Tetraquark molecular states in the $D_{s}\bar{D}_{s1}$ and
$D_{s}^{*}\bar{D}_{s0}^{*}$ mass spectrum

Zhi-Gang Wang 111E-mail<EMAIL_ADDRESS>, Xiao-Song Yang, Qi Xin

Department of Physics, North China Electric Power University, Baoding 071003,
P. R. China

###### Abstract

In the present work, we construct the color-singlet-color-singlet type four-
quark currents to investigate the $D_{s}\bar{D}_{s1}$ and
$D_{s}^{*}\bar{D}_{s0}^{*}$ tetraquark molecular states with the
$J^{PC}=1^{--}$ and $1^{-+}$ via the QCD sum rules, and obtain satisfactory
results. We can search for the $D_{s}\bar{D}_{s1}$ and
$D_{s}^{*}\bar{D}_{s0}^{*}$ tetraquark molecular states with the
$J^{PC}=1^{--}$ and $1^{-+}$ at the BESIII and Belle II in the future.

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark molecular states, QCD sum rules

##  1 Introduction

In 2008, the CLEO collaboration measured the cross sections of the processes
$e^{+}e^{-}\to D_{s}^{+}D_{s}^{-}$, $D_{s}^{*+}D_{s}^{-}$ and
$D_{s}^{*+}D_{s}^{*-}$ up to the center-of-mass energy $4.26\,\rm{GeV}$, and
observed no evidence of the $Y(4260)$ [1]. In 2010, the BaBar collaboration
measured the cross sections of the processes $e^{+}e^{-}\to
D_{s}^{+}D_{s}^{-}$, $D_{s}^{*+}D_{s}^{-}$ and $D_{s}^{*+}D_{s}^{*-}$ up to
the center-of-mass energy $6.2\,\rm{GeV}$ via initial-state radiation (ISR),
and observed no evidence of the $Y(4260)$ either [2]. Also in 2010, the Belle
collaboration measured the cross sections of the processes $e^{+}e^{-}\to
D_{s}^{+}D_{s}^{-}$, $D_{s}^{*+}D_{s}^{-}$ and $D_{s}^{*+}D_{s}^{*-}$ up to
the center-of-mass energy $5.0\,\rm{GeV}$ via initial-state radiation, and
observed that both the $e^{+}e^{-}\to D_{s}^{*+}D_{s}^{-}$ cross section and
$R$ ratio exhibit an obvious dip near the mass of the $Y(4260)$ [3]. In 2020,
the BESIII collaboration measured the cross sections of the processes
$e^{+}e^{-}\to D_{s}^{+}D_{s1}(2460)^{-}$ $+c.c.$ at the center-of-mass energy
$4.467\,\rm{GeV}-4.600\,\rm{GeV}$, and $e^{+}e^{-}\to
D_{s}^{\ast+}D_{s1}(2460)^{-}$ $+c.c.$ at the center-of-mass energy
$4.590\,\rm{GeV}-4.600\,\rm{GeV}$, and observed no obvious charmonium or
charmonium-like structure [4]. Recently, the BESIII collaboration measured the
cross sections of the processes $e^{+}e^{-}\to
D_{s}^{*+}D_{s0}^{*}(2317)^{-}+c.c.$ and $e^{+}e^{-}\to
D_{s}^{*+}D_{s1}(2460)^{-}+c.c.$ at the center-of-mass energy
$4.600\,\rm{GeV}-4.700\,\rm{GeV}$, and $e^{+}e^{-}\to
D_{s}^{*+}D_{s1}(2536)^{-}+c.c.$ at the center-of-mass energy
$4.660\,\rm{GeV}-4.700\,\rm{GeV}$, and observed no structure in either process
[5].

The assignments of the $Y(4260)$, such as the tetraquark state [6, 7, 8, 9],
tetraquark molecular state [10], hybrid state [11, 12], conventional
charmonium [13, 14], are still in hot debate. If there exist the color-
singlet-color-singlet type tetraquark states $c\bar{s}s\bar{c}$, irrespective
of weak bound states or higher resonances, they can decay to their
constituents, the $c\bar{s}$ and $s\bar{c}$ color-singlet clusters, through
the Okubo-Zweig-Iizuka super-allowed fall-apart mechanism saving feasible in
the phase-space. When the experimental data are accumulated, the BESIII and
Belle II collaborations maybe observe them in the $D_{s}\bar{D}_{s}$,
$D^{*}_{s}\bar{D}_{s}$, $D^{*}_{s}\bar{D}^{*}_{s}$, $D_{s}\bar{D}_{s1}$,
$D_{s}^{*}\bar{D}_{s1}$, $D^{*}_{s}\bar{D}_{s0}^{*}$, $\cdots$ invariant mass
spectrum in the $e^{+}e^{-}$ scattering processes, which can shed light on the
nature of the $X$, $Y$ and $Z$ states. It is necessary and important to
investigate the mass spectrum of the $D_{s}\bar{D}_{s}$,
$D_{s}\bar{D}_{s}^{*}$, $D_{s}^{*}\bar{D}_{s}^{*}$, $D_{s}\bar{D}_{s1}$,
$D_{s}^{*}\bar{D}_{s1}$, $D_{s}^{*}\bar{D}_{s0}^{*}$, $\cdots$ tetraquark
molecular states and make reliable predictions.

In Ref.[15], we accomplish the operator product expansion for the correlation
functions up to the vacuum condensates of dimension $10$ consistently and
investigate the ground state hidden-charm tetraquark molecular states without
strange, with strange and with hidden-strange, such as the $D\bar{D}^{*}$,
$D^{*}\bar{D}^{*}$, $D\bar{D}_{s}^{*}$, $D_{s}\bar{D}_{s}^{*}$,
$D^{*}\bar{D}_{s}^{*}$ and $D_{s}^{*}\bar{D}_{s}^{*}$ tetraquark molecular
states with the $J^{PC}=0^{++}$, $1^{++}$, $1^{+-}$ and $2^{++}$, via the QCD
sum rules comprehensively, and make possible assignments of the existing $X$,
$Y$ and $Z$ states, such as the $X_{c}(3872)$, $Z_{c}(3900)$,
$Z_{cs}(3985/4000)$, $Z_{c}(4020/4025)$.

In Ref.[16], we construct the color-singlet-color-singlet type four-quark
currents to explore tetraquark molecular states $D\bar{D}_{1}(2420)$ and
$D^{*}\bar{D}_{0}^{*}(2400)$ with the $J^{PC}=1^{--}$ and $1^{-+}$ via the QCD
sum rules by calculating the contributions of the vacuum condensates up to
dimension-10. The predictions only support assigning the $Y(4390)$ to be the
$D\bar{D}_{1}$ molecular state with the $J^{PC}=1^{--}$.

