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# Interpretation of the $\eta_{1}(1855)$ as a $K\bar{K}_{1}(1400)+$ c.c.
molecule

Xiang-Kun Dong  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences,
Zhong Guan Cun East Street 55, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Yong-Hui Lin  Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center
for Theoretical Physics,
Universität Bonn, D-53115 Bonn, Germany  Bing-Song Zou<EMAIL_ADDRESS>CAS
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences,
Zhong Guan Cun East Street 55, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
School of Physics, Central South University, Changsha 410083, China

###### Abstract

An exotic state with $J^{PC}=1^{-+}$, denoted by $\eta_{1}(1855)$, was
observed by BESIII collaboration recently in
$J/\psi\to\gamma\eta\eta^{\prime}$. The fact that its mass is just below the
threshold of $K\bar{K}_{1}(1400)$ stimulates us to investigate whether this
exotic state can be interpreted as a $K\bar{K}_{1}(1400)+$ c.c molecule or
not. Using the one boson exchange model, we show that it is possible for
$K\bar{K}_{1}(1400)$ with $J^{PC}=1^{-+}$ to bind together by taking the
momentum cutoff $\Lambda\gtrsim 2$ GeV and yield the same binding energy as
the experimental value when $\Lambda\approx 2.5$ GeV. In this molecular
picture, the predicted branch ratio
$\mathrm{Br}(\eta_{1}(1855)\to\eta\eta^{\prime})\approx 15\%$ is consistent
with the experimental results, which again supports the molecular explanation
of $\eta_{1}(1855)$. Relevant systems, namely $K\bar{K}_{1}(1400)$ with
$J^{PC}=1^{--}$ and $K\bar{K}_{1}(1270)$ with $J^{PC}=1^{-\pm}$, are also
investigated, some of which can be searched for in the future experiments.

###### pacs:

12.40.Vv,13.25.Jx,14.40.Cs
Keyword: hadronic molecule, exotic hadrons, one boson exchange

##  I Introduction

Many efforts have been put into searching for non-conventional hadrons other
than quark-antiquark mesons or 3-quark baryons since the quark model was
proposed by Gell-Mann Gell-Mann (1964) and Zweig Zweig (1964). Quantum
Chromodynamics (QCD), the most fundamental theory describing the strong
interaction, does not forbid the existence of multi-quark states, hybrid
states or glueballs, collectively called exotic hadrons. Actually, there were
several candidates for exotic hadrons before the beginning of this century but
none of them are identified unambiguously Klempt and Zaitsev (2007). For
example, even now there is not a widely accepted explanation for the nature of
$\Lambda(1405)$, which was ever discovered almost 60 years ago Dalitz and Tuan
(1959, 1960); Alston  _et al._ (1961), see the discussions in the most recent
reviews Hyodo and Niiyama (2021); Mai (2021).

Since the discovery of the charmonium-like state X(3872) Choi  _et al._
(2003), the last twenty years has witnessed the booming experimental evidence
of exotic states, see Refs. Chen  _et al._ (2016); Hosaka  _et al._ (2016);
Richard (2016); Lebed  _et al._ (2017); Esposito  _et al._ (2017); Guo  _et
al._ (2018); Ali  _et al._ (2017); Olsen  _et al._ (2018); Altmannshofer  _et
al._ (2019); Kalashnikova and Nefediev (2019); Cerri  _et al._ (2019); Liu
_et al._ (2019); Brambilla  _et al._ (2020); Guo  _et al._ (2020); Yang  _et
al._ (2020); Ortega and Entem (2021); Dong  _et al._ (2021a, b) for recent
reviews. To identify exotic states or their candidates, the following
guidelines may be helpful: a) In the charmonium and bottomonium sectors, quark
models give quite precise descriptions of the conventional mesons (see, e.g.,
Refs. Godfrey and Isgur (1985); Capstick and Isgur (1985)). If the observed
resonances do not fit such properties, they can be considered as candidates of
exotic states, such as X(3872) and Y(4260) Aubert  _et al._ (2005); b) In some
cases, the decay channels imply the resonances consist of multiquarks and
therefore, these states are exotic definitely, such as $Z_{c(s)}$ states
Ablikim  _et al._ (2013); Liu  _et al._ (2013); Ablikim  _et al._ (2021),
$P_{c(s)}$ states Aaij  _et al._ (2015, 2019, 2021a) and most recently
observed doubly charmed $T_{cc}$ state Aaij  _et al._ (2021b, c); c) Some
combinations of $J^{PC}$ for conventional mesons are forbidden, such as
$0^{--},(\rm{even})^{+-}$ and $(\rm{odd})^{-+}$. Therefore, a state with such
forbidden quantum numbers must be exotic. Up to now there are two well-
established states with exotic $J^{PC}$ in light quark sector,
$\pi_{1}(1400),\pi_{1}(1600)$ Zyla  _et al._ (2020) and one possible state
$\pi_{1}(2015)$ Kuhn  _et al._ (2004); Lu  _et al._ (2005)111See Ref. Meyer
and Swanson (2015) for a recent review on the experimental status of these
$\pi_{1}$’s., all of which have $J^{PC}=1^{-+}$, while no experimental signals
appear in the heavy quark sector.

