Prediction of possible exotic states in the $\eta \bar{K}K^*$ system

TL;DR

This paper predicts potential exotic meson states with specific quantum numbers by analyzing three-body interactions involving eta, K, and K* particles, using a fixed center approximation to the Faddeev equations.

Contribution

It introduces a novel application of the fixed center approximation to predict exotic states in the eta-K-K* system, identifying possible bound states with specific quantum numbers.

Findings

Evidence of a bound state around 1700 MeV with width 180 MeV.

Another bound state near 1680 MeV with width 160 MeV.

Identification of states with exotic quantum numbers in the meson spectrum.

Abstract

We investigate the $\eta \bar{K}K^*$ three body system in order to look for possible $I^G(J^{PC})=0^+(1^{-+})$ exotic states in the framework of the fixed center approximation to the Faddeev equation. We assume the scattering of $\eta$ on a clusterized system $\bar{K}K^*$, which is known to generate the $f_1(1285)$, or a $\bar{K}$ on a clusterized system $\eta K^*$, which is shown to generate the $K_1(1270)$. In the case of the $\eta$-$(\bar{K}K^*)_{f_1(1285)}$ scattering, we find evidence of a bound state $I^G(J^{PC})=0^+(1^{-+})$ below the $\eta{f_1(1285)}$ threshold with mass around 1700 MeV and width about 180 MeV. Considering the $\bar{K}$-$(\eta K^*)_{K_1(1270)}$ scattering, we obtain a bound state $I(J^{P})=0(1^{-})$ just below the $\bar{K}{K_1(1270)}$ threshold with a mass around 1680 MeV and a width about 160 MeV.

Equations

$$T_1$$

$$T_2$$

$$T$$

$$S^{(1)}=$$

$$S^{(2)}=$$

$$\times \frac{1}{\sqrt{2w_3}}\frac{1}{\sqrt{2w_3^{\prime }}}\frac{1}{\sqrt{2w_1}}\frac{1}{\sqrt{2w_1^{\prime }}}\frac{1}{\sqrt{2w_2}}\frac{1}{\sqrt{2w_2^{\prime }}}$$

$$\times \int \frac{d^{3}q}{(2\pi {)}^3}{F}_{\mathrm{cls}}(q)\frac{1}{{q^0}^2-|\vec{q}{|}^2-m_3^2+iϵ},$$

$$q^0(s)=\frac{s+m_3^2-m_{\mathrm{cls}}^2}{2\sqrt{s}},$$

$$F_{\mathrm{cls}}(q)=$$

$$\times \frac{1}{m_{\mathrm{cls}}-w_1(\vec{p})-w_2(\vec{p})}\frac{1}{2w_1(\vec{p}-\vec{q})}\frac{1}{2w_2(\vec{p}-\vec{q})}$$

$$\times \frac{1}{m_{\mathrm{cls}}-w_1(\vec{p}-\vec{q})-w_2(\vec{p}-\vec{q})},$$

$$\displaystyle N=\int_{|\vec{p}|<\Lambda}d^{3}\vec{p}~{}\Big{(}\frac{1}{2w_{1}(\vec{p})}\frac{1}{2w_{2}(\vec{p})}\frac{1}{m_{\rm{cls}}-w_{1}(\vec{p})-w_{2}(\vec{p})}\Big{)}^{2},$$

$$S=$$

$$\frac{1}{\sqrt{2w_3}}\frac{1}{\sqrt{2w_3^{\prime }}}\frac{1}{\sqrt{2w_{\mathrm{cls}}}}\frac{1}{\sqrt{2w_{\mathrm{cls}}^{\prime }}}.$$

$${\tilde{t}}_1=t_1\sqrt{\frac{2w_{\mathrm{cls}}}{2w_1}}\sqrt{\frac{2w_{\mathrm{cls}}^{\prime }}{2w_1^{\prime }}},{\tilde{t}}_2=t_2\sqrt{\frac{2w_{\mathrm{cls}}}{2w_2}}\sqrt{\frac{2w_{\mathrm{cls}}^{\prime }}{2w_2^{\prime }}}.$$

$$T=\frac{{\tilde{t}}_1+{\tilde{t}}_2+2{\tilde{t}}_1{\tilde{t}}_2{G}_0}{1-{\tilde{t}}_1{\tilde{t}}_2{G}_0^2},$$

