# Searching for bound states in the open strangeness systems

C. W. Xiao<EMAIL_ADDRESS>Department of Physics, Guangxi Normal
University, Guilin 541004, China  Guangxi Key Laboratory of Nuclear Physics
and Technology, Guangxi Normal University, Guilin 541004, China  School of
Physics, Central South University, Changsha 410083, China  J. J. Wu
<EMAIL_ADDRESS>School of Physical Sciences, University of Chinese Academy
of Sciences (UCAS), Beijing 100049, China  Southern Center for Nuclear-Science
Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences,
Huizhou 516000, China

###### Abstract

Inspired by the recent findings of $Z_{cs}$ and $P_{cs}$ states, we
investigate the strong interactions of the systems with open strangeness(es)
from the light sector to the heavy sector (no beauty quark), where the
interaction potential is derived from the vector meson exchange mechanism in
$t$\- and $u$-channels. In the current work, we discuss all of single channel
cases for the open strangeness in the systemic framework, where the resonances
$X_{0}(2866)$, $D^{*}_{s0}(2317)$ and $D_{s1}(2460)$ are dynamically
generated. Furthermore, there are many new exotics predicted. In addition, the
left-hand cut problem in $t$\- and $u$-channels is discussed in detail.

##  I Introduction

Searching for exotic states is a key issue to deeply understand the properties
of quantum chromodynamics (QCD), which attract much interest both in theory
and experiment. In 2021, a new tetraquark-like state $T_{cc}^{+}$ in the
$D^{0}D^{0}\pi^{+}$ invariant mass spectrum was discovered by the LHCb
Collaboration LHCb:2021vvq ; LHCb:2021auc , the mass and the width of which
were about 3875.1 MeV and 410 keV, respectively. This state looks like a
doubly charmed exotic state with spin-parity $J^{P}=1^{+}$ and constituent
quarks $cc\bar{u}\bar{d}$, and catches lots of theoretical attention
Meng:2021jnw ; Agaev:2021vur ; Ling:2021bir ; Chen:2021vhg ; Feijoo:2021ppq ;
Yan:2021wdl ; Dai:2021wxi ; Weng:2021hje ; Xin:2021wcr ; Fleming:2021wmk ;
Ren:2021dsi ; Hu:2021gdg ; Albaladejo:2021vln ; Abreu:2021jwm ;
Karliner:2021wju ; Du:2021zzh ; Ortega:2022efc ; Wang:2024vjc ; Guo:2023xyf .
Note that the mass of this state is very close to the first candidate of heavy
exotic state $X(3872)$, found by the Belle Collaboration in 2003 Belle:2003nnu
, and the first charge charmonium-like state $Z_{c}(3900)$, reported by the
BESIII and Belle Collaborations BESIII:2013ris ; Belle:2013yex  in 2013.
Especially, the $X(3872)$ is very close to the $D\bar{D}^{*}$ threshold, less
than 0.1 MeV compared to the $D^{0}\bar{D}^{*0}$ threshold, and is called
$\chi_{c1}(3872)$ now pdg2022  for its $J^{PC}=1^{++}$. But, its exotic
properties for a non-conventional $q\bar{q}$ state are still under debate,
such as its molecular nature Tornqvist:2004qy ; Swanson:2003tb ; Dong:2008gb ;
Gamermann:2009fv ; Gamermann:2009uq ; Guo:2014taa ; Song:2023pdq ;
Wang:2023ovj , see more discussions in the reviews of Particle Data Group
pdg2022  and Ref. Guo:2017jvc . For the $Z_{c}(3900)$, also around the
$D\bar{D}^{*}$ threshold, a challenge issue is to see if it is a molecular
state Guo:2013sya ; Wang:2013daa ; Aceti:2014uea ; Dong:2021juy , a tetraquark
candidate Dias:2013xfa ; Braaten:2013boa ; Wang:2013vex  or a cusp effect
Wang:2013cya ; Swanson:2014tra ; Szczepaniak:2015eza ; Liu:2015taa , see more
discussions in Ref. Voloshin:2013dpa . As done in Refs. Dong:2020hxe ;
Dong:2021juy , Ref. Dong:2021bvy  investigated the interactions of double
charm systems, some of which with the strangeness, and found many heavy-heavy
hadronic molecules, some of them consistent with the experimental findings.
Furthermore, recently a new structure with similar mass 3872.5 MeV in the
$D\bar{D}$ invariant mass distribution was report by the BESIII Collaboration
BESIII:2024ths , which is also nearby the $D\bar{D}^{*}$ threshold. As
discussed in Refs. Dong:2020hxe ; Dong:2021juy , many such new found resonant
structures have manifestly a feature that the masses of most of them are close
to the thresholds of a pair of hadrons, which are possibly caused by an
attractive interaction. In view of these discoveries, the picture of hadronic
molecule arises. Although in principle it is possible to have states near
threshold corresponding to non molecular states, this occurs with a very high
price of having extremely small scattering lengths and huge effective range
for the threshold channel, which in the case of the $X(3872)$ and
$T_{cc}(3875)$ have been shown to grossly disagree with experiment
Song:2023pdq ; Dai:2023kwv . Typically, for $J^{p}=1^{++}$ sector, a new state
$\chi_{c1}(4010)$ was discovered recently by the LHCb Collaboration
LHCb:2024vfz , which indicates that the main component of $X(3872)$ should be
$D\bar{D}^{*}$ while the charmonium state of $\chi_{c1}(2p)$ is possibly the
$\chi_{c1}(4010)$ Wang:2023ovj .

Now let us turn to the strangeness sector with the light quark replaced by the
the strangeness one, a new structure near the
$\bar{D}_{s}D^{*}/\bar{D}_{s}^{*}D$ thresholds, with the mass and the width as
about 3982.5 MeV and 12.8 MeV, respectively, was observed by the BESIII
Collaboration in 2020 BESIII:2020qkh , which was called as $Z_{cs}(3985)$.
Conversely, in 2021 the LHCb Collaboration reported two new resonances in the
$J/\psi K^{+}$ invariant mass distribution of the $B^{+}\to J/\psi\phi K^{+}$
decay LHCb:2021uow . The first one is named as $Z_{cs}(4000)$, with the mass
and the width about 4003 MeV and 131 MeV, respectively, which has a mass
analogous to the $Z_{cs}(3985)$ but with much larger width, and the second one
$Z_{cs}(4220)$ has the mass about 4216 MeV and the width 233 MeV. More results
on the experimental and theoretical status can be found in Refs. Chen:2016qju
; Esposito:2016noz ; Guo:2017jvc ; Olsen:2017bmm ; Brambilla:2019esw  for
reviews and references therein. Besides, the $D_{s0}^{*}(2317)$ was
dynamically generated from the $KD$ coupled channel interactions Guo:2006fu ;
Gamermann:2006nm ; Faessler:2007gv ; Cleven:2010aw ; Guo:2011dd , whereas, the
$D_{s1}(2460)$ from the $KD^{*}$ coupled channel interactions Gamermann:2007fi
; Guo:2006rp ; Faessler:2007us ; Cleven:2010aw ; Guo:2011dd .