Recently, BESIII Collaboration Ablikim  _et al._ (2022a, b) reported a
resonance, denoted as $\eta_{1}(1855)$, with $m=1855\pm 9^{+6}_{-1}$ MeV and
$\Gamma=188\pm 18^{+3}_{-8}$ MeV, in the invariant mass distribution of
$\eta\eta^{\prime}$ in $J/\psi\to\gamma\eta\eta^{\prime}$ with a significance
of 19 $\sigma$. Its quantum numbers $J^{PC}=1^{-+}$ ensure it to be an exotic
state, which immediately stimulates the explanation of $\eta_{1}(1875)$ being
an isoscalar hybrid candidate Chen  _et al._ (2022); Qiu and Zhao (2022).
Notice that this state locates at just around 40 MeV below the
$K\bar{K}_{1}(1400)$ threshold and therefore, it is natural to interpret it as
a molecule of $K\bar{K}_{1}(1400)+$ c.c.($K\bar{K}_{1}$ for short in the
following), which can decay into $\eta\eta^{\prime}$ via $K^{*}$ exchange on
the hadronic level.

In this paper, we investigate whether $K\bar{K}_{1}$ can form a bound state
via the interaction driven by one boson exchange as shown in Fig. 1.
Pseudoscalar exchange between $K\bar{K}_{1}$ is forbidden by parity
conservation. Hence it is expected that the $\sigma$ and vector meson
($\rho,\omega,\phi$) exchanges dominate the interaction. In the Review of
Particle Physics (RPP) Zyla  _et al._ (2020) there are two $K_{1}$’s,
$K_{1}(1270)$ with mass 1270 MeV and $K_{1}(1400)$ with mass 1403 MeV. These
two mass eigenstates have quite different decay behaviors and are considered
as the mixing of two flavor eigenstates from the ${}^{3}P_{1}$ and
${}^{1}P_{1}$ octets Burakovsky and Goldman (1997); Suzuki (1993); Cheng
(2003); Yang (2011); Hatanaka and Yang (2008); Tayduganov  _et al._ (2012);
Divotgey  _et al._ (2013); Zhang  _et al._ (2018). It was also explored in
Refs. Roca  _et al._ (2005); Geng  _et al._ (2007); Wang  _et al._ (2019) that
the $K_{1}(1270)$ may have a two-pole structure in vector-pseudoscalar
scattering. In this paper we adopt the former treatment to investigate their
interactions.

With the above considerations we find that a) the meson exchange potential
with reasonable cutoff is strong enough for $K\bar{K}_{1}(1400)$ to form a
bound state; b) The branch ratio of the predicted molecule decaying into
$\eta\eta^{\prime}$ is consistent with the experimental data of BESIII Ablikim
_et al._ (2022a, b),
Br$(J/\psi\rightarrow\gamma\eta_{1}(1855)\rightarrow\gamma\eta\eta^{\prime})=\left(2.70\pm
0.41_{-0.35}^{+0.16}\right)\times 10^{-6}$; c) Its $C-$parity partner are
predicted to have a similar binding energy and the $\pi\rho$, $\eta\omega$ and
$K\bar{K}$ channels are good places to search for it. The properties predicted
in this paper serve as guidance for further experimental explorations of these
states and future experiments can in turn test our molecular state assignment
for the observed $\eta_{1}(1855)$.

##  II Possible bound states of $K\bar{K}_{1}(1400)$

Due to the $U(3)_{V}$ flavor symmetry breaking effect derived from the mass
difference between $u/d$ and $s$ quarks, the axialvector $K_{1A}$
(${}^{3}P_{1}$ state) and pseudovector $K_{1B}$ (${}^{1}P_{1}$ state) can mix
with each other and generate the two physical resonances $K_{1}(1270)$ and
$K_{1}(1400)$. Following Ref. Divotgey  _et al._ (2013) the mixing is
parameterized as

$\displaystyle\left(\begin{array}[]{cc}{\left|K_{1}(1270)\right\rangle}\\\
{\left|K_{1}(1400)\right\rangle}\end{array}\right)=\left(\begin{array}[]{cc}{\cos\theta}&{-i\sin\theta}\\\
{-i\sin\theta}&{\cos\theta}\end{array}\right)\left(\begin{array}[]{c}{\left|K_{1A}\right\rangle}\\\
{\left|K_{1B}\right\rangle}\end{array}\right)$ (7)

where the mixing angle $\theta$ is determined to be $(56.4\pm 4.3)^{\circ}$.
The flavor wave functions of $|K\bar{K}_{1}\rangle$ with positive and negative
C-parity read

$\displaystyle
C=\pm:\frac{1}{\sqrt{2}}\left(|K\bar{K}_{1}\rangle\pm\mathcal{C}|KK_{1}\rangle\right)$
(8)

with the following conventions, $\mathcal{C}|K\rangle=|\bar{K}\rangle$,
$\mathcal{C}|K_{1A}\rangle=|\bar{K}_{1A}\rangle$ and
$\mathcal{C}|K_{1B}\rangle=-|\bar{K}_{1B}\rangle$. Here $\mathcal{C}$ is the
charge conjugation operator.

