# Faddeev fixed-center approximation to the $\eta K^{*}\bar{K}^{*}$, $\pi
K^{*}\bar{K}^{*}$ and $KK^{*}\bar{K}^{*}$ systems

Qing-Hua Shen<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of Sciences,Lanzhou 730000,China  School of Nuclear Sciences and
Technology, University of Chinese Academy of Sciences,Beijing 101408, China
Ju-Jun Xie<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy
of Sciences,Lanzhou 730000,China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences,Beijing 101408, China  Lanzhou
Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu
Province, Lanzhou University, Lanzhou, Gansu 730000, China

###### Abstract

The three-body $\eta K^{*}\bar{K}^{*}$, $\pi K^{*}\bar{K}^{*}$ and
$KK^{*}\bar{K}^{*}$ systems are investigated within the framework of fixed-
center approximation to the Faddeev equations, where $K^{*}\bar{K}^{*}$ is
treated as the scalar meson $f_{0}(1710)$. The interactions between $\pi$,
$\eta$, $K$ and $K^{*}$ are taking from the chiral unitary approach. By
scattering the $\eta$ meson on the clusterized
$(K^{*}\bar{K}^{*})_{f_{0}(1710)}$ system, we find a peak in the modulus
squared of the three-body scattering amplitude and it can be associated as a
bound state with quantum numbers $I^{G}(J^{PC})=0^{+}(0^{-+})$. Its mass and
width are around 2054 MeV and 60 MeV, respectively. This state could be
associated to the $\eta(2100)$ meson. For the
$\pi(K^{*}\bar{K}^{*})_{f_{0}(1710)}$ scattering, we find a bump structure
around 1900-2000 MeV with quantum numbers $1^{-}({0^{-+}})$. While for the
$K(K^{*}\bar{K}^{*})_{f_{0}(1710)}$ system, there are three structures. One of
them is much stable and its mass is about 2130 MeV. It is expected that these
theoretical predictions here could be tested by future experimental
measurements, such as by the BESIII, BelleII and LHCb collaborations.

##  I Introduction

It is known that quark models have achieved great success in studying the
properties of hadrons, especially for these ground states. Within the quark
models, it is commonly accepted that mesons are composed of quark and
antiquark ($q\bar{q}$) and baryons are composed of three quarks $(qqq)$.
Recently, the topic of meson-meson and meson-baryon states, with hadrons and
hadrons governed by strong interactions, has been well developed by the
combination of the chiral effective Lagrangians with nonperturbative unitary
techniques in coupled channels, which has been a very fruitful scheme to study
the nature of many hadronic states, both on light and heavy sectors Oset  _et
al._ (2016); Guo  _et al._ (2018); Dong  _et al._ (2021a, b). Some of them are
not easily explained by the classical quark models.

Those hadronic states with exotic quantum numbers cannot be descried by quark
models and they have more complex structures. For example, a state with exotic
quantum numbers $J^{PC}=1^{-+}$, denoted as $\eta_{1}(1855)$, is observed by
the BESIII collaboration Ablikim  _et al._ (2022a, b), which cannot be
explained by traditional $q\bar{q}$ picture. However, it can be easily
obtained in the molecular picture. In Refs. Yang and Huang (2022); Dong  _et
al._ (2022), the $\eta_{1}(1855)$ is interpreted as a $\bar{K}K(1400)$
molecular state. In Ref. Chen  _et al._ (2022), the $\eta_{1}(1855)$ state was
investigated by the method of QCD sum rules and the results support the
interpretation of the $\eta_{1}(1855)$ as a $\bar{s}sg$ hybrid meson. On the
other hand, those states with exotic quantum numbers can be dynamically
generated from the three-body interactions. In Ref. Zhang  _et al._ (2017),
the meson $\pi_{1}(1600)$ with exotic quantum numbers
$I^{G}(J^{PC})=1^{-}(1^{-+})$, is interpreted as a dynamically generated state
of the $\pi K^{*}\bar{K}$ system by using the fixed-center approximation
(FCA), where the two-body $\bar{K}K^{*}$ is fixed as $f_{1}(1285)$ state Lutz
and Kolomeitsev (2004); Roca  _et al._ (2005); Zhou  _et al._ (2014); Geng
_et al._ (2015); Xie  _et al._ (2020). Similarly, in Ref. Zhang and Xie
(2020), possible exotic states with mass around $1700$ MeV and quantum numbers
$I^{G}(J^{PC})=0^{+}(1^{-+})$ in the $\eta K^{*}\bar{K}$ system were
predicted.

Indeed, there is growing evidence that some existing and new observed hadronic
states could be interpreted in terms of resonances or bound states of three
hadrons Martinez Torres  _et al._ (2020); Wu  _et al._ (2022); Malabarba  _et
al._ (2022). Furthermore, some new states are also predicted in three-body
systems Luo  _et al._ (2022a, b); Ikeno  _et al._ (2022). For example, $DKK$
and $DK\bar{K}$ systems were studied in Ref. Debastiani  _et al._ (2017)
within FCA, where the evidence of a state with mass of about $2833$-$2858$ MeV
was found in the $DK\bar{K}$ system. In Ref. Ikeno  _et al._ (2022),
$D^{*}D^{*}\bar{K}^{*}$ is studied by FCA, and they obtain bound states with
isospin $I=1/2$ in total spin $J=0$, $1$, and $2$. These states are very
narrow. In Ref. Wu  _et al._ (2021), the $KD\bar{D}$ system is studied using
the Gaussian expansion method, minimizing the energy of the system. A new
hidden charm $K_{c}(4180)$ excited state was predicted. The same system is
also studied in Ref. Wei  _et al._ (2022) within the framework of FCA and the
results obtained are very similar. Besides, the FCA method have many
achievements on the study of three-body systems Martinez Torres  _et al._
(2008); Ren  _et al._ (2018); Dias  _et al._ (2017); Bayar  _et al._ (2011);
Xiao  _et al._ (2011); Bayar  _et al._ (2014); Liang  _et al._ (2013); Durkaya
and Bayar (2015); Yamagata-Sekihara  _et al._ (2010); Martinez Torres  _et
al._ (2019); Sanchez Sanchez  _et al._ (2018); Xie  _et al._ (2011a, b). Those
successes give us confidence in the FCA method that we use in the present work
to study the $\eta K^{*}\bar{K}^{*}$, $\pi K^{*}\bar{K}^{*}$, and
$KK^{*}\bar{K}^{*}$ systems. In the frame work of FCA, there is a cluster of
two bound particles ($K^{*}\bar{K}^{*}$ in this work) and the third one
($\eta$, $\pi$, or $K$ in this work) collides with the components of this
cluster without modifying its wave function. Certainly, if the third particle
is lighter than the constituents of the cluster, the approximation is better.
Note that the $\eta$, $\pi$, and $K$ mesons are much lighter than the
$K^{*}/\bar{K}^{*}$ meson.

