# Charmoniumlike resonant explanation on the newly observed $X(3960)$

Rui Chen1<EMAIL_ADDRESS>Qi Huang2111Corresponding author
<EMAIL_ADDRESS>1Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China
2Department of Physics, Nanjing Normal University, Nanjing 210023, China

###### Abstract

Stimulated by the observation of the newly $X(3960)$ observed by the LHCb
collaboration, we adopt the one-boson-exchange model and consider the $S-D$
wave mixing effects to study the
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ interactions with
$I(J^{PC})=0(0^{++})$. After producing the phase shifts of this coupled
channel systems, our results show that there can exist a charmoniumlike
resonance, whose obtained mass and width can both well match with the
experimental data of the newly observed $X(3960)$. We also find that the
$D^{*}\bar{D}^{*}$ system plays an important role in the formation of the
newly observed $X(3960)$ as a charmoniumlike resonance, and the
$D_{s}^{*}\bar{D}_{s}^{*}$ system makes a significant contribution to the
resonant width. As a byproduct, we perform a coupled channel analysis on the
$D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ interactions
with $I(J^{PC})=0(1^{+-})$, our results can predict the existence of the
$D_{s}\bar{D}_{s}^{*}$ molecule with $1^{+-}$ and the
$D_{s}^{*}\bar{D}_{s}^{*}$ molecule with $1^{+-}$. Their widths are around
several and several to several tens MeV, respectively. Experimental searches
for these two possible charmoniumlike molecular candidates can be helpful to
verify our proposal.

###### pacs:

12.39.Pn, 13.75.Lb, 14.40.Rt

##  I introduction

Recently, in a talk given at CERN, the LHCb collaboration reported their
observations of three new states from the $B$ decay processes Chen and Norella
. In this talk, apart from the two charged states, i.e., the first pentaquark
with $s$ quark content observed in the $J/\psi\Lambda$ invariant mass spectrum
of the $B^{-}\to J/\psi\Lambda\bar{p}$ process and the
$T_{c\bar{s}}^{a++(0)}(2900)$ shown in the $D_{s}^{+}\pi^{+/(-)}$ invariant
mass spectrum of the $B^{0(+)}\to D^{0(-)}D_{s}^{+}\pi^{+(-)}$ process, one
another neutral state, namely $X(3960)$, was also observed by the LHCb
Collaboration in the $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum of the $B$
decay process $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ Chen and Norella .

Usually, when a new neutral state is observed in the invariant mass spectrum
that composed by a pair of heavy and anti-heavy mesons, our first
consideration may often be that if this new state can be treated as a
conventional charmonia. Here, since the quantum number of the $X(3960)$ is
reported as $0^{++}$, the first idea of us is that if it is a new $\chi_{c0}$
state. When checking the theoretical results of the potential model Barnes et
al. (2005); Li and Chao (2009); Duan et al. (2020), the $\chi_{c0}(2P)$ state
has been denoted as $\chi_{c0}(3915)$ Workman and Others (2022) and the
position of $\chi_{c0}(3P)$ is around 4.2 GeV. For $\chi_{c0}(3915)$, although
its mass is close to the $X(3960)$, whose mass and width are measured as
$M=3955\pm 6\pm 22$ MeV and $\Gamma=48\pm 17\pm 10$ MeV respectively, due to
its mass is below the threshold of $D_{s}\bar{D}_{s}$, i.e., about 3938 MeV,
it is puzzled that it can be observed in the $D_{s}\bar{D}_{s}$ invariant mass
spectrum. While for the $\chi_{c0}(3P)$, its predicted mass is too far away
from $X(3960)$, thus denoting the $X(3960)$ as $\chi_{c0}(3P)$ may be not
appropriate.

Another reason that $X(3960)$ may not be a good candidate of charmonia is that
its decay property is a little different. In the talk Chen and Norella , the
LHCb Collaboration compared its decay widths to the $D^{+}D^{-}$ and
$D_{s}^{+}D_{s}^{-}$, and the measurement gave that Chen and Norella

$\displaystyle\frac{\Gamma(X(3960)\to D^{+}D^{-})}{\Gamma(X(3960)\to
D_{s}^{+}D_{s}^{-})}=0.29\pm 0.09\pm 0.10\pm 0.08,$ (1)

which means it is easier for $X(3960)$ to decay into $D_{s}^{+}D_{s}^{-}$
rather than $D^{+}D^{-}$. Since usually it is harder to excite a pair of
$s\bar{s}$ from vacuum compared with $u\bar{u}(d\bar{d})$, thus conventional
charmonia predominantly decay into a pair of $D$ meson, which implies the
exotic nature of this new state $X(3960)$ Chen and Norella .

Thus, the next thing we do is naturally to see if this $X(3960)$ can really be
assigned as an exotic state, in which a state composed by four valence quarks
may be the easist generalization. Since the position of $X(3960)$ is close to
the $D_{s}\bar{D}_{s}$ threshold, a consideration that it is related to some
molecular states can easily raise up. Actually, studies on the
$D_{s}\bar{D}_{s}$ molecular states has already done before the observation of
$X(3960)$ Prelovsek et al. (2021); Gamermann et al. (2007); Nieves and
Valderrama (2012); Hidalgo-Duque et al. (2013); Meng et al. (2021); Dong et
al. (2021), and it turns out that although a $0^{++}$ bound state that couples
strongly to $D_{s}^{+}D_{s}^{-}$ and weakly to $D^{+}D^{-}$ is found just
below $D_{s}^{+}D_{s}^{-}$ threshold, after carrying out a dynamic study of
$D\bar{D}$ and $D_{s}^{+}D_{s}^{-}$ in coupled channels, this bound state
disappears. Thus, recently, Ref. Bayar et al. (2022) reanalyzed this situation
and found that if the strength of $D\bar{D}\to D_{s}^{+}D_{s}^{-}$ transition
is slightly reduced, that missing state will appear and behave similarly on
the $D_{s}^{+}D_{s}^{-}$ invariant mass spectrum as the experimental
observation Chen and Norella ; Bayar et al. (2022).