$\rho,\omega,\phi,\sigma$$K(\bar{K})$$\bar{K}_{1}(K_{1})$$K_{1}(\bar{K}_{1})$$\bar{K}(K)$$q$$p_{1}$$p^{\prime}_{2}$$p_{2}$$p^{\prime}_{1}$$\rho,\omega,\phi,\sigma$$K(\bar{K})$$\bar{K}_{1}(K_{1})$$K(\bar{K})$$\bar{K}_{1}(K_{1})$$q$$p_{1}$$p^{\prime}_{1}$$p_{2}$$p^{\prime}_{2}$
Figure 1: Feynman diagrams for direct (top) and cross (bottom) meson exchange
between $K\bar{K}_{1}$.

Working with the chiral perturbation theory, the pseudoscalar-pseudoscalar-
vector coupling is described by Meissner (1988)

$\displaystyle\mathcal{L}_{\rm{PPV}}$ $\displaystyle=i\sqrt{2}\,G_{\rm
V}\,{\rm{Tr}}\left([\partial_{\mu}P,P]V^{\mu}\right)$ (9)

and analogously, we assume the following couplings

$\displaystyle\mathcal{L}_{{AAV}}$ $\displaystyle=i\sqrt{2}\,G_{\rm
V}^{\prime}\,{\rm{Tr}}\left([\partial_{\mu}A^{\nu},A_{\nu}]V^{\mu}\right)$
(10) $\displaystyle\mathcal{L}_{{BBV}}$ $\displaystyle=i\sqrt{2}\,G_{\rm
V}^{\prime}\,{\rm{Tr}}\left([\partial_{\mu}B^{\nu},B_{\nu}]V^{\mu}\right)$
(11) $\displaystyle\mathcal{L}_{APV}$ $\displaystyle=ia\ {\rm
Tr}(A_{\mu}[V^{\mu},P]),$ (12) $\displaystyle\mathcal{L}_{BPV}$
$\displaystyle=b\ {\rm Tr}(B_{\mu}\\{V^{\mu},P\\})$ (13)

with

$\displaystyle
P=\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right),$ (17)
$\displaystyle
V=\left(\begin{array}[]{ccc}\frac{\omega}{\sqrt{2}}+\frac{\rho^{0}}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&\frac{\omega}{\sqrt{2}}-\frac{\rho^{0}}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right),$ (21)

and

$\displaystyle A(B)=\left(\begin{array}[]{ccc}{*}&{*}&{K_{1A(B)}^{+}}\\\
{*}&{*}&{K_{1A(B)}^{0}}\\\
{K_{1A(B)}^{-}}&{\bar{K}_{1A(B)}^{0}}&*\end{array}\right).$ (25)

Here all irrelevant axialvector and pseudovector states are labeled as “*” in
the multiplet matrix Eq. (25). The coupling constant $G_{\rm V}\approx 3.0$
was estimated from the decay width of $\rho\to\pi\pi$ Zhang  _et al._ (2006)
and we assumed the magnitude of $G^{\prime}_{V}$ is comparable with $G_{\rm
V}$, i.e., $|G^{\prime}_{\rm V}|\approx G_{\rm V}$. Moreover, taking the
mixing angle obtained in Ref. Divotgey  _et al._ (2013) and fitting to the
experimental widths of $K_{1}(1270)$/$K_{1}(1400)$ decaying to $K\rho$, one
can determine the coupling constants as $a\approx 1.92\pm 0.09$ GeV and
$b\approx-2.47\pm 0.08$ GeV Dong and Zou (2021).

In our previous work Dong  _et al._ (2020), we have discussed the
$D\bar{D}_{1}+$ c.c. bound state via one boson exchange to understand the
nature of Y(4260). $KK_{1}$ system behaves similarly with $DD_{1}$ system in
the one boson exchange picture, just replacing the $c$ quarks in $DD_{1}$ with
$s$ quarks, except that $KK_{1}$ can couple to $\phi$ while $DD_{1}$ can not.
Therefore one can expect that they have the same Lagrangian except the
slightly different values of coupling constants. Note that the $DDV$ coupling
constant is 0.9 of the $KKV$ coupling Casalbuoni  _et al._ (1997); Isola  _et
al._ (2003). These observations make us confident that the Language in Eqs.
(10, 11) can give a good description of the $K_{1}K_{1}V$ vertices and also
indicate the following couplings,

