# Investigation of the possible $D\bar{D}^{*}$/$B\bar{B}^{*}$ and
$DD^{*}$/$\bar{B}\bar{B}^{*}$ molecule states

Meng-Jie Zhao  Physics Department, Ningbo University, Zhejiang 315211, China
Zhen-Yang Wang 111Corresponding author, e-mail<EMAIL_ADDRESS>Physics Department, Ningbo University, Zhejiang 315211, China Chao Wang
222e-mail<EMAIL_ADDRESS>Center for Ecological and Environmental
Sciences, Key Laboratory for Space Bioscience & Biotechnology, Northwestern
Polytechnical University, Xi’an 710072, China Xin-Heng Guo 333Corresponding
author, e-mail<EMAIL_ADDRESS>College of Nuclear Science and Technology,
Beijing Normal University, Beijing 100875, China

###### Abstract

In this work, we systematically study the $D\bar{D}^{*}$/$B\bar{B}^{*}$ and
$DD^{*}$/$\bar{B}\bar{B}^{*}$ systems with the Bethe-Salpeter equation in the
ladder and instantaneous approximations for the kernel. By solving the Bethe-
Salpeter equation numerically with the kernel containing the direct and
crossed one particle-exchange diagrams and introducing three different form
factors (monopole, dipole, and exponential form factors) at the vertices, we
find only the isoscalar $D\bar{D}^{*}$/$B\bar{B}^{*}$ and
$DD^{*}$/$\bar{B}\bar{B}^{*}$ systems can exist as bound states. This indicate
that the $X(3872)$ and $T^{+}_{cc}$ could be accommodated as
$I^{G}(J^{PC})=0^{+}(1^{++})$ $D\bar{D}^{*}$ and $(I)J^{P}=(0)1^{+}$ $DD^{*}$
molecular states while the molecular state explanations for $Z_{b}(10610)$ and
$Z_{c}(3900)$ are excluded.

###### pacs:

***

##  I introduction

Since the discovery of $X(3872)$ in 2003 Belle:2003nnu , more than 20 exotic
states candidates containing $c\bar{c}$ and $b\bar{b}$ quarks have been found
and studied by the LHCb, ATLAS, CMS, BES$\mathrm{III}$, Belle, $BABAR$, CDF
and D0 experiments pdg2020 ; review:paper . The structures of these exotic
states are more complex than the standard $q\bar{q}$ mesons. Recently, the
LHCb Collaboration reported the first doubly open charmed tetraquark state
$T^{+}_{cc}$ in proton-proton collisions with a signal significance over
10$\sigma$ LHCb:2021vvq ; LHCb:2021auc , with its mass and width being

$\begin{split}\delta_{m}&=m_{T^{+}_{cc}}-(m_{D^{\ast+}}+m_{D^{0}})=-273\pm
61\pm 5^{+11}_{-14}\ \mathrm{keV},\\\ \Gamma&=410\pm 165\pm 43^{+18}_{-38}\
\mathrm{keV},\end{split}$ (1)

respectively. This exotic state with a mass of about 3875 MeV manifests itself
as a narrow peak in the mass spectrum of $D^{0}D^{0}\pi^{+}$ mesons just below
the $D^{*+}D^{0}$ mass threshold, and is consistent with the ground isoscalar
$cc\bar{u}\bar{d}$ state with the spin-parity quantum numbers $J^{P}=1^{+}$.
Although there have been many studies on the exotic states, the puzzle about
the nature of these states still remains unsolved so far.

Up to now, three exotic states ($X(3872)$, $Z_{c}(3900)$, and $T^{+}_{cc}$)
with their masses close to the threshold of $DD^{\ast}$ have been found
experimentally. To describe these exotic states a variety of phenomenological
models have been proposed, including the chiral effective field theory
Xu:2017tsr ; Ohkoda:2012hv ; Li:2012cs ; Li:2012ss ; Ren:2021dsi ; Sun:2012zzd
; Feijoo:2021ppq ; Chen:2021vhg ; Liu:2019stu ; Sun:2011uh , the Bethe-
Salpeter approach He:2014nya ; Dong:2021bvy ; He:2015mja ; Ding:2020dio ;
Wallbott:2019dng , the constituent quark model Zhu:2019iwm ; Noh:2021lqs ;
Ortega:2021ufy ; Tan:2020ldi ; Luo:2017eub ; Yang:2017prf , QCD sum rules
Navarra:2007yw ; Xin:2021wcr ; Chen:2017dpy ; Azizi:2021aib ; Tang:2019nwv ,
and the relativized quark model Lu:2020rog ; Ebert:2007rn ; Wang:2018pwi ;
Anwar:2018sol , etc. In these models some physical pictures have also been
developed to understand the known exotic states, such as hadronic molecules
and tetraquark state. Since the masses of these exotic states are close to the
threshold of the two $S$-wave lowest lying standard mesons ($D$ and
$D^{\ast}$), one would naturally identify them as molecular states of standard
mesons. Therefore, it is very interesting to investigate whether this is
plausible for these exotic states, which will be very helpful to reveal the
structures of these three exotic states.

In this paper we will focus on the $D\bar{D}^{*}$ and $DD^{*}$ molecular
states and their $b$ partners. Our purpose is to investigate whether the bound
states of the $D\bar{D}^{*}$ and $DD^{*}$ systems and their $b$ partners via
the interact on through exchanging a scalar meson ($\sigma$), vector mesons
($\rho$ and $\omega$), and pseudoscalar mesons ($\pi$ and $\eta$) can exist.
As the relativistic equation describing the bound state of two particles, the
Bethe-Salpeter (BS) equation is an effective method to deal with
nonperturbative QCD effects and has been applied to many theoretical studies
concerning standard heavy hadrons Jin:1992mw ; Guo:1999ss ; Guo:1998ef  and
exotic states Guo:2007mm ; Xie:2010zza ; Ke:2013yka ; Wang:2020lua ;
Ke:2021iyh . The kernel of the BS equation can be derived from the relevant
effective Lagrangian, based on which numerical solutions for the BS wave
functions can be obtained in the covariant instantaneous approximation and the
ladder approximation, and these solutions can be used to judge whether these
bound states exist.

The remainder of this paper is organized as follows. In Sec. II we will
discuss the $D\bar{D}^{\ast}$ and $DD^{\ast}$ system hadronic wave function
and derive the BS equation for the vector and pseudoscalar mesons system. We
will also discuss the kernel derived from the relevant effective Lagrangian.
In Sec. III, we will present numerical solutions for $D\bar{D}^{*}$ and
$DD^{*}$ bound states and their $b$ partners with three different form
factors. Finally, Sec. IV contains a summary.