The molecular state interpretion of the $X(3960)$ is also supported by Refs.
Xin et al. (2022); Ji et al. (2022). In addition, apart from bound state
interpretion, Ref. Ji et al. (2022) pointed out that virtual state explanation
was also valid. Then, Ref. Xie et al. (2022) used the effective Lagrangian
approach to calculate the production rate of $X(3960)$ in the $B$ decays
utilizing triangle diagrams, and the results showed that both the bound and
virtual state interpretions can match the relevant experimental data.

Thus, for $X(3960)$ appeared in the $D_{s}^{+}D_{s}^{-}$ invariant mass
spectrum, Refs. Prelovsek et al. (2021); Gamermann et al. (2007); Nieves and
Valderrama (2012); Hidalgo-Duque et al. (2013); Meng et al. (2021); Dong et
al. (2021); Bayar et al. (2022); Xin et al. (2022); Ji et al. (2022); Xie et
al. (2022) explain it as an effect caused by a molecular state located below
the $D_{s}^{+}D_{s}^{-}$ threshold. However, considering the fact that its
measured mass is above the $D_{s}^{+}D_{s}^{-}$ threshold Chen and Norella ,
the resonant state explanation, in our view, is also possible, and this work
is to study this possibility.

In general, the resonances can be divided into two types, i.e., the shape-type
resonances and the Feshbach-type resonances, and the generation of these two
types of resonances is controlled by the potential barriers Chen and Huang
(2021). In addition, we want to emphasize here that the coupled channel effect
plays a very important role in producing the Feshbach-type resonances since
the mass gaps between the relevant channels will give additional contributions
to the potential barriers.

Thus, in this work, considering the measured mass and quantum number of
$X(3960)$, we perform an analysis that includes coupled channel effect and
$S-D$ wave mixing effect to see if the newly observed $X(3960)$ can be
interpreted as a resonance, the included channels are $D_{s}\bar{D}_{s}$,
$D^{\ast}\bar{D}^{\ast}$, and $D_{s}^{\ast}\bar{D}_{s}^{\ast}$.

This paper is organized as follows. After this introduction, we deduce the
coupled $D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$
interactions with $I(J^{PC})=0(0^{++})$ by using the OBE model in Sec. II. In
Sec. III, we present the corresponding numerical results by producing the
phase shifts and predict possible charmoniumlike structures from the isoscalar
$D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ interactions
with $J^{PC}=1^{+-}$. The paper ends with a summary in Sec. IV.

##  II Interactions

In the OBE model, the relevant effective potentials can be deduced as follows.
Firstly, we can write down the scattering amplitude by adopting the effective
Lagrangian approach. And then, one can derive the effective potentials based
on the approximation relation to the scattering amplitude,

$\displaystyle\mathcal{V}_{E}^{AB\to CD}(\bm{q})$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{M}(AB\to
CD)}{\sqrt{\prod_{i}2M_{i}\prod_{f}2M_{f}}},$ (2)

where $\mathcal{M}(AB\to CD)$ denotes the scattering amplitude for the $AB\to
CD$ process in $t-$channel. $M_{i}$ and $M_{f}$ are the masses of the initial
states and final states, respectively. Then we can finally obtain the
effective potentials in the coordinate space $\mathcal{V}(\bm{r})$ by
performing the Fourier transformation, i.e.,

$\displaystyle\mathcal{V}_{E}(\bm{r})=\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}_{E}(\bm{q})\mathcal{F}^{2}(q^{2},m_{E}^{2}).$
(3)

Here, $\mathcal{F}(q^{2},m_{E}^{2})$ is the form factor, it is introduced at
every interactive vertex to compensate the off-shell effect of the exchanged
meson. In this work, we take the monopole type form factor,
$\mathcal{F}(q^{2},m_{E}^{2})=(\Lambda^{2}-m_{E}^{2})/(\Lambda^{2}-q^{2})$,
where $\Lambda$, $m_{E}$ and $q$ are the cutoff, the mass and four-momentum of
the exchanged meson, respectively. According to the experience of the deuteron
Tornqvist (1994a, b), the reasonable cutoff value is taken around 1.00 GeV.

Based on the heavy quark symmetry and chiral symmetry Yan et al. (1992); Wise
(1992); Burdman and Donoghue (1992); Casalbuoni et al. (1997); Falk and Luke
(1992), the relevant effective Lagrangians are constructed as

$\displaystyle{\mathcal{L}}$ $\displaystyle=$ $\displaystyle
g_{\sigma}\left\langle
H^{(Q)}_{a}\sigma\overline{H}^{(Q)}_{a}\right\rangle+g_{\sigma}\left\langle\overline{H}^{(\bar{Q})}_{a}\sigma
H^{(\bar{Q})}_{a}\right\rangle$ (4) $\displaystyle+ig\left\langle
H^{(Q)}_{b}{\mathcal{A}}\\!\\!\\!/_{ba}\gamma_{5}\overline{H}^{\,({Q})}_{a}\right\rangle+ig\left\langle\overline{H}^{(\bar{Q})}_{a}{\mathcal{A}}\\!\\!\\!/_{ab}\gamma_{5}H^{\,(\bar{Q})}_{b}\right\rangle$
$\displaystyle+\left\langle iH^{(Q)}_{b}\left(\beta
v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})+\lambda\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ba}\overline{H}^{\,(Q)}_{a}\right\rangle$
$\displaystyle-\left\langle i\overline{H}^{(\bar{Q})}_{a}\left(\beta
v^{\mu}({\mathcal{V}}_{\mu}-\rho_{\mu})-\lambda\sigma^{\mu\nu}F_{\mu\nu}(\rho)\right)_{ab}H^{\,(\bar{Q})}_{b}\right\rangle.$

