^{3}

We study the compositeness of near-threshold states to investigate the internal structure of exotic hadron candidates. Within the framework of effective field theory, Weinberg's weak-binding relation is extended to more general cases by easing several preconditions. First, by evaluating the contribution from the decay channel, we obtain the generalized relation for unstable quasibound states. Next, we generalize the relation to include the nearby CDD (Castillejo–Dalitz–Dyson) pole contribution with the help of the Padé approximant. The validity of the estimation with the generalized weak-binding relations is examined by numerical calculations. A method to systematically evaluate the error in the weak-binding relation is presented. Finally, by applying the extended relation to

The investigation of the internal structure of hadrons is one of the most fundamental subjects in hadron physics. The discovery of many candidates for exotic hadrons, which are not assigned to simple

To investigate the hadron structure, a number of models have been constructed by reproducing the experimental data. However, there is an ambiguity in identifying the structure in model-dependent studies. When we construct a model to reproduce the experimental data, the contribution of the degrees of freedom that are excluded from the model space is taken into account by the model parameters. It is not clear whether the employed degrees of freedom are suitable, even when the model reproduces the experimental data well [

In contrast, the internal structure is determined without ambiguity in model-independent approaches in the weak-binding limit. One such method discusses the compositeness of the state with the weak-binding relation, which is derived by Weinberg [

To apply this method to the candidates for exotic hadrons, we have to extend the weak-binding relation to be more practical for various cases. First, because most of the exotic hadrons are unstable and have a finite decay width, we need a relation valid for an unstable state. Several approaches in this direction have been proposed. In Ref. [

Second, as already mentioned in Ref. [

Third, the weak-binding relations always contain higher-order correction terms, as we will see below. Although the terms are small for the near-threshold states, they cause uncertainty in the results of the weak-binding relations. For practical applications to the hadron structures, it is useful to develop a method to estimate the uncertainties of the weak-binding relations that arise from the correction terms.

In this paper, we directly extend the weak-binding relation to a near-threshold quasibound state with a lower-energy decay channel and to a state with a nearby CDD pole based on the effective field theory. In

First, we review the derivation of the weak-binding relation written in Ref. [

Using Noether's theorem for the phase symmetry of the corresponding Lagrangian, we obtain the following conservation laws of particle numbers:

Next we construct the scattering amplitude of

By using the Schrödinger equation (

Next we derive the scattering amplitude using Eq. (

With this solution of the

We normalize the bound state

Using Eqs. (

Now we discuss the role of the discrete state

Next assuming the binding energy of the eigenstate

Now we consider the estimation of

Now we assume the length scale of the ERE is smaller than the interaction range,

Using Eqs. (

Here we note that the compositeness

Up to this point, we have derived the weak-binding relation of Ref. [

In the actual applications to near-threshold hadrons, the magnitude of the higher-order term

With the weak-binding relation (

In this section, we extend the weak-binding relation to the quasibound state, which has a decay mode. In the first part of this section, we show the detailed derivation of the extended weak-binding relation written in Ref. [

We consider the nonrelativistic EFT with the field

Using Noether's theorem, the following particle numbers are conserved:

Next we derive the scattering amplitude of the

The unstable state cannot be normalized by the ordinary condition Eq. (

From the Schrödinger equation for the quasibound state, we obtain

Here we consider the case that the eigenenergy lies near the threshold energy of channel 1 and far from that of channel 2. In this case, we can derive the weak-binding relation between the compositeness of channel 1 and the observables.

As in Eq. (

Expanding the scattering length in Eq. (

By assuming again that the eigenstate pole is in the convergence region of the effective range expansion, the expansion of ^{1}

Thus, as in the stable bound state case, the compositeness of the quasibound state with small

Now we discuss the interpretation of the complex compositeness, utilizing some concrete examples. Because we can determine the quantity

For the bound state, the compositeness defined with Eq. (

To illustrate the problem, let us introduce three examples of compositeness. Suppose that we obtain the compositeness of a quasibound state

Examples of the values of

Case | |||||
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I | |||||

II | |||||

III |

In previous works, several interpretations of complex compositeness are proposed. For example, the absolute value of ^{2}

We now analyze the examples in more detail. For case (I), the real part satisfies

In Ref. [

Condition (1)

Condition (2)

Condition (3) When there is no cancelation in

Condition (4)