##  II the bethe-salpeter formalism for $D\bar{D}^{*}$ and $DD^{\ast}$ systems

The $D\bar{D}^{\ast}$ and $DD^{\ast}$ systems have isospin 0 or 1. The flavor
wave functions of the $D\bar{D}^{*}$ systems are Ding:2020dio ; Sun:2011uh ;
Zhao:2014gqa

$\begin{split}|X^{0}_{D\bar{D}^{\ast}}\rangle_{I=0}&=\frac{1}{2}\left[(|D^{\ast+}D^{-}\rangle+|D^{\ast
0}\bar{D}^{0}\rangle)+c(|D^{+}D^{\ast-}\rangle+|D^{0}\bar{D}^{\ast
0}\rangle)\right],\\\
|X^{0}_{D\bar{D}^{\ast}}\rangle_{I=1}&=\frac{1}{2}\left[(|D^{\ast}D^{-}\rangle-|D^{\ast}\bar{D}^{0}\rangle)+c(|D^{+}D^{\ast-}\rangle-|D^{0}\bar{D}^{\ast
0}\rangle)\right],\\\
|X^{+}_{D\bar{D}^{\ast}}\rangle_{I=1}&=\frac{1}{\sqrt{2}}\left(|D^{\ast+}\bar{D}^{0}\rangle+c|D^{+}\bar{D}^{\ast
0}\rangle\right),\\\
|X^{-}_{D\bar{D}^{\ast}}\rangle_{I=1}&=\frac{1}{\sqrt{2}}\left(|D^{\ast-}\bar{D}^{0}\rangle+c|D^{-}\bar{D}^{\ast
0}\rangle\right),\\\ \end{split}$ (2)

where $c=\pm 1$ correspond to $C$ parity $C=\mp$ respectively. The wave
functions for the $DD^{\ast}$ systems are

$\begin{split}|T_{cc}^{+}\rangle_{I=0}&=\frac{1}{\sqrt{2}}\left(|D^{+}D^{*0}\rangle-|D^{0}D^{*+}\rangle\right),\\\
|T_{cc}^{+}\rangle_{I=1}&=\frac{1}{\sqrt{2}}\left(|D^{+}D^{*0}\rangle+|D^{0}D^{*+}\rangle\right),\\\
|T_{cc}^{++}\rangle_{I=1}&=|D^{+}D^{*+}\rangle,\\\
|T_{cc}^{0}\rangle_{I=1}&=|D^{0}D^{*0}\rangle.\\\ \end{split}$ (3)

The wave functions of the hidden bottom and doubly bottom states can be
obtained analogously.

Since the bound states are composed of a vector meson ($D^{\ast}$) and a
pseudoscalar meson ($D$), we can define the BS wave function of the bound
state by

$\chi_{P}^{\alpha}(x_{1},x_{2},P)=\langle
0|T{D}^{*\alpha}(x_{1}){D}(x_{2})|P\rangle=e^{-iPX}\chi_{P}^{\alpha}(x),$ (4)

where $P=Mv$ is the total momentum of the bound state, and $v$ is its
velocity. ${D}^{*\alpha}(x_{1})$ and ${D}(x_{2})$ are the field operators of
the vector meson $D^{*}$ and the pseudoscalar meson $D$ at space coordinates
$x_{1}$ and $x_{2}$, respectively. Let us define $\lambda_{1}\equiv
m_{1}/(m_{1}+m_{2})$ and $\lambda_{2}\equiv m_{2}/(m_{1}+m_{2})$ with $m_{1}$
and $m_{2}$ being the masses of $D^{\ast}$ and $D$, respectively, and let $p$
be the relative momentum of the two constituents. The BS wave function in
momentum space is defined as

$\chi_{P}^{\alpha}(x_{1},x_{2},P)=e^{-iP\cdot
X}\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip\cdot x}\chi_{P}^{\alpha}(p),$ (5)

where $X\equiv\lambda_{1}x_{1}+\lambda_{2}x_{2}$ is the coordinate of the
center of mass and $x\equiv x_{1}-x_{2}$. The momentum of the vector meson is
$p_{1}=\lambda_{1}P+p$ and that of the pseudoscalar meson is
$p_{2}=-\lambda_{2}P+p$.

The BS equation for the bound state consisting of a vector and a pseudoscalar
mesons can be written in the following form:

$\chi_{P}^{\alpha}(p)=S^{\alpha\lambda}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{\lambda\tau}(P,p,q)\chi_{P}^{\tau}(q)S(p_{2}),$
(6)

where $S^{\alpha\lambda}(p_{1})$ and $S(p_{2})$ are the propagators of $D^{*}$
and $D$, respectively, and $K_{\lambda\tau}(P,p,q)$ is the four-point
truncated irreducible kernel. The $K_{\lambda\tau}(P,p,q)$ can be derived from
four-point Green function as following:

$\begin{split}G^{\alpha\beta}(x_{1},x_{2};y_{2},y_{1})=&G^{\alpha\beta}_{(0)}(x_{1},x_{2};y_{2},y_{1})\\\
&+\int
d^{4}u_{1}d^{4}u_{2}d^{4}v_{1}d^{4}v_{2}G^{\alpha\lambda}_{(0)}(x_{1},x_{2};u_{2},u_{1})\bar{K}_{\lambda\tau}(u_{1},u_{2};v_{2},v_{1})G^{\tau\beta}(v_{1},v_{2};y_{2},y_{1}),\end{split}$
(7)

where $G^{\alpha\beta}_{(0)}$ is related to the forward scattering
disconnected four-point amplitude,

$G^{\alpha\beta}_{(0)}(x_{1},x_{2};y_{2},y_{1})=S^{\alpha\beta}(x_{1},y_{1})S(x_{2},y_{2}),$
(8)

$S^{\alpha\beta}(x_{1},y_{1})$ and $S(x_{2},y_{2})$ are the propagators of
constituent particles $D^{\ast}$ and $D$ in coordinate Space, respectively.

For convenience, we define $p_{l}(=p\cdot v)$ and
$p_{t}^{\mu}(=p^{\mu}-p_{l}v^{\mu})$ to be the longitudinal and transverse
projections of the relative momentum ($p$) along the bound state momentum
($P$). Then the $D^{*}$ propagator has the form

$S^{\alpha\beta}(p_{1})=\frac{-i\left[g^{\alpha\beta}-(\lambda_{1}Mv+p_{l}v+p_{t})^{\alpha}(\lambda_{1}Mv+p_{l}v+p_{t})^{\beta}/m_{1}^{2}\right]}{(\lambda_{1}M+p_{l})^{2}-\omega_{1}^{2}+i\epsilon},$
(9)

and the propagator of $D$ meson has the form

$S(p_{2})=\frac{i}{(\lambda_{2}M-p_{l})^{2}-\omega_{2}^{2}+i\epsilon},$ (10)

where $\omega_{1(2)}=\sqrt{m_{1(2)}^{2}-p_{t}^{2}}$.

In general, for $D\bar{D}^{*}$ and $DD^{*}$ systems, $\chi^{\alpha}_{P}(p)$
can be written as

$\begin{split}\chi^{\alpha}_{P}(p)=&f_{1}(p)\varepsilon^{\alpha\beta\mu\nu}g_{\mu\nu}\epsilon_{\beta}(P)+f_{2}(p)\varepsilon^{\alpha\beta\mu\nu}P_{\mu}P_{\nu}\epsilon_{\beta}(P)+f_{3}(p)\varepsilon^{\alpha\beta\mu\nu}p_{\mu}P_{\nu}\epsilon_{\beta}(P)+f_{4}(p)\varepsilon^{\alpha\beta\mu\nu}p_{\mu}p_{\nu}\epsilon_{\beta}(P),\end{split}$
(11)

where $\epsilon_{\beta}(P)$ represents the polarization vector of the bound
state and $f_{i}$($i=1,2,3,4$) are Lorentz-scalar functions. With the
constraints imposed by parity and Lorentz transformations, it is easily to
prove that $\chi^{\alpha}_{P}(p)$ can be simplified as

$\chi_{P}^{\alpha}(p)=f(p)\varepsilon^{\alpha\beta\mu\nu}p_{\mu}P_{\nu}\epsilon_{\beta},$
(12)

where the function $f(p)$ contains all the dynamics and is a Lorentz-scalar
function of $p$.