In Ref.[17], we construct the color-singlet-color-singlet type tensor current
to explore the neutral $D_{s}^{*}\bar{D}_{s1}-D_{s1}\bar{D}_{s}^{*}$
tetraquark molecular states with the $J^{PC}=1^{-+}$ via the QCD sum rules,
and obtain the molecule mass $4.67\pm 0.08\,\rm{GeV}$, which is in very good
agreement with the mass of the $X(4630)$ observed later by the LHCb
collaboration, $M_{X(4630)}=4626\pm 16{}^{+18}_{-110}\,\rm{MeV}$ [18].

All in all, the QCD sum rules is a powerful theoretical approach in exploring
the masses and decay widths of the $X$, $Y$ and $Z$ states, and has achieved
many successful descriptions in the scenario of tetraquark states [19, 20, 21,
22, 23, 24, 25, 26, 27, 28] or tetraquark molecular states [29, 30, 31, 32,
33, 34].

In the present work, we extend our previous works to explore tetraquark
molecular states $D_{s}\bar{D}_{s1}$ and $D_{s}^{*}\bar{D}_{s0}^{*}$ with the
$J^{PC}=1^{--}$ and $1^{-+}$ via the QCD sum rules by accomplishing the
operator product expansion up to the vacuum condensates of dimension 10
consistently, and make possible predictions to be confronted to the
experimental data at the BESIII and Belle II in the future.

The article is arranged as follows: we obtain the QCD sum rules for the vector
tetraquark molecular states in section 2; in section 3, we present the
numerical results and discussions; section 4 is reserved for our conclusion.

##  2 QCD sum rules for the vector tetraquark molecular states

Let us write down the correlation functions $\Pi_{\mu\nu}(p)$ in the QCD sum
rules,

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{ip\cdot x}\langle
0|T\left\\{J_{\mu}(x)J_{\nu}^{\dagger}(0)\right\\}|0\rangle\,,$ (1)

where the color-singlet-color-singlet type four-quark currents
$J_{\mu}(x)=J_{\mu}^{1}(x),\,J_{\mu}^{2}(x),\,J_{\mu}^{3}(x),\,J_{\mu}^{4}(x)$,

$\displaystyle J^{1}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{\\{}\bar{s}(x)i\gamma_{5}c(x)\bar{c}(x)\gamma_{\mu}\gamma_{5}s(x)-\bar{s}(x)\gamma_{\mu}\gamma_{5}c(x)\bar{c}(x)i\gamma_{5}s(x)\Big{\\}}\,,$
$\displaystyle J^{2}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{\\{}\bar{s}(x)i\gamma_{5}c(x)\bar{c}(x)\gamma_{\mu}\gamma_{5}s(x)+\bar{s}(x)\gamma_{\mu}\gamma_{5}c(x)\bar{c}(x)i\gamma_{5}s(x)\Big{\\}}\,,$
$\displaystyle J^{3}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{\\{}\bar{s}(x)c(x)\bar{c}(x)\gamma_{\mu}s(x)+\bar{s}(x)\gamma_{\mu}c(x)\bar{c}(x)s(x)\Big{\\}}\,,$
$\displaystyle J^{4}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{\\{}\bar{s}(x)c(x)\bar{c}(x)\gamma_{\mu}d(x)-\bar{s}(x)\gamma_{\mu}c(x)\bar{c}(x)s(x)\Big{\\}}\,.$
(2)

Under charge conjugation transform $\widehat{C}$, the currents $J_{\mu}(x)$
have the properties,

$\displaystyle\widehat{C}J^{1/3}_{\mu}(x)\widehat{C}^{-1}$ $\displaystyle=$
$\displaystyle-J^{1/3}_{\mu}(x)\,,$
$\displaystyle\widehat{C}J^{2/4}_{\mu}(x)\widehat{C}^{-1}$ $\displaystyle=$
$\displaystyle+J^{2/4}_{\mu}(x)\,,$ (3)

the currents $J_{\mu}(x)$ are eigenstates of the charge conjugation.

In the present work, we choose the local color-singlet-color-singlet type
four-quark currents to interpolate the hidden-charm tetraquark states, which
have two color-singlet clusters. The color-singlet clusters have the same
quantum numbers as the charmed mesons, such as $D_{s}$, $D_{s}^{*}$,
$D^{*}_{s0}$ and $D_{s1}$, except for the masses, as those color-singlet
clusters are not necessary to be the physical mesons.

The physical mesons with two valence quarks are spatial extended objects and
have mean spatial sizes about the magnitude $\sqrt{\langle r^{2}\rangle}\sim
0.5\,\rm{fm}$. In the present work or in the QCD sum rules, though we refer
the color-singlet-color-singlet type tetraquark states as the tetraquark
molecular states, they are not the usually called molecular states. They have
the mean/average spatial sizes as that of the typical heavy mesons, and are
compact objects. The usually called molecular states are loosely bound states
consist of the physical mesons, the mean spatial sizes are proportional to the
inverse of the binding energies, about $1\,\rm{fm}$ or larger than
$1\,\rm{fm}$, which are too large to be interpolated by the local four-quark
currents.

The currents $\bar{s}(x)i\gamma_{5}c(x)$ and $\bar{s}(x)\gamma_{\mu}c(x)$ have
the spin-parity $J^{P}=0^{-}$ and $1^{-}$, respectively, and couple
potentially to the mesons $D_{s}$ and $D_{s}^{*}$, respectively. While the
currents $\bar{s}(x)i\gamma_{5}\gamma_{5}c(x)$ and
$\bar{s}(x)\gamma_{\mu}\gamma_{5}c(x)$ have the spin-parity $J^{P}=0^{+}$ and
$1^{+}$, respectively, and couple potentially to the mesons $D^{*}_{s0}$ and
$D_{s1}$, respectively, as multiplying $\gamma_{5}$ to the currents changes
their parity, the net effects of the relative P-wave are embodied implicitly
in the $\gamma_{5}$. In the heavy quark limit, the total angular momentum of a
heavy-light meson $\vec{J}=\vec{j}_{\ell}+\vec{S}_{Q}$, the light quark total
angular momentum $\vec{j}_{\ell}=\vec{L}+\vec{S}_{\bar{q}}$, the $\vec{L}$ is
the light quark angular momentum, the $\vec{S}_{Q}$ and $\vec{S}_{\bar{q}}$
are the heavy quark and light quark spins, respectively. There exist two
doublets $(0^{+},1^{+})$ and $(1^{+},2^{+})$ for $j_{\ell}=\frac{1}{2}$ and
$\frac{3}{2}$, respectively. On the other hand, the heavy-light mesons can
also be classified in terms of eigenvalues of the light quark angular momentum
$\left|{}^{2S+1}L_{J}\right>$, $\vec{J}=\vec{S}+\vec{L}$,
$\vec{S}=\vec{S}_{\bar{q}}+\vec{S}_{Q}$ is a sum of the intrinsic quark spins.