In the light quark sector, an exotic tetraquark state with $ud\bar{s}\bar{s}$
and $J^{P}=1^{-}$ was predicted around 1.6 GeV in Ref. Burns:2004wy , which
could decay to $K^{+}K^{0}$. Analogously, an axial-vector isoscalar tetraquark
state about 1.4 GeV with $ud\bar{s}\bar{s}$ and $J^{P}=1^{+}$ was claimed in
Ref. Kanada-Enyo:2005gga  within a flux-tube quark model, which was near the
$KK^{*}$ threshold and could decay to the $K^{+}K^{+}\pi^{-}$ final states,
and similar predictions were obtained in Ref Cui:2005az  with the constituent
quark model. Using a QCD sum rule, a tetraquark state of $ud\bar{s}\bar{s}$
with $J^{P}=0^{+}$ and isospin $I=1$ was obtained around 1.5 GeV in Ref.
Chen:2006hy . Furthermore, using the non-relativistic constituent quark model,
a possible tetraquark candidate below the $K^{*}K^{*}$ threshold was predicted
in Ref. Wang:2007kb , with also the $ud\bar{s}\bar{s}$ configuration but with
$J^{P}=1^{+}$ and $I=0$, and an analogous result but below the $KK^{*}$
threshold was found in Ref. Gao:2012zza  with the chiral quark model. Thus,
Ref. Liu:2008ck  suggested to search these tetraquark candidates (one can call
them as $T_{ss}$ states) of $J^{P}=0^{+},\ 1^{-},\ 1^{+}$ in the reaction
$pn\to\Lambda\Lambda X(ud\bar{s}\bar{s})\to\Lambda\Lambda K^{+}K^{0}$ or
$\Lambda\Lambda KK^{*}$. As we known, there is not much experimental results
for the $T_{ss}$-like states, which is the motivation for us to bring more
theoretical information to the experiments.

In summary, for the open strangeness system, there may exist many hadronic
molecular candidates. However, a systematic study of these states is still
missing. In the present work, we focus on the systems with open strangeness
both in the light and heavy sectors (but no beauty quark) to look for the
molecular candidates near the thresholds of certain channels. Our paper is
organized as follow. In the next section, the formalism of local hidden gauge
is presented. Then, our results are shown in the following section. Finally, a
short conclusion is made.

##  II Formalism

###  II.1 Diagrams of the interactions

In this work, we will consider the following diagrams for the hadron-hadron
interaction of $t$\- and $u$-channels, as shown in Fig. 1(a) and 1(b),
respectively, where $V$ stands for the vector meson, $P$ represents the
pseudoscalar meson, and the particles $V$ and $P$ are specified with their
corresponding momenta $p_{i}$ ($i=1,2,3,4$) or the mass of the exchanged
vector meson $m_{ex}$. For the $t$\- and $u$-channels, the vector meson
exchange mechanism is taken into account based on the local hidden gauge
formalism Bando:1984ej ; Bando:1987br ; Meissner:1987ge . Note that, we do not
consider the $s$-channel by exchanging vector meson, since it leads to a
$p$-wave amplitude for the $VV$ interactions under the constraint $L+S+I=$
even Molina:2008jw  and does not have any exchange meson in some open
strangeness and charm systems. In addition, for the $VV$ interactions, one
should also take into account the contribution of the contact term in the tree
level, as shown in Fig. 2. In fact, from the local hidden gauge Lagrangians,
there is no contact term for the $PP$ and $PV$ interactions, which are well
described by the chiral Lagrangians, see the one used in Ref. Roca:2005nm  for
the $PV$ interaction as discussed below. Of course, there is no $PPP$ vertex
in the local hidden gauge formalism, due to angular momentum and parity
conservation.

(a) $t$-channel

(b) $u$-channel

Figure 1: Diagrams for the $t$\- and $u$-channels with vector meson exchanged
mechanisms. Figure 2: Diagrams for the contact term.

###  II.2 Interaction Lagrangians

The local hidden gauge Lagrangians involving the vertexes of the exchanged
vector mesons are given by Bando:1984ej ; Bando:1987br ; Meissner:1987ge

$\displaystyle{\cal L}_{VVV}$ $\displaystyle=$ $\displaystyle
ig~{}\langle[V_{\nu},\partial_{\mu}V_{\nu}]V^{\mu}\rangle,$ (1)
$\displaystyle{\cal L}_{VPP}$ $\displaystyle=$ $\displaystyle-
ig~{}\langle[P,\partial_{\mu}P]V^{\mu}\rangle,$ (2)

and for the contact term of $VV$ interaction, the Lagrangian is written as
Bando:1984ej ; Bando:1987br ; Meissner:1987ge

${\cal
L}_{VVVV}=\frac{g}{2}\langle~{}[V_{\mu},~{}V_{\nu}]~{}V^{\mu}V^{\nu}\rangle,$
(3)

where the coupling is $g=m_{V}/(2f)$ with the pion decay constant $f=93$ MeV
and taking $m_{V}=m_{\rho}$. $P$ and $V_{\mu}$ are the SU(4) matrix of the
pseudoscalar and vector meson fields in the Lagrangian terms, respectively.
From Eqs. (1) and (2), one can see that there is a minus sign different from
them, which will lead to the same structure for the potentials of the $PP$,
$VP$, and $VV$ interactions by the extra factor
$\epsilon\cdot\epsilon^{\prime}=-\vec{\epsilon}\cdot\vec{\epsilon}\,^{\prime}=-1$
when taking the approximations of the external vector with $\epsilon^{0}=0$
and $\vec{p}_{V}=0$ (already implicit in Eq. (1)). Thus, we can make a general
formalism for the interactions of $PP$, $VP$, and $VV$ with the potentials
derived from Eq. (1) and (2), see more discussions later. Note that, the
formalism used in Ref. Roca:2005nm  is a bit different from the normal coupled
channel Bethe-Salpeter equation, see the discussions later.

###  II.3 Derived potentials

Applying the vector meson exchange mechanism for the systems with open
strangeness(es), the interaction potentials are given in Table 1, where we
define the potential notations with the exchange meson as

$V_{t(u)}^{ex}=\frac{-V_{t(u)}}{t(u)-m_{ex}^{2}+i\epsilon},$ (4)

with $V_{t}=(p_{1}+p_{3})\cdot(p_{2}+p_{4})=s-u$ and
$V_{u}=(p_{1}+p_{4})\cdot(p_{2}+p_{3})=s-t$ for the $t$\- and $u$-channels,
respectively. One should keep in mind that $p_{i}\ (i=1,\ 2,\ 3,\ 4)$ are the
corresponding four momenta of initial and final states, as shown in Fig. 1,
and $m_{ex}$ is the mass of the exchange vector meson. Note that, in Table 1,
there is a factor $\vec{\epsilon}\cdot\vec{\epsilon}\,^{\prime}$ for the
$VP\to VP$ transitions, and a factor
$\epsilon_{1}\cdot\epsilon_{3}\;\epsilon_{2}\cdot\epsilon_{4}$ and
$\epsilon_{1}\cdot\epsilon_{4}\;\epsilon_{2}\cdot\epsilon_{3}$ for the $t$\-
and $u$-channels, respectively, in the case of the $VV\to VV$ transitions,
which have been ignored. Furthermore, in the results of Table 1 the isospin
projection has already been made, and thus, the potentials are specified with
different isospins.