In order to obtain the interaction kernel $K_{\lambda\tau}(P,p,q)$ of
$D\bar{D}^{\ast}$ and $DD^{\ast}$ systems, the Lagrangian for heavy mesons
interacting with light mesons are needed, which can be described from the
heavy meson chiral perturbation theory as the following Ding:2008gr ;
Cheng:2004ru :

$\begin{split}\mathcal{L}_{{{DD}}\sigma}=&-2g_{\sigma}m_{D}{{D}}_{a}{{D}}_{a}^{\dagger}\sigma-2g_{\sigma}m_{D}\bar{D}_{a}\bar{D}_{a}^{\dagger}\sigma,\\\
{\mathcal{L}}_{{{D}}^{*}{{D}}^{*}\sigma}=&2g_{\sigma}m_{{D}^{*}}{\mathcal{D}}_{a}^{*\alpha}{{D}}_{\alpha
a}^{*\dagger}\sigma+2g_{\sigma}m_{{D}^{*}}{\bar{{D}}}_{a}^{*\alpha}{\bar{{D}}}_{\alpha
a}^{*\dagger}\sigma,\\\
\mathcal{L}_{{D}{D}^{*}\mathbb{P}}=&g_{DD^{*}\mathbb{P}}({{D}}_{b}{{D}}_{\alpha
a}^{*\dagger}+{{D}}_{\alpha
b}^{*}{{D}}_{a}^{\dagger})\partial^{\alpha}\mathbb{P}_{ba}+g_{\bar{D}\bar{D}^{*}\mathbb{P}}(\bar{{D}}_{\alpha
a}^{*\dagger}\bar{{D}}_{b}+\bar{{D}}_{a}^{\dagger}\bar{{D}}_{\alpha
b}^{*})\partial^{\alpha}\mathbb{P}_{ab},\\\
\mathcal{L}_{{{D}}{{D}}^{*}\mathbb{V}}=&-2ig_{{D}{D}^{*}\mathbb{V}}\varepsilon_{\mu\nu\alpha\beta}(\partial^{\mu}\mathbb{V}^{\nu})_{ba}[({{D}}_{a}^{\dagger}\partial^{\alpha}{{D}}_{b}^{*\beta}-{{D}}_{b}^{*\beta}\partial^{\alpha}{{D}}_{a}^{\dagger})+({{D}}_{a}^{*\beta\dagger}\partial^{\alpha}{\mathcal{D}}_{b}-{\mathcal{D}}_{b}\partial^{\alpha}{{D}}_{a}^{*\beta\dagger})]\\\
&-2ig_{\bar{D}\bar{D}^{*}\mathbb{V}}\varepsilon_{\mu\nu\alpha\beta}(\partial^{\mu}\mathbb{V}^{\nu})_{ab}[(\bar{{D}}_{a}^{\dagger}\partial^{\alpha}\bar{{D}}_{b}^{*\beta}-\bar{{D}}_{b}^{*\beta}\partial^{\alpha}\bar{{D}}_{a}^{\dagger})+(\bar{{D}}_{a}^{*\beta\dagger}\partial^{\alpha}\bar{{D}}_{b}-\bar{{D}}_{b}\partial^{\alpha}\bar{{D}}_{a}^{*\beta\dagger})],\\\
\mathcal{L}_{{{DD}}\mathbb{V}}=&ig_{{DD}\mathbb{V}}({{D}}_{b}\partial_{\alpha}{{D}}_{a}^{\dagger}-{{D}}_{a}^{\dagger}\partial_{\alpha}{{D}}_{b})(\mathbb{V}^{\alpha})_{ba}+ig_{\bar{D}\bar{D}\mathbb{V}}(\bar{{D}}_{b}\partial_{\alpha}\bar{{D}}_{a}^{\dagger}-\bar{{D}}_{a}^{\dagger}\partial_{\alpha}\bar{{D}}_{b})(\mathbb{V}^{\alpha})_{ab},\\\
\mathcal{L}_{{D}^{*}{D}^{*}\mathbb{V}}=&ig_{D^{*}D^{*}\mathbb{V}}({{D}}_{\beta,b}^{*}\partial^{\alpha}{{D}}_{a}^{*\beta\dagger}-{{D}}_{a}^{*\beta\dagger}\partial^{\alpha}{{D}}_{\beta,b}^{*})(\mathbb{V}_{\alpha})_{ba}-ig^{\prime}_{D^{*}D^{*}\mathbb{V}}{D}_{\alpha,a}^{*\dagger}{{D}}_{\beta,b}^{*}(\partial^{\alpha}\mathbb{V}^{\beta}-\partial^{\beta}\mathbb{V}^{\alpha})_{ba}\\\
&+ig_{\bar{D}^{*}\bar{D}^{*}\mathbb{V}}(\bar{{D}}_{\beta,b}^{*}\partial^{\alpha}\bar{{D}}_{a}^{*\beta\dagger}-\bar{{D}}_{a}^{*\beta\dagger}\partial^{\alpha}\bar{{D}}_{\beta,b}^{*})(\mathbb{V}_{\alpha})_{ab}-ig^{\prime}_{\bar{D}^{*}\bar{D}^{*}\mathbb{V}}\bar{{D}}_{\alpha,a}^{*\dagger}\bar{{D}}_{\beta,b}^{*}(\partial^{\alpha}\mathbb{V}^{\beta}-\partial^{\beta}\mathbb{V}^{\alpha})_{ab},\end{split}$
(13)

where $D$, $\bar{D}$, $D^{\ast}$, and $\bar{D}^{\ast}$ are heavy flavored
meson fields, with $D=(D^{0},D^{+},D_{s}^{+})$,
$\bar{D}=(\bar{D}^{0},D^{-},D_{s}^{-})$, $D^{\ast}=(D^{\ast
0},D^{\ast+},D_{s}^{\ast+})$, and $\bar{D}^{\ast}=(\bar{D}^{\ast
0},\bar{D}^{\ast-},\bar{D}_{s}^{\ast-})$, $a$ and $b$ denote the light quark
flavour indices, the octet pseudoscalar $\mathbb{P}$ and the nonet vector
$\mathbb{V}$ meson matrices are defined as

