Dynamically generated states from the $\eta K^*\bar{K}^*$, $\pi K^*\bar{K}^*$, and $K K^*\bar{K}^*$ systems within the fixed-center approximation

The three-body systems $\eta K^* \bar{K}^*$, $\pi K^* \bar{K}^*$, and $K K^* \bar{K}^*$ are further investigated within the framework of fixed-center approximation, where $K^* \bar{K}^*$ is treated as the fixed-center, corresponding to the possible scalar meson $a_0(1780)$ or the tensor meson $f_2'(1525)$. The interactions between $\eta$, $\pi$, $K$, and $K^*$ are taken from the chiral unitary approach. The resonance structures appear in the modulus squared of the three-body scattering amplitude and suggest that $\eta / \pi / K$-$(K^*\bar{K}^*)_{a_0(1780)/f'_2(1525)}$ hadron state can be formed. By scattering the $\eta$ meson on the fixed-center $(K^* \bar{K}^*)_{a_0(1780)}$, it is found that there is a distinct peak around 2100 MeV, as shown in the modulus squared of the three-body scattering amplitude, which can be associated with the meson $\pi(2070)$. For the scattering of the $\eta$ meson on the $(K^* \bar{K}^*)_{f_2'(1525)}$, a resonance structure around 1890 MeV is found and it can be associated with the $\eta_2(1870)$ meson. Other resonance structures are also found and can be associated with $\pi_2(1880)$ and $\eta(2010)$.


I. INTRODUCTION
With abundant observations of new hadronic states since 2003 [1][2][3][4][5][6][7][8][9], the search for exotic hadronic matter has become a frontier of particle physics, which is also an effective way to deepen our understanding of the non-perturbative behavior of the strong interaction.These novel phenomena have also stimulated extensive discussion of the interaction between hadrons.A typical example is that the characteristic mass spectrum of P c states in the Λ 0 b → K − J/ψp process [10] supports the hidden-charm molecular baryons composed of anti-charmed meson and charmed baryon [11][12][13][14].Thus, it is also the reason why hadronic molecular state is popular for deciphering the nature of these new hadronic states.
In the research field of the few-body problem, the treatment of the three-body system continues to attract the attention of theorists.Indeed, there is growing evidence that some existing and newly observed hadronic states could be interpreted in 1 Similar conclusions are found in Ref. [29], where these pseudoscalarpseudoscalar coupled channels were considered.The mass of a 0 (1780) obtained in Ref. [29] is smaller than that predicted in Refs.[19,20].The scalar meson a 0 (1780) has recently been observed by experiments [30,31].In view of the K * K * molecular state, the production of scalar mesons a 0 (1780) in D + s → π + K 0 S K 0 S and D + s → π 0 K + K 0 S was theoretically studied in Refs.[32][33][34], where the experimental measurements on the invariant mass distributions of the final states can be reproduced well.
In Ref. [64], the light flavor three-body ηK * K * , πK * K * , and KK * K * systems are partly investigated within the framework of FCA, where the fixed-center K * K * is treated as the scalar meson f 0 (1710).In fact, there exist other allowed combinations of the K * K * system.As in Ref. [19], an h 1 state with mass around 1800 MeV and an a 2 state with mass about 1570 MeV were also found, and they couple strongly to the K * K * channel.However, these two dynamically generated states from the vector meson-vector meson interaction cannot be clearly identified with any of the h 1 and a 2 states listed in the Review of particle physics (RPP) [65].In the present work, we further investigate ηK * K * , πK * K * and KK * K * three-body systems by the FCA, and we take the scalar meson a 0 (1780) and tensor meson f ′ 2 (1525) as K * K * molecular states [66][67][68][69][70], and then scatter the η, π and K mesons on the fixed-center of K * and K * .Since we consider only the s-wave interaction, the above three-body systems can have quantum numbers J P C = 0 −+ or 2 −+ , which indicates that we will investigate the pseudoscalar and pseudotensor lowlying excited states in the ηK * K * , πK * K * and KK * K * three-body systems.Besides, these states with exotic quantum numbers for the K(K * K * ) a0(1780) with the total isospin I = 3  2 sector are also investigated.For the two-body scattering, we take the interactions between pseudoscalar mesons and vector mesons as obtained with the chiral unitary approach [71,72].