Table 1: Interaction potential of the strangeness(es) systems. Systems | Coefficients $I=0$ | Coes. $I=1\,(\frac{1}{2})$
---|---|---
$KK$ | $\frac{1}{2}(-3V_{t}^{\rho}+V_{t}^{\omega}+2V_{t}^{\phi})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}+V_{t}^{\omega}+2V_{t}^{\phi})g^{2}$
| $-\frac{1}{2}(-3V_{u}^{\rho}+V_{u}^{\omega}+2V_{u}^{\phi})g^{2}$ | $+\frac{1}{2}(V_{u}^{\rho}+V_{u}^{\omega}+2V_{u}^{\phi})g^{2}$
$KK^{*}$ | $\frac{1}{2}(-3V_{t}^{\rho}+V_{t}^{\omega}+2V_{t}^{\phi})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}+V_{t}^{\omega}+2V_{t}^{\phi})g^{2}$
$K^{*}K^{*}$ | $\frac{1}{2}(-3V_{t}^{\rho}+V_{t}^{\omega}+2V_{t}^{\phi})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}+V_{t}^{\omega}+2V_{t}^{\phi})g^{2}$
| $-\frac{1}{2}(-3V_{u}^{\rho}+V_{u}^{\omega}+2V_{u}^{\phi})g^{2}$ | $+\frac{1}{2}(V_{u}^{\rho}+V_{u}^{\omega}+2V_{u}^{\phi})g^{2}$
$\bar{K}D$ | $\frac{1}{2}(-3V_{t}^{\rho}+V_{t}^{\omega}-2V_{u}^{D^{*}_{s}})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}+V_{t}^{\omega}+2V_{u}^{D^{*}_{s}})g^{2}$
$\bar{K}D^{*}$ | $\frac{1}{2}(-3V_{t}^{\rho}+V_{t}^{\omega})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}+V_{t}^{\omega})g^{2}$
$\bar{K}^{*}D$ | $\frac{1}{2}(-3V_{t}^{\rho}+V_{t}^{\omega})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}+V_{t}^{\omega})g^{2}$
$\bar{K}^{*}D^{*}$ | $\frac{1}{2}(-3V_{t}^{\rho}+V_{t}^{\omega}-2V_{u}^{D^{*}_{s}})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}+V_{t}^{\omega}+2V_{u}^{D^{*}_{s}})g^{2}$
$KD$ | $\frac{1}{2}(-3V_{t}^{\rho}-V_{t}^{\omega})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}-V_{t}^{\omega})g^{2}$
$KD^{*}$ | $\frac{1}{2}(-3V_{t}^{\rho}-V_{t}^{\omega})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}-V_{t}^{\omega})g^{2}$
$K^{*}D$ | $\frac{1}{2}(-3V_{t}^{\rho}-V_{t}^{\omega})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}-V_{t}^{\omega})g^{2}$
$K^{*}D^{*}$ | $\frac{1}{2}(-3V_{t}^{\rho}-V_{t}^{\omega})g^{2}$ | $\frac{1}{2}(V_{t}^{\rho}-V_{t}^{\omega})g^{2}$
$KD_{s}$ | – | $(V_{t}^{\phi}+V_{u}^{D^{*}})g^{2}$
$KD_{s}^{*}$ | – | $V_{t}^{\phi}g^{2}$
$K^{*}D_{s}$ | – | $V_{t}^{\phi}g^{2}$
$K^{*}D_{s}^{*}$ | – | $(V_{t}^{\phi}+V_{u}^{D^{*}})g^{2}$
$\bar{D}D_{s}$ | – | $-V_{t}^{J/\psi}g^{2}$
$\bar{D}D_{s}^{*}$ | – | $-V_{t}^{J/\psi}g^{2}$
$\bar{D}^{*}D_{s}$ | – | $-V_{t}^{J/\psi}g^{2}$
$\bar{D}^{*}D_{s}^{*}$ | – | $-V_{t}^{J/\psi}g^{2}$
$DD_{s}$ | – | $(V_{t}^{J/\psi}+V_{u}^{K^{*}})g^{2}$
$DD_{s}^{*}$ | – | $V_{t}^{J/\psi}g^{2}$
$D^{*}D_{s}$ | – | $V_{t}^{J/\psi}g^{2}$
$D^{*}D_{s}^{*}$ | – | $(V_{t}^{J/\psi}+V_{u}^{K^{*}})g^{2}$
$D_{s}D_{s}$ | $(V_{t}^{J/\psi}+V_{t}^{\phi})g^{2}$ | –
| $+(V_{u}^{J/\psi}+V_{u}^{\phi})g^{2}$ | –
$D_{s}D^{*}_{s}$ | $(V_{t}^{J/\psi}+V_{t}^{\phi})g^{2}$ | –
$D^{*}_{s}D^{*}_{s}$ | $(V_{t}^{J/\psi}+V_{t}^{\phi})g^{2}$ | –
| $+(V_{u}^{J/\psi}+V_{u}^{\phi})g^{2}$ | –

In addition, for the $VP\to VP$ transitions, we do not consider the
$u$-channel contribution, since this comes from the “Z” diagram with the
Lagrangian ${\cal L}_{VVP}$, the contribution of which was found to be small
and could be ignored as discussed in Refs. Nakamura:2015qga ; Dias:2021upl .
111We keep this problem for further investigation in future when the coupled
channel interaction is consider. Besides, for the $VP\to VP$ transitions, a
chiral Lagrangian was used in Ref. Roca:2005nm , which is different from the
vector meson exchange mechanism taken into account in the present work. But,
the equivalence these two ways was proved by a general derivation of the
interaction potential in the appendix of Ref. Dias:2021upl . Departing from
the extrapolated chiral dynamics for the $D\bar{D}$ interaction as done in
Ref. Gamermann:2006nm , Ref. Bayar:2022dqa  preferred to use the vector meson
exchange mechanism for the heavy flavor $PP$ interaction, which is out of the
constrain of chiral symmetry, normally suited to the light quark sector.