$\mathbb{P}=\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2\eta}{\sqrt{6}}\\\ \end{array}\right),$ (14)

and

$\mathbb{V}=\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\\\ \end{array}\right),$ (15)

respectively, and the coupling constants are given as

$\begin{split}g_{{D}{D}^{*}\mathbb{P}}&=-g_{\bar{D}\bar{D}^{*}\mathbb{P}}=-\frac{2g}{f_{\pi}}\sqrt{m_{D}m_{D^{*}}},\quad
g_{{D}{D}^{*}\mathbb{V}}=g_{\bar{D}\bar{D}^{*}\mathbb{V}}=\frac{\lambda
g_{\mathbb{V}}}{\sqrt{2}},\\\
g_{{D}{D}\mathbb{V}}&=-g_{\bar{D}\bar{D}\mathbb{V}}=-g_{{D}^{*}{D}^{*}\mathbb{V}}=g_{\bar{D}^{*}\bar{D}^{*}\mathbb{V}}=\frac{\beta
g_{\mathbb{V}}}{\sqrt{2}},\\\
g^{\prime}_{{D}^{*}{D}^{*}\mathbb{V}}&=-g^{\prime}_{\bar{D}^{*}\bar{D}^{*}\mathbb{V}}=-\sqrt{2}\lambda
g_{V}m_{D^{*}}\\\ g_{\mathbb{V}}&=\frac{m_{\rho}}{f_{\pi}},\quad
g_{\sigma}=\frac{g_{\pi}}{2\sqrt{6}},\quad g=0.59,\quad\beta=0.9,\\\
\lambda&=0.56\ {\rm GeV}^{-1},\quad f_{\pi}=132\ {\rm MeV},\quad
g_{\pi}=3.73.\end{split}$ (16)

In the so-called ladder approximation, the kernel $K_{\lambda\tau}(P,p,q)$ is
replaced by its lowest order form. Then with the Lagrangian for heavy mesons
interacting with light mesons, for the kernel of the bound states induced by
$\rho$, $\omega$, $\sigma$, $\pi$, and $\eta$ exchanges, we have

$\begin{split}\bar{K}^{\lambda\tau,\sigma}_{DD^{*}-direct}(p_{1},p_{2};q_{2},q_{1})=&(2\pi)^{4}\delta^{4}(q_{1}+q_{2}-p_{1}-p_{2})4c_{I}^{d}g_{\sigma}^{2}m_{D^{*}}m_{D}\Delta(k,m_{\sigma})g^{\lambda\tau},\\\
\bar{K}^{\lambda\tau,P}_{DD^{*}-crossed}(p_{1},p_{2};q_{2},q_{1})=&-(2\pi)^{4}\delta^{4}(q_{1}+q_{2}-p_{1}-p_{2})c_{I}^{c}g_{DD^{*}\mathbb{P}}^{2}k^{\lambda}k^{\tau}\Delta(k,m_{P}),\\\
\bar{K}^{\lambda\tau,V}_{DD^{*}-direct}(p_{1},p_{2};q_{2},q_{1})=&-(2\pi)^{4}\delta^{4}(q_{1}+q_{2}-p_{1}-p_{2})c_{I}^{d}g_{DDV}(p_{2}+q_{2})_{\nu}\Delta^{\mu\nu}(k,m_{V})\\\
&\times\left[g_{D^{*}D^{*}\mathbb{V}}(p_{1}+q_{1})_{\mu}g_{\lambda\tau}+g^{\prime}_{D^{*}D^{*}\mathbb{V}}(k_{\lambda}g_{\tau\mu}-k_{\tau}g_{\lambda\mu})\right],\\\
\bar{K}^{\lambda\tau,V}_{DD^{*}-crossed}(p_{1},p_{2};q_{2},q_{1})=&-(2\pi)^{4}\delta^{4}(q_{1}+q_{2}-p_{1}-p_{2})4c_{I}^{c}f_{{D}{D}^{*}\mathbb{V}}^{2}\epsilon^{\mu\beta\sigma\lambda}\epsilon^{\nu\rho\gamma\tau}\\\
&\times
k_{\mu}k_{\nu}(p_{1}+q_{2})_{\sigma}(q_{1}+p_{2})_{\gamma}\Delta_{\beta\rho}(k,m_{V}),\\\
\end{split}$ (17)

for the $DD^{*}$ bound state, and

$\begin{split}&\bar{K}^{\lambda\tau,\sigma}_{D\bar{D}^{*}-direct}(p_{1},p_{2};q_{2},q_{1})=\bar{K}^{\lambda\tau,\sigma}_{DD^{*}-direct}(p_{1},p_{2};q_{2},q_{1}),\\\
&\bar{K}^{\lambda\tau,P}_{D\bar{D}^{*}-crossed}(p_{1},p_{2};q_{2},q_{1})=-\bar{K}^{\lambda\tau,P}_{DD^{*}-crossed}(p_{1},p_{2};q_{2},q_{1}),\\\
&\bar{K}^{\lambda\tau,V}_{D\bar{D}^{*}-direct}(p_{1},p_{2};q_{2},q_{1})=\bar{K}^{\lambda\tau,V}_{DD^{*}-direct}(p_{1},p_{2};q_{2},q_{1}),\\\
&\bar{K}^{\lambda\tau,V}_{D\bar{D}^{*}-crossed}(p_{1},p_{2};q_{2},q_{1})=\bar{K}^{\lambda\tau,V}_{DD^{*}-crossed}(p_{1},p_{2};q_{2},q_{1}),\end{split}$
(18)

for the $D\bar{D}^{*}$ bound state, where $m_{\sigma}$, $m_{P}$ and $m_{V}$
represent the masses of the exchanged $\sigma$, the pseudoscalar light meson
and the vector light meson, respectively. $c_{I}^{d}$ and $c_{I}^{c}$ are the
isospin coefficients for the direct and crossed diagrams, and the values for
different exchange mesons are listed in Table 1. $\Delta^{\mu\nu}$ represents
the propagator for a vector meson and $\Delta$ represents the pseudoscalar or
scalar meson propagator, and they have the following forms:

$\begin{split}\Delta^{\mu\nu}(k,m_{V})&=\frac{-i}{k^{2}-m_{V}^{2}}\left(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m_{V}^{2}}\right),\\\
\Delta(k,m_{\sigma(P)})&=\frac{i}{k^{2}-m_{\sigma(P)}^{2}}.\\\ \end{split}$
(19)

Table 1: The isospin factors $c_{I}^{d}$ and $c_{I}^{c}$ for direct and crossed feynman diagrams with $I=0$ and $I=1$ . |  | $c_{I}^{d}$ | $c_{I}^{c}$
---|---|---|---
|  |  $\rho$ |  $\omega$ |  $\pi$ |  $\eta$ |  $\sigma$ |  $\rho$ |  $\omega$ |  $\pi$ |  $\eta$ |  $\sigma$
$D\bar{D}^{\ast}$ |  $I=0$ |  3/2 |  1/2 |  – |  – |  1 |  3c/2 |  c/2 |  3c/2 |  c/6 |  –
$I=1$ |  -1/2 |  1/2 |  – |  – |  1 |  -c/2 |  c/2 |  -c/2 |  c/6 |  –
$DD^{\ast}$ |  $I=0$ |  -3/2 |  1/2 |  – |  – |  1 |  3/2 |  -1/2 |  3/2 |  -1/6 |  –
$I=1$ |  1/2 |  1/2 |  – |  – |  1 |  1/2 |  1/2 |  1/2 |  1/6 |  –