For the $K^{*}\bar{K}^{*}$ subsystem, it has strong attraction between $K^{*}$
and $\bar{K}^{*}$. In fact, the well established scalar meson $f_{0}(1710)$
was proposed to be a state dynamically generated from the vector meson-vector
meson interactions in coupled channels Geng and Oset (2009); Geng  _et al._
(2009); Du  _et al._ (2018); García-Recio  _et al._ (2013); Wang and Zou
(2021). Within this picture, the $f_{0}(1710)$ resonance can be interpreted as
a $K^{*}\bar{K^{*}}$ molecular state, and the properties of $f_{0}(1710)$
resonance and its production have been investigated in Refs. Nagahiro  _et
al._ (2008); Branz  _et al._ (2010); Geng  _et al._ ; Martinez Torres  _et
al._ (2013); Xie and Oset (2014); Dai  _et al._ (2015); Molina  _et al._
(2020).

In this work, we investigate the three-body $\eta K^{*}\bar{K}^{*}$, $\pi
K^{*}\bar{K}^{*}$, and $KK^{*}\bar{K}^{*}$ systems by considering the
interactions of the three components among themselves, keeping in mind the
expected strong correlations of the $K^{*}\bar{K}^{*}$ subsystem to make the
$f_{0}(1710)$ state. Then in terms of two-body pseudoscalar meson-vector meson
scattering amplitudes, which were obtained from the chiral unitary approach
Lutz and Kolomeitsev (2004); Roca  _et al._ (2005), we solve the Faddeev
equations for the $\eta$, $\pi$, and $K$ scattering on the $K^{*}\bar{K}^{*}$
cluster in $S$-wave by using the fixed center approximation. In this way, it
is quite probable to be able to generate pseudoscalar mesons with three-body
nature. The above three-body systems have quantum numbers $J^{PC}=0^{-+}$ and
isospin $I=0$, $1$, and $1/2$, respectively.

The paper is organized as follows. In next section, we present the FCA method
to the three-body $\eta K^{*}\bar{K}^{*}$, $\pi K^{*}\bar{K}^{*}$, and
$KK^{*}\bar{K}^{*}$ systems. In Sec. III, our theoretical results and
discussions are presented. Finally, a short summary is followed.

##  II Formalism and ingredients

We are interested in three different systems constituted by three mesons:
$\eta K^{*}\bar{K}^{*}$, $\pi K^{*}\bar{K}^{*}$, and $KK^{*}\bar{K}^{*}$. In
order to study the dynamics of these systems we will solve the Faddeev
equations using the fixed-center approximation to obtain the total scattering
amplitudes. In this section we will summarize the derivation of the three-body
scattering amplitudes within the framework of the FCA.

###  II.1 Fixed-center approximation

We are going to use the fixed-center approximation formalism to study the
three-body $\eta K^{*}\bar{K}^{*}$, $\pi K^{*}\bar{K}^{*}$, and
$KK^{*}\bar{K}^{*}$ systems. In this frame work, we consider the
$K^{*}\bar{K}^{*}$ as a cluster, which we take it as the $f_{0}(1710)$ state,
and $\eta$, $\pi$ or $K$ interacts with it. The corresponding diagrams are
shown in Fig. 1. In the following we will label $K^{*}$, $\bar{K}^{*}$, and
$\eta$ ($\pi$ or $K$) as particle 1, 2, and 3, respectively.

Figure 1: Diagrammatic representation of FCA for the Faddeev equations for the
three-body $\eta K^{*}\bar{K}^{*}$, $\pi K^{*}\bar{K}^{*}$, and
$KK^{*}\bar{K}^{*}$ systems.

Following the formalism of Ref. Roca and Oset (2010), the three-body
scattering amplitude $T$ for $\eta$ ($\pi$ or $K$) collisions with the
$K^{*}\bar{K}^{*}$ cluster can be obtained by the sum of the partition
functions $T_{1}$, and $T_{2}$:

$\displaystyle T_{1}$ $\displaystyle=$ $\displaystyle t_{1}+t_{1}G_{0}T_{2},$
(1) $\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle
t_{2}+t_{2}G_{0}T_{1},$ (2) $\displaystyle T$ $\displaystyle=$ $\displaystyle
T_{1}+T_{2}=\frac{t_{1}+t_{2}+2t_{1}t_{2}G_{0}}{1-t_{1}t_{2}G^{2}_{0}},$ (3)

where $T_{1}$ stands for the sum of all the diagrams in Fig. 1 where the third
particle collides firstly with the particle $K^{*}$ in the cluster, while
$T_{2}$ is the sum of all the diagrams where the third particle collides
firstly with particle $\bar{K}^{*}$ of the cluster. The $t_{1}$ and $t_{2}$
represent the unitary two-body scattering amplitudes in coupled channels for
the interactions of particle 3 with particle 1 and 2, respectively. These two-
body scattering amplitudes will be discussed later on.

Besides, in the above equations, $G_{0}$ is the loop function for the particle
3 propagating between the $K^{*}$ and $\bar{K}^{*}$ of the cluster, which can
be written as,

$\displaystyle
G_{0}(s)=\frac{1}{2M_{f_{0}(1710)}}\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{F_{f_{0}(1710)}(q)}{q^{0^{2}}-\left\lvert\vec{q}\\!~{}\right\rvert^{2}-m_{3}^{2}+i\epsilon},$
(4)

where $s$ is the invariant mass square of the whole three-body system, and
$M_{f_{0}(1710)}$ is the mass of the bound state $f_{0}(1710)$. Besides,
$F_{f_{0}(1710)}(q)$ is the form factor of the $K^{*}\bar{K}^{*}$ subsystem,
which is treated as the $f_{0}(1710)$ state. The $q^{0}$ is the energy of
particle 3 with mass $m_{3}$ in the center-of-mass frame of particle 3 and the
$(K^{*}\bar{K}^{*})_{f_{0}(1710)}$ cluster, which is given by

$\displaystyle q^{0}(s)=\frac{s+m_{3}^{2}-M_{f_{0}(1710)}^{2}}{2\sqrt{s}}.$
(5)

Following the approach of Refs. Gamermann  _et al._ (2010); Yamagata-Sekihara
_et al._ (2011); Roca and Oset (2010), one can easily get the expression of
the form factor $F_{f_{0}(1710)}$ for the $S$-wave $K^{*}\bar{K}^{*}$ bound
state $f_{0}(1710)$ as

$\displaystyle F_{f_{1}(1710)}(q)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\int_{\left\lvert\vec{p}\\!~{}\right\rvert\leq\Lambda,\left\lvert\vec{p}-\vec{q}\\!~{}\right\rvert\leq\Lambda}d^{3}\vec{q}\frac{1}{2\omega_{1}(\vec{p}\\!~{})}\frac{1}{2\omega_{2}(\vec{p}\\!~{})}\times$
(6)
$\displaystyle\frac{1}{M_{f_{0}(1710)}-\omega_{1}(\vec{p}\\!~{})-\omega_{2}(\vec{p}\\!~{})}\times$
$\displaystyle\frac{1}{2\omega_{1}(\vec{p}-\vec{q}\\!~{})}\frac{1}{2\omega_{2}(\vec{p}-\vec{q}\\!~{})}\times$
$\displaystyle\frac{1}{M_{f_{1}(1710)}-\omega_{1}(\vec{p}-\vec{q}\\!~{})-\omega_{2}(\vec{p}-\vec{q}\\!~{})},$

where
$\omega_{1}(\vec{p}\\!~{})=\omega_{2}(\vec{p}\\!~{})=\sqrt{|\vec{p}\\!~{}|^{2}+m^{2}_{K^{*}}}$,
and the normalization factor $N$ is given by