For the potentials defined in Eq. (4), $V_{t(u)}^{ex}$, we need to do the
$s$-wave projection, i.e., the $t$-channel case, having

$\displaystyle\frac{1}{2}\int_{-1}^{+1}\text{d}\cos\theta\frac{-V_{t}}{t-m_{ex}^{2}+i\epsilon}\equiv\frac{1}{2}\int_{-1}^{+1}\text{d}\cos\theta\frac{-V_{t}(s,\cos\theta)}{t(s,\cos\theta)-m_{ex}^{2}+i\epsilon}$
(5) $\displaystyle=$
$\displaystyle-\frac{s}{\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}}\left(m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}-2s-m_{ex}^{2}\right)$
$\displaystyle\times\log\frac{m_{1}^{2}+m_{3}^{2}-\frac{\left(s+m_{1}^{2}-m_{2}^{2}\right)\left(s+m_{3}^{2}-m_{4}^{2}\right)}{2s}-\frac{\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}}{2s}-m_{ex}^{2}+i\epsilon}{m_{1}^{2}+m_{3}^{2}-\frac{\left(s+m_{1}^{2}-m_{2}^{2}\right)\left(s+m_{3}^{2}-m_{4}^{2}\right)}{2s}+\frac{\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}}{2s}-m_{ex}^{2}+i\epsilon}-1,$
$\displaystyle=$
$\displaystyle-1-\frac{s}{\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}}\left(s_{0}-2s-m_{ex}^{2}\right)$
$\displaystyle\times\log\frac{s\left(s_{0}-s-2m_{ex}^{2}\right)-\left(m_{1}^{2}-m_{2}^{2}\right)\left(m_{3}^{2}-m_{4}^{2}\right)-\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}+i\epsilon}{s\left(s_{0}-s-2m_{ex}^{2}\right)-\left(m_{1}^{2}-m_{2}^{2}\right)\left(m_{3}^{2}-m_{4}^{2}\right)+\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}+i\epsilon},$

with the Källén function $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$,
$s=(p_{1}+p_{2})^{2}$, and
$s_{0}\equiv\left(m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}\right)$, where
$m_{i}\ (i=1,\ 2,\ 3,\ 4)$ are the corresponding masses of initial and final
states. For the $u$-channel case, one just takes the replacement
$m_{3}\leftrightarrow m_{4}$.

For the contact term of $VV$ interaction, the interaction $A(1)+B(2)\to
C(3)+D(4)$, the amplitude derived from Eq. (3) gives rise to three products of
polarization vectors in the order of 1, 2, 3, 4, written as

$\epsilon_{\mu}\epsilon^{\mu}\epsilon_{\nu}\epsilon^{\nu},\qquad\epsilon_{\mu}\epsilon_{\nu}\epsilon^{\mu}\epsilon^{\nu},\qquad\epsilon_{\mu}\epsilon_{\nu}\epsilon^{\nu}\epsilon^{\mu},$
(6)

which will lead to the spin components of $J=0,\ 1,\ 2$ for the amplitude. As
done in Ref. Molina:2008jw , taking the approximation
$\frac{q^{2}}{M_{V}^{2}}=0$, one can apply the spin projectors to separate the
spin components of the amplitude, given by

$\displaystyle{\cal
P}^{(0)}=\frac{1}{3}\epsilon_{\mu}\epsilon^{\mu}\epsilon_{\nu}\epsilon^{\nu},$
(7) $\displaystyle{\cal
P}^{(1)}=\frac{1}{2}\left(\epsilon_{\mu}\epsilon_{\nu}\epsilon^{\mu}\epsilon^{\nu}-\epsilon_{\mu}\epsilon_{\nu}\epsilon^{\nu}\epsilon^{\mu}\right),$
$\displaystyle{\cal
P}^{(2)}=\left[\frac{1}{2}\left(\epsilon_{\mu}\epsilon_{\nu}\epsilon^{\mu}\epsilon^{\nu}+\epsilon_{\mu}\epsilon_{\nu}\epsilon^{\nu}\epsilon^{\mu}\right)-\frac{1}{3}\epsilon_{\mu}\epsilon^{\mu}\epsilon_{\nu}\epsilon^{\nu}\right],$

with the order of 1, 2, 3, 4 for the polarization vectors, which, in fact, is
analogous to the isospin projection operators of two-pion system Holz:2022smu
, and different from the method proposed in Ref. Gulmez:2016scm . Thus, with
these spin projectors, we obtain the results of the spin projection to the
$VV$ interaction amplitudes of the contact term as shown in Table 2, which are
consistent with the results of Ref. Molina:2010tx .

Furthermore, taking these spin projectors, one can easily find that the spin
components $J=0,\ 1,\ 2$ of the $VV$ interaction amplitudes with vector meson
exchanged are the same for the $t$-channel with the structure of the
polarization vectors
$\epsilon_{1}\cdot\epsilon_{3}\;\epsilon_{2}\cdot\epsilon_{4}$, whereas, only
the ones with spin $J=0,\ 2$ are equal for the $u$-channel with the structure
of the polarization vectors
$\epsilon_{1}\cdot\epsilon_{4}\;\epsilon_{2}\cdot\epsilon_{3}$, since the one
with spin $J=1$ has a minus sign.

###  II.4 Scattering equation

The scattering amplitude of the coupled channel interaction is evaluated from
the coupled channel Bethe-Salpeter equation with the on-shell description,
given by Oller:1997ti ; Oset:1997it

$T=[1-VG]^{-1}V,$ (8)

where the $V$ matrix is made from the transition potentials as discussed
above, and the diagonal $G$ matrix is constructed by the loop functions of two
intermediate mesons in a certain channel. Note that, as discussed above for
the $VP$ interactions, a different form of the Bethe-Salpeter equation was
taken in Ref. Roca:2005nm  due to the polarization vectors appearing in the
potential, where a slight correction $1+\frac{1}{3}\frac{q^{2}}{M_{V}^{2}}$
was also introduced to the loop function of the vector-meson propagator. As
discussed before, the transition potentials shown in Table 1 for the $VP$
interactions in fact have a ignored factor
$\vec{\epsilon}\cdot\vec{\epsilon}\,^{\prime}$, which means that a minus sign
from
$\epsilon\cdot\epsilon^{\prime}=-\vec{\epsilon}\cdot\vec{\epsilon}\,^{\prime}$
is already added to the potentials, see the discussions after Eq. (2).
Besides, as mentioned above, the approximation $\frac{q^{2}}{M_{V}^{2}}=0$ is
taken for the $VV$ interaction potential, and thus, taking the same
approximation, the corrected term $\frac{1}{3}\frac{q^{2}}{M_{V}^{2}}$ is
indeed small Roca:2005nm  and can be ignored safely in the loop function of
the $VP$ interaction. Therefore, for the consistency of our formalism, the
general form of Eq. (8) is applied for all three cases of the interactions,
$PP$, $VP$, and $VV$. The element of $G$ matrix in the $i$-th channel is given
by

$G_{ii}(s)=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{1}^{2}+i\varepsilon}\frac{1}{\left(p_{1}+p_{2}-q\right)^{2}-m_{2}^{2}+i\varepsilon}\text{
,}$ (9)

where $p_{1}$ and $p_{2}$ are the four-momenta of the two initial states,
respectively, and $m_{1}$, $m_{2}$ the masses of the two intermediate
particles for the $i$-th channel appearing in the loop. It should be mentioned
that the $G$ function is logarithmically divergent. To solve this singular
integral, one either uses the three-momentum cutoff method Oller:1997ti ,
where the analytic expression is given by Refs. Oller:1998hw ; Guo:2005wp , or
takes the dimensional regularization method Oller:2000fj . Utilizing the
cutoff method, Eq. (9) can be rewritten as Oller:1997ti

$G_{ii}(s)=\int_{0}^{q_{\max}}\frac{q^{2}dq}{(2\pi)^{2}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}\left[s-\left(\omega_{1}+\omega_{2}\right)^{2}+i\varepsilon\right]}\text{
,}$ (10)

with $q=|\vec{q}\,|$ and $\omega_{i}=\sqrt{\vec{q}^{\>2}+m_{i}^{2}}$, where
the cutoff $q_{max}$ is the free parameter.