$$G_0(s)=\frac{1}{2m_{\mathrm{cls}}}\int \frac{d^{3}q}{(2\pi {)}^3}\frac{F_{\mathrm{cls}}(q)}{{q^0}^2-|\vec{q}{|}^2-m_3^2+iϵ}.$$

$$s_1=m_3^2+m_1^2+\frac{(s-m_3^2-m_{\mathrm{cls}}^2)(m_{\mathrm{cls}}^2+m_1^2-m_2^2)}{2m_{\mathrm{cls}}^2},$$

$$s_2=m_3^2+m_2^2+\frac{(s-m_3^2-m_{\mathrm{cls}}^2)(m_{\mathrm{cls}}^2+m_2^2-m_1^2)}{2m_{\mathrm{cls}}^2},$$

$$|\bar{K}{K}^{\ast }{⟩}_{I=0}=\frac{1}{\sqrt{2}}|(\frac{1}{2},-\frac{1}{2})⟩-\frac{1}{\sqrt{2}}|(-\frac{1}{2},\frac{1}{2})⟩,$$

$$⟨\eta (\bar{K}{K}^{\ast })|t|\eta (\bar{K}{K}^{\ast })⟩=$$

$$=$$

$$\displaystyle+\Big{(}\frac{1}{\sqrt{2}}\Big{\langle}(\frac{1}{2}\frac{1}{2},-\frac{1}{2})\Big{|}-\frac{1}{\sqrt{2}}\Big{\langle}(\frac{1}{2}\!\!-\!\!\frac{1}{2},\frac{1}{2})\Big{|}\Big{)}t_{32}\Big{(}\frac{1}{\sqrt{2}}\Big{|}(\frac{1}{2}\!\!-\!\!\frac{1}{2},\frac{1}{2})\Big{\rangle}-\frac{1}{\sqrt{2}}\Big{|}(\frac{1}{2}\frac{1}{2},-\frac{1}{2})\Big{\rangle}\Big{)},$$

$$t_1=t_{\eta \bar{K}\to \eta \bar{K}}^{I=1/2},t_2=t_{\eta K^{\ast }\to \eta K^{\ast }}^{I=1/2}.$$

$$⟨\bar{K}(\eta K^{\ast })|t|\bar{K}(\eta K^{\ast })⟩=$$

$$\displaystyle\frac{1}{\sqrt{2}}\Big{(}\langle\frac{1}{2}\frac{1}{2}|\otimes\Big{\langle}(\frac{1}{2},-\frac{1}{2})\Big{|}-\langle\frac{1}{2}\!\!-\!\!\frac{1}{2}|\otimes\Big{\langle}(\frac{1}{2},\frac{1}{2})\Big{|}\Big{)}(t_{31}+t_{32})\frac{1}{\sqrt{2}}\Big{(}|\frac{1}{2}\frac{1}{2}\rangle\otimes\Big{|}(\frac{1}{2},-\frac{1}{2})\Big{\rangle}-|\frac{1}{2}\!\!-\!\!\frac{1}{2}\rangle\otimes\Big{|}(\frac{1}{2},\frac{1}{2})\Big{\rangle}\Big{)}$$

$$=$$

$$\displaystyle+\frac{1}{\sqrt{2}}\Big{(}\frac{1}{\sqrt{2}}\Big{\langle}(00,0)+\frac{1}{\sqrt{2}}\Big{\langle}(00,0)\Big{|}\Big{)}t_{32}\frac{1}{\sqrt{2}}\Big{(}\frac{1}{\sqrt{2}}\Big{|}(00,0)\Big{\rangle}+\frac{1}{\sqrt{2}}\Big{|}(00,0)\Big{\rangle}\Big{)}.$$

$$t_1=t_{\bar{K}\eta \to \bar{K}\eta }^{I=1/2},t_2=t_{\bar{K}{K}^{\ast }\to \bar{K}{K}^{\ast }}^{I=0}.$$

Full text

I Introduction

Exotic states cannot be described by the traditional quark model and may be of more complex structure allowed in QCD such as glueballs, hybrid mesons and multiquark states. The discovery of exotic states and the study of their structure will apparently extend our knowledge of the strong interaction dynamics Liu:2019zoy ; Guo:2017jvc ; Chen:2016spr .