Here, the super-fields
$H^{(Q)}_{a}=(1+{v}\\!\\!\\!/)(\mathcal{P}^{*(Q)\mu}_{a}\gamma_{\mu}-\mathcal{P}^{(Q)}_{a}\gamma_{5})/2$
and
$H^{(\overline{Q})}_{a}=(\bar{\mathcal{P}}^{*(\overline{Q})\mu}_{a}\gamma_{\mu}-\bar{\mathcal{P}}^{(\overline{Q})}_{a}\gamma_{5})(1-{v}\\!\\!\\!/)/{2}$
are expressed as the combinations of the $S-$wave charmed (anti-charmed)
mesons with $J^{P}=0^{-}$ and $1^{-}$ as they are in the same doublet in the
heavy quark limit. The conjugate field reads as
$\overline{H}=\gamma_{0}H^{\dagger}\gamma_{0}$. $\mathcal{P}^{{Q}(*)}$ stands
for the pseudoscalar (or vector) mesons fields
$\mathcal{P}^{{Q}(*)}=(D^{(*)+},~{}D^{(*)0},~{}D_{s}^{(*)+})^{T}$. $v^{\mu}$
is the four velocity. In the non-relativistic approximation, we take the form
of $v^{\mu}=(1,\bf{0})$.
$\mathcal{V}=\frac{1}{2}(\xi^{{\dagger}}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{{\dagger}})$
and
$A_{\mu}=\frac{1}{2}(\xi^{{\dagger}}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{{\dagger}})$
respectively, which stand for the vector and axial currents with
$\xi=\text{exp}(i{\mathbb{P}}/f_{\pi})$, $f_{\pi}=132$ MeV. And
$F_{\mu\nu}(\rho)=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+[\rho_{\mu},\rho_{\nu}]$,
with $\rho_{\mu}=ig_{V}{\mathcal{V}}_{\mu}/\sqrt{2}$.
$g_{V}=m_{\rho}/f_{\pi}=5.8$. The $\mathbb{P}$ and $\mathbb{V}$ denote the
light pseudoscalar meson and the light vector meson matrices, respectively,
which read as

$\displaystyle\left.\begin{array}[]{c}{\mathbb{P}}={\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right)},\\\
{\mathbb{V}}={\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right)}.\end{array}\right.$ (13)

Once expanding the effective Lagrangians in Eq. (4), we can obtain

$\displaystyle\mathcal{L}_{\mathcal{P}^{*}\mathcal{P}^{*}\sigma}$
$\displaystyle=$ $\displaystyle
2g_{s}\mathcal{P}_{a}^{*}\cdot\mathcal{P}_{a}^{*{\dagger}}\sigma+2g_{s}\bar{\mathcal{P}}_{a}^{*{\dagger}}\cdot\bar{\mathcal{P}}_{a}^{*}\sigma,$
(14) $\displaystyle\mathcal{L}_{\mathcal{P}^{*}\mathcal{P}^{*}\mathbb{P}}$
$\displaystyle=$
$\displaystyle-i\frac{2g}{f_{\pi}}v^{\beta}\varepsilon_{\beta\mu\alpha\nu}\mathcal{P}_{b}^{*\mu}\mathcal{P}_{a}^{*\nu{\dagger}}\partial^{\alpha}\mathbb{P}_{ba}$
(15)
$\displaystyle+i\frac{2g}{f_{\pi}}v^{\beta}\varepsilon_{\beta\mu\alpha\nu}\bar{\mathcal{P}}_{a}^{*\mu{\dagger}}\bar{\mathcal{P}}_{b}^{*\nu}\partial^{\alpha}\mathbb{P}_{ab},$
$\displaystyle\mathcal{L}_{\mathcal{P}^{*}\mathcal{P}^{*}\mathbb{V}}$
$\displaystyle=$ $\displaystyle\sqrt{2}\beta
g_{V}\left(\mathcal{P}_{b}^{*}\cdot\mathcal{P}_{a}^{*{\dagger}}\right)(v\cdot\mathbb{V}_{ba})$
(16) $\displaystyle-i2\sqrt{2}\lambda
g_{V}\mathcal{P}_{b}^{*\mu}\mathcal{P}_{a}^{*\nu{\dagger}}\left(\partial_{\mu}\mathbb{V}_{\nu}-\partial_{\nu}\mathbb{V}_{\mu}\right)_{ba}$
$\displaystyle-\sqrt{2}\beta
g_{V}\left(\bar{\mathcal{P}}_{a}^{*{\dagger}}\cdot\bar{\mathcal{P}}_{b}^{*}\right)(v\cdot\mathbb{V}_{ab})$
$\displaystyle-i2\sqrt{2}\lambda
g_{V}\bar{\mathcal{P}}_{a}^{*\mu{\dagger}}\bar{\mathcal{P}}_{b}^{*\nu}\left(\partial_{\mu}\mathbb{V}_{\nu}-\partial_{\nu}\mathbb{V}_{\mu}\right)_{ab}.$

In the above Lagrangians, the $\sigma$ meson coupling $g_{s}=2.82$ is
estimated from the quark model Machleidt et al. (1987); Wang et al. (2019).
For the $\pi-$exchange coupling, $g=0.59$ is extracted from the decay width of
$D^{*}\to D\pi$ Casalbuoni et al. (1997). Using the vector meson dominance
Isola et al. (2003), $\beta$ is fixed as $\beta=$0.9. $\lambda=$ 0.56 GeV-1 is
determined through a comparison of the form factor between the theoretical
result and the lattice QCD.

The flavor wave functions $|I=0,I_{3}=0\rangle$ for the isoscalar
$D_{s}^{(*)}\bar{D}_{s}^{(*)}$ and $D^{*}\bar{D}^{*}$ are constructed as
$|D_{s}^{(*)+}{D}_{s}^{(*)-}\rangle$ and
$(|D^{*0}\bar{D}^{*0}\rangle+|D^{*+}D^{*-}\rangle)/\sqrt{2}$, respectively.
When we consider the $S$-$D$ wave mixing effects, the spin-orbit wave
functions for the $D_{(s)}^{*}\bar{D}_{(s)}^{*}$ systems with $0^{++}$ are
$|{}^{1}\mathbb{S}_{0}\rangle$ and $|{}^{5}\mathbb{D}_{0}\rangle$. In the
isoscalar $D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled
channel analysis, the OBE effective potentials can be expresses as

$\displaystyle V$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}V^{D_{s}\bar{D}_{s}\to
D_{s}\bar{D}_{s}}&V^{D^{*}\bar{D}^{*}\to
D_{s}\bar{D}_{s}}&V^{D_{s}^{*}\bar{D}_{s}^{*}\to D_{s}\bar{D}_{s}}\\\
V^{D_{s}\bar{D}_{s}\to D^{*}\bar{D}^{*}}&V^{D^{*}\bar{D}^{*}\to
D^{*}\bar{D}^{*}}&V^{D_{s}^{*}\bar{D}_{s}^{*}\to D^{*}\bar{D}^{*}}\\\
V^{D_{s}\bar{D}_{s}\to D_{s}^{*}\bar{D}_{s}^{*}}&V^{D_{s}\bar{D}_{s}\to
D^{*}\bar{D}^{*}}&V^{D_{s}^{*}\bar{D}_{s}^{*}\to
D_{s}^{*}\bar{D}_{s}^{*}}\end{array}\right).$ (20)