In the present case, the two doublets are
$(D_{s0}^{*}(2317),D_{s1}^{\prime}(2460))$ and
$(D_{s1}(2536),D_{s2}^{*}(2573))$. We usually choose the currents
$\bar{s}(x)c(x)$ and $\bar{s}(x)\gamma_{\mu}\gamma_{5}c(x)$, which have the
total angular momenta, $\vec{J}=\vec{S}+\vec{L}$, i.e.
$\vec{0}=\vec{1}+\vec{1}$ and $\vec{1}=\vec{1}+\vec{1}$ respectively, to
interpolate the $D_{s0}^{*}(2317)$ and $D_{s1}^{\prime}(2460)$ respectively
[35]. In fact, there exists mixing effect between the $j_{\ell}=\frac{1}{2}$
and $\frac{3}{2}$ states with the spin-parity $J^{P}=1^{+}$,

$\displaystyle\left({\begin{array}[]{*{20}{c}}{\left|{{1^{+}},j_{\ell}={{\frac{1}{2}}}}\right\rangle}\\\
{\left|{{1^{+}},j_{\ell}={{\frac{3}{2}}}}\right\rangle}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}{\cos\theta}&{-\sin\theta}\\\
{\sin\theta}&{\cos\theta}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{\left|{{}^{3}{P_{1}}}\right\rangle}\\\
{\left|{{}^{1}{P_{1}}}\right\rangle}\\\ \end{array}}\right)\,,$ (10)

where the mixing angle $\tan\theta=\frac{1}{\sqrt{2}}$ in the heavy quark
limit [36]. So we cannot exclude the coupling between the current
$\bar{s}(x)\gamma_{\mu}\gamma_{5}c(x)$ and meson $D_{s1}(2536)$. In the
present work, we will not distinguish the $D_{s1}^{\prime}(2460)$ and
$D_{s1}(2536)$, and use the notation $D_{s1}$ to represent the color-singlet
cluster with the spin-parity $J^{P}=1^{+}$.

At the hadron side of the correlation functions $\Pi_{\mu\nu}(p)$, we isolate
the contributions of the ground state vector tetraquark molecular states $Y$,

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{Y}^{2}}{M_{Y}^{2}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,\,,$
(11) $\displaystyle=$
$\displaystyle\Pi(p^{2})\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,\,,$

where the pole residues $\lambda_{Y}$ are defined by $\langle
0|J_{\mu}(0)|Y(p)\rangle=\lambda_{Y}\,\varepsilon_{\mu}$, the
$\varepsilon_{\mu}$ are the polarization vectors.

We accomplish the operator product expansion up to the vacuum condensates of
dimension-10 consistently and assume vacuum saturation for the higher
dimensional vacuum condensates, and write the correlation functions
$\Pi(p^{2})$ at the QCD side in the form,

$\displaystyle\Pi(p^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{4m_{c}^{2}}^{\infty}ds\frac{{\rm
Im}\Pi(s)}{s-p^{2}}\,,$ (12)

through dispersion relation. In calculations, we contract the $s$ and $c$
quark fields in the correlation functions with Wick theorem, and substitute
the quark lines with the full $s$ and $c$ quark propagators $S_{ij}(x)$ and
$C_{ij}(x)$, respectively,

$\displaystyle S_{ij}(x)$ $\displaystyle=$
$\displaystyle\frac{i\delta_{ij}\\!\not\\!{x}}{2\pi^{2}x^{4}}-\frac{\delta_{ij}m_{s}}{4\pi^{2}x^{2}}-\frac{\delta_{ij}\langle\bar{s}s\rangle}{12}+\frac{i\delta_{ij}\\!\not\\!{x}m_{s}\langle\bar{s}s\rangle}{48}-\frac{\delta_{ij}x^{2}\langle\bar{s}g_{s}\sigma
Gs\rangle}{192}+\frac{i\delta_{ij}x^{2}\\!\not\\!{x}m_{s}\langle\bar{s}g_{s}\sigma
Gs\rangle}{1152}$ (13)
$\displaystyle-\frac{ig_{s}G^{a}_{\alpha\beta}t^{a}_{ij}(\\!\not\\!{x}\sigma^{\alpha\beta}+\sigma^{\alpha\beta}\\!\not\\!{x})}{32\pi^{2}x^{2}}-\frac{1}{8}\langle\bar{s}_{j}\sigma^{\mu\nu}s_{i}\rangle\sigma_{\mu\nu}+\cdots\,,$

$\displaystyle C_{ij}(x)$ $\displaystyle=$
$\displaystyle\frac{i}{(2\pi)^{4}}\int d^{4}ke^{-ik\cdot
x}\left\\{\frac{\delta_{ij}}{\\!\not\\!{k}-m_{c}}-\frac{g_{s}G^{n}_{\alpha\beta}t^{n}_{ij}}{4}\frac{\sigma^{\alpha\beta}(\\!\not\\!{k}+m_{c})+(\\!\not\\!{k}+m_{c})\sigma^{\alpha\beta}}{(k^{2}-m_{c}^{2})^{2}}\right.$
$\displaystyle\left.-\frac{g_{s}^{2}(t^{a}t^{b})_{ij}G^{a}_{\alpha\beta}G^{b}_{\mu\nu}(f^{\alpha\beta\mu\nu}+f^{\alpha\mu\beta\nu}+f^{\alpha\mu\nu\beta})}{4(k^{2}-m_{c}^{2})^{5}}+\cdots\right\\}\,,$
$\displaystyle f^{\alpha\beta\mu\nu}$ $\displaystyle=$
$\displaystyle(\\!\not\\!{k}+m_{c})\gamma^{\alpha}(\\!\not\\!{k}+m_{c})\gamma^{\beta}(\\!\not\\!{k}+m_{c})\gamma^{\mu}(\\!\not\\!{k}+m_{c})\gamma^{\nu}(\\!\not\\!{k}+m_{c})\,,$
(14)

to facilitate the cumbersome tasks, where $t^{n}=\frac{\lambda^{n}}{2}$, the
$\lambda^{n}$ is the Gell-Mann matrix [37]. In the full $s$-quark propagator,
see Eq.(13), the $s$-quark mass $m_{s}$ is taken as a mall quantity and is
treated perturbatively, direct calculations indicate that such a perturbative
treatment of the $s$-quark mass does not modify the dispersion relation
comparing to the massless light quarks. For more technical details, one can
consult Refs.[26, 27, 28, 33, 34].

Then we implement the quark-hadron duality below the continuum thresholds
$s_{0}$ and accomplish Borel transform in regard to the variable
$P^{2}=-p^{2}$ to acquire the QCD sum rules,

$\displaystyle\lambda^{2}_{Y}\,\exp\left(-\frac{M^{2}_{Y}}{T^{2}}\right)=\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{QCD}(s)\,\exp\left(-\frac{s}{T^{2}}\right)\,,$
(15)

the explicit expressions of the QCD spectral densities $\rho_{QCD}$ are
available via contacting the corresponding author by E-mail.