$N=\int_{\left\lvert\vec{p}\\!~{}\right\rvert\leq\Lambda}d^{3}\vec{p}\left(\frac{1}{4\omega^{2}_{1}(\vec{p}\\!~{})}\frac{1}{M_{f_{0}(1710)}-2\omega_{1}(\vec{p}\\!~{})}\right)^{2}.$
(7)

The cutoff parameter $\Lambda$ is needed to regularize the vector meson-vector
meson loop functions in the chiral unitary approach Geng and Oset (2009); Geng
_et al._ (2009). In the present work, the upper integration limit of $\Lambda$
has the same value of the cutoff used in Refs. Geng and Oset (2009); Geng  _et
al._ (2009). With the values of $\Lambda$, one can get the $f_{0}(1710)$ state
in the vector meson-vector meson interactions in coupled channels.

In Fig. 2, we show numerical results of the form factor $F_{f_{0}(1710)}$ as a
function of $q=|\vec{q}\\!~{}|$, where the solid and dash curves are obtained
with $\Lambda=945$ MeV and $1125$ MeV, respectively. Note that with these
values for cutoff parameter $\Lambda$, one can get the $f_{0}(1710)$ state in
the vector-vector interactions in coupled channels. Besides, the integration
condition $|\vec{p}-\vec{q}\\!~{}|<\Lambda$ implies that the form factor of
$F_{f_{0}(1710)}$ is exactly zero when $q>2\Lambda$.

Figure 2: Form factor of $(K^{*}\bar{K}^{*})_{f_{0}(1710)}$ cluster with
$\Lambda=945$ MeV (solid line) and $\Lambda=1125$ MeV (dash line).

###  II.2 Unitarized $K^{*}\bar{K}^{*}$ interaction for the $f_{0}(1710)$
state and the loop function $G_{0}$

The $K^{*}\bar{K}^{*}$ interaction in $S$-wave has been analyzed within the
formalism developed in Ref. Geng and Oset (2009) for studying the interaction
of the nonet of vector mesons among themselves. In Fig. 3, the module square
of $t_{K^{*}\bar{K}^{*}\to K^{*}\bar{K}^{*}}$ obtained from the chiral unitary
approach in $\rho\rho$, $\omega\omega$, $\omega\phi$, $\phi\phi$, and
$K^{*}\bar{K}^{*}$ coupled channels for isospin $I=0$ sector, are shown, where
one can see that there is a clear peak for $f_{0}(1710)$ state around $1704$
MeV with $\Lambda=1125$ MeV. While the peak moves to $1733$ MeV if one take
$\Lambda=945$ MeV. Note that in the calculations of these two-body loop
functions of $G_{\rho\rho}$ and $G_{K^{*}\bar{K}^{*}}$, the widths of vector
mesons $\rho$ and $K^{*}$ are taken into account, since they have large total
decay widths. On the other hand, in the calculations for the form factor of
$F_{f_{0}(1710)}(q)$, the mass of cluster $f_{0}(1710)$ is taken as $1704$ MeV
when $\Lambda=1125$ MeV, while $M_{f_{0}(1710)}=1733$ if one takes
$\Lambda=945$ MeV.

It is worthy to mention that the “our evaluation” value in the 2020 version
and 2021 updated of the Review of Particle Physics (RPP) Zyla  _et al._ (2020)
for the mass of $f_{0}(1710)$ resonance is $1704\pm 12$ MeV, while the “our
average” value for it is $1733^{+8}_{-7}$ MeV. In fact, even the $f_{0}(1710)$
resonance is well established in the RPP, there are still many doubts about
its nature Zhu  _et al._ (2022). We refer the Review of Particle Physics Zyla
_et al._ (2020) for more details.

Figure 3: Modulus squared of $t_{K^{*}\bar{K}^{*}\to K^{*}\bar{K}^{*}}$ as a
function of the invariant mass of $K^{*}\bar{K}^{*}$ system with $\Lambda=945$
MeV (dash line) and $\Lambda=1125$ MeV (solid line).

Next we present the numerical results for the loop function $G_{0}$, which is
the $\eta$ ($\pi$ and $K$) propagator between the $K^{*}$ and $\bar{K}^{*}$ of
the cluster $f_{0}(1710)$. Note that $G_{0}$ is dependent on the invariant
mass $\sqrt{s}$ of the three-body system. In this work, there are three
$G_{0}$ functions for $\eta$-$(K^{*}\bar{K}^{*})_{f_{0}(1710)}$,
$\pi$-$(K^{*}\bar{K}^{*})_{f_{0}(1710)}$, and
$K$-$(K^{*}\bar{K}^{*})_{f_{0}(1710)}$ systems. In Fig. 4, we show the real
and imaginary parts of the $G_{0}$ function for $I_{K^{*}\bar{K}^{*}}=0$ as a
function of the total three body $\eta$-$K^{*}$-$\bar{K}^{*}$ system invariant
mass. The other two cases can be easily obtained by the replacements of
$\eta\to\pi$ and $\eta\to K$, respectively. They are shown in Figs. 5 and 6.

Figure 4: The real (solid line) and imaginary (dash line) parts of the loop
function $G_{0}$ for $\eta K^{*}\bar{K}^{*}$ system with the cutoff parameter
$\Lambda=945$ MeV (red line) and $\Lambda=1125$ MeV (blue line). Figure 5: As
in Fig. 4 but for the $\pi K^{*}\bar{K}^{*}$ system. Figure 6: As in Fig. 4
but for the $KK^{*}\bar{K}^{*}$ system.

###  II.3 Contributions for the $\eta$, $\pi$, and $K$ interaction with the
$K^{*}\bar{K}^{*}$ system

According to Fig. 1 (a) and (e), the single-scattering contribution from
$t_{1}$ and $t_{2}$ are appropriate the combination of the two-body unitarized
scattering amplitudes. For example, let us firstly consider the $\eta
K^{*}\bar{K}^{*}$ system with the $K^{*}\bar{K}^{*}$ cluster in $I=0$ for the
$f_{0}(1710)$ state. The $K^{*}\bar{K}^{*}$ isospin state is written as

$\displaystyle\ket{K^{*}\bar{K}^{*}}_{I=0}=\frac{1}{\sqrt{2}}(\ket{\frac{1}{2},-\frac{1}{2}}-\ket{-\frac{1}{2},\frac{1}{2}}),$
(8)

where the kets on the right side of the above equation represent
$\ket{I_{z}^{K^{*}},I_{z}^{\bar{K}^{*}}}$ for $K^{*}\bar{K}^{*}$.