Considering the size effects of constituent particles, we introduce the
monopole, exponential and dipole form factors at each vertex. The form factors
are respectively defined as

$\begin{split}F_{M}(k^{2})&=\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}-k^{2}},\\\
F_{E}(k^{2})&=e^{(k^{2}-m^{2})/\Lambda^{2}},\\\
F_{D}(k^{2})&=\frac{(\Lambda^{2}-m^{2})^{2}}{(\Lambda^{2}-k^{2})^{2}}.\\\
\end{split}$ (20)

In the following we will drive explicitly the integral equation for the BS
scalar wave function. Substituting Eqs. (9), (10), (12), and (17) or (18) into
the BS equation (6), we obtain the following form for the BS equation

$\begin{split}f(p)&=\frac{-i}{\left[(\lambda_{1}M+p_{l})^{2}-w_{1}^{2}+i\epsilon\right]\left[(\lambda_{2}M-p_{l})^{2}-w_{2}^{2}+i\epsilon\right]}\int\frac{d^{4}q}{(2\pi)^{4}}\Bigg{\\{}4c_{I}^{d}g_{\sigma}^{2}m_{D^{\ast}}m_{D}\\\
&\times\bigg{\\{}\frac{\left[(\lambda_{1}M+p_{l})^{2}-\mathrm{\mathbf{p}_{t}}^{2}\right]}{3m_{1}^{2}}-\frac{(\lambda_{1}M+p_{l})\left[p_{l}(\lambda_{1}M+p_{l})-\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}}\right]}{3m_{1}^{2}p_{l}}-1\bigg{\\}}F^{2}(\mathrm{\mathbf{k}_{t}}^{2},m_{\sigma})\\\
&-c_{I}^{c}g_{DD^{\ast}P}^{2}\left[\frac{1}{3}(\mathrm{\mathbf{p}_{t}}-\mathrm{\mathbf{q}_{t}})^{2}+\frac{(\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}}-\mathrm{\mathbf{p}_{t}}^{2})^{2}}{3m_{1}^{2}}\right]F^{2}(\mathrm{\mathbf{k}_{t}}^{2},m_{P})\\\
&+\Bigg{\\{}c_{I}^{d}g_{DDV}g_{D^{\ast}D^{\ast}V}\Bigg{\\{}\frac{\left[(\lambda_{1}M+p_{l})^{2}-\mathrm{\mathbf{p}_{t}}^{2}\right]\left[4(\lambda_{1}M+p_{l})(\lambda_{2}M-p_{l})-(\mathrm{\mathbf{p}_{t}}-\mathrm{\mathbf{q}_{t}})^{2}\right]}{3m_{1}^{2}}\\\
&-\frac{(\lambda_{1}M+p_{l})\left[p_{l}(\lambda_{1}M+p_{l})-\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}}\right]\left[4(\lambda_{1}M+p_{l})(\lambda_{2}M-p_{l})+(\mathrm{\mathbf{p}_{t}}-\mathrm{\mathbf{q}_{t}})^{2}\right]}{3m_{1}^{2}p_{l}}\\\
&-4(\lambda_{1}M+p_{l})(\lambda_{2}M-p_{l})-(\mathrm{\mathbf{p}_{t}}-\mathrm{\mathbf{q}_{t}})^{2}-\frac{(\mathrm{\mathbf{p}_{t}}^{2}-\mathrm{\mathbf{q}_{t}}^{2})^{2}(\lambda_{1}M+p_{l})\left[p_{l}(\lambda_{1}M+p_{l})-\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}}\right]}{3m_{1}^{2}m_{V}^{2}p_{l}}\\\
&+\frac{(\mathrm{\mathbf{p}_{t}}^{2}-\mathrm{\mathbf{q}_{t}}^{2})^{2}\left[(\lambda_{1}M+p_{l})-\mathrm{\mathbf{p}_{t}}^{2}\right]}{3m_{1}^{2}m_{V}^{2}}-\frac{(\mathrm{\mathbf{p}_{t}}^{2}-\mathrm{\mathbf{q}_{t}}^{2})^{2}}{m_{V}^{2}}\bigg{\\}}\\\
&-c_{I}^{d}g_{DDV}g^{\prime}_{D^{\ast}D^{\ast}V}\Bigg{\\{}\frac{2(\lambda_{2}M-p_{l})(\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}}-\mathrm{\mathbf{p}_{t}}^{2})\left[p_{l}(\lambda_{1}M+p_{l})-\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}}\right]}{3m_{1}^{2}p_{l}}\\\
&-\frac{2(\lambda_{2}M-p_{l})(\mathrm{\mathbf{q}_{t}}^{2}-\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}})}{3p_{l}}\Bigg{\\}}\Bigg{\\}}F^{2}(\mathrm{\mathbf{k}_{t}}^{2},m_{V})\\\
&-g_{DD^{\ast}P}^{2}\frac{M^{2}\left[2m_{1}^{2}(\mathrm{\mathbf{p}_{t}}-\mathrm{\mathbf{q}_{t}})^{2}+\mathrm{\mathbf{p}_{t}}^{2}\mathrm{\mathbf{q}_{t}}^{2}-(\mathrm{\mathbf{p}_{t}}\cdot\mathrm{\mathbf{q}_{t}})^{2}\right]}{3m_{1}^{2}}F^{2}(\mathrm{\mathbf{k}_{t}}^{2},m_{V})\Bigg{\\}}f(p),\end{split}$
(21)

where
$\mathrm{\mathbf{k}_{t}}=\mathrm{\mathbf{p}_{t}}-\mathrm{\mathbf{q}_{t}}$ and
we have made explicit use of the covariant instantaneous approximation (in
which the energy exchanged between the constituent particles of the binding
system is neglected.).

In Eq. (21) there are poles in $p_{l}$ at $-\lambda_{1}M-w_{1}+i\epsilon$,
$-\lambda_{1}M+w_{1}-i\epsilon$, $\lambda_{2}M+w_{2}-i\epsilon$, and
$\lambda_{2}M-w_{2}+i\epsilon$. By choosing the appropriate contour, we
integrate over $p_{l}$ on both sides of Eq. (21) and obtain the three-
dimensional integral equation. Then the scalar BS wave function is
rotationally invariant, and $f(\mathrm{\mathbf{p}_{t}})$
($f(\mathrm{\mathbf{p}_{t}})=\int dp_{l}f(p)$) only depends on the norm of the
three-momentum, $\mathrm{\mathbf{p}_{t}}$. Therefore, after completing the
azimuthal integration, the above BS equation becomes a one-dimensional
integral equation, which is

$f(|\mathrm{\mathbf{p}_{t}}|)=\int
d|\mathrm{\mathbf{p}_{t}}|A(|\mathrm{\mathbf{p}_{t}}|,|\mathrm{\mathbf{q}_{t}}|)f(|\mathrm{\mathbf{q}_{t}}|),$
(22)

where $f(|\mathrm{\mathbf{p}_{t}}|)$ is the one-dimensional Lorentz-scalar BS
function. The propagators and kernels after one-dimensional simplification are
included in $A(|\mathrm{\mathbf{p}_{t}}|,|\mathrm{\mathbf{q}_{t}}|)$.