To end this introduction, we would like to mention that the fixed center approximation is an effective and practical way to study three-body systems, widely accepted in the literatures [62,63,[73][74][75][76][77][78][79][80].It assumes the existence of a bound state of two particles that interact strongly with each other, and that the wave function of the bound state is not significantly changed by the interaction of an outside particle with it.This occurs when the third particle is lighter than the two particles in the bound state, or when the third particle has low energy, causing it to have minimal impact on the wave function of the bound state.Moreover, the FCA is technically much simpler than performing full calculations of the Faddeev equations.Therefore, it is important to utilize the FCA in various threebody systems, especially in the light flavor sector, where rich and available experimental data exists.However, it is important to note that limitations of using the FCA in three-meson studies were discussed in Ref. [81].
This paper is organized as follows.In Sec.II, we present the FCA method to the three-body ηK * K * , πK * K * and KK * K * systems.In Sec.III, the numerical theoretical re-sults and discussions are presented.Finally, a short summary is followed.

II. FORMALISM AND INGREDIENTS
We are interested in the three-body systems: ηK * K * , πK * K * , and KK * K * .To study the dynamics of these threebody systems, we obtain the three-body scattering amplitudes using the fixed-center approximation method.A basic feature of the FCA is that one has a fixed-center bound of two particles and one allows the multiple scattering of the third particle with this bound state, which should not be changed by the interaction of the third particle.In addition, the interaction of a particle with a bound state of a pair of particles at very low energies or below the threshold can be studied efficiently and accurately by means of the FCA for the threeparticle system [57,58,60].In the present work, we extend this formalism to include states above three-body mass threshold and apply it to π(K * K * ) a0(1780)/f ′ 2 (1525) systems.In this section, we summarize the deduction of the three-body scattering amplitudes in the FCA framework.
A. Fixed-center approximation to the three-body system We will use the fixed-center approximation formalism to study the three-mesons system ηK * K * , πK * K * , and KK * K * , where we consider the K * K * as the fixed center, and treat it as a a 0 (1780) or f ′ 2 (1525) state.Then the η, π or K meson interacts with it.The corresponding diagrams are shown in Fig. 1.In the following, we will refer to K * , K * and η (π or K) as particles 1, 2 and 3 respectively.
Following the formalism of Ref. [61], the three-body scattering amplitude T for η (π or K) collisions with the fixedcenter K * K * can be obtained by the sum of the partition functions T 1 and T 2 : with T 1 (T 2 ) the sum of all the diagrams in Fig. 1, where the particle 3 collides firstly with the K * ( K * ) in the fixedcenter.The t 1 (t 2 ) represents the unitary two-body scattering amplitudes in coupled channels for the interactions of particle 3 with K * ( K * ).
In addition, G 0 is the loop function for particle 3 propagating between the K * and K * inside the bound state (B * ), which can be written as where M B * and F B * (q) are the mass and the form factor of the K * K * bound state, respectively.The is treated in this paper as the scalar meson a 0 (1780) or the tensor meson f ′ 2 (1525), so M B * is the mass of a 0 (1780) or f ′ 2 (1525).In addition, m 3 is the mass of the π/η/K meson.The q 0 is the energy of particle 3 with mass m 3 in the centerof-mass frame of particle 3 and K * K * bound state, which is given by where s is the invariant mass square of the whole three-body system.
One of the ingredients in the calculation is the form factor for the assumed two-body K * K * bound state, the scalar meson a 0 (1780) and the tensor meson f ′ 2 (1525).Following the procedures as in Refs.[61,82,83], one can obtain the expression of the form factor F B * for the s-wave K * K * bound state a 0 (1780) or f ′ 2 (1525) as where , and the normaliza-tion factor N is given by The cutoff parameter Λ is used to regularize the vector meson-vector meson loop functions in the chiral unitary approach [19,23].In this work, the upper integration limit of Λ has the same value as the cutoff used in Refs.[19,23], with which one can obtain the scalar meson a 0 (1780) or the tensor meson f ′ 2 (1525) state in the vector meson-vector meson interactions in coupled channels.