Then the single scattering contributions to the total amplitude of $\bra{\eta
K^{*}\bar{K}^{*}}\hat{t}\ket{\eta K^{*}\bar{K}^{*}}$ can be easily obtained in
terms of unitary two-body transition amplitudes $t_{\eta K^{*}\to\eta K^{*}}$
and $t_{\eta\bar{K}^{*}\to\eta\bar{K}^{*}}$ derived in Ref. Roca  _et al._
(2005). Here we write explicitly the case of
$\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}\to\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}$,

$\displaystyle\bra{\eta(K^{*}\bar{K}^{*})}\hat{t}\ket{\eta(K^{*}\bar{K}^{*})}$
(9)
$\displaystyle=(\bra{A_{1}}+\bra{A_{2}})(\hat{t}_{31}+\hat{t}_{32})(\ket{A_{1}}+\ket{A_{2}})$
$\displaystyle=\bra{A_{1}}\hat{t}_{31}\ket{A_{1}}+\bra{A_{2}}\hat{t}_{32}\ket{A_{2}},$

where $\ket{A_{1}}$ stands for the state combined with $\eta$ and $K^{*}$,
while $\ket{A_{2}}$ is the state of $\eta$ and $\bar{K}^{*}$. They are given
by,

$\displaystyle\ket{A_{1}}$
$\displaystyle=\frac{1}{\sqrt{2}}\ket{\frac{1}{2}\frac{1}{2},-\frac{1}{2}}-\frac{1}{\sqrt{2}}\ket{\frac{1}{2}-\frac{1}{2},\frac{1}{2}},$
(10) $\displaystyle\ket{A_{2}}$
$\displaystyle=\frac{1}{\sqrt{2}}\ket{\frac{1}{2}-\frac{1}{2},\frac{1}{2}}-\frac{1}{\sqrt{2}}\ket{\frac{1}{2}\frac{1}{2},-\frac{1}{2}}.$

Thus, we have

$\displaystyle t_{1}$ $\displaystyle=\bra{A_{1}}\hat{t}_{31}\ket{A_{1}}$ (11)
$\displaystyle=\frac{1}{2}t^{I=\frac{1}{2}}_{\eta K^{*}\rightarrow\eta
K^{*}}+\frac{1}{2}t^{I=\frac{1}{2}}_{\eta K^{*}\rightarrow\eta K^{*}},$
$\displaystyle=t^{I=\frac{1}{2}}_{\eta K^{*}\rightarrow\eta K^{*}}$
$\displaystyle t_{2}$ $\displaystyle=\bra{A_{2}}\hat{t}_{32}\ket{A_{2}}$
$\displaystyle=\frac{1}{2}t^{I=\frac{1}{2}}_{\eta\bar{K}^{*}\rightarrow\eta\bar{K}^{*}}+\frac{1}{2}t^{I=\frac{1}{2}}_{\eta\bar{K}^{*}\rightarrow\eta\bar{K}^{*}}$
$\displaystyle=t^{I=\frac{1}{2}}_{\eta\bar{K}^{*}\rightarrow\eta\bar{K}^{*}}.$

.

Similarly, for the case of $\pi K^{*}\bar{K}^{*}$, we have

$\displaystyle t_{1}$ $\displaystyle=\bra{A_{1}}\hat{t}_{31}\ket{A_{1}}$ (12)
$\displaystyle=\frac{2}{3}t_{\pi K^{*}\rightarrow\pi
K^{*}}^{I=\frac{3}{2}}+\frac{1}{3}t_{\pi K^{*}\rightarrow\pi
K^{*}}^{I=\frac{1}{2}},$ $\displaystyle t_{2}$
$\displaystyle=\bra{A_{2}}\hat{t}_{32}\ket{A_{2}}$
$\displaystyle=\frac{2}{3}t_{\pi\bar{K}^{*}\rightarrow\pi\bar{K}^{*}}^{I=\frac{3}{2}}+\frac{1}{3}t_{\pi\bar{K}^{*}\rightarrow\pi\bar{K}^{*}}^{I=\frac{1}{2}}.$

Next, for the case of $KK^{*}\bar{K}^{*}$, we can get following results,

$\displaystyle t_{1}$ $\displaystyle=$
$\displaystyle\bra{A_{1}}\hat{t_{31}}\ket{A_{1}}$ (13) $\displaystyle=$
$\displaystyle\frac{3}{4}t_{KK^{*}\to KK^{*}}^{I=1}+\frac{1}{4}t_{KK^{*}\to
KK^{*}}^{I=0},$ $\displaystyle t_{2}$ $\displaystyle=$
$\displaystyle\bra{A_{2}}\hat{t_{32}}\ket{A_{32}}$ (14) $\displaystyle=$
$\displaystyle\frac{3}{4}t_{K\bar{K}^{*}\to
K\bar{K}^{*}}^{I=1}+\frac{1}{4}t_{K\bar{K}^{*}\to K\bar{K}^{*}}^{I=0}.$

On the other hand, following the approach developed in Refs. Roca and Oset
(2010); Yamagata-Sekihara  _et al._ (2010), we need to give a weight to
$t_{1}$ and $t_{2}$ such that we have the right normalization for the fields
of mesons. This is achieved by replacing

$\displaystyle t_{1}\to\tilde{t_{1}}$
$\displaystyle=\frac{M_{f_{0}(1710)}}{m_{K^{*}}}t_{1},$ (15) $\displaystyle
t_{2}\to\tilde{t_{2}}$
$\displaystyle=\frac{M_{f_{0}(1710)}}{m_{\bar{K}^{*}}}t_{2}.$

Finally, we have the three-body scattering amplitude:

$\displaystyle
T=\frac{\tilde{t_{1}}+\tilde{t_{2}}+2\tilde{t_{1}}\tilde{t_{2}}G_{0}}{1-\tilde{t_{1}}\tilde{t_{2}}G_{0}^{2}}.$
(16)

It is worth to mention that the three-body total scattering amplitude $T$ is a
function of the total invariant mass $\sqrt{s}$ of the three-body system.
While the two-body scattering amplitudes $t_{1}$ and $t_{2}$ depend on the
invariant masses $\sqrt{s_{1}}$ and $\sqrt{s_{2}}$, which are the invariant
masses of $\eta$ ($\pi$ or $K$) and particle $K^{*}$ ($\bar{K}^{*}$) inside
the cluster of $f_{0}(1710)$. The arguments $s_{1}$ and $s_{2}$ are obtained
as

$\displaystyle s_{1}$
$\displaystyle=m_{\eta/\pi/K}^{2}+m_{K^{*}}^{2}+\frac{s-m_{\eta/\pi/K}^{2}-M_{f_{0}(1710)}^{2}}{2}$
(17) $\displaystyle s_{2}$
$\displaystyle=m_{\eta/\pi/K}^{2}+m_{\bar{K}^{*}}^{2}+\frac{s-m_{\eta/\pi/K}^{2}-M_{f_{0}(1710)}^{2}}{2}$

where we have used $m_{K^{*}}=m_{\bar{K}^{*}}$.