A meson with quantum numbers $J^{PC}=1^{-+}$ which is excluded by the traditional quark model with $q\bar{q}$ picture is an exotic state Amsler:2004ps . Interestingly, three isovector $J^{PC}=1^{-+}$ exotic candidates, namely $\pi_{1}(1400)$, $\pi_{1}(1600)$, and $\pi_{1}(2015)$ have been reported experimentally Tanabashi:2018oca . On the theoretical side, the isovector exotic states are interpreted as hybrid mesons within a few different theoretical approaches such as the flux tube model Isgur:1984bm ; Close:1994hc ; Page:1998gz , ADS/QCD model Kim:2008qh ; Bellantuono:2014lra , and Lattice QCD Dudek:2010wm ; Dudek:2009qf ; Bernard:2003jd . In addition, the hybrid meson decay properties are studied within the framework of QCD sum rules in Refs. Huang:2017pzh ; Huang:2016upt ; Huang:2014hya ; Chen:2010ic . Some works suggest that the isovector exotic state might be a fourquark state Chen:2008qw or a molecule/four-quark mixing state Narison:2009vj . On the other hand, the three body system can also carry the quantum numbers $J^{PC}=1^{-+}$. In Ref. Zhang:2016bmy , by keeping the strong interactions of $\bar{K}K^{*}$ which generate the $f_{1}(1285)$ resonance Roca:2005nm ; Lutz:2003fm , the $\pi\bar{K}K^{*}$ three body system was investigated within the framework of the fixed center approximation (FCA) to the Faddeev equation, where the $\pi_{1}(1600)$ could be interpreted as a dynamically generated state from $\pi$-$(\bar{K}K^{*})_{f_{1}(1285)}$ system.

In principle, an isoscalar exotic state is also possible, though not observed experimentally Page:1998gz ; Bernard:2003jd . In fact, these isoscalar exotic states were studied with the QCD sum rules analysis using the tetraquark currents Chen:2008ne , where the obtained mass is around $1.8\sim 2.1$ GeV, and the decay width is about 150 MeV.

In this paper we study the $\eta\bar{K}K^{*}$ three body system in order to look for possible $I^{G}(J^{PC})=0^{+}(1^{-+})$ exotic states within the FCA approach, which has been used to investigate the interaction of $K^{-}d$ at threshold Gal:2006cw ; Barrett:1999cw ; Kamalov:2000iy . Within the FCA approach, a possible state in three body system $K^{-}pp$ according to the calculation done within the framework of FCA approach Bayar:2011qj ; Bayar:2012hn , has been supported by the J-PARC experiments Ajimura:2018iyx . In Ref. Xie:2011uw the $\Delta_{5/2^{+}}(2000)$ puzzle is solved in the study of the $\pi$-$(\Delta\rho)$ interaction, and in Ref. Xie:2010ig , a peak is found around 1920 MeV indicating a $NK\bar{K}$ state with $I=1/2$ around that energy, which supports the existence of a $N^{*}$ resonance with $J^{P}=1/2^{+}$ around 1920 MeV obtained in Refs. Jido:2008kp ; MartinezTorres:2009cw ; MartinezTorres:2008kh ; MartinezTorres:2010zv , where the full Faddeev calculations were done. Recently, the predictions of several heavy flavor resonance states in three body system have been carried out within the framework of FCA approach like $\bar{K}^{(*)}B^{(*)}\bar{B}^{(*)}$ Ren:2018qhr , $D^{(*)}B^{(*)}\bar{B}^{(*)}$ Dias:2018iuy , $\rho B^{*}\bar{B}^{*}$ Bayar:2015zba , $\rho D^{*}\bar{D}^{*}$ Bayar:2015oea ; Xiao:2012dw , $DKK$ $(DK\bar{K})$ Debastiani:2017vhv , $BDD$ $(BD\bar{D})$ Dias:2017miz , and $K\bar{D}D^{*}$ Ren:2018pcd . While the $DDK$ system is investigated in coupled channels by solving the Faddeev equations using the two-body inputs in Ref. MartinezTorres:2018zbl , where it is found that an isospin $1/2$ state is formed at 4140 MeV when the $D^{*}_{s0}(2317)$ is generated in the $DK$ subsystems. And, such a result is compatible with the one found in Ref. SanchezSanchez:2017xtl where the system $D$-$D^{*}_{s0}(2317)$ was studied without considering explicit three-body dynamics. In a more recent work Wu:2019vsy , where the Gaussian expansion method was used, the existence of these $DDK$ states has been further confirmed. These above examples show that the results of the FCA prove to be rather reasonable. However, as important as it is to know the success of the FCA, there are problems for the FCA to study the $\phi K\bar{K}$ system MartinezTorres:2010ax (more details about the limits to the FCA can be found there), for which the $\phi(2175)$ can be reproduced by full Faddeev equation calculations with the $\phi K\bar{K}$ system MartinezTorres:2008gy .