The important ingredients in the calculation of the total scattering amplitude for the ηK * K * , πK * K * , and KK * K * systems using the FCA are the two-body π/η/K-K * , and K * K * unitarized s-wave interactions from the chiral unitary approach.Although the form of these interactions has been detailed elsewhere, we will briefly revisit them below for the case of K * K * .This will allow us to review the general procedure for calculating the two-body amplitudes entering the FCA equations.
In Fig. 2, the module squared of the transition amplitude |t K * K * →K * K * | 2 obtained from the chiral unitary approach in the coupled channels [ρρ, ρω, ρϕ, K * K * for Fig. 2 (a) and K * K * , ρρ, ωω, ωϕ, ϕϕ for Fig. 2 (b)] are shown.In these calculations, we use the cutoff regularization for the two-body vector meson-vector meson loop functions of G V V , and the width of the vector meson ρ and K * are taken into account.Furthermore, we take the same cutoff parameter Λ = 1100 (1009) MeV for all the channels in isospin=1 and spin=0 (isospin=0 and spin=2) sector.In addition, the obtained masses are 1769 and 1517 MeV for a 0 (1780) and f ′ 2 (1525), respectively, which are consistent with their masses quoted in the RPP [65].   2 (a), one can see a bump structure around 1780 MeV that can be assigned to the a 0 (1780) state.As discussed in the introduction part, the scalar meson a 0 (1780) was first observed by the BaBar Collaboration [30] in 2021 and recently confirmed by the BESIII Collaboration [31].
It can be seen that, in Fig. 2 (b), the narrow peak around 1517 MeV can be associated with the tensor meson f ′ 2 (1525).Comparing with the numerical results shown in Fig. 2 (a), it is found that the strength of |t K * K * →K * K * | 2 for f ′ 2 (1525) is much larger than that for the case of a 0 (1780).This indicates that the s-wave K * K * interaction in the isospin=0 and spin=2 sector is much stronger than that for the case of isospin=1 and spin=0.Furthermore, comparing the line shapes in Fig. 2 (a) with Fig. 2 (b), it is found that the width obtained for the tensor meson f ′ 2 (1525) is much narrower than that for the scalar meson a 0 (1780).
Next, in Fig. 3, we show the numerical results for the respective form factors of a 0 (1780) (solid curve) and f ′ 2 (1525) (dashed curve) as a function of q = |⃗ q|, where the theoretical results are obtained for the a 0 (1780)[f ′ 2 (1525)] with Λ = 1100(1009) MeV.The condition |⃗ p − ⃗ q| < Λ implies that the form factor F B * (q) is exactly zero for q > 2Λ.As discussed earlier, with these values of the cutoff parameter Λ, one can obtain a 0 (1780) and f ′ 2 (1525) resonances in vector mesons-vector meson coupled channel interactions as in Refs.[19,23].According to Fig. 1 (a) and (e), the single-scattering contributions of t 1 and t 2 are the appropriate combination of the two-body unitarized scattering amplitudes.For example, let us first consider the ηK * K * system with the fixed-center K * K * as the a 0 (1780) state (denoted by a 0 ): with where the kets on the right side of the above equation represent Then the single-scattering contributions to the total amplitude of ⟨η(K * K * ) a0 | t |η(K * K * ) a0 ⟩ can be easily obtained in terms of the unitary two-body transition amplitudes t ηK * →ηK * and t η K * →η K * : Here, |A 1 ⟩ stands for the state combined with η and K * , while |A 2 ⟩ is the state of η and K * .They are given by The kets of the above equations represent . So, we have Using the same procedures, one can easily obtain all the amplitudes for the single-scattering contribution in the present calculation which are shown in table I for the case of η/π/K-(K * K * ) a0(1780) and η/π/K-(K * K * ) f ′ 2 (1525) configurations with different total isospins.