Moreover, the mass and the decay width of the state generated in the coupled
channel interaction can be determined just by looking for the pole of the
scattering amplitude in the complex Riemann sheets. Thus, one needs to
extrapolate analytically the scattering amplitude in the complex energy plane,
where the $G$ function should be extrapolated to the second Riemann sheet by
the continuity condition, given by

$G_{ii}^{(II)}\left(s\right)=G_{ii}^{(I)}\left(s\right)-2i\operatorname{Im}G_{ii}^{(I)}\left(s\right)=G_{ii}^{(I)}\left(s\right)+\frac{i}{4\pi}\frac{p_{cmi}(s)}{\sqrt{s}}\textbf{
,}$ (11)

for $\text{Re}(\sqrt{s})>\sqrt{s_{th}},\ \text{Im}\;p_{cmi}>0$ (see Ref.
Roca:2005nm ), where the loop function of the first Riemann sheet,
$G_{ii}^{(I)}(s)$, is given by Eq. (10), and the three momentum in center-of-
mass (CM) frame is given by

$p_{cmi}(s)=\frac{\sqrt{\lambda(s,m_{1}^{2},m_{2}^{2})}}{2\sqrt{s}}\,,$ (12)

with the usual Källén triangle function $\lambda(a,b,c)$, defined above.

###  II.5 Discussions for the left-hand cut problem

The numerator and denominator of the $\log$ in Eq. (5) maybe zero, which is
equivalent to $t-m_{ex}^{2}=0$ at $\cos\theta=\pm 1$. It is known as left-hand
cut. In fact, we can rewrite $t$ and $u$ as the functions of $s$ and
$\cos\theta$, i.e., $t\equiv t[s,\cos\theta]$, and thus, the left-hand cut
appears at the solution of the constraint $t[s,\cos\theta]\equiv m_{ex}^{2}$.
In other word, it is crucial to understand the left-hand cut in view of the
analytical behaviours of the function
$f[s,\cos\theta]=t[s,\cos\theta]-m_{ex}^{2}$. Then following Ref. Lutz:2015lca
, we can define the contour functions $c_{\pm}^{t}(t^{\prime})$ as follows,

$t[c_{\pm}^{t}(t^{\prime}),\pm 1]=t^{\prime},$ (13)

with the well-known roots, given by

$c_{\pm}^{t}(t^{\prime})=\frac{1}{2}\left[s_{0}-t^{\prime}-\frac{1}{t^{\prime}}\left(m_{1}^{2}-m_{3}^{2}\right)\left(m_{2}^{2}-m_{4}^{2}\right)\pm\frac{\sqrt{\lambda\left(t^{\prime},m_{1}^{2},m_{3}^{2}\right)\lambda\left(t^{\prime},m_{2}^{2},m_{4}^{2}\right)}}{t^{\prime}}\right],$
(14)

which is kept to discuss the analytical properties of Eq. (5) later, and
similar for the case of $u$-channel.

Taking the approximation, $t(u)\to 0$, as done in Refs. Molina:2008jw ;
Geng:2008gx , one can obtain

$V_{t(u)}^{ex}\to\frac{1}{m_{ex}^{2}}V_{t(u)},$ (15)

where the part $V_{t}$ should be projected to the $s$-wave, having

$\displaystyle V_{t}$
$\displaystyle\to\frac{1}{2}\left[3s-\left(m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}\right)-\frac{1}{s}\left(m_{1}^{2}-m_{2}^{2}\right)\left(m_{3}^{2}-m_{4}^{2}\right)\right],$
(16)
$\displaystyle=\frac{1}{2}\left[3s-s_{0}-\frac{1}{s}\left(m_{1}^{2}-m_{2}^{2}\right)\left(m_{3}^{2}-m_{4}^{2}\right)\right].$

Whereas, for the $u$-channel one just needs to change $m_{3}\leftrightarrow
m_{4}$.

This approximation is fine for the heavy cases, such as the transition
$\bar{D}D_{s}\to\bar{D}D_{s}$, see Fig. 3 for the comparison of Eqs. (5) and
(15). In fact, in this case, the singularity from the left-hand cut is far
away from the threshold.

Figure 3: Inverse of the Potentials of $\bar{D}D_{s}\to\bar{D}D_{s}$ with Eqs.
(5) [Ours], (15) [Oset].

For the potential of Eq.(5), one should be careful with the $\log$ function.
As found in Ref. Wang:2023aza  for the non-diagonal channel of the $VV\to PP$
transitions, a discontinuity appears below the threshold of the bound channel,
which in fact is given by the condition

$m_{1}^{2}+m_{3}^{2}-\frac{\left(s+m_{1}^{2}-m_{2}^{2}\right)\left(s+m_{3}^{2}-m_{4}^{2}\right)}{2s}-m_{ex}^{2}\equiv
0,$ (17)

because for this case
$\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}$
is a purely imaginary and thus the $\log$ function behaves as the $\arctan$
function, see a complicated case in Ref. Holz:2015tcg . For the present case
of the single channel, which is the diagonal one, we do not face this problem.
But, for the bound state that we look for, which is located below the
threshold and leads to
$\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}$
being real, a serious singularity of the left-hand cut will be happened at the
energy under the constraint

$m_{1}^{2}+m_{3}^{2}-\frac{\left(s+m_{1}^{2}-m_{2}^{2}\right)\left(s+m_{3}^{2}-m_{4}^{2}\right)}{2s}\pm\frac{\sqrt{\lambda\left(s,m_{1}^{2},m_{2}^{2}\right)\lambda\left(s,m_{3}^{2},m_{4}^{2}\right)}}{2s}-m_{ex}^{2}\equiv
0,$ (18)

of which the solutions are given by Eq. (14), having
$s=c_{\pm}^{t}(m_{ex}^{2})$.