In the present work, we take account of the vacuum condensates
$\langle\bar{s}s\rangle$, $\langle\frac{\alpha_{s}GG}{\pi}\rangle$,
$\langle\bar{s}g_{s}\sigma Gs\rangle$, $\langle\bar{s}s\rangle^{2}$,
$g_{s}^{2}\langle\bar{s}s\rangle^{2}$,
$\langle\bar{s}s\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$,
$\langle\bar{s}s\rangle\langle\bar{s}g_{s}\sigma Gs\rangle$,
$\langle\bar{s}g_{s}\sigma Gs\rangle^{2}$ and
$\langle\bar{s}s\rangle^{2}\langle\frac{\alpha_{s}GG}{\pi}\rangle$. The four-
quark condensate $g_{s}^{2}\langle\bar{s}s\rangle^{2}$ originates from the
matrix elements
$\langle\bar{s}\gamma_{\mu}t^{a}sg_{s}D_{\eta}G^{a}_{\lambda\tau}\rangle$,
$\langle\bar{s}_{j}D^{\dagger}_{\mu}D^{\dagger}_{\nu}D^{\dagger}_{\alpha}s_{i}\rangle$
and $\langle\bar{s}_{j}D_{\mu}D_{\nu}D_{\alpha}s_{i}\rangle$, rather than
originates from the radiative corrections of the $\langle\bar{s}s\rangle^{2}$,
the strong fine structure constant $\alpha_{s}=\frac{g_{s}^{2}}{4\pi}$ appears
at the tree level. We adopt the truncations $n\leq 10$ and $k\leq 1$
consistently, the operators of the orders $\mathcal{O}(\alpha_{s}^{k})$ with
$k>1$ are discarded [26, 27, 28, 33, 34]. The condensates $\langle
g_{s}^{3}GGG\rangle$, $\langle\frac{\alpha_{s}GG}{\pi}\rangle^{2}$,
$\langle\frac{\alpha_{s}GG}{\pi}\rangle\langle\bar{s}g_{s}\sigma Gs\rangle$
have the dimensions of mass 6, 8, 9 respectively, but they are vacuum
expectations of the operators of the order $\mathcal{O}(\alpha_{s}^{3/2})$,
$\mathcal{O}(\alpha_{s}^{2})$, $\mathcal{O}(\alpha_{s}^{3/2})$ respectively,
and are discarded for a consistent treatment.

We differentiate Eq.(15) in regard to $\tau=\frac{1}{T^{2}}$, and eliminate
the pole residues $\lambda_{Y}$ to acquire the QCD sum rules for the molecule
masses,

$\displaystyle
M^{2}_{Y}=\frac{-\frac{d}{d\tau}\int_{4m_{c}^{2}}^{s_{0}}ds\rho_{QCD}(s)e^{-\tau
s}}{\int_{4m_{c}^{2}}^{s_{0}}ds\rho_{QCD}(s)e^{-\tau s}}\,.$ (16)

##  3 Numerical results and discussions

At the QCD side, we adopt the standard values of the vacuum condensates
$\langle\bar{q}q\rangle=-(0.24\pm 0.01\,\rm{GeV})^{3}$,
$\langle\bar{s}s\rangle=(0.8\pm 0.1)\langle\bar{q}q\rangle$,
$\langle\bar{s}g_{s}\sigma Gs\rangle=m_{0}^{2}\langle\bar{s}s\rangle$,
$m_{0}^{2}=(0.8\pm 0.1)\,\rm{GeV}^{2}$,
$\langle\frac{\alpha_{s}GG}{\pi}\rangle=(0.33\,\rm{GeV})^{4}$ at the energy
scale $\mu=1\,\rm{GeV}$ [37, 38, 39, 40], and take the $\overline{MS}$ masses
$m_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}$ and
$m_{s}(\mu=2\,\rm{GeV})=(0.095\pm 0.005)\,\rm{GeV}$ from the Particle Data
Group [41]. In addition, we take account of the energy-scale dependence of the
quark condensates, mixed quark condensates and $\overline{MS}$ masses in
regard to the renormalization group equation [42],

$\displaystyle\langle\bar{s}s\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{s}s\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$ $\displaystyle
m_{c}(\mu)$ $\displaystyle=$ $\displaystyle
m_{c}(m_{c})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle m_{s}(\mu)$ $\displaystyle=$ $\displaystyle m_{s}({\rm
2GeV})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}({\rm
2GeV})}\right]^{\frac{12}{33-2n_{f}}}\,,$ $\displaystyle\alpha_{s}(\mu)$
$\displaystyle=$
$\displaystyle\frac{1}{b_{0}t}\left[1-\frac{b_{1}}{b_{0}^{2}}\frac{\log
t}{t}+\frac{b_{1}^{2}(\log^{2}{t}-\log{t}-1)+b_{0}b_{2}}{b_{0}^{4}t^{2}}\right]\,,$
(17)

as the strong fine-structure constant $\alpha_{s}$ already appears at the tree
level, where $t=\log\frac{\mu^{2}}{\Lambda^{2}}$,
$b_{0}=\frac{33-2n_{f}}{12\pi}$, $b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$,
$b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$,
$\Lambda=213\,\rm{MeV}$, $296\,\rm{MeV}$ and $339\,\rm{MeV}$ for the quark
flavors $n_{f}=5$, $4$ and $3$, respectively [41]. As we investigate the
tetraquark molecular states with hidden-charm and hidden-strange, it is
naturel to choose the quark flavors $n_{f}=4$, and evolve the QCD spectral
densities $\rho_{QCD}(s)$ to the suitable energy scales $\mu$ to extract the
molecule masses.

In the present work, we take the energy scale formula
$\mu=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{c})^{2}}$ with the updated value of
the effective $c$-quark mass ${\mathbb{M}}_{c}=1.85\,\rm{GeV}$ to acquire the
suitable energy scales of the QCD spectral densities [16, 33, 34]. We
introduce the effective heavy quark mass $\mathbb{M}_{c}$ and divide the
tetraquark molecular states into both the heavy degrees of freedom
$2{\mathbb{M}}_{c}$ and light degrees of freedom
$\mu=\sqrt{M^{2}_{X/Y/Z/T}-(2{\mathbb{M}}_{c})^{2}}$ by setting
$m_{u}=m_{d}=0$. We can also consider the light flavor $SU(3)$ breaking
effects, and acquire the light degrees of freedom
$\mu=\sqrt{M^{2}_{X/Y/Z/T}-(2{\mathbb{M}}_{c})^{2}}-2\,m_{s}(\mu)$, in other
words, $\mu+2\,m_{s}(\mu)=\sqrt{M^{2}_{X/Y/Z/T}-(2{\mathbb{M}}_{c})^{2}}$.