On the other hand, following the approach developed in Refs.[61,62], we need to give a weight to the two-body scat-tering amplitudes t 1 and t 2 so that we have the correct normalization for the meson fields.This is achieved by replacing In addition, we also consider the effect of single-scattering above the mass threshold of particle 3 and the bound state B * .Following Refs.[38,84], we need to project the form factor into the s-wave.Then we have 2 with The form factor F F S was taken to be unity in Ref. [64], since for the η(K * K * ) f 0 (1710) and K(K * K * ) f 0 (1710) systems, only states below threshold were found.While for the π(K * K * ) f 0 (1710) system, there are uncertainties of about 20 MeV for the peak position of the modulus squared of the three-body scattering amplitudes, which is a small effect.Therefore, the main conclusions there are unchanged when the form factor F F S is taken into account.
It is worth noting that the total three-body scattering amplitude T is a function of the total invariant mass √ s of the three-body system.While the two-body scattering amplitudes t 1 and t 2 depend on the invariant masses √ s 1 and √ s 2 , which are the invariant masses of η (π or K) and the particle K * ( K * ) within the bound state of a 0 (1780) or f ′ 2 (1525).The s 1 and s 2 are:

III. NUMERICAL RESULTS
In this section, we will show the theoretical numerical results obtained for the scattering amplitude modulus square of the three-body systems ηK * K * , πK * K * , and KK * K * , respectively, and we evaluate the three-body scattering amplitude T and associate the peaks or bumps in the modulus squared of |T | 2 with resonances.
A. There-body system with the K * K * subsystem as a0(1780) For the ηK * K * system, its total isospin is one because K * K * has isospin one and η meson is zero.In order to obtain the three-body scattering amplitudes, one needs to obtain these two-body scattering amplitudes t 1 and t 2 .While t 1 and t 2 can be obtained with these scattering amplitudes of ηK * and η K * , which are taken from these previous works as in Refs.[71,72,85].Furthermore, we also consider the width of the vector meson [85] and the effect of the η ′ meson as done in Refs.[86,87].With these model parameters as used in Refs.[72,85], the two-body scattering amplitude of t ηK * →ηK * can be easily obtained and one can find that the interaction between η and K * is strong.
In fact, it is expected that the ηK * K * three-body system could be bound, since these interactions between η, K * and K * are all strong and attractive.The modulus squared scattering amplitude T η(K * K * ) a 0 (1780) →η(K * K * ) a 0 (1780) is shown in Fig. 5 (a), showing a clear peak structure around 2122 MeV, which can be associated with the π(2070).There is some evidence for this state in a combined partial wave analysis of pp annihilation channels [88].It is also cited as further state in RPP [65], and its mass and width are about 2070 ± 35 MeV and 310 +100 −50 MeV, respectively.Here, we explain the π(2070) meson as a η(K * K * ) a0(1780) molecular state.Improved experimental data are desirable to draw more firm conclusions.
2. Three-body π(K * K * ) a 0 (1780) system In the case of a three-body πK * K * system, its total isospin could be zero, one or two.We need two-body coupledchannels scattering amplitudes t πK * →πK * in the I πK * = 1 2 and 3 2 sectors.Within the formula and theoretical parameters as in Refs.[85], one can easily obtain the two-body scattering amplitude t πK * →πK * in coupled channels.And then we can calculate the three-body scattering amplitude T π(K * K * ) a 0 (1780) →π(K * K * ) a 0 (1780) .
For the case of I = 0, one can see a bump structure located at 1988 MeV in Fig. 5 (b), which can be interpreted as a η excited state η(2010) in RPP [65].It is found in a combining fit to data on pp annihilation [89], with mass 2010 +35 −60 MeV and width 270 ± 60 MeV.For the case of I = 1, as shown in Fig. 5 (c), a bump structures is located around 2030 MeV, and it may be associated with the π(2070) state, which was not quoted in the summary table of RPP [65].However, the π(2070) state was required in a combined partial wave analysis of the pp annihilation channels as studied in Ref. [89].It is hoped that further experimental measurements will test this prediction.
Finally, no special structures are found for the case of I = 2, because the interactions of πK * and π K * in the isospin I = 3  2 sector are rather small.