The corresponding subpotentials read as

$\displaystyle V^{D_{s}\bar{D}_{s}\to D_{s}\bar{D}_{s}}$ $\displaystyle=$
$\displaystyle-
g_{s}^{2}Y(\Lambda,m_{\sigma},r)-\frac{1}{2}\beta^{2}g_{V}^{2}Y(\Lambda,m_{\phi},r),$
(21) $\displaystyle V^{D_{s}\bar{D}_{s}\to D^{*}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{3}\frac{g^{2}}{f_{\pi}^{2}}\mathcal{Z}^{12}_{\Lambda,m_{K}}-\frac{2\sqrt{2}}{3}\lambda^{2}g_{V}^{2}\mathcal{Z}^{\prime
12}_{\Lambda,m_{K^{*}}},$ (22) $\displaystyle V^{D_{s}\bar{D}_{s}\to
D_{s}^{*}\bar{D}_{s}^{*}}$ $\displaystyle=$
$\displaystyle\frac{2}{9}\frac{g^{2}}{f_{\pi}^{2}}\mathcal{Z}^{13}_{\Lambda,m_{\eta}}-\frac{2}{3}\lambda^{2}g_{V}^{2}\mathcal{Z}^{\prime
13}_{\Lambda,m_{\phi}},$ (23) $\displaystyle V^{D^{*}\bar{D}^{*}\to
D^{*}\bar{D}^{*}}$ $\displaystyle=$ $\displaystyle-
g_{s}^{2}\mathcal{Y}^{22}_{\Lambda,m_{\sigma}}-\frac{1}{2}\frac{g^{2}}{f_{\pi}^{2}}\mathcal{Z}^{22}_{\Lambda,m_{\pi}}-\frac{1}{18}\frac{g^{2}}{f_{\pi}^{2}}\mathcal{Z}^{22}_{\Lambda,m_{\eta}}$
(24)
$\displaystyle-\frac{3}{4}\beta^{2}g_{V}^{2}\mathcal{Y}^{22}_{\Lambda,m_{\rho}}+\lambda^{2}g_{V}^{2}\mathcal{Z}^{\prime
22}_{\Lambda,m_{\rho}}$
$\displaystyle-\beta^{2}g_{V}^{2}\mathcal{Y}^{22}_{\Lambda,m_{\omega}}+\frac{1}{3}\lambda^{2}g_{V}^{2}\mathcal{Z}^{\prime
22}_{\Lambda,m_{\omega}},$ $\displaystyle V^{D^{*}\bar{D}^{*}\to
D_{s}^{*}\bar{D}_{s}^{*}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{3}\frac{g^{2}}{f_{\pi}^{2}}\mathcal{Z}^{13}_{\Lambda,m_{K}}-\frac{\beta^{2}g_{V}^{2}}{\sqrt{2}}\mathcal{Y}^{13}_{\Lambda,m_{K^{*}}}$
(25) $\displaystyle+\frac{2\sqrt{2}}{3}\lambda^{2}g_{V}^{2}\mathcal{Z}^{\prime
13}_{\Lambda,m_{K^{*}}},$ $\displaystyle V^{D_{s}^{*}\bar{D}_{s}^{*}\to
D_{s}^{*}\bar{D}_{s}^{*}}$ $\displaystyle=$ $\displaystyle-
g_{s}^{2}\mathcal{Y}^{33}_{\Lambda,m_{\sigma}}-\frac{2}{9}\frac{g^{2}}{f_{\pi}^{2}}\mathcal{Z}^{33}_{\Lambda,m_{\eta}}-\frac{\beta^{2}g_{V}^{2}}{2}\mathcal{Y}^{33}_{\Lambda,m_{\phi}}$
(26) $\displaystyle+\frac{2}{3}\lambda^{2}g_{V}^{2}\mathcal{Z}^{\prime
33}_{\Lambda,m_{\phi}}.$

Here, we define several useful functions as follows, i.e.,

$\displaystyle Y(\Lambda,m,{r})$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r}(e^{-mr}-e^{-\Lambda r})-\frac{\Lambda^{2}-m^{2}}{8\pi\Lambda}e^{-\Lambda
r},$ (27) $\displaystyle\mathcal{Y}^{ij}_{\Lambda,m_{a}}$ $\displaystyle=$
$\displaystyle\mathcal{D}_{ij}Y(\Lambda,m_{a},r),$ (28)
$\displaystyle\mathcal{Z}^{ij}_{\Lambda,m_{a}}$ $\displaystyle=$
$\displaystyle\left(\mathcal{E}_{ij}\nabla^{2}+\mathcal{F}_{ij}r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)Y(\Lambda,m_{a},r),$ (29)
$\displaystyle\mathcal{Z}^{\prime ij}_{\Lambda,m_{a}}$ $\displaystyle=$
$\displaystyle\left(2\mathcal{E}_{ij}\nabla^{2}-\mathcal{F}_{ij}r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)Y(\Lambda,m_{a},r).$ (30)