^{3}

Geometric illustration of

We note that Eq. (^{4}^{5}

Finally we construct a method to evaluate the error of the compositeness of the quasibound states. In contrast to the stable bound states in

To estimate the effect of the higher-order terms, we first introduce a complex quantity ^{6}

In the derivation of the weak-binding relation in the previous sections, we have assumed the convergence of the effective range expansion (ERE) at the eigenenergy

The validity of the ERE is related to the magnitude of

We first show an alternative derivation of the weak-binding relation (

Now we examine the cutoff-dependence of the coupling constant

In this derivation, we notice that there are two expansions, both of which contain the higher-order correction terms. Equation (

(i)

(ii) the convergence region of ERE reaches the bound state pole:

(iii)

Equation (

When a CDD pole lies near the threshold, the convergence region of the effective range expansion may not reach the bound state pole. Here we use the Padé approximant method to describe

From the discussion in

We consider

We show the ratio of the estimation to the exact value

The estimated compositeness normalized by the exact value in the square-well potential model. The results with Eqs. (

In this subsection, we consider a case with nonzero elementariness using the field theoretical model with contact interaction as introduced in

From Eqs. (

Case 1:

Case 2:

Case 3:

The estimated values of the normalized elementariness as functions of

The estimated elementariness

One may wonder why the estimation with

The same calculation as in

The ratio of estimated elementariness

The ratio of the estimated elementariness

In

We first consider the square-well potential model used in the previous subsection. As mentioned above, we vary the potential depth

Compositeness

Next, we consider a contact interaction model with Hamiltonian (

Compositeness

Finally, we examine the case with an elementary-dominant bound state. To achieve this, we consider the model without the four-point interaction (

Compositeness

In this section, we have shown the results of the numerical model calculations to examine the weak-binding relations from various viewpoints. In all cases studied, we confirm that the compositeness can be appropriately determined in the weak-binding limit. With the help of the extended relation, the determination works well even if the CDD pole lies near the threshold.

On the other hand, it is also important to consider the case away from the strict weak-binding limit, because the effect of the higher-order correction terms is not negligible for actual hadron resonances. Based on the model calculations with both composite-dominant and elementary-dominant bound states, we find that the exact values are consistent with the uncertainty estimation developed in

At this point, it is instructive to compare the determination of the compositeness by the weak-binding relation with the evaluation of the compositeness at the pole position using Eq. (

In the weak-binding relation approach, the compositeness can be estimated with only a few observable quantities. Moreover, the result is model independent, as long as the weak-binding limit is concerned. Nevertheless, in practical applications, the estimation always contains uncertainty from the higher-order correction terms, because the compositeness is evaluated by the expansion of the amplitude. At the same time, we have to keep in mind that the applicability of the weak-binding relation is limited to

One may consider that the uncertainty in the weak-binding relation is a reflection of the model dependence of the compositeness, because the origin of the uncertainty is the model-dependent higher-order terms. As we have demonstrated here, it is possible to estimate the magnitude of this uncertainty/model-dependence for the weak-binding

In this way, the two approaches are in some sense complementary. Motivated by these observations, in the next section, we discuss the determination of the compositeness of physical hadron resonances with the weak-binding relations, in comparison with the evaluations at the pole position in theoretical amplitudes.

^{7}^{8}

Properties and results for the higher-energy pole of

Set 1 [ |
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Set 2 [ |
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Set 3 [ |
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Set 4 [ |
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Set 5 [ |

We first estimate the magnitude of the higher-order terms in the weak-binding relation. Using the eigenenergies in ^{9}

For each set, we have calculated the compositeness

With the method constructed in

The results of error evaluation of the compositeness

The results of error evaluation of the compositeness

Set 1 [ |
0.17 | 0.14 | |

Set 2 [ |
0.10 | 0.03 | |

Set 3 [ |
0.16 | 0.11 | |

Set 4 [ |
0.10 | 0.03 | |

Set 5 [ |
0.12 | 0.14 | 0.8^{+\,0.2}_{-\,0.2} |

To investigate the CDD pole contribution to the ^{10}

In Refs. [

The results of the evaluation of the compositeness of

Ref. | Amplitude | Prescription | Other components | |
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[ |
[ |
complex |
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[ |
[ |
complex |
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[ |
[ |
Re |
0.795 | 0.205 |