##  III numerical solutions for the BS wave function

In this section we solve the integral equation (22). The method is to
discretize the integration region into sufficiently large $n$ pieces by the
Gaussian quadrature method. In this way, the BS scalar function
$f(|\mathrm{\mathbf{p}_{t}}|)$ becomes $n$ dimensional vector and the integral
equation becomes an eigenvalue equation. The matrix equation obtained in this
way can be written in the form $f^{(n)}(|\mathrm{\mathbf{p}_{t}}|)=A^{n\times
n}(|\mathrm{\mathbf{p}_{t}}|,|\mathrm{\mathbf{q}_{t}}|)f^{(n)}(|\mathrm{\mathbf{q}_{t}}|)$.

There are we two parameters in our model, the cutoff $\Lambda$ and the binding
energy $E_{b}$. The cutoff $\Lambda$ contains the information about the
nonpoint interaction of the hadrons which is nonperturbative. Therefore, the
cutoff $\Lambda$ cannot be determined exactly. Fortunately, the cutoff was
found in the study of the deuteron to be around 1 GeV. In this work, we vary
the cutoff $\Lambda$ over a much wider range (0.8 - 5) GeV to find all
possible solutions of the $D\bar{D}^{*}$/$B\bar{B}^{*}$ and
$DD^{*}$/$\bar{B}\bar{B}^{*}$ bound states. The binding energy $E_{b}$ is
defined as $E_{b}=E-m_{1}-m_{2}$, and we will vary $E_{b}$ from -30 to 0 MeV.
Fixing a value of the cutoff $\Lambda$ and varying the binding energy $E_{b}$
we will obtain a series of the trial eigenvalues. For some (not all) values of
the cutoff, we could find that the binding energy in the range of -30 to 0 MeV
correspond to the eigenvalue closest to 1.0. Our task is to find out all these
cutoff values.

In the present paper, we will systematically study the $S$-wave
$D\bar{D}^{*}$/$B\bar{B}^{*}$ and $DD^{*}$/$\bar{B}\bar{B}^{*}$ molecular
states. As studied in Ref. Ding:2020dio , the $D\bar{D}^{*}$/$B\bar{B}^{*}$
carry different $G$ and $C$ parities for the isoscalar and isovector states,
respectively. From the effective Lagrangian listed in Eq. (13), we can find
the kernels in the bottom sector which have the same form as those in the
charm sector. The numerical results from our calculation for the charm and
bottom systems are listed in Tables 2 and 3, respectively.

From the results in Tables 2 and 3, we can draw three main conclusions. First,
only the bound states with isospin $I=0$ exist. Secondly, for the same binding
energy $E_{b}$, the values of the cutoff $\Lambda$ is smaller in the bottom
sector than those in the charm sector, which means the interactions in the
bottom molecular states are stronger than the interactions in the charm sector
as expected. Finally, the values of the cutoff $\Lambda$ for the dipole form
factor are larger than the corresponding values for the monopole and
exponential form factors for both the charm states or bottom states.

Table 2: The numerical results for the $D\bar{D}^{*}$ and $DD^{*}$ systems.  | ${D}\bar{D}^{*}$ | ${D}{D}^{*}$
---|---|---
| $G=1$ | $G=-1$ |  |  |  |  |  |
| $I=0$ | $I=1$ | $I=0$ | $I=1$ | $I=0$ | $I=1$
$E_{b}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$
-5 | 1056 | 1013 | 1456 | —- | —- | —- | 1340 | 1250 | 1823 | —- | —- | —- | 1093 | 1097 | 1540 | —- | —- | —-
-10 | 1117 | 1100 | 1562 | —- | —- | —- | 1402 | 1337 | 1928 | —- | —- | —- | 1146 | 1177 | 1635 | —- | —- | —-
-15 | 1165 | 1166 | 1643 | —- | —- | —- | 1448 | 1401 | 2006 | —- | —- | —- | 1184 | 1234 | 1701 | —- | —- | —-
-20 | 1208 | 1224 | 1714 | —- | —- | —- | 1488 | 1455 | 2072 | —- | —- | —- | 1214 | 1279 | 1755 | —- | —- | —-
-25 | 1247 | 1276 | 1779 | —- | —- | —- | 1523 | 1502 | 2130 | —- | —- | —- | 1240 | 1318 | 1800 | —- | —- | —-
-30 | 1284 | 1324 | 1839 | —- | —- | —- | 1555 | 1544 | 2182 | —- | —- | —- | 1264 | 1352 | 1840 | —- | —- | —-
Table 3: The numerical results for the $B\bar{B}^{*}$ and $\bar{B}\bar{B}^{*}$ systems.  | ${B}\bar{B}^{*}$ | $\bar{B}\bar{B}^{*}$
---|---|---
| $G=1$ | $G=-1$ |  |  |  |  |  |
| $I=0$ | $I=1$ | $I=0$ | $I=1$ | $I=0$ | $I=1$
$E_{b}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$ | $\Lambda_{M}$ | $\Lambda_{E}$ | $\Lambda_{D}$
-5 | 861 | 754 | 1126 | —- | —- | —- | 1130 | 964 | 1469 | —- | —- | —- | 871 | 799 | 1165 | —- | —- | —-
-10 | 894 | 817 | 1195 | —- | —- | —- | 1151 | 1019 | 1522 | —- | —- | —- | 906 | 865 | 1238 | —- | —- | —-
-15 | 920 | 865 | 1249 | —- | —- | —- | 1172 | 1062 | 1567 | —- | —- | —- | 933 | 914 | 1291 | —- | —- | —-
-20 | 943 | 905 | 1294 | —- | —- | —- | 1192 | 1099 | 1607 | —- | —- | —- | 955 | 953 | 1335 | —- | —- | —-
-25 | 964 | 940 | 1335 | —- | —- | —- | 1210 | 1131 | 1642 | —- | —- | —- | 975 | 987 | 1374 | —- | —- | —-
-30 | 984 | 973 | 1373 | —- | —- | —- | 1228 | 1161 | 1675 | —- | —- | —- | 992 | 1017 | 1407 | —- | —- | —-