We can rewrite the energy scale formula in the form,

$\displaystyle M^{2}_{X/Y/Z}$ $\displaystyle=$ $\displaystyle\mu^{2}+{\rm
Constants}\,,$ (18)

where the Constants have the value $4{\mathbb{M}}_{c}^{2}$ and fitted by the
QCD sum rules [27, 28, 33, 34], the predicted tetraquark (molecule) masses and
the pertinent/suitable energy scales of the QCD spectral densities have a
Regge-trajectory-like relation [43]. In calculations, we take account of the
light-flavor $SU(3)$ mass-breaking effects by subtracting a small $s$-quark
mass to obtain the modified energy scale formula
$\mu=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{c})^{2}}-2m_{s}(\mu)$. Analysis of
the $J/\psi$ and $\Upsilon$ with the famous Cornell potential, i.e. the
Coulomb-potential-plus-linear-potential, leads to the constituent quark masses
$m_{c}=1.84\,\rm{GeV}$ and $m_{b}=5.17\,\rm{GeV}$ [44]. We can set the
effective $c$-quark mass ${\mathbb{M}}_{c}=m_{c}=1.84\,\rm{GeV}$, which is
consistent with the updated value ${\mathbb{M}}_{c}=1.85\,\rm{GeV}$. In
numerical calculations, we add an uncertainty $\delta\mu=\pm 0.1\,\rm{GeV}$
considering the uncertainty of the ${\mathbb{M}}_{c}$.

At the beginning, we tentatively set the masses of the molecular states to be
the sum of the physical masses of the two charmed mesons, which correspond to
the two color-singlet clusters inside the molecular states, and obtain the
energy scales $\mu$ through the modified energy scale formula. Then we
calculate the molecule masses with the QCD sum rules by searching for the best
Borel parameters $T^{2}$ and continuum threshold parameters $s_{0}$ via trial
and error, and examine whether or not the modified energy scale formula is
satisfied. We vary the molecule masses therefore the energy scales $\mu$
slowly and steadily until reach the satisfactory results, and acquire the
Borel parameters, continuum threshold parameters, pole contributions and
optimal energy scales, which are shown plainly in Table 1.

From Table 1, we can see that the central values of the pole contributions are
larger than $50\%$, the pole dominance criterion can be satisfied very well.
In calculations, we observe that in the Borel windows, the dominant
contributions come from the perturbative terms, the contributions come from
the vacuum condensates of dimension $10$, the highest dimensional vacuum
condensates, are much less than $1\%$, the convergent behaviors of the
operator product expansion are very good.

Now we take account of all uncertainties of the input parameters, and obtain
the values of the masses and pole residues of the vector tetraquark molecular
states with hidden-charm and hidden-strange, which are shown plainly in Fig.1
or Table 1. From Table 1, we can infer that the threshold parameters and the
predicted masses satisfy the relation $\sqrt{s_{0}}=M_{Y}+(0.4\sim
0.6)\,\rm{GeV}$, which is consistent with our naive expectation, in addition,
we can also infer that the modified energy scale formula is well satisfied. In
Fig.1, we plot the tetraquark molecule masses with variations of the Borel
parameters at much larger intervals than the Borel windows, where the regions
between the two short vertical lines are the Borel windows. From the figure,
we can see clearly that there appear platforms really in the Borel windows,
and we expect to make reasonable predictions, which can be confronted to the
experimental data in the future.

From Table 1, we can see that the mass-splitting for the two molecular states
with the $J^{PC}=1^{--}$ is rather large, while the mass-splitting for the two
molecular states with the $J^{PC}=1^{-+}$ is rather small. We simplify the
analysis to explore the origination of the mass-splittings by taking the limit
$m_{s}\to 0$. For the $D_{s}\bar{D}_{s1}$ and $D_{s}^{*}\bar{D}_{s0}^{*}$
molecular states with the $J^{PC}=1^{--}$, the contributions of the quark
condensate $\langle\bar{s}{s}\rangle$ are zero, the contributions of the mixed
condensate $\langle\bar{s}g_{s}\sigma G{s}\rangle$ are of the same magnitude
but opposite sign. For the $D_{s}\bar{D}_{s1}$ and $D_{s}^{*}\bar{D}_{s0}^{*}$
molecular states with the $J^{PC}=1^{-+}$, the contributions of the vacuum
condensates $\langle\bar{s}{s}\rangle$ and $\langle\bar{s}g_{s}\sigma
G{s}\rangle$ are canceled out with each other severely, the net contributions
are very small. The different contributions of the vacuum condensates
$\langle\bar{s}{s}\rangle$ and $\langle\bar{s}g_{s}\sigma G{s}\rangle$ lead to
the different behaviors of the mass-splittings of the molecular states with
the $J^{PC}=1^{--}$ and $1^{-+}$.

In Table 2, we also present the predictions for the masses of the hidden-charm
tetraquark molecular states with one P-wave constituent (or color-singlet
cluster) in our previous works [16, 17, 45]. The predictions
$M_{D\bar{D}_{1}(1^{--})}=4.36\pm 0.08\,\rm{GeV}$ and
$M_{D_{s}^{*}\bar{D}_{s1}(1^{-+})}=4.67\pm 0.08\,\rm{GeV}$ are consistent with
the experimental data $M_{Y(4390)}=4391.6\pm 6.3\pm 1.0\,\rm{MeV}$ and
$M_{X(4630)}=4626\pm 16{}^{+18}_{-110}\,\rm{MeV}$ respectively within
uncertainties [18, 46], and support assigning the $Y(4390)$ and $X(4630)$ to
be the $D\bar{D}_{1}(1^{--})$ and $D_{s}^{*}\bar{D}_{s1}(1^{-+})$ tetraquark
molecular states, respectively.