In above effective potentials (21)-(26), $\mathcal{D}_{ij}$,
$\mathcal{E}_{ij}$, and $\mathcal{F}_{ij}$ stand for the operators for the
spin-spin interactions and the tensor forces, respectively. For example,
$\mathcal{E}_{12}=\mathcal{E}_{13}=\bm{\epsilon}_{3}^{{\dagger}}\cdot\bm{\epsilon}_{4}^{{\dagger}}$,
$\mathcal{F}_{12}=\mathcal{F}_{13}=S(\hat{r},\bm{\epsilon}_{3}^{{\dagger}},\bm{\epsilon}_{4}^{{\dagger}})$,
$\mathcal{D}_{22}=\mathcal{D}_{33}=(\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{3}^{{\dagger}})(\bm{\epsilon}_{2}\cdot\bm{\epsilon}_{4}^{{\dagger}})$,
$\mathcal{E}_{22}=\mathcal{E}_{23}=\mathcal{E}_{33}=(\bm{\epsilon}_{1}\times\bm{\epsilon}_{3}^{{\dagger}})\cdot(\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})$,
$\mathcal{F}_{22}=\mathcal{F}_{23}=\mathcal{F}_{33}=S(\hat{r},\bm{\epsilon}_{1}\times\bm{\epsilon}_{3}^{{\dagger}},\bm{\epsilon}_{2}\times\bm{\epsilon}_{4}^{{\dagger}})$.
In the numerical calculations, these operators $\mathcal{O}$ are replaced by
several nonzero matrix elements by employing $\langle f|\mathcal{O}|i\rangle$,
where the $|i\rangle$ and $\langle f|$ stand for the spin-orbit wave functions
for the initial and final states, respectively. For the $J^{PC}=0^{++}$
channel, $\langle f|\mathcal{E}_{12}|i\rangle\mapsto(-\sqrt{3},0)$, $\langle
f|\mathcal{F}_{12}|i\rangle\mapsto(0,\sqrt{6})$, $\langle
f|\mathcal{D}_{22}|i\rangle\mapsto\text{diag}(1,1)$, $\langle
f|\mathcal{E}_{22}|i\rangle\mapsto\left(\begin{array}[]{cc}2&0\\\
0&-1\end{array}\right)$, and $\langle
f|\mathcal{F}_{22}|i\rangle\mapsto\left(\begin{array}[]{cc}0&\sqrt{2}\\\
\sqrt{2}&2\end{array}\right)$.

##  III Numerical results

###  III.1 Newly $X(3960)$ as the isoscalar
$\mathcal{D}\overline{\mathcal{D}}-$type charmoniumlike resonance with
$J^{P}=0^{++}$

In general, the scattering cross section can be expressed as a function of the
energy of the incident particle and the phase shift, i.e.,
$\sigma(E)=({4\pi}/{2mE})\sum_{l=0}^{\infty}(2l+1)\text{sin}^{2}\delta_{l}$.
The resonance appears at the the cross section $\sigma(E)$ takes the maximum
$\sigma(E_{0})$, where the phase shift $\delta_{l}$ satisfies
$\delta=(n+1/2)\pi$ with $n=0,1,2,\ldots$. One can extract the resonance
parameters from the phase shift $\delta$ dependence of the scattering energy
$E$, i.e., $M_{r}=E_{0}(\delta=(n+1/2)\pi)$ with $n=0,1,2,\ldots$, and
$\Gamma=2/\left(\frac{d\delta(E)}{{d}E}\right)_{E_{0}}$.

In present work, we produce the phase shifts for the isoscalar
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems
with $J^{PC}=0^{++}$ to check the possibility of the newly $X(3960)$ as the
charmoniumlike resonance. In our OBE effective potentials, the cutoff
$\Lambda$ is a free parameter. Here, we attempt to obtain resonant parameters
by varying the cutoff value around the reasonable region $1.00\leq\Lambda\leq
2.00$ GeV.

Figure 1: The cutoff $\Lambda$ dependence of the obtained resonant parameters
(the mass $M$ and decay width $\Gamma$) for the isoscalar
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems
with $J^{PC}=0^{++}$. Here, the shallow area corresponds to the reported
experimental mass and width for the newly $X(3960)$ including the experimental
uncertainty.

In Figure 1, we present the cutoff $\Lambda$ dependence of the resonant mass
$M$ and width $\Gamma$ for the isoscalar
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems
with $J^{PC}=0^{++}$. Here, we can obtain a charmoniumlike resonance with the
cutoff around 1.55 GeV. It just locates below the $D^{*}\bar{D}^{*}$ mass
threshold, its width is around 10 MeV. When increasing the cutoff value, the
OBE effective potentials become more and more attractive, which makes the
resonant mass become smaller and smaller. Meanwhile, the decay width becomes
much larger. In particular, when the cutoff is taken larger than 1.65 GeV, we
can reproduce the mass of the newly observed $X(3960)$. In the cutoff region
from 1.57 GeV to 1.68 GeV, our results of the decay width is taken from 21 MeV
to 70 MeV, which is consistent with the experimental data of the newly
$X(3960)$ with the experimental uncertainties. Therefore, we can
simultaneously reproduce the mass and the decay width of the newly observed
$X(3960)$ in the same cutoff region $\Lambda\geq 1.65$ GeV. Since the cutoff
is close to the reasonable value Tornqvist (1994a, b), we can conclude that
the newly $X(3960)$ can be explained as the isoscalar charmoniumlike
$\mathcal{D}\overline{\mathcal{D}}-$type resonance with $J^{P}=0^{++}$, where
notation $\mathcal{D}$ stands for the charmed or charmed-strange mesons.

In order to explore the roles of the $D^{*}\bar{D}^{*}$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ channels in generating the $X(3960)$ as the
charmoniumlike $\mathcal{D}\overline{\mathcal{D}}-$type resonant, we further
analyze the phase shifts for the $D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}$ coupled
systems with $J^{P}=0^{++}$ and the
$D_{s}\bar{D}_{s}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems with
$J^{P}=0^{++}$. In Figure 2, we present the cutoff dependence of the obtained
masses and widths for the resonances in the
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}$ coupled systems with $J^{P}=0^{++}$ and
the $D_{s}\bar{D}_{s}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems with
$J^{P}=0^{++}$, respectively. Here, we can see that

  1. 1.

For the $D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}$ coupled systems with
$J^{P}=0^{++}$, we can obtain a charmoniumlike resonance in the cutoff region
$1.60\leq\Lambda\leq 1.9$ GeV. If we take the same resonance mass for the
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems
with $J^{PC}=0^{++}$, the cutoff here is a little larger. Therefore, the
$D_{s}^{*}\bar{D}_{s}^{*}$ channels can help to bind this resonance. However,
the obtained width here is very small compared to the experimental data for
the newly $X(3960)$, it is just several MeV or even smaller. Therefore, we can
conclude that the $D_{s}^{*}\bar{D}_{s}^{*}$ channels play a very important
role in generating the width of the $X(3960)$ as the charmoniumlike resonance.
In fact, as presented in Figure 2, there exists a much larger phase space for
the $D_{s}^{*}\bar{D}_{s}^{*}$ decaying to the $D_{s}D_{s}$ final state in
comparison with the $D^{*}\bar{D}^{*}$ decaying to the $D_{s}D_{s}$, the small
width here may be understandable as the result of the much larger phase space.
Anyway, if the newly $X(3960)$ can be regarded as the isoscalar charmoniumlike
$\mathcal{D}\overline{\mathcal{D}}-$type resonance with $J^{P}=0^{++}$, the
contribution from the $D_{s}^{*}\bar{D}_{s}^{*}$ cannot be ignored.