For the $D\bar{D}^{*}$ bound state, there are two relevant states $X(3872)$
and $Z_{c}(3900)$ observed by the experiments. The quantum numbers of
$X(3872)$ and $Z_{c}(3900)$ are $I^{G}(J^{PC})=0^{+}(1^{++})$ and
$I^{G}(J^{PC})=1^{+}(1^{+-})$, respectively pdg2020 . From our calculations,
there are no $I=1$ $D\bar{D}^{*}$ bound states existing with the cutoff
$\Lambda$ in the range (0.8 - 5) GeV. Therfore, $Z_{c}(3900)$ cannot exist as
an $I=0$ $D\bar{D}^{*}$ molecular state in our model. Our result consistent
with many other studies. In Ref. Aceti:2014uea , using the local hidden gauge
approach, the authors found a state with 3900 MeV could not be easily
interpreted as a $D\bar{D}^{\ast}$ ($\bar{D}D^{\ast}$) molecular state.
Albaladejo et al. Albaladejo:2016jsg  found the $Z_{c}(3900)$ cannot be a
$D\bar{D}^{\ast}$ bound state considering that the $Z_{c}$ enhancement was
originated from a resonance with a mass around the $D\bar{D}^{\ast}$ threshold
or produced by a virtual state which must have a hadronic molecular nature.
From the BS approach with quasipotential approximation He:2014nya , no bound
state was induced from the interactions of $D\bar{D}^{\ast}$ in the isovector
sector, which also suggested that the molecular state explanation for
$Z_{c}(3900)$ was excluded. In the framework of the one-boson exchange model
Zhao:2014gqa , the results showed that the momentum-related corrections was
unfavorable for the formation of the molecular state in the $I=0$,
$J^{PC}=1^{+-}$ channel in the $D\bar{D}^{\ast}$ system. Based on the lattice
QCD the CLQCD Collaboration Chen:2014afa  studied the low-energy scattering of
the $(D\bar{D}^{\ast})^{\pm}$ meson system. It was found that the
$D\bar{D}^{\ast}$ interaction was weakly repulsive, hence the results did not
support the possibility of a shallow bound state for the two mesons for the
pion mass values studied. We conclude that the $X(3872)$ can be a isoscalar
$D\bar{D}^{*}$ molecular state with the cutoff $\Lambda$ = 1043 MeV, 993 MeV,
and 1432 MeV for the monopole, exponential and dipole form factors,
respectively. The properties of $X(3872)$ as a $D\bar{D}^{*}$ molecular state
have been studied in our previous work Wang:2017dcq .

For the $DD^{*}$ bound state, there is the recently experimentally discovered
$T_{cc}^{+}$ state could be associated with it. In our model, the $T^{+}_{cc}$
can be $I=0$ $DD^{*}$ bound state with the cutoff $\Lambda$ taking the values
of 957 MeV, 885 MeV, and 1292 MeV for the monopole, exponential and dipole
form factors, respectively. Theoretically, the $DD^{*}$ system has been
studied within a variety of approaches and different models. In Ref.
Carames:2011zz , the authors studied doubly charmed exotic states by solving
the scattering problem of two $D$ mesons. Their results pointed to the
existence of a stable isoscalar doubly charmed bound state with the quantum
numbers $(I)J^{P}=(0)1^{+}$. By solving the coupled channel
Schr$\mathrm{\ddot{o}}$dinger equations, Ohkoda et al. Ohkoda:2012hv  found
the $DD^{*}$ system could be a deeply $(I)J^{P}=(0)1^{+}$ bound state, and no
isovector bound state could exist. Via solving the single channel BS equation,
the authors of Ref. Dong:2021bvy  found the $I(J^{P})=0(1^{+})$ $DD^{\ast}$
system could be the double charm tetraquark $T^{+}_{cc}$ observed by LHCb
observation with a reasonable cutoff regularizing the loop integral. However,
in Ref. Xu:2017tsr , they studied the $DD^{\ast}$ system up to
$O(\epsilon^{2})$ at the one-loop level within the framework of heavy meson
chiral effective field theory. No bound state was found in the $I=0$ channel
within a wide range of the cutoff parameter, while there existed a bound state
in the $I=1$ channel as the cutoff is near $m_{\rho}$. In the bottom sector,
only the $Z_{b}(10610)$ reported by the Belle Collaboration in 2011 is close
to the $B\bar{B}^{\ast}$ threshold Belle:2011aa . A later analysis for the
same experiment allowed for an amplitude analysis where the quantum numbers
$I^{G}(J^{P})=1^{+}(1^{+})$ were strongly favored for $Z_{b}(10610)$
Belle:2014vzn . In our model, no isovector $B\bar{B}^{\ast}$ was found, which
disfavors the molecular state explanation for the $Z_{b}(10610)$.

##  IV summary

In this work, we applied the BS equation to systematically study the
$D\bar{D}^{*}$/$B\bar{B}^{*}$ and $DD^{*}$/$\bar{B}\bar{B}^{*}$ systems with
the ladder approximation and the instantaneous approximation, try to find the
possible bound states of these systems. In our calculations, both direct and
cross diagrams were considered for the kernel induced by $\rho$, $\omega$,
$\pi$, $\eta$, and $\sigma$ exchanges. Since the constituent particles and the
exchanged particles in the $D\bar{D}^{*}$/$B\bar{B}^{*}$ or
$DD^{*}$/$\bar{B}\bar{B}^{*}$ systems are not pointlike, we introduced three
different form factors (monopole, exponential and dipole form factors) which
all contain a cutoff $\Lambda$ that reflects the effects of the structure of
these particles. Since $\Lambda$ is controlled by nonperturbative QCD and
cannot be determined exactly, we let it vary in a reasonable range within
which we tried to find possible bound states of the
$D\bar{D}^{*}$/$B\bar{B}^{*}$ and $DD^{*}$/$\bar{B}\bar{B}^{*}$ systems.

The results of our studies showed that only the $S$-wave
$D\bar{D}^{*}$/$B\bar{B}^{*}$ and $DD^{*}$/$\bar{B}\bar{B}^{*}$ systems with
$I=0$ could exist as bound states. We also found that for the same binding
energy $E_{b}$ the values of the cutoff $\Lambda$ in the bottom sector are
smaller than those in the charm sector, and that the values of the cutoff
$\Lambda$ is larger for the dipole form factor than those for the exponential
and dipole form factors in both charm and bottom sector. From our results, we
can see that the experimentally observed $X(3872)$ and $T_{cc}^{+}$ can be
assigned as $I=0$ $D\bar{D}^{*}$ and $I=0$ $DD^{*}$ molecular states,
respectively, corresponding to $\Lambda$ = 1043 MeV, 993 MeV, and 1432 MeV and
$\Lambda$ = 957 MeV, 885 MeV, and 1538 MeV for the monopole, exponential and
dipole form factors, respectively. No bound state was found for isovector from
$S$-wave $D\bar{D}^{*}$/$B\bar{B}^{*}$ and $DD^{*}$/$\bar{B}\bar{B}^{*}$
systems, which disfavors the molecular state explanation for $Z_{c}(3900)$ and
$Z_{b}(10610)$.

The possible $S$-wave $D\bar{D}^{*}$/$B\bar{B}^{*}$ and
$DD^{*}$/$\bar{B}\bar{B}^{*}$ bound states studied in our work are helpful in
explaining the structure of experimentally discovered exotic states and the
discovery of unobserved exotic states. In some cases the theoretical
explanations of the structures for the experimentally observed exotic states
and the existence of theoretical predictions of the possible molecular states
remains controversial. Therefore more precise experimental studies of the
exotic states will be needed to check the results of theoretical studies and
to improve theoretical model.

###### Acknowledgements.

One of the authors (Z.-Y. Wang) thanks Professor Jun He, Ning Li, and Yin-Jie
Zhang for helpful discussions. This work was supported by National Natural
Science Foundation of China (Projects No. 12105149, No. 11947001, No. 11775024
and No.11605150), the Natural Science Foundation of Zhejiang province (no.
LQ21A050005), and the Fundamental Research Funds for the Central Universities
of China (Project No. 31020170QD052).