##  III Numerical results

In this section we shown the theoretical numerical results obtained for the
scattering amplitude square of the three-body $\eta K^{*}\bar{K}^{*}$, $\pi
K^{*}\bar{K}^{*}$, and $KK^{*}\bar{K}^{*}$ systems with total isospin $I=0$,
$1$, and $1/2$, respectively. We evaluate the three body scattering amplitude
$T$ and associate the peaks in the modulus squared $|T|^{2}$ to resonances. In
addition, the masses of the particles considered in this work are shown in
Table 1. These values are taken from the review of particle physics Zyla  _et
al._ (2020). For the mass of $f_{0}(1710)$ state, both ”evaluation mass”
(eva.) and ”average mass” (ave.) are shown.

Table 1: Particle masses (in MeV) used in this work. $m_{K}$ | $m_{\pi}$ | $m_{\eta}$ | $m_{\eta^{\prime}}$ | $\Gamma_{K^{*}}$ |  $M_{f_{0}(1710)}$ (eva.)
---|---|---|---|---|---
495.6455 | 138.04 | 547.862 | 957.78 | 49.1 | 1704
$m_{K^{*}/\bar{K}^{*}}$ | $m_{\rho}$ | $m_{\phi}$ | $m_{\omega}$ | $\Gamma_{\rho}$ |  $M_{f_{1}(1710)}$ (ave.)
893.1 | 775.26 | 1019.461 | 782.66 | 149.1 | 1733

###  III.1  Three-body $\eta K^{*}\bar{K}^{*}$ system

In order to obtain two-body system scattering amplitude $t_{1}$ and $t_{2}$,
we need the scattering amplitude of $\eta K^{*}$ and $\eta\bar{K}^{*}$, which
were studied in Ref. Roca  _et al._ (2005); Geng  _et al._ (2007). In this
work, we have considered also the affect of $\eta^{\prime}$ meson as done in
Refs. Guo  _et al._ (2006); Sun  _et al._ (2022). Within the model parameters
as used in Refs. Roca  _et al._ (2005); Geng  _et al._ (2007): $\mu=900$ MeV
and $\alpha_{\mu}=-1.85$, one can easily obtain the modulus squared of
$t_{\eta K^{*}\to\eta K^{*}}$, which is shown in Fig. 7. One can see that
there is a clear peak for the $K_{1}(1270)$ state Geng  _et al._ (2007), and
the interaction of $\eta K^{*}$ is strong. Note that the width of these vector
mesons are taken into account in the calculations for the loop functions Geng
and Oset (2009), and the affects of those channels including the
$\eta^{\prime}$ meson are very small and could be safely neglected.

Figure 7: Modulus squared of $t_{\eta K^{*}\rightarrow\eta K^{*}}$ as a
function of the invariant mass of $\eta K^{*}$ system

On the other hand, one can also get the transition amplitude
$t_{\eta\bar{K}^{*}\to\eta\bar{K}^{*}}$. Since the $K^{*}\bar{K}^{*}$, $\eta
K^{*}$ and $\eta\bar{K}^{*}$ interactions are attractive and strong enough to
generate bound states, it is natural to expect the existence of multi-hadron
states composed of $K^{*}\bar{K}^{*}$ and $\eta$.

Then, we obtain the total three body scattering amplitude
$T_{\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}\to\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}}$.
In Fig. 8 we show the modulus squared
$|T_{\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}\to\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}}|^{2}$
for the
$\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}\to\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}$
scattering amplitude. The picture shows a clear peak around $2100$ MeV with
cutoff parameter $\Lambda=945$ MeV. If one take $\Lambda=1125$ MeV, the peak
moves to $2070$ MeV. This peak with quantum numbers
$I^{G}(J^{PC})=0^{+}(0^{-+})$ could be interpreted as $\eta(2100)$ that have
mass $2050_{-24-26}^{+30+75}$ MeV and width $250_{-30-164}^{+36+181}$ MeV. In
addition, it is found that the peak is not sensitive to the cutoff parameter
$\Lambda$.

On the experimental side, the $\eta(2100)$ meson was observed firstly by DM2
collaboration in 1988 Bisello  _et al._ (1989). Then it is also observed by
BESIII collaboration in a partial wave analysis of the
$J/\Phi\rightarrow\gamma\phi\phi$ decay with 22 $\sigma$ significance Ablikim
_et al._ (2016). However, this state is not listed in the summary meson tables
of the RPP Zyla  _et al._ (2020), which indicates more analysis about it are
needed. Besides, in Ref. Wang  _et al._ (2017), the $\eta(2100)$ meson is
interpreted as a candidate for $\eta(4S)$ of the fourth pseudoscalar meson
nonet.

Figure 8: The modulus squared of scattering amplitude for $\eta
K^{*}\bar{K}^{*}$ three-body system with $\Lambda=945$ MeV (solid line) and
$\Lambda=1125$ MeV (dash line).

###  III.2 Three-body $\pi K^{*}\bar{K}^{*}$ system

In the case of $\pi K^{*}\bar{K}^{*}$ system, we need the two-body $\pi
K^{*}\to\pi K^{*}$ scattering amplitude as inputs. Within the formula and
theoretical parameters of Ref. Roca  _et al._ (2005), one can easily get the
two-body scattering amplitude $t_{\pi K^{*}\to\pi K^{*}}$ in coupled channels.
Note that we use the cutoff regularization scheme for the loop functions of
the intermediate vector-pseduoscalar mesons, and the widths of the vector
mesons are also considered. Furthermore, we take cutoff parameter
$\Lambda_{\pi K^{*}}=1000$ MeV to regularize the loop function containing a
vector and a pseudoscalar meson.

The modulus squared of $t_{\pi K^{*}\to\pi K^{*}}$ in isospin $I=1/2$ is shown
in Fig. 9, from where one can see that there is a clear peak around $1150$
MeV. As discussed in Ref. Roca  _et al._ (2005), it could be the lower pole of
the two $K_{1}(1270)$ states.

The interaction of $\pi K^{*}$ in isospin $I=3/2$ is much weaker then the one
with $I=1/2$. The numerical results of $t_{\pi K^{*}\to\pi K^{*}}$ with
$I=3/2$ are shown in Fig. 10. On can see that the strength of $t^{I=3/2}_{\pi
K^{*}\to\pi K^{*}}$ is much smaller than the one of $t^{I=1/2}_{\pi
K^{*}\to\pi K^{*}}$ and there is no clear bump structure of
the$|t^{I=3/2}_{\pi K^{*}\to\pi K^{*}}|^{2}$ in a wide energy region.

Figure 9: The modulus squared of scattering amplitude of $t_{\pi
K^{*}\rightarrow\pi K^{*}}$ in isospin $I=1/2$ sector as the function of $\pi
K^{*}$ system. Figure 10: The modulus squared of scattering amplitude of
$t_{\pi K^{*}\rightarrow\pi K^{*}}$ in isospin $I=3/2$ sector as the function
of $\pi K^{*}$ system.