If this left-hand cut is just close to the energy region that we are
interested in, it will be affected the results seriously as found in Ref.
Du:2018gyn . In contrast, for the case of the left-hand cut far away from the
concerned energy region as discussed in Ref. Wang:2022pin , one can safely
ignore them, as discussed above and shown in Fig. 3. Indeed, the singularity
of the left-hand cut in Eq.(5) is difficult to normalize with the help of the
term $i\epsilon$ due to the fact $\epsilon\to 0$ too. And thus, to avoid the
singularity, one also can take an improved approximation for Eq. (4) by
ignoring the angle part in the propagator, writing

$\displaystyle V_{t(u)}^{ex}$
$\displaystyle\to\frac{-V_{t(u)}}{m_{1}^{2}+m_{3}^{2}-\frac{\left(s+m_{1}^{2}-m_{2}^{2}\right)\left(s+m_{3}^{2}-m_{4}^{2}\right)}{2s}-m_{ex}^{2}}$
(19)
$\displaystyle=\frac{-V_{t(u)}}{\frac{1}{2}\left[\left(s_{0}-s\right)-\frac{1}{s}\left(m_{1}^{2}-m_{2}^{2}\right)\left(m_{3}^{2}-m_{4}^{2}\right)\right]-m_{ex}^{2}},$

with $V_{t(u)}$ projected to the $s$ wave as done in Eq. (16). The compared
results of Eqs. (5) [Ours-1], (15) [Oset] and (19) [Ours-2] are shown in Fig.
4 for the case of $KD$ channel, which are also compared with the real part of
the $G$ function.

Figure 4: Inverse of the Potentials with Eqs. (5) [Ours-1], (15) [Oset] and
(19) [Ours-2] for the channel $KD$ compared with the $G$ function taking
$q_{max}=875$ MeV.

Note that, as discussed in Ref. Geng:2016pmf , the singularity of the left-
hand cut found in the $\rho\rho$ interaction Gulmez:2016scm  was unphysical
and the authors avoided the factorization of the potential in the loops by
looking at the actual loops with four propagators, where one could define an
effective potential from the one-loop level diagrams to obtain the full Bethe-
Salpeter series under the “on-shell” factorization. We have performed the
evaluation of the box diagram of the one-loop level diagrams, i.e. for the
$D^{*}K^{*}$ channel with two $\rho$, $D^{*}$ and $K^{*}$ in the loop, and try
to extract the “effective” potential by extending the way suggested in the
appendix of Ref. Geng:2016pmf . Due to the contribution (the real part) of the
box diagram quite small, as found in Refs. Molina:2008jw ; Geng:2016pmf , we
do not introduce the “effective” potential in our formalism to avoid the
confusion, where more details can be found in Ref. Geng:2016pmf . We just
focus on the following potentials, Eqs. (5), (15) and (19), only contributed
from the tree level diagram.

Table 2: The contact terms for the VV interactions. Systems | Qua. Num. | Coefficients $I=0$ | Coes. $I=1,\ (\frac{1}{2})$
---|---|---|---
|  | Spins $S=0,\ 1,\ 2$ | Spins $S=0,\ 1,\ 2$
$K^{*}K^{*}$ | $S=2,\ C=0$ | 0 | $-8g^{2},\ 0,\ 4g^{2}$
$\bar{K}^{*}D^{*}$ | $S=-1,\ C=1$ | $4g^{2},\ 0,\ -2g^{2}$ | $-4g^{2},\ 0,\ 2g^{2}$
$K^{*}D^{*}$ | $S=1,\ C=1$ | $4g^{2},\ 6g^{2},\ -2g^{2}$ | 0
$K^{*}D_{s}^{*}$ | $S=2,\ C=1$ | 0 | $-4g^{2},\ 0,\ 2g^{2}$
$\bar{D}^{*}D_{s}^{*}$ | $S=1,\ C=0$ | 0 | $2g^{2},\ 3g^{2},\ -1g^{2}$
$D^{*}D_{s}^{*}$ | $S=1,\ C=2$ | 0 | $-4g^{2},\ 0,\ 2g^{2}$
$D^{*}_{s}D^{*}_{s}$ | $S=2,\ C=2$ | $-8g^{2},\ 0,\ 4g^{2}$ | 0

##  III Results

Note that, from the potentials listed in Table 1, one can see that some of
them are repulsive with positive values, especially for the isospin $I=1$
sectors, which can be easily seen from Eqs. (4) and (15), and thus, these
channels with certain isospin can not lead to any bound states and are out of
our concern. For the $D^{*}D^{*}_{s}$ system with the repulsive potential as
shown in Table 1, which is in fact that only for the spin $J=0,\ 2$ when one
does the spin projection for the $u$-channel potential as discussed above.
However, since the potential of $J=1$ had a minus sign different from the ones
with $J=0,\ 2$ and became attractive, see the results of Ref. Molina:2010tx ,
a strong cusp around the threshold was found in Ref. Dai:2021vgf  for the
$D^{*}D^{*}_{s}$ with $J=1$. The interactions of $KD_{s}$ and $K^{*}D_{s}^{*}$
are also repulsive as found in Refs. Guo:2009ct  and Molina:2010tx ,
respectively. Thus, the sectors $(S=2,C=1)$, $(S=1,C=2)$ and $(S=2,C=2)$, and
some repulsive channels in the other sectors are not taken into account for
searching for bound states. Furthermore, from Table 1, we can see that the
potential of the $KK$ (with $I=0$) channel are slightly attractive due to the
$\rho$ exchange term partly cancelled with the contributions of $\omega$ and
$\phi$ exchanged, and the ones of the $\bar{D}^{(*)}D_{s}^{(*)}$ (with
$I=\frac{1}{2}$) channels only come from the heavy $J/\psi$ exchange. Thus,
the systems $KK$ (with $I=0$) and $\bar{D}^{(*)}D_{s}^{(*)}$ (the whole $S=1$,
$C=0$ sector with $I=\frac{1}{2}$) are too weak to bind a state. At last, in
the present work, only the $K^{(*)}D^{(*)}$ and $\bar{K}^{(*)}D^{(*)}$
channels with isospin $I=0$ sector may lead to bound states.

As discussed above, there is a free parameter in our formalism, the cutoff
$q_{max}$ in the loop function. Since we do not consider the coupled channels
interactions for the systems with open strangeness(es) without beauty quarks,
we just take one cutoff for all of the systems to reduce the uncertainties. We
try to determine a proper value of the cutoff by some known resonance(s). In
the all of systems that we are concerned, there are two systems that generate
the well known resonances. Recalling that, the $D^{*}_{s0}(2317)$ state was
dynamically reproduced in the $KD$ interactions with its coupled channels in
Refs. Guo:2006fu ; Gamermann:2006nm , in principle it can also be generated
from the single $KD$ interaction. Besides, Ref. Molina:2020hde  took the
interaction of the $\bar{K}^{*}D^{*}$ channel to dynamically generate the
$X_{0}(2866)$ state, found recently by the LHCb collaboration LHCb:2022sfr
and confirmed by the later work LHCb:2024vfz . Thus, based on these two
resonances, $D^{*}_{s0}(2317)$ and $X_{0}(2866)$, we try to determine the
value of the cutoff $q_{max}$.