There are two possible scattering cases for the $\eta\bar{K}K^{*}$ three body system since the $\bar{K}K^{*}$ and $\eta K^{*}$ system lead to the formation of two dynamically generated resonances, $f_{1}(1285)$ and $K_{1}(1270)$, respectively. Based on the two body $\eta\bar{K}$, $\eta K^{*}$, and $\bar{K}K^{*}$ scattering amplitudes obtained from the chiral unitary approach Roca:2005nm ; Geng:2006yb ; Guo:2005wp , we perform an analysis of the $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ and $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ scattering amplitude, which will allow us to predict the possible exotic states with quantum numbers $I^{G}(J^{PC})=0^{+}(1^{-+})$.

This paper is organized as follows. In Sec. II we present the FCA formalism and ingredients to analyze the $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ and $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ systems. In Sec. III, numerical results and discussions are shown. Finally, a short summary is given in Sec. IV.

II Formalism and ingredients

II.1 Fixed center approximation formalism

Within the framework of FCA, we consider $\bar{K}K^{*}(\eta K^{*})$ as a cluster and $\eta(\bar{K})$ interacts with the components of the cluster. The total three body scattering amplitude $T$ can be simplified as the summation of the two partition functions $T_{1}$ and $T_{2}$, summing for all the diagrams of Fig. 1 starting with the interaction of particle 3 with particle 1(2) of the cluster. Then the FCA equations can be written in terms of $T_{1}$ and $T_{2}$, which sum up to the total scattering amplitude $T$, and read Barrett:1999cw ; Deloff:1999gc ; Kamalov:2000iy

[TABLE]

where the amplitudes $t_{1}$ and $t_{2}$ represent the unitary scattering amplitudes with coupled channels for the interactions of particle 3 with particle 1 and 2, respectively. The function $G_{0}$ in the above equations is the propagator for the particle 3 between the particle 1 and 2 components of the cluster, which we will discuss below.

We will calculate the total scattering amplitude $T$ in the low energy regime, close to the threshold of the $\eta\bar{K}K^{*}$ systme or below, where the FCA is a good approximation, then the on-shell approximation for the three particles are also used.

Following the field normalization of Refs. Roca:2010tf ; YamagataSekihara:2010qk , we can write down the $S$-matrix for the single scattering term [Fig. 1(a) and 1(e)] as 111In principle, there are two parts of the $S$-matrix: non-interaction part and interaction part. Here, we consider only the interesting part of the $S$-matrix that is the part due to interactions.

[TABLE]

where $\mathcal{V}$ stands for the volume of a box in which the states are normalized to unity, while the momentum $k(k^{\prime})$ and the on-shell energy $w(w^{\prime})$ refer to the initial (final) particles, respectively.

The double scattering contributions are from Fig. 1(b) and 1(f). The expression for the $S$-matrix for the double scattering [$S_{2}^{(2)}=S_{1}^{(2)}$] is given by

[TABLE]

where the $F_{\rm{cls}}(q)$ is the form factor of the cluster which is a bound state of particles 1 and 2. The information on the bound state is encoded in the form factor $F_{\rm{cls}}(q)$ appearing in Eq. (5), which represents essentially the Fourier transform of the cluster wave function. The variable $q^{0}$ is the energy carried by the particle 3 in the center of mass frame of the particle 3 and the cluster, which is given by

[TABLE]

with $s$ the invariant mass squared of the $\eta\bar{K}K^{*}$ system.

For the form factor $F_{\rm{cls}}(q)$, We take the following form only for $s$-wave bound states, as it was discussed in Refs. Roca:2010tf ; YamagataSekihara:2010qk ; YamagataSekihara:2010pj :

[TABLE]

where the normalization factor $N$ is

[TABLE]

with $m_{\rm{cls}}$ the mass of the cluster. Note that the width of $K^{*}$ should be also included in the $F_{\rm{cls}}(q)$ Xie:2011uw . However, as will see later, the masses of $f_{1}(1285)$ and $K_{1}(1270)$ are below the threshold of $\bar{K}K^{*}$ and $\eta K^{*}$, the effect of the width of $K^{*}$ is small and can be neglected.