$\displaystyle\mathcal{L}_{KK\sigma}$
$\displaystyle=-2g_{\sigma}m_{K}\bar{K}K\sigma,$ (26)
$\displaystyle\mathcal{L}_{K_{1}K_{1}\sigma}$
$\displaystyle=-2g^{\prime\prime}_{\sigma}m_{K_{1}}\bar{K^{\mu}_{1}}K_{1\mu}\sigma,$
(27) $\displaystyle\mathcal{L}_{K_{1}K\sigma}$
$\displaystyle=-\frac{2\sqrt{6}}{3}\frac{h_{\sigma}^{\prime}}{f_{\pi}}\sqrt{m_{K}m_{K_{1}}}\left(\bar{K}_{1}^{\mu}K+\bar{K}K_{1}^{\mu}\right)\partial_{\mu}\sigma$
(28)

with $f_{\pi}=132\ \rm{MeV}$ the pion decay constant. As a rough
approximation, we take $g_{\sigma}=g^{\prime\prime}_{\sigma}=-0.76$,
$h^{\prime}_{\sigma}=0.35$ Bardeen  _et al._ (2003). We have ignored the
mixing nature of $K_{1}(1270)$ and $K_{1}(1400)$ in Eqs.(27, 28), which does
not matter here.

The potential of $K\bar{K}_{1}$ in the non-relativistic limit reads

$\tilde{V}(\bm{q})=-\frac{\mathcal{M}}{4m_{K}m_{K_{1}}}$ (29)

with $\mathcal{M}$ the invariant scattering amplitude of $K\bar{K}_{1}\to
K\bar{K}_{1}$. The potentials in momentum space for the direct and cross
scatterings, see the top and bottom diagram in Fig. 1, are written as

$\displaystyle\tilde{V}_{\rm V,d}({\bf q})$
$\displaystyle=f_{\mathrm{I}}g_{\rm{V,d}}g^{\prime}_{\rm{V,d}}\frac{1}{|{\bf
q}|^{2}+m_{\rm V}^{2}},$ (30) $\displaystyle\tilde{V}_{\rm{V,c}}({\bf q})$
$\displaystyle=\frac{f_{\mathrm{I}}g^{2}_{\rm{V,c}}}{4m_{\rm{K}}m_{{\rm{K}_{1}}}}\frac{\left(1+{\bm{\epsilon}}_{1}\cdot\bm{q\epsilon}_{2}^{*}\cdot\bm{q}/m_{\rm{V}}^{2}\right)}{|{\bm{q}}|^{2}+\tilde{m}_{\rm
V}^{2}},$ (31) $\displaystyle\tilde{V}_{\sigma,\rm{d}}({\bf q})$
$\displaystyle=-g_{\sigma}g^{\prime\prime}_{\sigma}\frac{1}{|{\bf
q}|^{2}+m_{\sigma}^{2}},$ (32) $\displaystyle\tilde{V}_{\sigma,\rm{c}}({\bf
q})$
$\displaystyle=\frac{-2h^{{}^{\prime}2}_{\sigma}}{3f_{\pi}^{2}}\frac{{\bm{\epsilon}}_{1}\cdot\bm{q\epsilon}_{2}^{*}\cdot\bm{q}}{|{\bm{q}}|^{2}+\tilde{m}_{\sigma}^{2}}$
(33)

with $\tilde{m}_{\rm{V}/\sigma}^{2}=m_{\rm
V/\sigma}^{2}-(m_{\rm{K}_{1}}-m_{\rm{K}})^{2}$. The coupling constants are

$\displaystyle g_{\rho,\rm d}$ $\displaystyle=g_{\omega,\rm
d}=-\frac{1}{\sqrt{2}}g_{\phi,\rm d}=G_{\rm V}$ (34) $\displaystyle
g^{\prime}_{\rho,\rm d}$ $\displaystyle=g^{\prime}_{\omega,\rm
d}=-\frac{1}{\sqrt{2}}g^{\prime}_{\phi,\rm d}=G^{\prime}_{\rm V}$ (35)
$\displaystyle g^{2}_{\rho,\rm c}$ $\displaystyle=g^{2}_{\omega,\rm
c}=\frac{1}{2}g^{2}_{\phi,\rm c}=\frac{1}{2}(a\sin\theta\pm b\cos\theta)^{2}$
(36)

with “$+$” for $K_{1}(1400)$ and “$-$” for $K_{1}(1270)$ and the isospin
factors, $f_{\mathrm{I}}$, are

$\displaystyle f_{0}=\left\\{\begin{array}[]{cc}{3}&{\rho}\\\ {1}&{\omega}\\\
{1}&{\phi}\end{array}\right.\ \ \ \
f_{1}=\left\\{\begin{array}[]{cc}{-1}&{\rho}\\\ {1}&{\omega}\\\
{1}&{\phi}\end{array}\right..$ (43)

A form factor should be introduced at each vertex to account for the finite
size of the involved mesons. Here we take the commonly used monopole form
factor,

$F(q,m,\Lambda)=\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}-q^{2}},$ (44)

which in position space can be looked upon as a spherical source of the
exchanged mesons Tornqvist (1994). The potentials in position space can be
obtained by Fourier transformation of Eqs.(30, 31), together with the form
factor Eq. (44),