  2. 2.

For the $D_{s}\bar{D}_{s}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems with
$J^{P}=0^{++}$, the resonance appears at the cutoff larger than 2.29 GeV. It
is a little away from the cutoff value in the
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems
with $J^{PC}=0^{++}$, which shows that the attractions of the OBE effective
potentials in this coupled channel systems are a little weaker. Thus, the
$D^{*}\bar{D}^{*}$ system discarded here provides important dynamic
interactions in forming the $X(3960)$ as the charmoniumlike resonance. In
addition, when the obtained resonance mass overlaps to the reported mass of
the $X(3960)$, our theoretical calculations of the width are consistent with
the experimental data. This results again indicate that the
$D_{s}^{*}\bar{D}_{s}^{*}$ channel affects the width for the $X(3960)$ a lot.

Thus, from the current numerical results, both the $D^{*}\bar{D}^{*}$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ systems are very important in formation of the
$X(3960)$ as the charmoniumlike resonance.

Figure 2: The cutoff $\Lambda$ dependence of the obtained resonant parameters
(the mass $M$ and decay width $\Gamma$) for the
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}$ coupled systems with $J^{P}=0^{++}$ and
the $D_{s}\bar{D}_{s}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems with
$J^{P}=0^{++}$. Here, the red dots and the blue error bars correspond to the
obtained masses and widths $\Gamma$ for the discussed coupled channels,
respectively. The pink dash lines and the shallow areas label the center mass
$M=3955$ MeV and center width $\Gamma=48$ MeV for the reported $X(3960)$,
respectively.

###  III.2 Possible charmoniumlike states from the isoscalar
$D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ interactions
with $J^{PC}=1^{+-}$

Then, we extend our study on the isoscalar
$D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ interactions
with $J^{PC}=1^{+-}$ by using the same model. After considering the $S-D$ wave
mixing effects, the corresponding wave functions can be expanded as

$\displaystyle|\Psi(1^{+-})\rangle$ $\displaystyle=$
$\displaystyle\psi_{1}(r)D^{*}\bar{D}^{*}|^{3}S_{1}\rangle+\psi_{2}(r)D^{*}\bar{D}^{*}|^{3}D_{1}\rangle$
(31) $\displaystyle+$
$\displaystyle\psi_{3}(r)D^{*}\bar{D}^{*}|^{5}D_{1}\rangle+\psi_{4}(r)D_{s}\bar{D}_{s}^{*}|^{3}S_{1}\rangle$
$\displaystyle+$
$\displaystyle\psi_{5}(r)D_{s}\bar{D}_{s}^{*}|^{3}D_{1}\rangle+\psi_{6}(r)D_{s}^{*}\bar{D}_{s}^{*}|^{3}S_{1}\rangle$
$\displaystyle+$
$\displaystyle\psi_{7}(r)D_{s}^{*}\bar{D}_{s}^{*}|^{3}D_{1}\rangle+\psi_{8}(r)D_{s}^{*}\bar{D}_{s}^{*}|^{5}D_{1}\rangle.$

Their OBE effective potentials are written as

$\displaystyle V$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}V^{D^{*}\bar{D}^{*}\to
D^{*}\bar{D}^{*}}&V^{D_{s}\bar{D}_{s}^{*}\to
D^{*}\bar{D}^{*}}&V^{D_{s}^{*}\bar{D}_{s}^{*}\to D^{*}\bar{D}^{*}}\\\
V^{D^{*}\bar{D}^{*}\to D_{s}\bar{D}_{s}^{*}}&V^{D_{s}\bar{D}_{s}^{*}\to
D_{s}\bar{D}_{s}^{*}}&V^{D_{s}^{*}\bar{D}_{s}^{*}\to D_{s}\bar{D}_{s}^{*}}\\\
V^{D^{*}\bar{D}^{*}\to D_{s}^{*}\bar{D}_{s}^{*}}&V^{D_{s}\bar{D}_{s}^{*}\to
D_{s}^{*}\bar{D}_{s}^{*}}&V^{D_{s}^{*}\bar{D}_{s}^{*}\to
D_{s}^{*}\bar{D}_{s}^{*}}\end{array}\right).$ (35)

Here, the subpotentials read as

$\displaystyle V^{D_{s}\bar{D}_{s}^{*}\to D_{s}\bar{D}_{s}^{*}}$
$\displaystyle=$ $\displaystyle-
g_{s}^{2}\mathcal{Y}^{14}_{\Lambda,m_{\sigma}}-\frac{2}{9}\frac{g^{2}}{f_{\pi}^{2}}\mathcal{Z}^{41}_{\Lambda_{0},m_{\eta
0}}-\frac{\beta^{2}g_{V}^{2}}{2}\mathcal{Y}^{14}_{\Lambda,m_{\phi}}$ (36)
$\displaystyle+\frac{2}{3}\lambda^{2}g_{V}^{2}\mathcal{Z}^{\prime
41}_{\Lambda_{0},m_{\phi 0}},$ $\displaystyle V^{D_{s}\bar{D}_{s}^{*}\to
D^{*}\bar{D}^{*}}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\frac{g^{2}}{f_{\pi}^{2}}(\mathcal{Z}^{24}_{\Lambda_{1},m_{K1}}+\mathcal{Z}^{42}_{\Lambda_{1},m_{K1}})$
(37)
$\displaystyle+\frac{2}{3}\lambda^{2}g_{V}^{2}(\mathcal{X}^{24}_{\Lambda_{1},m_{K^{*}1}}+\mathcal{X}^{42}_{\Lambda_{1},m_{K^{*}1}}),$
$\displaystyle V^{D_{s}\bar{D}_{s}^{*}\to D_{s}^{*}\bar{D}_{s}^{*}}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{9}\frac{g^{2}}{f_{\pi}^{2}}(\mathcal{Z}^{34}_{\Lambda_{2},m_{\eta
2}}+\mathcal{Z}^{43}_{\Lambda_{2},m_{\eta 2}})$ (38)
$\displaystyle+\frac{\sqrt{2}}{3}\lambda^{2}g_{V}^{2}(\mathcal{X}^{34}_{\Lambda_{2},m_{\phi
2}}+\mathcal{X}^{43}_{\Lambda_{2},m_{\phi 2}}),$