Similarly, the full $S$ matrix for the scattering of particle 3 with the cluster will be given by

[TABLE]

By comparing Eqs. (4), (5), and (8), we see that we have to give a wight to $t_{1}$ and $t_{2}$ such that Eqs. (4) and (5) the weight factors that appear in Eq. (8). This is achieved by,

[TABLE]

Then, one can quickly solve Eqs. (3) and obtain

[TABLE]

where $G_{0}$ depends on the invariant mass of the $\eta\bar{K}K^{*}$ system and it is given by

[TABLE]

II.2 Single scattering contribution

On the other hand, it is worth noting that the argument of the total scattering amplitude $T$ is regarded as a function of the total invariant mass $\sqrt{s}$ of the three body system, while the arguments of two body scattering amplitudes $t_{1}$ and $t_{2}$ depend on the two body invariant mass $\sqrt{s_{1}}$ and $\sqrt{s_{2}}$, respectively. The $s_{1}$ and $s_{2}$ are the invariant masses squared of the external particle $3$ with momentum $k_{3}$ and particle 1 (2) inside the cluster with momentum $k_{1}$ ($k_{2}$), which are given by

[TABLE]

where $m_{l}$ $(l=1,2,3)$ are the masses of the corresponding particles in the three body system.

It is worth mentioning that in order to evaluate the two body scattering amplitudes $t_{1}$ and $t_{2}$, the isospin of the cluster should be considered. For the case of $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ system, the cluster of $\bar{K}K^{*}$ has isospin $I_{\bar{K}K^{*}}=0$. Therefore, we have

[TABLE]

where the kets in the right-hand side indicate the $I_{z}$ components of the particles $\bar{K}$ and $K^{*}$, $|(I_{z}^{\bar{K}},I_{z}^{K^{*}})\rangle$. For the case of the total isospin $I_{\eta(\bar{K}K^{*})}=0$, the single scattering amplitude is written as Zhang:2016bmy

[TABLE]

where the notation followed in the last term for the states is $|(I_{\eta\bar{K}}I_{\eta\bar{K}}^{z},I_{K^{*}}^{z})\rangle$ for $t_{31}$, while $|(I_{\eta K^{*}}I_{\eta K^{*}}^{z},I_{\bar{K}}^{z})\rangle$ for $t_{32}$. This leads to the following amplitudes for the single scattering contribution [Fig. 1(a) and 1(e)] in the $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ system,

[TABLE]

In the $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ system, the cluster $\eta K^{*}$ can only have isospin $I_{\eta K^{*}}=1/2$. Therefore, for the total isospin $I_{\bar{K}(\eta K^{*})}=0$, the scattering amplitude is written as Zhang:2016bmy

[TABLE]

This leads to the following amplitudes for the single scattering contribution in the $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ system,

[TABLE]

We see that only the $\bar{K}K^{*}\to\bar{K}K^{*}$ in $I=0$ transition has contribution, since we want to make the total isospin $I_{\bar{K}(\eta K^{*})}=0$, and the $\eta$ meson has isospin zero.

II.3 Unitarized $\eta K^{*}$ and $\bar{K}K^{*}$ interactions

The important ingredients in the calculation of the total scattering amplitude for the $\eta\bar{K}K^{*}$ system using the FCA are the two body $\eta K$, $\eta K^{*}$, and $\bar{K}K^{*}$ unitarized $s$ wave interactions from the chiral unitary approach. These two body scattering amplitudes are studied with the dimensional regularization procedure, and they depend on the subtraction constants $a_{\eta K}$, $a_{\eta K^{*}}$ and $a_{\bar{K}K^{*}}$, and also the regularization scale $\mu$. Note that there is only one parameter for the dimensional regularization procedure, since any change in $\mu$ is reabsorbed by a change in $a(\mu)$ through $a(\mu^{\prime})-a(\mu)={\rm ln}(\frac{\mu^{\prime 2}}{\mu^{2}})$ so that the scattering amplitude is scale independent. In this work, we take these parameters as used in Refs. Roca:2005nm ; Geng:2006yb ; Guo:2005wp : $a_{\eta K}=-1.38$ and $\mu=m_{K}$ for $I_{\eta K}=1/2$; $a_{\eta K^{*}}=-1.85$ and $\mu=1000$ MeV for $I_{\eta K^{*}}=1/2$; $a_{\bar{K}K^{*}}=-1.85$ and $\mu=1000$ MeV for $I_{\bar{K}K^{*}}=0$. With those parameters, one can get the mass of $f_{1}(1285)$ and $K_{1}(1270)$ at their estimated values. 222More details for the two body scattering can be found in Refs. Roca:2005nm ; Geng:2006yb ; Guo:2005wp .