$\displaystyle V_{\rm{V,d}}\left(\mathbf{r}\right)$
$\displaystyle=f_{\mathrm{I}}g_{\rm{V,d}}g^{\prime}_{\rm{V,d}}{\Big{(}}m_{\rm{V}}Y(m_{\rm{V}}r)-\Lambda
Y(\Lambda r)$ $\displaystyle\ \ \ \
-\frac{1}{2}(\Lambda^{2}-m_{\rm{V}}^{2})rY(\Lambda r)\Big{)},$ (45)
$\displaystyle V_{\rm{V,c}}\left(\mathbf{r}\right)$
$\displaystyle=\frac{f_{\mathrm{I}}g^{2}_{\rm{V,c}}}{4m_{\rm{K}}m_{{\rm{K}_{1}}}}\Big{\\{}\Big{[}\tilde{m}_{\rm{V}}Y(\tilde{m}_{\rm{V}}r)-\tilde{\Lambda}Y(\tilde{\Lambda}r)$
$\displaystyle\ \ \ \
-\frac{1}{2}(\tilde{\Lambda}^{2}-\tilde{m}^{2}_{\rm{V}})rY(\tilde{\Lambda}r)\Big{]}$
$\displaystyle\ \ \ \
-\frac{1}{3m_{\rm{V}}^{2}}\Big{[}\tilde{m}^{2}_{\rm{V}}\left(\tilde{m}_{\rm{V}}Y(\tilde{m}_{\rm{V}}r)-\tilde{\Lambda}Y(\tilde{\Lambda}r)\right)$
$\displaystyle\ \ \ \
-\frac{1}{2}\lambda^{2}(\tilde{\Lambda}^{2}-\tilde{m}^{2}_{\rm{V}})rY(\tilde{\Lambda}r)\Big{]}\Big{\\}}$
(46) $\displaystyle V_{\rm{\sigma,d}}\left(\mathbf{r}\right)$
$\displaystyle=-g_{\sigma}g^{\prime\prime}_{\sigma}\Big{(}m_{\sigma}Y(m_{\sigma}r)-\Lambda
Y(\Lambda r)$ $\displaystyle\ \ \ \
-\frac{1}{2}(\Lambda^{2}-m_{\sigma}^{2})rY(\Lambda r)\Big{)},$ (47)
$\displaystyle V_{\sigma,\rm{c}}\left(\mathbf{r}\right)$
$\displaystyle=\frac{2h^{{}^{\prime}2}_{\sigma}}{9f_{\pi}^{2}}\Big{\\{}\Big{[}\tilde{m}^{2}_{\sigma}\left(\tilde{m}_{\sigma}Y(\tilde{m}_{\sigma}r)-\tilde{\Lambda}Y(\tilde{\Lambda}r)\right)$
$\displaystyle\ \ \ \
-\frac{1}{2}\lambda^{2}(\tilde{\Lambda}^{2}-\tilde{m}^{2}_{\sigma})rY(\tilde{\Lambda}r)\Big{]}\Big{\\}}$
(48)

where $Y(x)=e^{-x}/4\pi x$ and $\tilde{\Lambda}^{2}=\Lambda^{2}-(m_{\rm
K_{1}}-m_{\rm K})^{2}.$ Note that the Fourier transformation of Eqs.(31,33)
contains a $\delta$ function, which is omitted in some works, see the
discussion in Ref. Burns and Swanson (2019). The $\delta$ potential is from
short distance physics, which we are blind to, and may contain contributions
from the exchange of heavier mesons not considered here. In Ref. Yalikun  _et
al._ (2021), the $\delta$ contribution is varied in the range of $0\sim 1$ to
take into account the uncertainty of the unknown short distance interaction in
the formation of $P_{c}$ states. Here we consider the marginal cases where the
$\delta$ function is kept and omitted, corresponding to
$\lambda=\tilde{\Lambda}$ and $\lambda=\tilde{m}_{\rm V}$ in Eqs. (46, 48),
respectively. We take the real part of potentials when they are complex. The
total potential for positive or negative C-parity $K\bar{K}_{1}$ system reads

$\displaystyle V^{{I},C=\pm}=\left(V_{\rm V,d}^{I}+V_{\sigma,\rm
d}\right)\pm\left(V_{\rm V,c}^{I}+V_{\sigma,\rm c}\right)$ (49)

with $I=0,1$. We take the following masses from RPP Zyla  _et al._ (2020) to
perform numerical calculations, $m_{{K}}=0.495\ \rm{GeV}$,
$m_{{K_{1}}(1400)}=1.403\ \rm{GeV}$, $m_{{K_{1}}(1270)}=1.272\ \rm{GeV}$,
$m_{\rho}=0.775\ \rm{GeV}$, $m_{\omega}=0.783\ \rm{GeV}$, $m_{\phi}=1.019\
\rm{GeV}$ and $m_{\sigma}=0.600\ \rm{GeV}$.