The $X(4630)$ was observed in the $J/\psi\phi$ invariant mass spectrum in the
$B^{+}\to J/\psi\phi K^{+}$ decays by the LHCb collaboration and has the
quantum numbers $J^{PC}=1^{-+}$ [18]. The $Y(4630)$ was observed in the
$\Lambda_{c}^{+}\Lambda_{c}^{-}$ invariant mass spectrum in the exclusive
process $e^{+}e^{-}\to\gamma_{ISR}\Lambda_{c}^{+}\Lambda_{c}^{-}$ by the Belle
collaboration, and has the mass $M_{Y}=4634^{+8}_{-7}{}^{+5}_{-8}\,\rm{MeV}$
and width $\Gamma_{Y}=92^{+40}_{-24}{}^{+10}_{-21}\,\rm{MeV}$, respectively
[47], which are consistent with that of the charmonium-like state $Y(4660)$
within errors [41]. However, precise measurement of the
$e^{+}e^{-}\to\Lambda_{c}^{+}\Lambda_{c}^{-}$ cross section near the threshold
by the BESIII collaboration indicates that there maybe exist a bound state
below the $\Lambda_{c}^{+}\Lambda_{c}^{-}$ threshold, which differs from the
$Y(4630/4660)$ remarkably [48, 49]. In Ref.[50], we take the $Y(4660)$ as the
tetraquark state with the $J^{PC}=1^{--}$ and study the strong decays
$Y(4660)\to J/\psi f_{0}(980)$, $\eta_{c}\phi$, $\chi_{c0}\phi$,
$D_{s}\bar{D}_{s}$, $D_{s}^{*}\bar{D}^{*}_{s}$, $D_{s}\bar{D}^{*}_{s}$,
$D_{s}^{*}\bar{D}_{s}$, $\psi^{\prime}\pi^{+}\pi^{-}$, $J/\psi\phi$ with the
QCD sum rules based on rigorous quark-hadron quality, and observe that the
decay to $J/\psi\phi$ is greatly suppressed or forbidden. The predicted width
$\Gamma(Y(4660))=74.2^{+29.2}_{-19.2}\,{\rm{MeV}}$ supports assigning the
$Y(4660)$ to be the
$[sc]_{P}[\bar{s}\bar{c}]_{A}-[sc]_{A}[\bar{s}\bar{c}]_{P}$ type tetraquark
state with the $J^{PC}=1^{--}$. The $X(4630)$,
$\Lambda_{c}^{+}\Lambda_{c}^{-}$ resonance and $Y(4660)$ are different
particles.

At the present time, there are no experimental candidates for the
$D_{s}\bar{D}_{s1}$ and $D_{s}^{*}\bar{D}_{s0}^{*}$ tetraquark molecular
states with the $J^{PC}=1^{--}$ and $1^{-+}$. In the scenario of tetraquark
molecular states, we can assign the $X(3872)$, $Z_{c}(3900/3885)$,
$Z_{cs}(3985/4000)$, $Z_{c}(4020/4025)$, $Y_{c}(4390)$, $X_{c}(4630)$ and
$Z_{b}(10610/10650)$ to be the tetraquark molecular states tentatively based
on the predicted masses from our previous QCD sum rules calculations [15, 16,
17, 33, 34],

$\displaystyle X(3872)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(D\overline{D}^{*}-D^{*}\overline{D}\right)\,\,\,({\rm
with}\,\,\,1^{++})\,,$ $\displaystyle Z_{c}(3900/3885)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(D\overline{D}^{*}+D^{*}\overline{D}\right)\,\,\,({\rm
with}\,\,\,1^{+-})\,,$ $\displaystyle Z_{c}(4020/4025)$ $\displaystyle=$
$\displaystyle D^{*}\overline{D}^{*}\,\,\,({\rm with}\,\,\,1^{+-})\,,$
$\displaystyle Z_{cs}(3985/4000)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(D\overline{D}_{s}^{*}\mp
D^{*}\overline{D}_{s}\right)\,\,\,({\rm with}\,\,\,1^{+\pm})\,,$
$\displaystyle Y_{c}(4390)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(D\bar{D}_{1}-D_{1}\bar{D}\right)\,\,\,({\rm
with}\,\,\,1^{--})\,,$ $\displaystyle X_{c}(4630)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(D_{s}^{*}\bar{D}_{s1}-D_{s1}\bar{D}_{s}^{*}\right)\,\,\,({\rm
with}\,\,\,1^{-+})\,,$ $\displaystyle Z_{b}(10610)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(B\overline{B}^{*}+B^{*}\overline{B}\right)\,\,\,({\rm
with}\,\,\,1^{+-})\,,$ $\displaystyle Z_{b}(10650)$ $\displaystyle=$
$\displaystyle B^{*}\overline{B}^{*}\,\,\,({\rm with}\,\,\,1^{+-})\,,$ (19)

the $D_{s}\bar{D}_{s1}$ molecular states were also discussed in Ref.[51]
recently.

| $T^{2}(\rm{GeV}^{2})$ | $\sqrt{s_{0}}(\rm{GeV})$ | pole | $\mu(\rm{GeV})$ | $M_{Y}(\rm{GeV})$ | $\lambda_{Y}(10^{-2}\rm{GeV}^{5})$
---|---|---|---|---|---|---
$D_{s}\bar{D}_{s1}$ ($1^{--}$) | $3.3-3.7$ | $5.0\pm 0.1$ | $(45-64)\%$ | $2.3$ | $4.48\pm 0.08$ | $4.47\pm 0.61$
$D_{s}\bar{D}_{s1}$ ($1^{-+}$) | $3.6-4.0$ | $5.2\pm 0.1$ | $(44-63)\%$ | $2.7$ | $4.71\pm 0.10$ | $5.89\pm 0.74$
$D_{s}^{*}\bar{D}_{s0}^{*}$ ($1^{--}$) | $3.9-4.3$ | $5.3\pm 0.1$ | $(44-62)\%$ | $2.9$ | $4.80\pm 0.08$ | $7.46\pm 0.87$
$D_{s}^{*}\bar{D}_{s0}^{*}$ ($1^{-+}$) | $4.0-4.4$ | $5.3\pm 0.1$ | $(43-61)\%$ | $2.9$ | $4.79\pm 0.08$ | $7.49\pm 0.86$
Table 1:  The Borel parameters, continuum threshold parameters, pole
contributions, energy scales, masses and pole residues of the vector
tetraquark molecular states.

Figure 1:  The masses with variations of the Borel parameters $T^{2}$, where the $A$, $B$, $C$ and $D$ correspond to the tetraquark molecular states $D_{s}\bar{D}_{s1}(1^{--})$, $D_{s}\bar{D}_{s1}(1^{-+})$, $D_{s}^{*}\bar{D}_{s0}^{*}(1^{--})$ and $D_{s}^{*}\bar{D}_{s0}^{*}(1^{-+})$, respectively. The regions between the two short vertical lines are the Borel windows.  Molecule | $M_{Y}(\rm{GeV})$ | Assignment | References
---|---|---|---
$D\bar{D}_{1}$ ($1^{--}$) | $4.36\pm 0.08$ | ? $Y(4390)$ | [16]
$D^{*}\bar{D}_{0}^{*}$ ($1^{--}$) | $4.78\pm 0.07$ |  | [16]
$D\bar{D}_{1}$ ($1^{-+}$) | $4.60\pm 0.08$ |  | [16]
$D^{*}\bar{D}_{0}^{*}$ ($1^{-+}$) | $4.73\pm 0.07$ |  | [16]
$D_{s}\bar{D}_{s1}$ ($1^{--}$) | $4.48\pm 0.08$ |  | This work
$D_{s}^{*}\bar{D}_{s0}^{*}$ ($1^{--}$) | $4.80\pm 0.08$ |  | This work
$D_{s}\bar{D}_{s1}$ ($1^{-+}$) | $4.71\pm 0.10$ |  | This work
$D_{s}^{*}\bar{D}_{s0}^{*}$ ($1^{-+}$) | $4.79\pm 0.08$ |  | This work
$D_{s}^{*}\bar{D}_{s1}$ ($1^{-+}$) | $4.67\pm 0.08$ | ? $X(4630)$ | [17]
$D\bar{D}_{s0}^{*}$ ($0^{-}$) | $4.61\pm 0.10$ |  | [45]
$D^{*}\bar{D}_{s1}$ ($0^{-}$) | $4.60\pm 0.07$ |  | [45]
Table 2:  Assignments of the tetraquark molecular states with one P-wave
cluster.