In Figs. 2 (a) and (b) we show the numerical results for $|t_{\bar{K}K^{*}\to\bar{K}K^{*}}^{I=0}|^{2}$ and $|t_{\eta K^{*}\to\eta K^{*}}^{I=1/2}|^{2}$, respectively, from where we see clear peaks for $f_{1}(1285)$ and $K_{1}(1270)$ states.

II.4 Form factors $F_{\rm{cls}}(q)$ and propagator $G_{0}(s)$

To connect with the dimensional regularization procedure we choose the cutoff $\Lambda$ such that the value of the two body loop function at threshold coincides in both methods. Thus we take $\Lambda=990$ MeV such that the $f_{1}(1285)$ is obtained in Refs. Xie:2015lta ; Aceti:2015pma , while for $K_{1}(1270)$ we take $\Lambda=1000$ MeV. The cut-off is tuned to get a pole at $1288-i74$ for the $K_{1}(1270)$ state.

In Figs. 3 and 4 we show the respective form factors for the $f_{1}(1285)$ and $K_{1}(1270)$, respectively, where we take $m_{\rm{cls}}=1281.3$ MeV for $f_{1}(1285)$ and 1284 MeV for $K_{1}(1270)$ as obtained in Ref. Geng:2006yb . In the FCA, we keep the wave function of the cluster unchanged by the presence of the third particle. In order to estimate uncertainties of the FCA due to this “frozen” condition we admit that the wave function of the cluster could be modified by the presence of the third particle. For doing this, we perform calculations with different cut offs. These results shown in Figs. 3 and 4 are obtained with cutoff $\Lambda=890$, $990$ and $1090$ MeV for the case of $f_{1}(1285)$, while for the case of $K_{1}(1270)$, we take $\Lambda=900$, $1000$ and $1100$ MeV.

Next, in Fig. 5, we show the real (solid curves) and imaginary (dashed curves) parts of the $G_{0}$ as a function of the invariant mass for the $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ system with cutoff $\Lambda=890$, $990$ and $1090$ MeV.

The results of the $G_{0}$ function for the $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ system are shown in Fig. 6, we show the real (solid curves) and imaginary (dashed curves) parts , $\Lambda=900$, $1000$ and $1100$ MeV.

From Figs. 5 and 6 one can see that the imaginary part of $G_{0}(s)$ is not sensitive the values of cutoff, while the real part changes a bit with the changing of cutoff.

III Numerical results and discussion

For the numerical evaluation of the three body amplitude, we shall need the calculation of two body interaction amplitudes of $\eta\bar{K}$, $\eta K^{*}$, and $\bar{K}K^{*}$, which were investigated by the chiral dynamics and unitary coupled channels approach in Refs. Roca:2005nm ; Geng:2006yb ; Guo:2005wp . Then we calculate the total scattering amplitude $T$ and associate the peaks or bumps in the modulus squared $|T|^{2}$ to resonances.

In Fig. 7 we show the modulus squared $|T|^{2}$ for the $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ scattering with total isospin $I=0$. One can see that there is a clear bump structure which is below the $\eta{f_{1}(1285)}$ threshold with mass around 1700 MeV and width about 180 MeV. Furthermore, taking $\sqrt{s}=1700$ MeV, we get $\sqrt{s_{1}}=927$ MeV and $\sqrt{s_{2}}=1315$ MeV. At this energy point, the interactions of $\eta\bar{K}$ and $\eta K^{*}$ are strong.

