the dynamical component of the eigenstate is small if a CDD zero exists near the eigenstate pole. We show that the line shape of the resonance is distorted from the Breit-Wigner form as an observable consequence of the nearby CDD zero. Finally, studying the positions of poles and CDD zeros of the

Given the recent findings of many candidates of exotic hadrons

One step in this direction has been made by the pole counting method

Alternatively, several studies focus on the Castillegio-Dalitz-Dyson (CDD) zero

The CDD zero is often referred to as “CDD pole”, as it represents the pole of the inverse amplitude. In this paper, to avoid the confusion with the eigenstate pole, we call it CDD zero.

The general form of the hadron scattering amplitude including the CDD contribution is discussed in Refs.In this paper, we show that the distance between the eigenstate pole and the CDD zero is related to the structure of the state. To reveal the origin of the eigenstate, we consider the zero coupling limit (ZCL), which is the limit of turning off the couplings among different coupled channels. By analyzing the behavior of CDD zeros and poles in the ZCL, we discuss the relation between the internal structure of the eigenstate and the existence of a nearby CDD zero.

Our aim is to clarify the dynamical origin of the eigenstate expressed by a pole of the coupled-channel scattering amplitude in a given partial wave. For this purpose, we focus on one of the coupled channels, say channel

To elucidate the origin of the eigenstate, we consider the zero coupling limit (ZCL)

If a pole remains in two or more components in the exact ZCL, the degenerate eigenstates must exist. In this case, there must be a symmetry which relates different channels. If the eigenstate is generated not by the interaction of a specific channel but purely generated by the channel coupling effect, the pole cannot remain in any of the components. In this case, the pole should move away to infinity as

Thus, we can classify the behavior of the pole in a specific channel into two cases:

The pole remains in the amplitude in the ZCL.

The pole decouples from the amplitude in the ZCL.

First, we consider a single-channel scattering problem coupled to a bare state, utilizing the nonrelativistic effective field theory introduced in Ref.

We consider the case where the system has a discrete eigenstate. The zero of the denominator of Eq.

Let us examine the behavior of the pole

Next let us consider the CDD zero

Next, we consider the system with two scattering channels. We introduce fields

Again, we consider the case with one eigenstate at

The position of the CDD zero is determined from Eq.

It is also instructive to comment on the realization of the pole counting rule

Let us shortly summarize the discussion of this part. We have studied the behavior of the pole and zero of the amplitude in the ZCL in two models. In both cases, we find that the pole representing an eigenstate and the CDD zero annihilate each other in the ZCL if the origin of the eigenstate is attributed to the dynamics of the hidden channel. It is also shown that the distance between the pole and the CDD zero is small when the channel coupling is small.

Here we show that the annihilation of the pole and zero is, in fact, a consequence of the general property of the scattering amplitude. Let

Here we assume that all the poles and zeros are simple. Singularities with multiplicity can appear only with a fine tuning of the parameters, and its existence is not stable against the small perturbation of the parameters.

For instance, ifIf we continuously vary the parameters of the system (e.g. reducing the channel couplings toward the ZCL), the poles and zeros of the scattering amplitude move continuously in the complex

This feature of the scattering amplitude is particularly important for the fate of the pole in the ZCL.

The above argument can be applied to a single component of coupled-channel amplitude

If there is no CDD zero near a pole in

If a pole is accompanied by a nearby CDD zero, the origin of the eigenstate is not in channel

Here we focus on the zeros of the diagonal component of the scattering amplitude. However, when the effect of the multichannel (multicomponent) is important, the production amplitude obtained in an experiment is the mixture of the different terms. The position of the zero of this production amplitude can be different from that of the diagonal scattering amplitude. Therefor, in order to discuss the structure of the eigenstates, it is necessary to extract the coupled channel scattering amplitude by detailed analysis of the experimental data and to search the position of zero of the diagonal scattering amplitude.

The existence of the CDD zero near the pole causes yet another consequence for the near-threshold states. The effective range expansion (ERE) is often introduced in the analysis of near-threshold energy region of the scattering amplitude. However, the CDD zero near the threshold harms the convergence of the ERE as pointed out in Refs.

Because the analytic structure of the scattering amplitude is utilized to determine the origin of the eigenstate, the method constructed above has similarity with the pole counting method

While our method draws qualitative conclusion on the origin of the eigenstate, one may want to discuss the structure quantitatively. For this purpose, we can calculate Weinberg’s

In this section, we consider the influence of the existence of a CDD zero near the pole. Again, we take an example of coupled-channel scattering model with the Lagrangian in Eq.

Taking the model parameters as

The scattering amplitudes of the

The

The limited applicability of the Breit-Wigner fit for the near-threshold states is pointed out in Ref.

Let us now consider the physical

To study the

Searching for the

Positions of the poles (squares) and CDD zeros (circles) in the

In the ETW model, the ZCL is achieved by suppressing the off-diagonal interaction

Trajectories of poles (solid line) and CDD zeros (dashed line) in the

Now let us consider the trajectories of the CDD zeros. In the

We have proposed a useful method to study the origin of hadron resonances. It is shown that the eigenstate pole should be accompanied by a nearby CDD zero, if the resonance originates in the dynamics of a hidden channel. The existence of the zero is robust, as it is topologically guaranteed by Eq.

We summarize how to apply our method in practice. One way is to find the distortion of the peak of the invariant mass distribution. As discussed in Sec.

This work is in part supported by JSPS KAKENHI Grants No. JP17J04333 and No. JP16K17694 and by the Yukawa International Program for Quark-Hadron Sciences (YIPQS).