Figure 2: The binding energies of the isoscalar $K\bar{K}_{1}(1400)$ bound
states.

The Schrödinger equations for both $K\bar{K}_{1}(1400)$ and
$K\bar{K}_{1}(1270)$ systems are solved numerically. For the isovector cases,
no bound states are found since contributions from $\rho$ and $\omega$
exchanges cancel with each other. Now we focus on the isoscalar cases. The
binding energies for $K\bar{K}_{1}(1400)$ with different cutoffs are shown in
Fig. 2. We can see that the $1^{-+}$ system can form a bound state when
$\Lambda\gtrsim 1.9$ GeV. We need $\Lambda\approx 2.5$ GeV to produce a
binding energy of 40 MeV, the experimental value of the newly observed exotic
state by BESIII. With the same cutoff, the binding energy of the $1^{--}$
$K\bar{K}_{1}(1400)$ state is predicted to be $10\sim 30$ MeV where the
uncertainty comes from whether the $\delta$ potential is included or not. Due
to the large coupling of $K_{1}(1270)KV$, the potential between
$K\bar{K}_{1}(1270)$ is repulsive for $C=+$ and therefore, no bound states are
expected. On the other hand, the potential is too attractive for $C=-$ to form
a molecule with a reasonable binding energy if we use a cutoff about $2.5$
GeV. Even when $\Lambda\sim 1.5$ GeV the binding energy is larger than 100
MeV, which is not acceptable for a shallow bound state, for which our previous
treatments hold valid.

##  III Decay properties of $K\bar{K}_{1}$ molecules

Figure 3: Two-body and three-body decays of molecules composed of
$K\bar{K}_{1}$.

With reasonable cutoff we have obtained the possible bound states of
$K\bar{K}_{1}(1400)$ with $J^{PC}=1^{-\pm}$, denoted as $X$ temporarily. It is
now desirable to estimate the decay patterns of the predicted molecular
states, especially the $\eta\eta^{\prime}$ channel, to compare with BESIII
observation and provide guidance to further experimental investigations. It is
natural that such molecules decay through their components, as illustrated in
Fig. 3. Since $K_{1}$’s have a quite large decay width, we assume that the
three-body decays of the molecules are dominated by the process shown in the
right diagram in Fig. 3, where $P,V=\pi,K^{*}$ or $K,\rho/\omega$. Two-body
strong decay channels considered here are listed in Table 1.

The coupling between a hadronic molecule and its components can be constructed
as

$\displaystyle\mathcal{L}_{XKK_{1}}=yX^{\mu}K^{\dagger}K^{\dagger}_{1\mu}+h.c.$
(50)

where $y$ is the coupling constant and $X$ denotes the field of the molecule.
For a loosely bound state, one can estimate the coupling constant in a model-
independent way by means of Weinberg compositeness criterion Weinberg (1965);
Baru  _et al._ (2004); Guo  _et al._ (2018), which is developed to estimate
the amount of compact and molecular components in a given state, where the
molecular components consist of two stable (or narrow 222“narrow” is relative
to the binding energy of that molecular state, that is, the widths of the
molecular components should be much smaller than its binding energy.) compact
states. However, $K_{1}(1400)$ has a width of 174 MeV, much larger than the
obtained binding energy in our case. Strictly speaking, one should consider
the dynamics between the sub-ingredients of $K_{1}(1400)$ and kaon, such as
$K\bar{K}^{*}\pi$ three-body system, like what were done for $X(3872)$ Baru
_et al._ (2011) and $T_{cc}^{+}$ Du  _et al._ (2022). On the other hand, the
decay width of $K_{1}(1400)$ is much smaller than the masses of exchanged
$\rho$ and $\omega$, and hence it should not jeopardize the molecular
interpretation of $\eta_{1}(1855)$ as argued in Ref. Guo and Meissner (2011).

To obtain a better estimate of the coupling $y$, we consider in the following
the pole position of the $X$ on the complex $\sqrt{s}$ plane by assuming that
the $X$ is a purely molecular state of $K\bar{K}_{1}(1400)$. The dominant
decay channel of $K_{1}(1400)$ is $K^{*}\pi$ with a branching ratio of 94% and
for simplicity, we take it to be 100% by ignoring all other channels. The pole
position of $K\bar{K}(1400)$ system with coupled channel effects ignored is
determined by

$1-VG=0$ (51)

where $V$, assumed to be constant, is the interaction strength of
$K\bar{K}(1400)$ and $G$ is the scalar two particle loop propagator,

$\displaystyle G(s)=\int\frac{{\rm
d}^{4}l}{(2\pi)^{4}}\frac{1}{l^{2}-m_{K}^{2}+i\epsilon}\frac{1}{(P-l)^{2}-m_{K_{1}}^{2}+im_{K_{1}}\Gamma_{K_{1}}}.$
(52)

with $P$ the total momentum. Note that we have ignored the spin structures of
the intermediate $K_{1}$, the corrections from which, as argued in Refs Roca
_et al._ (2005); Molina  _et al._ (2008), are at the order of $l_{\rm
on}^{2}/m_{K_{1}}^{2}$ with $l_{\rm on}$ the on-shell 3-momentum of
$K\bar{K}_{1}$. A hard cutoff, $\Lambda$, of the 3-momentum $\bm{l}$ will be
introduced to regularize the UV divergence. The running width of $K_{1}$ reads