[ |
[ |
0.81 | 0.19 |

In these studies, Refs. [

Next we consider the scalar mesons

With the analyses in Refs. [

From these Flatté amplitudes, the eigenenergies of ^{11}

Properties and results for

Ref. | |||||
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[ |
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[ |
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[ |
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[ |
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[ |
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[ |

We summarize the results for

The results of ^{12}

Properties and results for

Ref. | |||||
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[ |
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[ |
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[ |
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[ |

We also evaluate the errors of

The results of error evaluation of the compositeness

The results of error evaluation of the compositeness

Ref | |||
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[ |
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[ |
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[ |
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[ |
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[ |
|||

[ |

The results of error evaluation of the compositeness

Ref. | |||
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[ |
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[ |
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[ |
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[ |

In Ref. [

For these scalar mesons, the compositeness is also calculated with the eigenenergy and the coupling constant in specific models obtained by phase shift analysis. We summarize the results of

The results of the evaluation of the compositeness of

Ref. | Amplitude | Prescription | Other components | |
---|---|---|---|---|

[ |
[ |
complex |
||

[ |
[ |
complex |
||

[ |
[ |
complex |
||

[ |
[ |
0.65 | 0.35 |

The results of the model evaluation of the compositeness of

Ref. | Amplitude | Prescription | Other components | |
---|---|---|---|---|

[ |
[ |
complex |
||

[ |
[ |
complex |

In Ref. [

We have discussed the compositeness of unstable states around a two-body threshold in the framework of nonrelativistic effective field theory. The weak-binding relation is generalized to the unstable quasibound states, showing that the compositeness of the near-threshold unstable states is also determined from complex observables as long as the contribution from the decay mode can be neglected. The interpretation of the complex compositeness is discussed by carefully examining the condition of the probabilistic interpretation. We have suggested a reasonable prescription to interpret complex compositeness with the real-valued compositeness

We have presented another derivation of the weak-binding relation by separating the expansion in terms of

We study the validity of the estimation of the compositeness with the weak-binding relation using models in which the exact values can be calculated. We verify that the deviation of the estimation from the exact value is of the order of the neglected terms. We also see that the generalized relation including the CDD pole contribution gives a good estimation of the compositeness even for the near-threshold state with a nearby CDD pole. Furthermore, the method of error evaluation is examined by using solvable models with both the composite-dominant and elementary-dominant states. The results show that the exact values of the compositeness are included well within the estimated uncertainty bands, and the qualitative conclusion remains unchanged if the magnitude of the higher-order terms is small.

Finally we have applied the extended weak-binding relation for quasibound states to physical hadron resonances. From the threshold parameters by means of chiral SU(3) dynamics, it is concluded that

We emphasize again that the weak-binding relations discussed in this paper connect the internal structure of near-threshold hadron resonances with observable quantities. Evaluation of the compositeness at the pole position, which is a commonly adopted approach in the literature, is only possible when a reliable theoretical amplitude is established with the help of a sufficient amount of experimental data, and the renormalization dependence of the compositeness is in principle unavoidable. In contrast, the weak-binding relation can be used with a few observables (eigenenergy, scattering length, and so on), and the renormalization dependence is suppressed thanks to the large length scale. The applicability is however limited to near-threshold states and the result is associated with the uncertainties by the higher-order terms. We thus consider that the two approaches are complementary, and both will help future investigation of the nature of hadron resonances.

The authors thank Eulogio Oset for a valuable discussion. This work is supported in part by JSPS KAKENHI Grant No. 24740152 and No. 16K17694 and by the Yukawa International Program for Quark-Hadron Sciences (YIPQS).

Open Access funding: SCOAP^{3}.

^{1} It is a basic assumption in EFT that there is no accidental fine tuning that causes an order difference in the parameters.

^{2} In Ref. [

^{3} In Ref. [

^{4} A nice feature of the definition in Ref. [

^{5} With this inequality, we can also show that the difference between

^{6} We note that

^{7} We do not consider the compositeness of the state associated with the lower-energy pole, because the weak-binding relation is derived for the closest pole to the threshold.

^{8} We thank Jose Antonio Oller and Maxim Mai for correspondences.

^{9} We do not use the

^{10} In the coupled-channel scattering, each component can have a CDD pole individually. This is in contrast to the pole of the amplitude representing the eigenstate, which is determined by

^{11} Here we do not use the most recent analysis of

^{12} As a reference, using the analysis of Ref. [