In Fig. 8, we show the results of the $|T|^{2}$ for the $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ system. A strong resonant structure around 1680 MeV with a width about 160 MeV shows up, which indicates that a $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ state can be formed. The mass of the state is below the $\bar{K}$ and ${K_{1}(1270)}$ mass threshold. The strength of $|T|^{2}$ at the peak is much larger than that of Fig. 7 for the case of $\eta f_{1}(1285)\to\eta f_{1}(1285)$ scattering. Thus, it is clear that the preferred configuration is $\bar{K}K_{1}(1270)$. However, the $\bar{K}$ will keep interacting with the $K^{*}$ and sometimes can also make $f_{1}(1285)$.

From Figs. 7 and 8, it can be seen that the peak positions and widths of $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ and $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ systems are quite stable with small variations of the cutoff $\Lambda$ parameter. 333The strength of $|T|^{2}$ changes a bit with the changing of the cutoff parameter. This gives confidence that the $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ and $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ bound states can be formed. In fact, the $\eta f_{1}(1285)$ configuration would mix with the $\bar{K}K_{1}(1270)$ configuration. However, since the strength of the $\bar{K}$-$(\eta K^{*})_{K_{1}(1270)}$ scattering is much larger than the one of $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ scattering, the interference between the two configurations should be small, and both configurations peak around the similar energy, thus, it is expected that the peak around this energy of any mixture of the state is guaranteed.

The $\eta\bar{K}K^{*}$ bound state with quantum numbers $I(J^{P})=0(1^{-})$ has a dominant $\bar{K}K_{1}(1270)$ component. Since $K_{1}(1270)$ decays into $K\pi\pi$ mostly [49], the dominant decay mode of the proposed state should be $\bar{K}K\pi\pi$, and one hope that the future experimental measurements can test our model calculations.

We should mention that there are two $K_{1}(1270)$ states are obtained in Ref. Geng:2006yb . The higher one with mass 1284 MeV couples more strongly to the $\eta K^{*}$ and $K\rho$ channels, while the lower one with mass 1195 MeV mainly couples to the $\pi K^{*}$ channel, and it couples to the $\eta K^{*}$ very weak. Thus, one can expect that the lower $K_{1}(1270)$ of Ref. Geng:2006yb will not affect the calculations here.

IV Summary

In this work, we have used the FCA to the Faddeev equations in order to look for possible $I^{G}(J^{PC})=0^{+}(1^{-+})$ exotic states generated from $\eta\bar{K}K^{*}$ three body interactions. We first select a cluster of $\bar{K}K^{*}$, which is known to generate the $f_{1}(1285)$ in $I=0$, and then let the $\eta$ meson interact with $\bar{K}$ and $K^{*}$. In the modulus squared of $\eta$-$(\bar{K}K^{*})_{f_{1}(1285)}$ scattering amplitude, we find evidence of a bound state below the $\eta{f_{1}(1285)}$ threshold with mass around 1700 MeV and width about 100 MeV. In the case of $\bar{K}$ scattering with the cluster $\eta K^{*}$, which is shown to generate the $K_{1}(1270)$ in $I=1/2$, we obtain a bound state $I(J^{P})=0(1^{-})$ just below the $\bar{K}{K_{1}(1270)}$ threshold with mass around 1680 MeV and width about 160 MeV. In addition, the simplicity of the present approach also allows for a transparent interpretation of the results, which is not easy to see when one uses the full Faddeev equations. In the present study, it is easy to know that the $\bar{K}K_{1}(1270)$ is the dominant one, where the $\bar{K}K^{*}$ subsystem can still couple to the $f_{1}(1285)$ resonance. Yet, one may think that we should rely on the full Faddeev calculations where all the scattering processes can be summed up to infinite order, for example, as pointed in the Refs. Miyagawa:2012xz ; Miyagawa:2018xge for the study of $K^{-}d\to\pi\Sigma n$ reaction. Such calculations are welcome and we intend to address this issue in a future study.

The predictions of existence of possible exotic states have been made within the framework of flux tube model Page:1998gz , Lattice QCD Bernard:2003jd and QCD sum rule Chen:2008ne . The results obtained here provide a different theoretical approach for a devoted investigation of these exotic states.

Acknowledgments

We would like to thank Prof. Li-Sheng Geng for useful discussions. This work is partly supported by the National Natural Science Foundation of China under Grant Nos. 11735003, 1191101015, 11475227, and by the Youth Innovation Promotion Association CAS (No. 2016367).