In Fig. 11, we show the obtained modulus squared of scattering amplitude
$|T_{\pi(K^{*}\bar{K}^{*})_{f_{0}(1710)}\to\pi(K^{*}\bar{K}^{*})_{f_{0}(1710)}}|^{2}$
of the three-body $\pi K^{*}\bar{K}^{*}$ system. It is found that there is a
wide bump structure around $1900$-$2000$ MeV, which may be associated to the
$\pi(2070)$ state, which was not quoted in the RPP Zyla  _et al._ (2020).
Around that energy region, the $\pi(2070)$ state was needed in a combined
partial wave analysis of $\bar{p}p$ annihilation channels in Ref. Anisovich
_et al._ (2001), and its mass and width are about $2070\pm 35$ MeV and
$310^{+100}_{-50}$ MeV, respectively. The obtained width of $\pi(2070)$ state
is rather wide. Thus the mass of $\pi(2070)$ state is needed to be precisely
measured by further experiment since its large width will lead to difficulties
for the measurements.

Figure 11: The modulus squared of scattering amplitude for $\pi
K^{*}\bar{K}^{*}$ three-body system with $\Lambda=945$ MeV (solid line) and
$\Lambda=1125$ MeV (dash line)

###  III.3 Three-body $KK^{*}\bar{K}^{*}$ system

The interaction between $K$ and $\bar{K}^{*}$ is strong, and the $f_{1}(1285)$
state is dynamically generated from the $K\bar{K}^{*}$ interaction in $S$-wave
and isospin $I=0$ sector Roca  _et al._ (2005). While in the $I=1$ sector, the
$a_{1}(1260)$ and $b_{1}(1235)$ can be also obtained Roca  _et al._ (2005).
Within the same parameters as used in Ref. Roca  _et al._ (2005), the obtained
modulus squared of $t_{K\bar{K}^{*}\to K\bar{K}^{*}}$ in $I=0$ and $I=1$ are
shown in Figs. 12 and 13, respectively. From these figures, one can see that
there are clearly the peaks for the $f_{1}(1285)$ and
$a_{1}(1260)/b_{1}(1235)$ states, and the former peak is much narrower, yet,
the later one is wide and the peak is superposition of $a_{1}(1260)$ and
$b_{1}(1235)$, since in the calculations, we don not distinguish their
$G$-parity. Furthermore, the interaction of $K\bar{K}^{*}$ in $I=1$ sector is
much weaker than that in the $I=0$ sector.

Figure 12: Modulus squared of $t_{K\bar{K}^{*}\rightarrow K\bar{K}^{*}}$ in
$I=0$ as a function of the invariant mass of $K\bar{K}^{*}$ system  Figure 13:
Modulus squared of $t_{K\bar{K}^{*}\rightarrow K\bar{K}^{*}}$ in $I=1$ as a
function of the invariant mass of $K\bar{K}^{*}$ system

On the other hand, the interaction between $K$ and $K^{*}$ is even weaker, and
the two body scattering amplitude of $t_{KK^{*}\to KK^{*}}$ in $I=0$ is zero.
In the $I=1$ sector, the scattering amplitude squared of $t_{KK^{*}\to
KK^{*}}$ is shown in Fig. 14. The strength of $|t_{KK^{*}\to KK^{*}}|^{2}$ is
much smaller than those shown in Figs. 12 and 13 for the case of
$K\bar{K}^{*}$ scattering.

Figure 14: Modulus squared of $t_{KK^{*}\rightarrow KK^{*}}$ in $I=1$ as a
function of the invariant mass of $KK^{*}$ system

In Fig. 15, the modulus squared of scattering amplitude of three-body
$KK^{*}\bar{K}^{*}$ system is shown, where there are three peaks. The one
around $2130$ MeV is not sensitive to the cut off parameter, while the other
two are much dependent on the value of cut off parameter $\Lambda$. The lowest
one is located at about 2072 MeV, and the highest one is about 2219 MeV.

Figure 15: The modulus squared of scattering amplitude for $KK^{*}\bar{K}^{*}$
three-body system with $\Lambda=945$ MeV (solid line) and $\Lambda=1125$ MeV
(dash line).

On the experimental side, there is a meson with strangeness around 2062 MeV
and named $X(2075)$. It was observed by BESII collaboration in the
$p\bar{\Lambda}$ invariant mass spectrum of the decay $J/\psi\rightarrow
K^{-}p\bar{\Lambda}$ Ablikim  _et al._ (2004). Its Breit-Wigner mass and width
are $2075\pm 12({\rm stat.})\pm 5({\rm syst.})$ MeV and $90\pm 35({\rm
stat.})\pm 9({\rm syst.})$ MeV, respectively. The spin of $X(2075)$ state
could be 0 or 1. Note that, as pointed in Ref. Ablikim  _et al._ (2004), its
interpretation as a conventional $K^{*}$ meson would be disfavored.

In addition, a similar near-threshold enhancement in the $p\bar{\Lambda}$
system was also observed in $B^{+}\rightarrow p\bar{\Lambda}D^{0}$ by Belle
collaboration Chen  _et al._ (2011). In fact, one can notice that there are a
large blank space between 2000-3000 MeV in RPP for those strangeness states
Zyla  _et al._ (2020). We look forward to further experiment can give us more
information about those mesons with a strange quark in this energy region.

##  IV Summary

In the framework of fixed-center approximation, we have performed a
calculation for the three-body $\eta(K^{*}\bar{K}^{*})_{f_{0}(1710)}$,
$\pi(K^{*}\bar{K}^{*})_{f_{0}(1710)}$, and $K(K^{*}\bar{K}^{*})_{f_{0}(1710)}$
systems treating $f_{0}(1710)$ as a $K^{*}\bar{K}^{*}$ bound state as found in
previous studies of the vector-vector interactions. The total three-body
scattering amplitude can be obtained in terms of the two-body interactions
which are taken from the chiral unitary approach. We associate the peaks in
the modulus squared of the three-body scattering amplitude to hadronic states.

Our theoretical results for the obtained mass of $f_{0}(1710)$ state and these
peak positions of squared of three-body scattering amplitudes with different
values of the cutoff parameter $\Lambda$ are listed in Table 2. The peak
positions are associated to the mass of the three-body state. The state found
in the $\eta K^{*}\bar{K}^{*}$ system could be interpreted as $\eta(2100)$
meson. For the $\pi K^{*}\bar{K}^{*}$ system, we find a bump structure between
1900 and 2000 MeV. While for the $KK^{*}\bar{K}^{*}$ system, there are three
peaks appeared. One of them is not sensitive to the cutoff parameter, and its
mass is about 2130 MeV. The third one is disappeared if the cutoff parameter
less than $855$ MeV. It is expected the theoretical calculations here could be
tested by future experiments.