3. Three-body K(K * K * ) a 0 (1780) system For the three-body K(K * K * ) a0(1780) system, its total isospin can be either I = 1  2 or I = 3 2 .The modulus squared of the scattering amplitudes |T K(K * K * ) a 0 (1780) →K(K * K * ) a 0 (1780) | 2 are shown in Fig. 5.For the case of I = 1 2 , a resonance can be found around 2156 MeV as shown in Fig. 5 (d).It should be mixed with a structure that is found in Ref. [64] with mass around 2130 MeV.
For the case of I = 3 2 , one can also find a peak structure around 2130 MeV in Fig. 5 (e).We look forward to observing this low-lying exotic mesonic state with isospin I = 3  2 in future experimental measurements.Since the scalar meson a 0 (1780) is not well established in the RPP [65], and its mass has some uncertainties, we should also consider the effects of the mass of a 0 (1780).We do this by adjusting the cutoff Λ from 1000 to 1500 MeV.For different cutoffs Λ, we can get different masses of a 0 (1780) and peak positions of the squared of the three-body scattering amplitude, which are shown in Table II.We find that with these numerical results for the mass of a 0 (1780) one can always find peak structures in the squared of the three-body scattering amplitude.For the ηK * K * and πK * K * systems with I = 1, they are associated with π(2070), and for the πK * K * sysmtem with the total isospin I = 0, they can be associated with η(2010).We also find two unobserved states in the K(K * K * ) a0(1780) system, one of which carries exotic quantum numbers I(J P ) = 3  2 (0 − ).B. There-body system with K * K * as f ′ 2 (1525) We show the modulus squared of the scattering amplitude for T η(K * K * ) f ′ 2 (1525) →η(K * K * ) f ′ 2 (1525) in Fig. 5 (f) with Λ = 1009 MeV.A distinct peak structure is found around 1896 MeV.It can be interpreted as η 2 (1870) with quantum numbers J P C = 2 −+ .Although the η 2 (1870) meson has been confirmed by several experiments [65], there are difference theoretical explanations for this controversial state.It is very interesting that the η 2 (1870) is labeled as a no-q q state in RPP [65].Its experimentally measured mass is between 1835 MeV [90] and 1881 MeV [91], and the average mass is 1842 ± 8 MeV and its average width is 225 ± 14 MeV [65].In addition, the η 2 (1870) state has been interpreted as a hybrid state, mixed by the ss( 1 D 2 ) and q q(2 1 D 2 ) states.Its mass, calculated by different models, is listed in Table III, where GI stands for the Godfrey-Isgur quark model and VFV for the Vijande-Fernandez-Valcarce quark model.ss( 1 D2) q q(2 1 D2) RPP [65] GI [93] VFV [94] GI [93] VFV [94] Mass 1896 1900 1890 1853 2130 1863 1842 ± 8 A major problem with the η 2 (1870) meson is that it did not appear in the K * K decay channel, and it is difficult to confirm the η 2 (1870) state as a conventional ss state [95] (see this reference for more details).On the other hand, if the η 2 (1870) state is explained as a q q(2 1 D 2 ) state, its mass is too low and the theoretical branching ratio is [96] However, the η 2 (1870) was not seen in the KK * channel, and the η 2 (1870) → f 2 (1270)η decay mode has been observed in many experiments.In this work, it is found that the η 2 (1870) can be interpreted as a η-(K * K * ) f ′ 2 (1525) threebody resonance.
In Fig. 5 (h), we show the modulus squared scattering amplitude for T 2 (1525) .The numerical results are calculated with Λ = 1009 MeV.It can be seen that there are three distinct peaks located at 1914, 1975, and 2072 MeV.However, none of them can be associated with the four excited K 2 states quoted in RPP [65] as discussed above.Further experimental and theoretical works is needed in this direction.