$\displaystyle\Gamma_{K_{1}}(s)=g_{K_{1}}^{2}\frac{|{\bm{q}}|}{16\pi
m_{K_{1}}^{2}}\left(3+\frac{{\bm{q}}^{2}}{m_{K^{*}}^{2}}\right)$ (53)

with $g_{K_{1}}=3.65$ GeV the coupling constant of $K_{1}K^{*}\pi$ and
$\bm{q}$, depending on $s$, the 3-momentum of $K^{*}$ in the center of mass
frame of $K_{1}$.

The pole trajectory when varying $V$ is shown in Fig. 4. As expected, the
partial width of $\eta_{1}(1855)\to K(\bar{K}_{1}(1400)\to\bar{K}^{*}\pi)$
gets smaller when the binding energy becomes larger and the pole will touch
the $K\bar{K}_{1}(1400)$ cut, around $1900-i87$ MeV, if the binding energy
goes to zero. For the measured $\eta_{1}$ mass, the three body decay width of
$X$ is around $105$ MeV, which weakly depends on the hard cutoff $\Lambda$ in
the phenomenological reasonable range of $0.6\sim 1$ GeV. By matching the
three body decay width of $X$ to the obtained pole position, we have an
estimate of the coupling constant,

$y=13.6\ \rm GeV.$ (54)

Figure 4: Pole trajectory of the $K\bar{K}_{1}$ molecule in the $S$-wave
$K\bar{K}_{1}$ single channel model. Red line shows the pole trajectory with
$\Lambda=0.8$ GeV and red shadow is the uncertainties caused by varying the
hard momentum cut from 0.6 GeV to 1.0 GeV. Open blue circle denotes the branch
point corresponding to the $K\bar{K}_{1}$ threshold and the blue line
indicates the corresponding branch cut. The vertical black dot-dashed line is
the $K\bar{K}^{*}\pi$ three-body threshold. The orange band shows the observed
$\eta_{1}$ mass.

We refer to our previous work for other relevant couplings Dong and Zou
(2021). In particular, a non-zero $D$-wave component is introduced in the
vertex $K_{1}K^{*}\pi$ and $K_{1}K^{*}\eta$ as similar with $D_{1}D^{*}\pi$
interaction used in Ref. Dong and Zou (2021). The ratio of $D$-wave component
over the $S$-wave from RPP Zyla  _et al._ (2020) is $0.04\pm 0.01$. When
performing the loop integral in the triangle diagram of the two-body decays, a
Gaussian regulator $f_{1}$ (at the vertex of $XK\bar{K}_{1}$) and a multipole
form factor $f_{2}$ (at one of the vertices where the exchanged particle $A$
is involved), formulated as follows, are included to regularize the
ultraviolet divergence.

$\displaystyle
f_{1}(\bm{p}^{2}/\Lambda_{0}^{2})={\rm{exp}}(-\bm{p}^{2}/\Lambda_{0}^{2}),$
(55) $\displaystyle
f_{2}(q^{2})=\frac{\Lambda_{1}^{4}}{(m^{2}-q^{2})^{2}+\Lambda_{1}^{4}},$ (56)

where $\bm{p}$ are the three dimensional space part of momentum of $K$ or
$\bar{K}_{1}$, $m$ and $q$ is the mass and momentum of the exchanged particle.

Table 1: Partial widths of the isoscalar $K\bar{K}_{1}(1400)$ molecular states
with quantum numbers $1^{--}$ and $1^{-+}$ with $\Lambda_{0}=1.3$ GeV and
$\Lambda_{1}=2.5$ GeV. All the decay widths are in unit of $\mathrm{MeV}$.
Here, we assume $5\%$ $D$-wave contribution in the $K_{1}K^{*}\pi$ and
$K_{1}K^{*}\eta$ vertex.

Mode | Widths ($\mathrm{MeV}$)
---|---
3 |  |
| $1^{--}~{}E_{B}=20~{}\mathrm{MeV}$ | $1^{-+}~{}E_{B}=40~{}\mathrm{MeV}$
$K^{*}\bar{K}^{*}$ | 38.1 | 26.3
$K\bar{K}$ | 0.5 | 0
$K\bar{K}^{*}$ | 1.0 | 0.9
$a_{1}\pi$ | 0 | 9.2
$f_{1}\eta$ | 0 | 0.2
$\eta\eta^{\prime}$ | 0 | 26.9
$\sigma\omega$ | 0.2 | 0
$\rho\rho$ | 0 | 0.04
$\pi\rho$ | 6.4 | 0
$\eta\omega$ | 0.4 | 0
$\omega\omega$ | 0 | 0.01
$\omega\phi$ | 0 | 0.4
$K\bar{K}^{*}\pi$ | 130.0 | 105.0
2-body | 46.5 | 64.0
Total | 176.5 | 169.0
|  |