Table 2: Peak positions of squared of three-body scattering amplitude corresponding to different value of cut off parameter $\Lambda$. Unit in MeV. $\Lambda$ | 855 | 945 | 1035 | 1125 | 1215
---|---|---|---|---|---
$M_{f_{0}(1710)}$ | 1747 | 1733 | 1718 | 1704 | 1690
$M_{\eta K^{*}\bar{K}^{*}}$ | 2104 | 2092 | 2070 | 2054 | 2038
$M_{\pi K^{*}\bar{K}^{*}}$ | 1981 | 1963 | 1948 | 1929 | 1913
$M_{KK^{*}\bar{K}^{*}}$ | state 1 | 2092 | 2083 | 2072 | 2062 | 2053
state 2 | 2140 | 2134 | 2130 | 2130 | 2134
state 3 | $--$ | 2233 | 2219 | 2209 | 2211

###### Acknowledgements.

We thank Xu Zhang for useful discussions. This work is partly supported by the
National Natural Science Foundation of China under Grant Nos. 12075288,
11735003, and 11961141012. It is also supported by the Youth Innovation
Promotion Association CAS.

## References

  * Oset  _et al._ (2016) E. Oset _et al._ , Int. J. Mod. Phys. E 25, 1630001 (2016), arXiv:1601.03972 [hep-ph] .
  * Guo  _et al._ (2018) F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. 90, 015004 (2018), [Erratum: Rev.Mod.Phys. 94, 029901 (2022)], arXiv:1705.00141 .
  * Dong  _et al._ (2021a) X.-K. Dong, F.-K. Guo, and B.-S. Zou, Progr. Phys. 41, 65 (2021a), arXiv:2101.01021 [hep-ph] .
  * Dong  _et al._ (2021b) X.-K. Dong, F.-K. Guo, and B.-S. Zou, Commun. Theor. Phys. 73, 125201 (2021b), arXiv:2108.02673 [hep-ph] .
  * Ablikim  _et al._ (2022a) M. Ablikim _et al._ (BESIII), Phys. Rev. D 106, 072012 (2022a), arXiv:2202.00623 [hep-ex] .
  * Ablikim  _et al._ (2022b) M. Ablikim _et al._ (BESIII), Phys. Rev. Lett. 129, 192002 (2022b), arXiv:2202.00621 [hep-ex] .
  * Yang and Huang (2022) F. Yang and Y. Huang, (2022), arXiv:2203.06934 [hep-ph] .
  * Dong  _et al._ (2022) X.-K. Dong, Y.-H. Lin, and B.-S. Zou, Sci. China Phys. Mech. Astron. 65, 261011 (2022), arXiv:2202.00863 [hep-ph] .
  * Chen  _et al._ (2022) H.-X. Chen, N. Su, and S.-L. Zhu, Chin. Phys. Lett. 39, 051201 (2022), arXiv:2202.04918 .
  * Zhang  _et al._ (2017) X. Zhang, J.-J. Xie, and X. Chen, Phys. Rev. D 95, 056014 (2017), arXiv:1612.02613 [hep-ph] .
  * Lutz and Kolomeitsev (2004) M. Lutz and E. Kolomeitsev, Nuclear Physics A 730, 392 (2004).
  * Roca  _et al._ (2005) L. Roca, E. Oset, and J. Singh, Phys. Rev. D 72, 014002 (2005), arXiv:hep-ph/0503273 .
  * Zhou  _et al._ (2014) Y. Zhou, X.-L. Ren, H.-X. Chen, and L.-S. Geng, Phys. Rev. D 90, 014020 (2014), arXiv:1404.6847 [nucl-th] .
  * Geng  _et al._ (2015) L.-S. Geng, X.-L. Ren, Y. Zhou, H.-X. Chen, and E. Oset, Phys. Rev. D 92, 014029 (2015), arXiv:1503.06633 [hep-ph] .
  * Xie  _et al._ (2020) J.-J. Xie, G. Li, and X.-H. Liu, Chin. Phys. C 44, 114104 (2020), arXiv:1907.12202 [hep-ph] .
  * Zhang and Xie (2020) X. Zhang and J.-J. Xie, Chin. Phys. C 44, 054104 (2020), arXiv:1906.07340 [nucl-th] .
  * Martinez Torres  _et al._ (2020) A. Martinez Torres, K. P. Khemchandani, L. Roca, and E. Oset, Few Body Syst. 61, 35 (2020), arXiv:2005.14357 [nucl-th] .
  * Wu  _et al._ (2022) T.-W. Wu, Y.-W. Pan, M.-Z. Liu, and L.-S. Geng, Sci. Bull. 67, 1735 (2022), arXiv:2208.00882 [hep-ph] .
  * Malabarba  _et al._ (2022) B. B. Malabarba, K. P. Khemchandani, and A. M. Torres, Eur. Phys. J. A 58, 33 (2022), arXiv:2103.09978 [hep-ph] .
  * Luo  _et al._ (2022a) S.-Q. Luo, T.-W. Wu, M.-Z. Liu, L.-S. Geng, and X. Liu, Phys. Rev. D 105, 074033 (2022a), arXiv:2111.15079 [hep-ph] .
  * Luo  _et al._ (2022b) S.-Q. Luo, L.-S. Geng, and X. Liu, Phys. Rev. D 106, 014017 (2022b), arXiv:2206.04586 [hep-ph] .
  * Ikeno  _et al._ (2022) N. Ikeno, M. Bayar, and E. Oset, (2022), arXiv:2208.03698 [hep-ph] .
  * Debastiani  _et al._ (2017) V. R. Debastiani, J. M. Dias, and E. Oset, Phys. Rev. D 96, 016014 (2017), arXiv:1705.09257 [hep-ph] .
  * Wu  _et al._ (2021) T.-W. Wu, M.-Z. Liu, and L.-S. Geng, Phys. Rev. D 103, L031501 (2021), arXiv:2012.01134 [hep-ph] .
  * Wei  _et al._ (2022) X. Wei, Q.-H. Shen, and J.-J. Xie, Eur. Phys. J. C 82, 718 (2022), arXiv:2205.12526 [hep-ph] .
  * Martinez Torres  _et al._ (2008) A. Martinez Torres, K. P. Khemchandani, L. S. Geng, M. Napsuciale, and E. Oset, Phys. Rev. D 78, 074031 (2008), arXiv:0801.3635 [nucl-th] .
  * Ren  _et al._ (2018) X.-L. Ren, B. B. Malabarba, L.-S. Geng, K. Khemchandani, and A. M. Torres, Physics Letters B 785, 112 (2018).
  * Dias  _et al._ (2017) J. M. Dias, V. R. Debastiani, L. Roca, S. Sakai, and E. Oset, Phys. Rev. D 96, 094007 (2017), arXiv:1709.01372 [hep-ph] .
  * Bayar  _et al._ (2011) M. Bayar, J. Yamagata-Sekihara, and E. Oset, Phys. Rev. C 84, 015209 (2011), arXiv:1102.2854 [hep-ph] .