## References

  * (1) S. K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003).
  * (2) Particle Data Group, P. Zyla et al., Review of particle physics, Prog. Theor. Exp. Phys. 6, 083C01 (2020).
  * (3) H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rept. 639, 1-121 (2016); A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept. 668, 1-97 (2017); Y. Dong, A. Faessler and V. E. Lyubovitskij, Prog. Part. Nucl. Phys. 94, 282-310 (2017); N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, Phys. Rept. 873, 1-154 (2020); S. L. Olsen, T. Skwarnicki and D. Zieminska, Rev. Mod. Phys. 90, no.1, 015003 (2018); F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no.1, 015004 (2018).
  * (4) R. Aaij et al. [LHCb], [arXiv:2109.01038 [hep-ex]].
  * (5) R. Aaij et al. [LHCb], [arXiv:2109.01056 [hep-ex]].
  * (6) H. Xu, B. Wang, Z. W. Liu and X. Liu, Phys. Rev. D 99, 014027 (2019).
  * (7) S. Ohkoda, Y. Yamaguchi, S. Yasui, K. Sudoh and A. Hosaka, Phys. Rev. D 86, 034019 (2012).
  * (8) N. Li and S. L. Zhu, Phys. Rev. D 86, 074022 (2012).
  * (9) N. Li, Z. F. Sun, X. Liu and S. L. Zhu, Phys. Rev. D 88, 114008 (2013).
  * (10) H. Ren, F. Wu and R. Zhu, [arXiv:2109.02531 [hep-ph]].
  * (11) Z. F. Sun, Z. G. Luo, J. He, X. Liu and S. L. Zhu, Chin. Phys. C 36, 194-204 (2012).
  * (12) A. Feijoo, W. H. Liang and E. Oset, [arXiv:2108.02730 [hep-ph]].
  * (13) R. Chen, Q. Huang, X. Liu and S. L. Zhu, [arXiv:2108.01911 [hep-ph]].
  * (14) M. Z. Liu, T. W. Wu, M. Pavon Valderrama, J. J. Xie and L. S. Geng, Phys. Rev. D 99, 094018 (2019).
  * (15) Z. F. Sun, J. He, X. Liu, Z. G. Luo and S. L. Zhu, Phys. Rev. D 84, 054002 (2011).
  * (16) J. He, Phys. Rev. D 90, 076008 (2014).
  * (17) X. K. Dong, F. K. Guo and B. S. Zou, Commun. Theor. Phys. 73, 125201 (2021).
  * (18) J. He, Phys. Rev. D 92, 034004 (2015).
  * (19) Z. M. Ding, H. Y. Jiang and J. He, Eur. Phys. J. C 80, 1179 (2020).
  * (20) P. C. Wallbott, G. Eichmann and C. S. Fischer, Phys. Rev. D 100, 014033 (2019).
  * (21) R. Zhu, X. Liu, H. Huang and C. F. Qiao, Phys. Lett. B 797, 134869 (2019).
  * (22) S. Noh, W. Park and S. H. Lee, Phys. Rev. D 103, 114009 (2021).
  * (23) P. G. Ortega, D. R. Entem and F. Fernandez, [arXiv:2111.02475 [hep-ph]].
  * (24) Y. Tan, W. Lu and J. Ping, Eur. Phys. J. Plus 135, 716 (2020).
  * (25) S. Q. Luo, K. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Eur. Phys. J. C 77, 709 (2017).
  * (26) Y. C. Yang, Z. Y. Tan, J. Ping and H. S. Zong, Eur. Phys. J. C 77, no.9, 575 (2017) doi:10.1140/epjc/s10052-017-5137-6 [arXiv:1703.09718 [hep-ph]].
  * (27) F. S. Navarra, M. Nielsen and S. H. Lee, Phys. Lett. B 649, 166-172 (2007).
  * (28) Q. Xin and Z. G. Wang, [arXiv:2108.12597 [hep-ph]].
  * (29) W. Chen, H. X. Chen, X. Liu, T. G. Steele and S. L. Zhu, Phys. Rev. D 96, 114017 (2017).
  * (30) K. Azizi and U. Özdem, Phys. Rev. D 104, 114002 (2021).
  * (31) L. Tang, B. D. Wan, K. Maltman and C. F. Qiao, Phys. Rev. D 101, 094032 (2020).
  * (32) Q. F. Lü, D. Y. Chen and Y. B. Dong, Phys. Rev. D 102, 034012 (2020).
  * (33) D. Ebert, R. N. Faustov, V. O. Galkin and W. Lucha, Phys. Rev. D 76, 114015 (2007).
  * (34) G. J. Wang, X. H. Liu, L. Ma, X. Liu, X. L. Chen, W. Z. Deng and S. L. Zhu, Eur. Phys. J. C 79, 567 (2019).
  * (35) M. N. Anwar, J. Ferretti and E. Santopinto, Phys. Rev. D 98, 094015 (2018).
  * (36) H. Y. Jin, C. S. Huang and Y. B. Dai, Z. Phys. C 56, 707-712 (1992).
  * (37) X. H. Guo, A. W. Thomas and A. G. Williams, Phys. Rev. D 61, 116015 (2000).
  * (38) X. H. Guo, A. W. Thomas and A. G. Williams, Phys. Rev. D 59, 116007 (1999).
  * (39) X. H. Guo and X. H. Wu, Phys. Rev. D 76, 056004 (2007).
  * (40) Z. X. Xie, G. Q. Feng and X. H. Guo, Phys. Rev. D 81, 036014 (2010).
  * (41) H. W. Ke, T. Liu and X. Q. Li, Phys. Rev. D 89, 017501 (2014).
  * (42) Z. Y. Wang, J. J. Qi, J. Xu and X. H. Guo, Phys. Rev. D 102, 036008 (2020).
  * (43) H. W. Ke, X. Han, X. H. Liu and Y. L. Shi, Eur. Phys. J. C 81, 427 (2021).
  * (44) G. J. Ding, Phys. Rev. D 79, 014001 (2009).
  * (45) H. Y. Cheng, C. K. Chua and A. Soni, Phys. Rev. D 71, 014030 (2005).
  * (46) F. Aceti, M. Bayar, E. Oset, A. Martinez Torres, K. P. Khemchandani, J. M. Dias, F. S. Navarra and M. Nielsen, Phys. Rev. D 90, 016003 (2014).
  * (47) M. Albaladejo, P. Fernandez-Soler and J. Nieves, Eur. Phys. J. C 76, 573 (2016).
  * (48) L. Zhao, L. Ma and S. L. Zhu, Phys. Rev. D 89, 094026 (2014).
  * (49) Y. Chen, M. Gong, Y. H. Lei, N. Li, J. Liang, C. Liu, H. Liu, J. L. Liu, L. Liu and Y. F. Liu, et al. Phys. Rev. D 89, 094506 (2014).
  * (50) Z. Y. Wang, J. J. Qi, X. H. Guo and C. Wang, Phys. Rev. D 97, 016015 (2018).
  * (51) T. F. Carames, A. Valcarce and J. Vijande, Phys. Lett. B 699, 291-295 (2011).
  * (52) A. Bondar et al. [Belle], Phys. Rev. Lett. 108, 122001 (2012).
  * (53) A. Garmash et al. [Belle], Phys. Rev. D 91, 072003 (2015).