Figure 5: Parameter sensitivity of the $K^{*}\bar{K}^{*}$ (top) and
$\eta\eta^{\prime}$ (bottom) two-body partial widths. All curvatures indicate
the dependence of $K^{*}\bar{K}^{*}$ and $\eta\eta^{\prime}$ widths on the
parameter $\Lambda_{0}$ with the $\Lambda_{1}$ and the ratio of $D$-wave
component fixed. Solid/Dashed/DotDashed lines show the results with
$5\%$/$4\%$/$3\%$ $D$-wave component and red/orange/blue lines show those with
$\Lambda_{1}=1.5$/$2.0$/$2.5$ GeV.

The decay patterns of $1^{--}$ and $1^{-+}$ $K\bar{K}_{1}(1400)$ molecules for
$\Lambda_{0}=1.3$ GeV and $\Lambda_{1}=2.5$ GeV are shown in Table 1. Our
results show that the three-body $K\bar{K}^{*}\pi$ is the largest decay
channel for both $1^{--}$ and $1^{-+}$ $K\bar{K}_{1}$ molecules. And their
dominant two-body decay channel is also the same, i.e., $K^{*}\bar{K}^{*}$.
Moreover, the $1^{-+}$ exotic state also couples strongly to the
$\eta\eta^{\prime}$ channel. It is quite similar with the properties of
$D\bar{D}_{1}$ molecules where the $D^{*}\bar{D}^{*}$ and $\eta\eta_{c}$
channels contribute dominantly to the two-body decay widths Dong  _et al._
(2020). In general, one can find that the molecular scenario always seems to
support the strong couplings with the open-flavor channels. In addition, we
find that the $1^{-+}$ molecule has also a strong coupling with $a_{1}\pi$
channel, which is also reflected in a recent lattice simulation Woss  _et al._
(2021). In that work, the exotic $1^{-+}$ hybird meson with negative
$G$-parity decays dominantly into $b_{1}\pi$ channel which translates to
$a_{1}\pi$ channel for the positive $G$-parity $1^{-+}$ state. The parameter
sensitivity of partial widths of two dominant two-body decay channels for the
$1^{-+}$ $K\bar{K}_{1}(1400)$ molecule, namely $K^{*}\bar{K}^{*}$ and
$\eta\eta^{\prime}$, is presented in Fig. 5.

##  IV Summary

We have used the one boson exchange potential between the $K\bar{K}_{1}$, for
both $J^{PC}=1^{--}$ and $J^{PC}=1^{-+}$ systems, to investigate if it is
possible for them to form bound states. It turns out that with a momentum
cutoff $\Lambda\approx 2.5$ GeV, the attractive force between
$K\bar{K}_{1}(1400)$ with $J^{PC}=1^{-+}$ is strong enough to form a bound
state with a binding energy of around 40 MeV, which can be related to the
newly observed $\eta_{1}(1855)$. The C-parity partner of this molecule is
predicted to exist with a binding energy around 10 $\sim$ 30 MeV. While for
the $K_{1}(1270)$, the strong coupling between $K_{1}(1270)KV$ results in a
repulsive force for $C=+$ and a much deeply attractive force for $C=-$,
neither of which is expected to form a molecular state.

The decay properties of the two possible molecules of $K\bar{K}_{1}(1400)$ are
investigated. For both the $1^{--}$ and $1^{-+}$ states, the three body
channel $K\bar{K}^{*}\pi$ dominates due to the large decay width of
$K_{1}(1400)\to K^{*}\pi$. The $K^{*}\bar{K}^{*}$ and $\eta\eta^{\prime}$
channels have the largest contributions among the two body decays. The
presented decay pattern of $1^{-+}$ $K\bar{K}_{1}(1400)$ molecule is
compatible with BESIII’s measurements.

In summary, it is shown that the newly observed exotic state $\eta_{1}(1855)$
can be explained as a $K\bar{K}_{1}(1400)$ molecule. This interpretation can
be checked by searching for $\eta_{1}(1855)$ in $\omega\phi$ channel and its
$C$-partner in $\pi\rho$, $\eta\omega$ and $K\bar{K}$ channels, in addition to
$K^{*}\bar{K}^{*}$ channel.

We thank Feng-Kun Guo, Xiao-Yu Li, Jia-Jun Wu and Mao-Jun Yan for useful
discussions. This work is supported by the NSFC and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the funds
provided to the Sino German Collaborative Research Center TRR110 Symmetries
and the Emergence of Structure in QCD (NSFC Grant No.12070131001, DFG Project-
ID 196253076-TRR 110), by the NSFC Grants No. 11835015 and No. 12047503, and
by the Chinese Academy of Sciences (CAS) under Grant No.XDB34030000.

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