with $\Lambda_{i}^{2}=\Lambda^{2}-q_{i}^{2}$,
$m_{ai}^{2}=m_{a}^{2}-q_{i}^{2}$, $i=0,1,2$, and
$q_{0}=m_{D_{s}^{*}}-m_{D_{s}}$,
$q_{1}=(m_{D_{s}^{*}}^{2}-m_{D_{s}}^{2})/4m_{D^{*}}$,
$q_{2}=(m_{D_{s}^{*}}^{2}-m_{D_{s}}^{2})/4m_{D_{s}^{*}}$. Here, we define

$\displaystyle\mathcal{X}^{ij}_{\Lambda,m_{a}}$ $\displaystyle=$
$\displaystyle\left(-2\mathcal{E}_{ij}\nabla^{2}-(\mathcal{F}_{ij}^{\prime}-\mathcal{F}_{ij}^{\prime\prime})r\frac{\partial}{\partial
r}\frac{1}{r}\frac{\partial}{\partial r}\right)Y(\Lambda,m_{a},r).$

In Table 1, we summary the corresponding matrix elements
$\langle{}^{2s^{\prime}+1}L^{\prime}_{J^{\prime}}|\mathcal{O}|{}^{2s+1}L_{J}\rangle$
for the spin-spin interactions and tensor force interactions operators in Eqs.
(36)-(38).

Table 1: Nonzero matrix elements $\langle{}^{2s^{\prime}+1}L^{\prime}_{J^{\prime}}|\mathcal{O}|{}^{2s+1}L_{J}\rangle$ in various channels for the spin-spin interactions and tensor force interactions operators in Eqs. (36)-(38). Here, $\mathcal{D}_{14}=\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{3}^{{\dagger}}$, $\mathcal{E}_{41}=\bm{\epsilon}_{1}\cdot\bm{\epsilon}_{4}^{{\dagger}}$, $\mathcal{F}_{41}=S(\hat{r},\bm{\epsilon}_{1},\bm{\epsilon}_{4}^{{\dagger}})$, $\mathcal{E}_{24}=\mathcal{E}_{34}=i\bm{\epsilon}_{3}^{{\dagger}}\cdot(\bm{\epsilon}_{4}^{{\dagger}}\times\bm{\epsilon}_{2})$, $\mathcal{F}_{24}=\mathcal{F}_{34}=iS(\hat{r},\bm{\epsilon}_{3}^{{\dagger}},\bm{\epsilon}_{4}^{{\dagger}}\times\bm{\epsilon}_{2})$, $\mathcal{F}_{24}^{\prime}=\mathcal{F}_{34}^{\prime}=iS(\hat{r},\bm{\epsilon}_{4}^{{\dagger}},\bm{\epsilon}_{2}\times\bm{\epsilon}_{3}^{{\dagger}})$, $\mathcal{F}_{24}^{\prime\prime}=\mathcal{F}_{34}^{\prime\prime}=iS(\hat{r},\bm{\epsilon}_{2},\bm{\epsilon}_{4}^{{\dagger}}\times\bm{\epsilon}_{3}^{{\dagger}})$, $\mathcal{E}_{42}=\mathcal{E}_{43}=i\bm{\epsilon}_{4}^{{\dagger}}\cdot(\bm{\epsilon}_{3}^{{\dagger}}\times\bm{\epsilon}_{1})$, $\mathcal{F}_{42}=\mathcal{F}_{43}=iS(\hat{r},\bm{\epsilon}_{4}^{{\dagger}},\bm{\epsilon}_{3}^{{\dagger}}\times\bm{\epsilon}_{1})$, $\mathcal{F}_{42}^{\prime}=\mathcal{F}_{43}^{\prime}=iS(\hat{r},\bm{\epsilon}_{3}^{{\dagger}},\bm{\epsilon}_{1}\times\bm{\epsilon}_{4}^{{\dagger}})$, $\mathcal{F}_{42}^{\prime\prime}=\mathcal{F}_{43}^{\prime\prime}=iS(\hat{r},\bm{\epsilon}_{1},\bm{\epsilon}_{3}^{{\dagger}}\times\bm{\epsilon}_{4}^{{\dagger}})$. $\mathcal{D}_{14}=\mathcal{E}_{41}$ | $\mathcal{F}_{41}$ | $\mathcal{E}_{24}=\mathcal{E}_{34}=-\mathcal{E}_{42}=-\mathcal{E}_{43}$
---|---|---
$\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$ | $\left(\begin{array}[]{cc}0&-\sqrt{2}\\\ -\sqrt{2}&1\end{array}\right)$ | $\left(\begin{array}[]{cc}\sqrt{2}&0\\\ 0&\sqrt{2}\\\ 0&0\end{array}\right)$
$\mathcal{F}_{24}=\mathcal{F}_{34}$ | $\mathcal{F}_{24}^{\prime}=\mathcal{F}_{34}^{\prime}$ | $\mathcal{F}_{24}^{\prime\prime}=\mathcal{F}_{34}^{\prime\prime}$
$-\mathcal{F}_{42}^{\prime}=-\mathcal{F}_{43}^{\prime}$ | $-\mathcal{F}_{42}=-\mathcal{F}_{43}$ | $-\mathcal{F}_{42}^{\prime\prime}=-\mathcal{F}_{43}^{\prime\prime}$
$\left(\begin{array}[]{cc}0&1\\\ 1&-\frac{1}{\sqrt{2}}\\\ \sqrt{3}&\sqrt{\frac{3}{2}}\end{array}\right)$ | $\left(\begin{array}[]{cc}0&1\\\ 1&-\frac{1}{\sqrt{2}}\\\ -\sqrt{3}&-\sqrt{\frac{3}{2}}\end{array}\right)$ | $\left(\begin{array}[]{cc}0&2\\\ 2&-\sqrt{2}\\\ 0&0\end{array}\right)$
Figure 3: The cutoff $\Lambda$ dependence of the resonant parameters (mass $M$
and decay width $\Gamma$) for the isoscalar
$D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D^{*}\bar{D}^{*}$ interactions with
$J^{PC}=1^{+-}$.