  * Xiao  _et al._ (2011) C. W. Xiao, M. Bayar, and E. Oset, Phys. Rev. D 84, 034037 (2011), arXiv:1106.0459 [hep-ph] .
  * Bayar  _et al._ (2014) M. Bayar, W. H. Liang, T. Uchino, and C. W. Xiao, Eur. Phys. J. A 50, 67 (2014), arXiv:1312.2869 [hep-ph] .
  * Liang  _et al._ (2013) W. Liang, C. W. Xiao, and E. Oset, Phys. Rev. D 88, 114024 (2013), arXiv:1309.7310 [hep-ph] .
  * Durkaya and Bayar (2015) B. Durkaya and M. Bayar, Phys. Rev. D 92, 036006 (2015).
  * Yamagata-Sekihara  _et al._ (2010) J. Yamagata-Sekihara, L. Roca, and E. Oset, Phys. Rev. D 82, 094017 (2010), [Erratum: Phys.Rev.D 85, 119905 (2012)], arXiv:1010.0525 [hep-ph] .
  * Martinez Torres  _et al._ (2019) A. Martinez Torres, K. P. Khemchandani, and L.-S. Geng, Phys. Rev. D 99, 076017 (2019), arXiv:1809.01059 [hep-ph] .
  * Sanchez Sanchez  _et al._ (2018) M. Sanchez Sanchez, L.-S. Geng, J.-X. Lu, T. Hyodo, and M. P. Valderrama, Phys. Rev. D 98, 054001 (2018), arXiv:1707.03802 [hep-ph] .
  * Xie  _et al._ (2011a) J.-J. Xie, A. Martinez Torres, and E. Oset, Phys. Rev. C 83, 065207 (2011a), arXiv:1010.6164 [nucl-th] .
  * Xie  _et al._ (2011b) J.-J. Xie, A. Martinez Torres, E. Oset, and P. Gonzalez, Phys. Rev. C 83, 055204 (2011b), arXiv:1101.1722 [nucl-th] .
  * Geng and Oset (2009) L. S. Geng and E. Oset, Phys. Rev. D 79, 074009 (2009), arXiv:0812.1199 [hep-ph] .
  * Geng  _et al._ (2009) L. S. Geng, E. Oset, R. Molina, and D. Nicmorus, PoS EFT09, 040 (2009), arXiv:0905.0419 [hep-ph] .
  * Du  _et al._ (2018) M.-L. Du, D. Gülmez, F.-K. Guo, U.-G. Meißner, and Q. Wang, Eur. Phys. J. C 78, 988 (2018), arXiv:1808.09664 [hep-ph] .
  * García-Recio  _et al._ (2013) C. García-Recio, L. S. Geng, J. Nieves, L. L. Salcedo, E. Wang, and J.-J. Xie, Phys. Rev. D 87, 096006 (2013), arXiv:1304.1021 [hep-ph] .
  * Wang and Zou (2021) Z.-L. Wang and B.-S. Zou, Phys. Rev. D 104, 114001 (2021), arXiv:2107.14470 [hep-ph] .
  * Nagahiro  _et al._ (2008) H. Nagahiro, L. Roca, E. Oset, and B. S. Zou, Phys. Rev. D 78, 014012 (2008), arXiv:0803.4460 [hep-ph] .
  * Branz  _et al._ (2010) T. Branz, L. S. Geng, and E. Oset, Phys. Rev. D 81, 054037 (2010), arXiv:0911.0206 [hep-ph] .
  * (46) L. S. Geng, F. K. Guo, C. Hanhart, R. Molina, E. Oset, and B. S. Zou, Eur. Phys. J. A 44, 305, arXiv:0910.5192 [hep-ph] .
  * Martinez Torres  _et al._ (2013) A. Martinez Torres, K. P. Khemchandani, F. S. Navarra, M. Nielsen, and E. Oset, Phys. Lett. B 719, 388 (2013), arXiv:1210.6392 [hep-ph] .
  * Xie and Oset (2014) J.-J. Xie and E. Oset, Phys. Rev. D 90, 094006 (2014), arXiv:1409.1341 [hep-ph] .
  * Dai  _et al._ (2015) L.-R. Dai, J.-J. Xie, and E. Oset, Phys. Rev. D 91, 094013 (2015), arXiv:1503.02463 [hep-ph] .
  * Molina  _et al._ (2020) R. Molina, L. R. Dai, L. S. Geng, and E. Oset, Eur. Phys. J. A 56, 173 (2020), arXiv:1909.10764 [hep-ph] .
  * Roca and Oset (2010) L. Roca and E. Oset, Phys. Rev. D 82, 054013 (2010), arXiv:1005.0283 [hep-ph] .
  * Gamermann  _et al._ (2010) D. Gamermann, J. Nieves, E. Oset, and E. Ruiz Arriola, Phys. Rev. D 81, 014029 (2010), arXiv:0911.4407 [hep-ph] .
  * Yamagata-Sekihara  _et al._ (2011) J. Yamagata-Sekihara, J. Nieves, and E. Oset, Phys. Rev. D 83, 014003 (2011), arXiv:1007.3923 [hep-ph] .
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Zhu  _et al._ (2022) X. Zhu, D.-M. Li, E. Wang, L.-S. Geng, and J.-J. Xie, Phys. Rev. D 105, 116010 (2022), arXiv:2204.09384 [hep-ph] .
  * Geng  _et al._ (2007) L. S. Geng, E. Oset, L. Roca, and J. A. Oller, Phys. Rev. D 75, 014017 (2007), arXiv:hep-ph/0610217 .
  * Guo  _et al._ (2006) F.-K. Guo, R.-G. Ping, P.-N. Shen, H.-C. Chiang, and B.-S. Zou, Nucl. Phys. A 773, 78 (2006), arXiv:hep-ph/0509050 .
  * Sun  _et al._ (2022) B.-X. Sun, Y.-Y. Fan, and Q.-Q. Cao, (2022), arXiv:2206.02961 [hep-ph] .
  * Bisello  _et al._ (1989) D. Bisello _et al._ (DM2), Phys. Rev. D 39, 701 (1989).
  * Ablikim  _et al._ (2016) M. Ablikim _et al._ (BESIII), Phys. Rev. D 93, 112011 (2016), arXiv:1602.01523 [hep-ex] .
  * Wang  _et al._ (2017) L.-M. Wang, S.-Q. Luo, Z.-F. Sun, and X. Liu, Phys. Rev. D 96, 034013 (2017), arXiv:1705.00549 [hep-ph] .
  * Anisovich  _et al._ (2001) A. V. Anisovich, C. A. Baker, C. J. Batty, D. V. Bugg, V. A. Nikonov, A. V. Sarantsev, V. V. Sarantsev, and B. S. Zou, Phys. Lett. B 517, 261 (2001), arXiv:1110.0278 [hep-ex] .
  * Ablikim  _et al._ (2004) M. Ablikim _et al._ (BES), Phys. Rev. Lett. 93, 112002 (2004), arXiv:hep-ex/0405050 .
  * Chen  _et al._ (2011) P. Chen _et al._ (Belle), Phys. Rev. D 84, 071501 (2011), arXiv:1108.4271 [hep-ex] .