IV. SUMMARY
Within the framework of the fixed-center approximation, we further study the ηK * K * , πK * K * and KK * K * threebody system where we view the K * K * subsystem as scalar meson a 0 (1780) and tensor meson f ′ 2 (1525).In terms of the two-body interactions, η(π/K)K * ( K * ) and K * K * provided by the chiral unitary approach, we describe the η/π/K-(K * K * ) systems by using the FCA.By analysis of the η/π/K-(K * K * ) a0(1780)/f ′ 2 (1525) scattering amplitudes, one can study those dynamically generated resonances from the above three-body systems.It is found that the η(2010) meson can be interpreted as π(K * K * ) a0(1780) with I = 0, and the π(2070) can be explained with the π(K * K * ) a0(1780) with I = 1 and η(K * K * ) a0(1780) .Two resonances with masses around 2150 MeV are predicted in K(K * K * ) a0(1780) with I = 1 2 and 3 2 .It is important to observe such an exotic light flavor state with isospin I = 3  2 by future experiments.Furthermore, the generated resonances with quantum numbers J P = 2 − can be unambiguously assigned to experimental states.The η 2 (1870) meson can be interpreted as η(K * K * ) f ′ 2 (1525) , while π 2 (1880) can be interpreted as π(K * K * ) f ′ 2 (1525) .This assignment provides a natural explanation to these states.Actually the possibility of providing a theoretical explanation of such resonances was the main motivation for our study since its description is clearly out of the scope of the classical q q model.Finally, we summarize the theoretical results obtained here about ηK * K * , πK * K * , and KK * K * three-body system in Table V.It is expected these theoretical calculations could be tested by future experiments, such as the BESIII, BellII, and LHCb.TABLE V. Summary about the theoretical results obtained in this work for the ηK * K * , πK * K * and KK * K * three-body systems.The question mark "?" stands for a non-cataloged state in the Review of particle physics [65].

Fig. 2 (
Fig.2(a) shows the results obtained in the isospin=1 and spin=0 sector with cut off parameter Λ = 1100 MeV, and Fig.2 (b)shows the results obtained in the isospin=0 and spin=2 sector with Λ = 1009 MeV.From Fig.2(a), one can see a bump structure around 1780 MeV that can be assigned to the a 0 (1780) state.As discussed in the introduction part, the scalar meson a 0 (1780) was first observed by the BaBar Collaboration[30] in 2021 and recently confirmed by the BESIII Collaboration[31].It can be seen that, in Fig.2(b), the narrow peak around 1517 MeV can be associated with the tensor meson f ′ 2 (1525).Comparing with the numerical results shown in Fig.2(a), it is found that the strength of |t K * K * →K * K * | 2 for f ′ 2 (1525) is much larger than that for the case of a 0 (1780).This indicates that the s-wave K * K * interaction in the isospin=0 and spin=2 sector is much stronger than that for the case of isospin=1 and spin=0.Furthermore, comparing the line shapes in Fig.2 (a)with Fig.2 (b), it is found that the width obtained for the tensor meson f ′ 2 (1525) is much narrower than that for the scalar meson a 0 (1780).Next, in Fig.3, we show the numerical results for the respective form factors of a 0 (1780) (solid curve) and f ′ 2 (1525) (dashed curve) as a function of q = |⃗ q|, where the theoretical results are obtained for the a 0 (1780)[f ′ 2 (1525)] with Λ = 1100(1009) MeV.The condition |⃗ p − ⃗ q| < Λ implies that the form factor F B * (q) is exactly zero for q > 2Λ.As discussed earlier, with these values of the cutoff parameter Λ, one can obtain a 0 (1780) and f ′ 2 (1525) resonances in vector mesons-vector meson coupled channel interactions as in Refs.[19,23].

TABLE I .
Three body single scattering amplitudes in terms of the unitarized two-body scattering amplitudes.Here, I denotes the total isospin of the discussed three-body sysmtems.

TABLE II .
The calculated mass of a0(1780) and the peak positions (M ηK * K * , M πK * K * , and M KK * K * ) of squared of three-body scattering amplitude corresponding to different values of the cutoff parameter Λ.Here, all values are in units of MeV.

TABLE III .
Mass of the η2(1870) meson calculated by different model.Here, all masses are given in units of MeV.

TABLE IV .
Mass of π2(1880) calculated by different model.Here, all mass values listed in the second row are in units of MeV.