With these prepared OBE effective potentials, we produce the phase shifts for
the isoscalar $D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$
interactions with $J^{PC}=1^{+-}$. As shown in Figure 3, we present the cutoff
$\Lambda$ dependence of the resonant parameters (mass $M$ and decay width
$\Gamma$) for the isoscalar
$D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled
systems with $J^{PC}=1^{+-}$. When we tune the cutoff at $\Lambda=$1.50 GeV,
we can obtain a possible charmoniumlike resonant candidate (labeled as
$R_{1}$) with mass $M=4079.58$ MeV, which just locates below the
$D_{s}\bar{D}_{s}^{*}$ threshold $M_{D_{s}\bar{D}_{s}^{*}}=4080.54$ MeV. The
width is 1.37 MeV. When the cutoff reaches at $\Lambda=$1.72 GeV, the
attractions of the OBE effective potentials become stronger, the mass and
decay width for the $R_{1}$ state turn into 4069.57 MeV and 2.80 MeV,
respectively. In addition, we find another possible charmoniumlike resonant
candidate (labeled as $R_{2}$) slightly located below the
$D_{s}^{*}\bar{D}_{s}^{*}$ threshold, whose width is 5.93 MeV. As shown in
Figure 3, the width for the $R_{2}$ resonance is larger than that for the
$R_{1}$ resonance. Especially, in the cutoff region $1.70\leq\Lambda\leq 2.00$
GeV, the width for the $R_{2}$ resonance can reach around 20 MeV. And the
decay width for the $R_{1}$ resonance is still less than 3.00 MeV.

Here, we also find that the cutoff values are close to those in the case of
the newly observec $X(3960)$ as the isoscalar
$\mathcal{D}\overline{\mathcal{D}}-$type charmoniumlike resonance with
$J^{P}=0^{++}$ as shown in Figure 1. Therefore, if the $X(3960)$ can be
assigned as a charmoniumlike resonance, the $R_{1}$ and $R_{2}$ can be also
possible charmoniumlike resonant candidates. Their masses are very close to
the $D_{s}\bar{D}_{s}^{*}$ and $D_{s}^{*}\bar{D}_{s}^{*}$ thresholds,
respectively. The close threshold properties remind us the predictions in our
previous paper Wang et al. (2021), when we systematically study the
interactions between a charmed (charmed-strange) meson and an anti-charmed
(anti-charm-strange) meson by using the OBE model, we find the
$D_{s}\bar{D}_{s}^{*}$ state with $J^{PC}=1^{+-}$ and the
$D_{s}^{*}\bar{D}_{s}^{*}$ state with $J^{PC}=1^{+-}$ can be possible
molecular candidates. And the coupled channel effects play an important role
in generating the $D_{s}\bar{D}_{s}^{*}$ molecular state with $J^{PC}=1^{+-}$.
In comparison with the previous results, we can further conclude that these
two possible charmoniumlike resonance obtained in this paper are not new
structures but the $D_{s}\bar{D}_{s}^{*}$ molecule with $J^{PC}=1^{+-}$ and
the $D_{s}^{*}\bar{D}_{s}^{*}$ molecule with $J^{PC}=1^{+-}$.

##  IV Summary

Very recent, the LHCb Collaboration observed a neutral state $X(3960)$ in the
$D_{s}^{+}D_{s}^{-}$ invariant mass spectrum of the $B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}$ process Chen and Norella . According to the mass and
decay properties, the $X(3960)$ is very likely to be a charmoniumlike exotic
state. Up to now, the inner structure for the newly $X(3960)$ is still open to
discuss. In this work, we propose the $X(3960)$ as the isoscalar
$\mathcal{D}\overline{\mathcal{D}}-$type charmoniumlike resonance with
$J^{P}=0^{++}$, where $\mathcal{D}$ stands for the $S-$wave charmed and
charmed-strange mesons.

In order to examine our proposal, we analyze the phase shifts for the
$D_{s}\bar{D}_{s}/D^{*}\bar{D}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ coupled systems
with $J^{PC}=0^{++}$ after adopting the OBE effective potentials and
considering the $S-D$ wave mixing effects. Our results show that there can
exist a possible charmoniumike resonance in the reasonable cutoff input. The
obtained resonant mass and width are consistent with the experimental data of
the newly $X(3960)$. Here, we also find that both the $D^{*}\bar{D}^{*}$ and
$D_{s}^{*}\bar{D}_{s}^{*}$ channels will play important roles in binding and
impacting on the decay width of the $X(3960)$ as the charmoniumlike resonance,
respectively.

In addition, we adopt the same OBE model to study the isoscalar
$D^{*}\bar{D}^{*}/D_{s}\bar{D}_{s}^{*}/D_{s}^{*}\bar{D}_{s}^{*}$ interactions
with $J^{PC}=1^{+-}$. Finally, we obtain two possible charmoniumlike
structures, which can correspond to the $D_{s}\bar{D}_{s}^{*}$ molecular state
with $J^{PC}=1^{+-}$ and the $D_{s}^{*}\bar{D}_{s}^{*}$ molecular state with
$J^{PC}=1^{+-}$ Wang et al. (2021), whose widths are in orders of magnitudes
of several and several to several tens MeV, respectively. The $\eta_{c}\phi$
and $J/\psi\eta^{(\prime)}$ can be the important two-body hidden-charm decay
channels for these two bound states. The $D_{s}\bar{D}_{s}^{*}$ channel is
also the only open-charm decay mode for the $D_{s}^{*}\bar{D}_{s}^{*}$
molecular state with $J^{PC}=1^{+-}$.

The experimental progress, especially the improvement of experimental
techniques and the accumulation of the experimental data, will provide us a
good chance to explore the underlying mechanism or inner structures for the
new exotic states. We look forward to the further experiment to verify our
proposal.

## ACKNOWLEDGMENTS

R. C. is supported by the Xiaoxiang Scholars Programme of Hunan Normal
University.

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