The results are shown in Table 3, where we take Eq. (5) for the evaluation of
the potentials and compare the results with Eq. (15). From the results of
Table 3, we find that it is impossible to use only one cutoff for generating
both of the $D^{*}_{s0}(2317)$ and $X_{0}(2866)$ states in this framework.
Thus, two sets of results are shown in Table 3 with two different cutoffs to
reproduce them one by one, where $q_{max}=875$ MeV is fixed from the mass of
the $X_{0}(2866)$ state, and $q_{max}=756$ MeV from the mass of the
$D^{*}_{s0}(2317)$ state. Similarly, for the results by using Oset’s
approximation, see Eq. (15), two different cutoffs are used, $q_{max}=1100$
MeV and $q_{max}=843$ MeV, for reproducing the $X_{0}(2866)$ and
$D^{*}_{s0}(2317)$ states, respectively. One can see that when we obtain the
$D^{*}_{s0}(2317)$ state from the $KD$ interaction using $q_{max}=756$ MeV,
the $D_{s1}(2460)$ is also dynamically generated from the $KD^{*}$ interaction
as found in Refs. Gamermann:2007fi ; Guo:2006rp ; Faessler:2007us ;
Cleven:2010aw ; Guo:2011dd  (see more discussions in recent reviews
Liu:2024uxn ), where the result with Oset’s approximation ($q_{max}=843$ MeV)
is consistent with what we have, a pole at 2454.71 MeV compared to 2456.29
MeV. It should be mentioned that the $D^{*}_{s2}(2573)$ was identified as a
$K^{*}D^{*}$ molecule with $J=2$ Molina:2010tx  for its pole more bound than
the ones with $J=0,1$, which is consistent with the results obtained with
Oset’s approximation, a pole at 2600.70 MeV or 2732.68 MeV, see Table 3.

Besides, it looks like the results for the $\bar{K}D^{*}$ interactions do not
have stable pole except for the result with Eq. (15) and $q_{max}=1100$ MeV.
From Table 3, in the case of $q_{max}=875$ MeV, only the systems of $\bar{K}D$
(with $I=0$) and $\bar{K}^{*}D^{*}$ (with $I=0$, $J=0,\ 1$) are stably bound,
which are consistent with the results of Eq. (15). Our results with
$q_{max}=875$ MeV for the $\bar{K}^{*}D$ (with $I=0$), $\bar{K}^{*}D^{*}$
(with $I=0$, $J=2$), and the whole $S=1,\ C=1$ sector become abnormal, where
the poles of the bound states in the first Riemann sheet for the single
channel interaction have a complex width. The first point of view that we
assume is that the poles are already close to the energy region of the
unphysical left-hand cut as discussed above and shown in Fig. 4, where we try
to make further analysis for the reason discussed next. The results taking
$q_{max}=756$ MeV seem a bit better, where the $\bar{K}^{*}D$ (with $I=0$)
system becomes bound stably in addition to the ones $KD$ and $KD^{*}$, as the
results obtained with Eq. (15), and the channels $\bar{K}^{*}D^{*}$ (with
$I=0$, $J=2$), $K^{*}D$ (with $I=0$) and $K^{*}D^{*}$ (with $I=0$ and
different $J$) are affected by the unphysical left-hand cut with abnormal
poles. Indeed, the results with Eq. (15) are always bound due to the left-hand
cut being totally removed.

Table 3: Poles in the first Riemann sheet (1RS) (Unit: MeV). Our results of Eq. (5) are compared with the results of Eq. (15). Systems | Ours | Oset’s | Ours-2 | Oset’s-2 | Exp.
---|---|---|---|---|---
$I=0\,(\frac{1}{2})$ | ($q_{max}=875$) | ($q_{max}=1100$) | ($q_{max}=756$) | ($q_{max}=843$) |
$\bar{K}D$ (2362.89) | 2354.97 | 2346.74 | 2362.49 | 2362.82 |
$\bar{K}D^{*}$ (2504.20) | 2284.28 (?) | 2502.68 | – | – |
$\bar{K}^{*}D$ (2760.86) | $2710.32+i37.23$ | 2689.01 | 2748.04 | 2741.24 |
$\bar{K}^{*}D^{*}$, $J=0$ (2902.17) | 2865.24 | 2866.38 | 2898.16 | 2897.02 | $X_{0}(2866)$
$\bar{K}^{*}D^{*}$, $J=1$ (2902.17) | 2895.65 | 2859.98 | 2902.10 | 2895.50 |
$\bar{K}^{*}D^{*}$, $J=2$ (2902.17) | $2821.08-i103.27$ | 2745.76 | $2829.93-i56.36$ | 2837.49 |
$KD$ (2362.89) | $2223.28\pm i22.32$ | 2262.15 | 2317.34 | 2317.14 | $D^{*}_{s0}(2317)$
$KD^{*}$ (2504.20) | $2364.55-i38.24$ | 2400.22 | 2454.71 | 2456.29 | $D_{s1}(2460)$
$K^{*}D$ (2760.86) | $2673.25-i183.79$ | 2504.39 | $2682.17\pm i131.15$ | 2620.27 |
$K^{*}D^{*}$, $J=0$ (2902.17) | $2851.45-i153.33$ | 2713.15 | $2848.10+i105.99$ | 2804.40 |
$K^{*}D^{*}$, $J=1$ (2902.17) | $2866.56+i132.57$ | 2748.72 | $2857.90\pm i88.87$ | 2826.76 |
$K^{*}D^{*}$, $J=2$ (2902.17) | $2795.35-i199.80$ | 2600.70 | $2811.04+i145.12$ | 2732.68 | $D_{s2}^{*}(2573)$

Therefore, trying to remove the left-hand cut, we take the approximation form
of Eq. (19) for the calculation of the potentials, where the results are given
in Table 4 compared with the results of Eq. (15) too. In this approximation,
it takes the new tuned cutoffs $q_{max}=883$ MeV and $q_{max}=760$ MeV, for
generating the $X_{0}(2866)$ and $D^{*}_{s0}(2317)$ states, respectively. For
the case of $q_{max}=883$ MeV, the systems that are obtained abnormal results
with Eq. (5) before, except for the ones $KD$ and $KD^{*}$ (with $I=0$) become
stable now as expected, similar results as before are still obtained. When we
check the potentials of Eq. (19) for these systems in detail, we find that the
inverse of the potentials with minus slope, see the short-dash (blue) line of
Fig. 3, do not cross with the the real part of the loop functions. Thus, there
is no bound pole, which means that the poles we find now with widths in the
first Riemann sheet are in fact virtual states. For the other case of
$q_{max}=760$ MeV, which is just a bit different from the former one
$q_{max}=756$ MeV, the results are not much different than before. Now we know
that these abnormal results, except for the ones $KD$ and $KD^{*}$ (with
$I=0$), are in fact the poles of virtual states, which means that these
systems could be also bound when we tune the cutoff $q_{max}$ and be searched
for the possible bound state in future experiments.