Now let us perform Fierz re-arrangement for the four-quark currents $J_{\mu}$
both in the color space and Dirac-spinor space,

$\displaystyle 2\sqrt{2}J_{\mu}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}i\bar{s}\gamma_{\mu}s\,\bar{c}c-\frac{1}{3}i\bar{s}s\,\bar{c}\gamma_{\mu}c-\frac{1}{3}\bar{s}\gamma^{\beta}\gamma_{5}s\,\bar{c}\sigma_{\mu\beta}\gamma_{5}c+\frac{1}{3}\bar{s}\sigma_{\mu\beta}\gamma_{5}s\,\bar{c}\gamma^{\beta}\gamma_{5}c+\cdots\,,$
$\displaystyle 2\sqrt{2}J_{\mu}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bar{s}\sigma_{\mu\beta}s\,\bar{c}\gamma^{\beta}c+\frac{1}{3}\bar{s}\gamma^{\beta}s\,\bar{c}\sigma_{\mu\beta}c-\frac{1}{3}\bar{s}i\gamma_{5}s\,\bar{c}\gamma_{\mu}\gamma_{5}c-\frac{1}{3}\bar{s}\gamma_{\mu}\gamma_{5}s\,\bar{c}i\gamma_{5}c+\cdots\,,$
$\displaystyle 2\sqrt{2}J_{\mu}^{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\bar{s}\gamma_{\mu}s\,\bar{c}c-\frac{1}{3}\bar{s}s\,\bar{c}\gamma_{\mu}c-\frac{1}{3}i\bar{s}\gamma^{\beta}\gamma_{5}s\,\bar{c}\sigma_{\mu\beta}\gamma_{5}c-\frac{1}{3}i\bar{s}\sigma_{\mu\beta}\gamma_{5}s\,\bar{c}\gamma^{\beta}\gamma_{5}c+\cdots\,,$
$\displaystyle 2\sqrt{2}J_{\mu}^{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}i\bar{s}\sigma_{\mu\beta}s\,\bar{c}\gamma^{\beta}c+\frac{1}{3}i\bar{s}\gamma^{\beta}s\,\bar{c}\sigma_{\mu\beta}c+\frac{1}{3}i\bar{s}i\gamma_{5}s\,\bar{c}\gamma_{\mu}\gamma_{5}c-\frac{1}{3}i\bar{s}\gamma_{\mu}\gamma_{5}s\,\bar{c}i\gamma_{5}c+\cdots\,,$
(20)

to illustrate the two-body strong decays. The components $\bar{s}\Gamma
s\,\bar{c}\Gamma^{\prime}c$ with
$\Gamma,\Gamma^{\prime}=1,\,\gamma_{\mu},\,\gamma_{\mu}\gamma_{5},\,\cdots$
couple potentially to a series of $c\bar{c}$-$s\bar{s}$-type meson-pairs, or
$c\bar{c}s\bar{s}$-type tetraquark molecular states, which decay to their
components via the Okubo-Zweig-Iizuka super-allowed fall-apart mechanism. We
can investigate the $D_{s}\bar{D}_{s1}$ and $D_{s}^{*}\bar{D}_{s0}^{*}$
tetraquark molecular states with the $J^{PC}=1^{--}$ and $1^{-+}$ through the
two-body strong decays,

$\displaystyle D_{s}\bar{D}_{s1}(1^{--})$ $\displaystyle\to$
$\displaystyle\chi_{c0}\phi\,,\,J/\psi f_{0}(980)\,,\,J/\psi
f_{1}(1285)\,,\,J/\psi\eta\,,\,h_{c}\eta\,,\,\eta_{c}\phi\,,\,\chi_{c1}\phi\,,\,\eta_{c}h_{1}(1415)\,,$
$\displaystyle D_{s}\bar{D}_{s1}(1^{-+})$ $\displaystyle\to$ $\displaystyle
J/\psi\phi\,,\,J/\psi
h_{1}(1415)\,,\,h_{c}\phi\,,\,\chi_{c1}\eta\,,\,\eta_{c}\eta\,,\,\eta_{c}f_{1}(1285)\,,$
$\displaystyle D_{s}^{*}\bar{D}_{s0}^{*}(1^{--})$ $\displaystyle\to$
$\displaystyle\chi_{c0}\phi\,,\,J/\psi f_{0}(980)\,,\,J/\psi
f_{1}(1285)\,,\,J/\psi\eta\,,\,h_{c}\eta\,,\,\eta_{c}\phi\,,\,\chi_{c1}\phi\,,\,\eta_{c}h_{1}(1415)\,,$
$\displaystyle D_{s}^{*}\bar{D}_{s0}^{*}(1^{-+})$ $\displaystyle\to$
$\displaystyle J/\psi\phi\,,\,J/\psi
h_{1}(1415)\,,\,h_{c}\phi\,,\,\chi_{c1}\eta\,,\,\eta_{c}\eta\,,\,\eta_{c}f_{1}(1285)\,,$
(21)

besides the final states $D_{s}\bar{D}_{s1}$, $D_{s1}\bar{D}_{s}$,
$D_{s}^{*}\bar{D}_{s0}^{*}$ and $D_{s0}^{*}\bar{D}_{s}^{*}$.

##  4 Conclusion

In this article, we construct the color-singlet-color-singlet type four-quark
currents to investigate the $D_{s}\bar{D}_{s1}$ and
$D_{s}^{*}\bar{D}_{s0}^{*}$ tetraquark molecular states with the
$J^{PC}=1^{--}$ and $1^{-+}$ via the QCD sum rules. We accomplish the operator
product expansion up to the vacuum condensates of dimension-10 consistently,
take account of the $SU(3)$ mass-breaking effects and adopt the modified
energy scale formula to choose the best energy scales of the QCD spectral
densities, then extract the masses and pole residues of the tetraquark
molecular states in the suitable Borel windows, which satisfy the two
fundamental criteria of the QCD sum rules. We can search for the
$D_{s}\bar{D}_{s1}$ and $D_{s}^{*}\bar{D}_{s0}^{*}$ tetraquark molecular
states with the $J^{PC}=1^{--}$ and $1^{-+}$ at the BESIII and Belle II in the
future, and confront the present predictions to the experimental data.

## Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number
11775079.