Table 4: Poles in the first Riemann sheet (1RS) (Unit: MeV). Our results of Eq. (19) are compared with the results of Eq. (15). Systems | Ours-new | Oset’s | Ours-new2 | Oset’s-2 | Exp.
---|---|---|---|---|---
| ($q_{max}=883$) | ($q_{max}=1100$) | ($q_{max}=760$) | ($q_{max}=843$) |
$\bar{K}D$ (2362.89) | 2354.10 | 2346.74 | 2362.40 | 2362.82 |
$\bar{K}D^{*}$ (2504.20) | – | 2502.68 | – | – |
$\bar{K}^{*}D$ (2760.86) | $2694.17+i42.94$ | 2689.01 | 2747.40 | 2741.24 |
$\bar{K}^{*}D^{*}$, $J=0$ (2902.17) | 2866.14 | 2866.38 | 2897.82 | 2897.02 | $X_{0}(2866)$
$\bar{K}^{*}D^{*}$, $J=1$ (2902.17) | 2894.66 | 2859.98 | 2902.06 | 2895.50 |
$\bar{K}^{*}D^{*}$, $J=2$ (2902.17) | $2813.41-i135.96$ | 2745.76 | $2807.55\pm i66.51$ | 2837.49 |
$KD$ (2362.89) | 2258.22 | 2262.15 | 2317.36 | 2317.14 | $D^{*}_{s0}(2317)$
$KD^{*}$ (2504.20) | 2391.78 | 2400.22 | 24555.06 | 2456.29 | $D_{s1}(2460)$
$K^{*}D$ (2760.86) | $2698.42-i237.63$ | 2504.39 | $2684.60\pm i170.50$ | 2620.27 |
$K^{*}D^{*}$, $J=0$ (2902.17) | $2874.93+i192.31$ | 2713.15 | $2849.62+i135.30$ | 2804.40 |
$K^{*}D^{*}$, $J=1$ (2902.17) | $2866.91+i164.04$ | 2748.72 | $2857.40\pm i112.55$ | 2826.76 |
$K^{*}D^{*}$, $J=2$ (2902.17) | $2824.87+i259.85$ | 2600.70 | $2817.27+i189.09$ | 2732.68 | $D_{s2}^{*}(2573)$

##  IV Discussions

Before closing with conclusions, we make some discussions beyond the current
formalism. In the present framework, we only find that the bound states would
appear in the $K^{(*)}D^{(*)}$ and $\bar{K}^{(*)}D^{(*)}$ channel with isospin
$I=0$ sector. However, there are two limitations, no $\pi$ exchange and no
coupled channels. Then various interactions will be missing, for example, for
the $K^{(*)}K^{(*)}$ channel, $KK^{*}$ could exchange $\pi$ in the $u$-channel
interaction, while the $KD^{*}\to K^{*}D$ process could happen through $\pi$
exchange, which will lead to coupled channels effects between the $KD^{*}$ and
$K^{*}D$ channels. From the study of the $Z_{c}(3900)$, we found that the
interaction from the vector meson exchange is not enough to bind two hadron
systems. To generate the $Z_{c}(3900)$ state from the $D\bar{D}^{*}$
interaction, except for the vector meson exchange considered, the pseudoscalar
and two-pion exchanges were also taken into account in Ref. Aceti:2014uea .
Here we will make more discussion on the $\bar{D}^{(*)}D_{s}^{(*)}$ system.

Note that, as discussed in the introduction, in the experiments the states
$Z_{cs}(3985)$ and $Z_{cs}(4000)$ are found near the threshold of the
$\bar{D}_{s}^{*}D_{s}+\bar{D}_{s}D_{s}^{*}$ channel. But, from the interaction
of vector meson exchange, where only the heavy $J/\psi$ is allowed, we can not
reproduce any bound state in the similar $\bar{D}^{(*)}D_{s}^{(*)}$ systems.
Indeed, using the same mechanism in Ref. Ikeno:2020mra , they did not find the
pole of bound state for the $Z_{cs}(3985)$ and concluded it as a virtual state
from a strong cusp structure in the threshold of $\bar{D}_{s}D^{*}$, which was
a threshold effect. Furthermore, using the heavy quark spin symmetry formalism
in Ref. Yang:2020nrt , the $Z_{cs}(3985)$ and other predicted $Z_{cs}^{*}$
states, as the strange molecular partners of the $Z_{c}(3900)$ and
$Z_{c}(4020)$ states, could be either a virtual state or a resonance. Within
the one boson exchange model, Ref. Yan:2021tcp  reproduced the $Z_{cs}(3985)$
by the interaction potentials from the axial, scalar, isovector, and vector
meson exchanges, where analogously the one boson exchange formalism was also
applied in Ref. Ding:2021igr  to explain the $Z_{cs}(3985)$ state with the
light and heavy meson exchanges. For this reason, more investigations should
be done in the future. One should keep in mind that in the present work we
only consider the single channel interaction with the interaction potential
from the vector meson exchange in $t$\- and $u$-channels.

For the further investigation, we will take into account the coupled channels
interactions for these systems with attractive potentials and the other
interaction dynamics, such as the pseudoscalar meson exchanges.

##  V Conclusions

In the present work, under the local hidden gauge formalism, we derive the
interaction potential with the vector meson exchange mechanism of the $t$\-
and $u$-channels, and study the single channel interaction of the systems with
open strangeness(es) from the light sector to the heavy one (without beauty
quark). We face the left-hand cut singularity when we do the $s$-wave
projection to the $t$\- and $u$-channels potentials, which looks like a
unavoided problem for us when the singularities happen to appear at the energy
region that we are concerned with. Thus, we make a simplified approximation to
this left-hand cut problem, where some stable results are obtained. In our
results, we successfully generate the states $X_{0}(2866)$, $D^{*}_{s0}(2317)$
and $D_{s1}(2460)$ in the interaction $\bar{K}^{*}D^{*}$, $KD$ and $KD^{*}$,
respectively. Furthermore, we also find some other bound systems in different
sectors, which will be further investigated in our future work. At last, from
the first results with the single channel interaction with the vector meson
exchange, some loose bound systems are necessary to include pseudoscalar
exchange and coupled channels effects in further study.

## Acknowledgments

We acknowledge Prof. Eulogio Oset for useful discussions and careful reading
of the manuscript, and Drs. Mao-Jun Yan and Ming-Zhu Liu for helpful comments.
One of us, C.W.X., acknowledges the hospitality of the University of Chinese
Academy of Sciences where part of this work was done. This work is supported
by the Natural Science Foundation (NSF) of Changsha under Grant No. kq2208257,
the NSF of Hunan province under Grant No. 2023JJ30647, the NSF of Guangxi
province under Grant No. 2023JJA110076, and the National NSF of China under
Grant No. 12365019, 12175239, and 12221005, and also by National Key Research
and Development Program of China under Contracts 2020YFA0406400, and also by
Chinese Academy of Sciences under Grant No. YSBR-101, and also by Xiaomi
Foundation / Xiaomi Young Talents Program.

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