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Proposed Knowledge Models for Intents and Services sec:know-models An intent
has been defined as “a set of operational goals (that a network should meet)
and outcomes (that a network is supposed to deliver), defined in a declarative
manner without specifying how to achieve or implement them” intentdefsrfc9315.
An intent can essentially be treated as a knowledge object with an associated
lifecycle. This knowledge must also be provided to the system in a way that
automated reasoning processes can convert it into a suitable system action.
This means that learning about the consumer’s expectations must be
communicated, disseminated, and regulated in a standardized manner. The
knowledge required to be expressed in intent is described as follows:

  * •

Functional requirements: consist of the type of service to be delivered along
with the expected functionality for the users;

  * •

Non-functional requirements: consist of the information required to explain
expectations of the required service e.g., kpi and associated metrics;

The representation of intent knowledge must follow a structured approach, and
knowledge graphs [know-graphs] provide an excellent information organization
tool.

##  Knowledge Graphs in ibn

Intents are considered knowledge objects with well-defined expressions based
on intrinsic vocabulary [intentdefsrfc9315] and semantic roles for different
entities like requirements, objectives, and associated constraints. A key goal
in any intent model is the standardization of the intent model to provide
interoperability for multi-domain and application use cases. The standardized
ICM [tmforum-intent-common-model-tr290] is a basic intent description
template, represented in RDF, that consists of knowledge entities necessary to
describe any type of intent ranging from domain-specific to multi-domain high-
level intents. We modify the icm with additional information and implement it
as a knowledge graph depicted in Figure •. The proposed extensions are
explained in the following subsections.

[width=0.49]Resources/Images/kg-combined-labeled2.png

Figure : Proposed intent model with service, resource and KPI extensions.

##  Proposed Service and Resource, and Service KPI Extension Models

A communication service is expected to have a distinct set of associated
deliverable metrics and kpi from a variety of standard well-defined
performance metrics [3gpp-5gkpis-28-554]. Hence, a service model consists of a
set of expectations that need to be fulfilled by the service provider for the
service consumers as per sla. These sets of expectations are defined as
distinct slo mapped onto a set of parameters such as availability,
reliability, latency, and bandwidth. We utilize icm as the basis for our
proposed knowledge base consisting of an implementation of intent extension
models for the 5g mc and nmc services listed in Table LABEL:tab:services. The
proposed extension models ensure compatibility by complementing several
information classes including icm:Expectation, icm:Parameter and, icm:Target
from the icm. These extended information classes are utilized to incorporate
domain-specific data for the services and resources for the intent deployment
in the network. The rdf classes in the extension models in Figure • are
described as follows:

###  Service and Resource Model

The resource model is extended via the icm:Target class. This is the intended
target for the deployment of an intent objective (e.g., a service). In the
proposed framework, this represents the necessary resource type (i.e.,
targetResource:GBR and targetResource:NGBR) for a specific category of
services as listed in Table LABEL:tab:services. Two possible subclasses are
defined in the service extension models for mc and nmc services requiring gbr
and ngbr resources (e.g., service:McpttGBRService and
service:NonMcpttNGBRService).

###  Service KPI Model

We utilize latency, packet error rate, priority, and 5G QoS identifier (5GQI)
as service kpi parameters included in the icm:PropertyParameter subclass of
the icm rdf model. These parameters contain the values as rdf literal terms
for different services requiring the gbr and ngbr resources in subclasses
kpi:latency, kpi:packeterrorrate, met:priority, and met:qi5G, respectively.
The kpi extension models for the mc and nmc services, shown in Figure •, are
based on the kpi from the 3gpp Table-5.7.4-1 [3gpp-5gsystem-23-501].
./Resources/Tablesservices The values in the extension models can be extracted
using SPARQL [sparql-w3c] queries during the processing and translation of
intents. Moreover, the information regarding expected resource type is also
modeled in the service models in order to enable the orchestration of relevant
resources from the available network resources. A generic framework for intent
processing using the proposed knowledge base is presented in Section
LABEL:sec:framework.

##  Querying the Knowledge Base

SPARQL is one of the three fundamental enablers alongside rdf and owl designed
for querying graph-based data. In SPARQL, the queries are focused on what the
user wants to know about the data rather than on the structure of the data.
The SPARQL syntax utilizes specific keywords to manipulate RDF graphs and
information stored in the knowledge base. For example, The keyword SELECT
matches the required data values from the RDF graphs, WHERE describes
information (relations and properties) about the queried data, and FILTER
modifies the data query. [language=SPARQL, frame=top, frame=bottom,
captionpos=b, caption=Sample query template for extraction of service KPIs
from the knowledge base where param refers to the service parameter and serv
refers to the service being queried., label=lst:serv-kpi-query, ] PREFIX icm:
¡http://tio.models.tmforum.org/tio/ v2.0.0/IntentCommonModel/¿ SELECT
?parameter ?value WHERE ?service ?property [ icm:valueBy [ ?parameter ?value ]
] . FILTER (?parameter = —param— & ?service = —serv—) These common features
between RDF graphs and SPARQL query structure motivate us to use SPARQL syntax
to perform queries for the extraction of relevant information from the
knowledge base. This is done successively for the extraction of intent
templates, services, and resource information from the knowledge base, as well
as the reported intent compliance information. A sample of the SPARQL query to
retrieve service specific kpi from the service extension model for a generic
service serv with associated kpi param is shown in Listing LABEL:lst:serv-kpi-
query.

##  Applicability of the Proposed Extension Models

A major advantage of the proposed domain-independent knowledge base is the
feasibility to initialize new knowledge graphs to represent different concepts
for the ibn management and service orchestration. Thus, the proposed service
and kpi models can be extended for any type of service utilizing the icm and
having well-defined performance metrics as well as resource requirements. The
process for expanding the proposed intent model with new service extensions is
intuitive and described as:

  1. 1.

The resource type is specified by defining new subclasses of icm:Target. For
instance, a new subclass could be delay critical gbr from the 5g system
specifications [3gpp-5gsystem-23-501].

  2. 2.

The service kpi is specified by defining the subclass of icm:PropertyParameter
for each new kpi along with respective definitions for the icm: Expectation
class.

  3. 3.

A new service could then be defined using the new resource target subclass of
the icm:Target and the service kpi. This step consists of the recommended
values for the literal bounds for each kpi parameter for different service
types.

  4. 4.

The initialization of new information sources in the knowledge base is
completed with the introduction of newly added parameters and service
expectations in the reporting subclasses of the icm namely
icm:RequirementReporter.

The described extensibility is matched by SPARQL, which provides an efficient
interface to the knowledge base for retrieval of information since one can
design queries that remain valid even when new services or resources are added
to the network. It is also intuitive to utilize SPARQL queries to introduce
performance metrics of different deployed services in the form of
icm:IntentReport in order to make them accessible in the knowledge base.

[width=1]Resources/Images/ibn-framework2.png

Figure : Proposed intent processing framework with knowledge-based ibn model.