The decay widths of the $Z_{cs}(3985/4000)$ based on rigorous quark-hadron
duality

Zhi-Gang Wang 111E-mail<EMAIL_ADDRESS>
Department of Physics, North China Electric Power University, Baoding 071003,
P. R. China

###### Abstract

In this work, we explore the hadronic coupling constants $G_{ZJ/\psi K}$,
$G_{Z\eta_{c}K^{*}}$, $G_{ZD^{*}\bar{D}_{s}}$ of the exotic states
$Z_{cs}(3985/4000)$ both in the pictures of the tetraquark states and
molecular states with the tentative assignments $J^{PC}=1^{+-}$ based on the
rigorous quark-hadron duality. Then we obtain the total widths
$\Gamma_{Z_{cs}}^{T}=15.31\pm 2.06\,\rm{MeV}$ and
$\Gamma_{Z_{cs}}^{M}=83.51\pm 21.09\,\rm{MeV}$, which are consistent with the
experimental data $13.8^{+8.1}_{-5.2}\pm 4.9\,\rm{MeV}$ from the BESIII
collaboration and $131\pm 15\pm 26\,\rm{MeV}$ from the LHCb collaboration,
respectively, and support assigning the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ to
be the hidden-charm tetraquark state and molecular state with the
$J^{PC}=1^{+-}$, respectively.

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark state, QCD sum rules

##  1 Introduction

In 2013, the BESIII and Belle collaborations explored the process
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$, and observed a structure
$Z_{c}^{\pm}(3900)$ in the $\pi^{\pm}J/\psi$ mass spectrum [1, 2]. Also in
2013, the BESIII collaboration investigated the process $e^{+}e^{-}\to\pi
D\bar{D}^{*}$, and observed a structure $Z_{c}^{\pm}(3885)$ in the
$(D\bar{D}^{*})^{\pm}$ mass spectrum [3]. The $Z_{c}(3900/3885)$ have the
spin-parity $J^{P}=1^{+}$ [3, 4].

In 2020, the BESIII collaboration observed a structure $Z_{cs}^{-}(3985)$ in
the $K^{+}$ recoil-mass spectrum in the processes of the $e^{+}e^{-}\to
K^{+}(D_{s}^{-}D^{*0}+D^{*-}_{s}D^{0})$ [5]. The Breit-Wigner mass and width
are $3985.2^{+2.1}_{-2.0}\pm 1.7\,\rm{MeV}$ and $13.8^{+8.1}_{-5.2}\pm
4.9\,\rm{MeV}$, respectively [5].

In 2021, the LHCb collaboration observed two new exotic states
$Z_{cs}^{+}(4000)$ and $Z_{cs}^{+}(4220)$ in the $J/\psi K^{+}$ mass spectrum
in the process $B^{+}\to J/\psi\phi K^{+}$ [6]. The most significant state,
$Z_{cs}^{+}(4000)$, has the Breit-Wigner mass and width $4003\pm
6{}^{+4}_{-14}\,\rm{MeV}$ and $131\pm 15\pm 26\,\rm{MeV}$, respectively, and
the spin-parity $J^{P}=1^{+}$ [6].

The $Z_{c}(3900/3885)$ and $Z_{cs}(3985/4000)$ have analogous decay modes,

$\displaystyle Z^{\pm}_{c}(3900)$ $\displaystyle\to$ $\displaystyle
J/\psi\pi^{\pm}\,,$ $\displaystyle Z^{+}_{cs}(4000)$ $\displaystyle\to$
$\displaystyle J/\psi K^{+}\,,$ (1) $\displaystyle Z_{c}^{\pm}(3885)$
$\displaystyle\to$ $\displaystyle(D\bar{D}^{*})^{\pm}\,,$ $\displaystyle
Z_{cs}^{-}(3985)$ $\displaystyle\to$ $\displaystyle
D_{s}^{-}D^{*0}\,,\,D^{*-}_{s}D^{0}\,,$ (2)

which lead to the possible outcome that they should have analogous quark
structures. Although the Particle Data Group takes it for granted that the
$Z_{c}(3900)$ and $Z_{c}(3885)$ are the same particle according to the
analogous masses and widths [7], however, it is difficult to explain the ratio
[3],

$\displaystyle R_{exp}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Z_{c}(3885)\to D\bar{D}^{*})}{\Gamma(Z_{c}(3900)\to
J/\psi\pi)}=6.2\pm 1.1\pm 2.7\,,$ (3)

in a satisfactory way. On the other hand, the widths of the $Z^{-}_{cs}(3985)$
and $Z_{cs}^{+}(4000)$ are inconsistent with each other. The
$Z_{c}(3900/3885)$ and $Z_{cs}(3985/4000)$ may be four distinguished particles
indeed.

The exotic states $Z_{cs}(3985/4000)$ lie above the $D_{s}^{-}D^{*0}$ and
$D^{*-}_{s}D^{0}$ thresholds $3975.2\,\rm{MeV}$ and $3977.0\,\rm{MeV}$,
respectively, if they are molecular states indeed, we should introduce the
coupled-channel effects to account for the mass gaps and decay widths [8, 9,
10, 11, 12], or just re-scattering effects [13, 14]. It is more natural to
reproduce their masses in the diquark-antidiquark type tetraquark pictures
than in the color-singlet-color-singlet type tetraquark pictures, if we choose
the theoretical framework of potential quark models [15, 16].

In the QCD sum rules, we usually choose the color antitriplet-triplet type and
singlet-singlet type four-quark currents to interpolate the hidden-charm
exotic states, and can reproduce the masses of the $Z_{cs}(3985/4000)$ both in
the pictures of the tetraquark states [17, 18, 19, 20, 21] and molecular
states [17, 21, 22, 23, 24, 25]. There maybe exist two $Z_{cs}$ states, an
antitriplet-triplet type tetraquark state and singlet-singlet type molecular
state, or only one $Z_{cs}$ state, which has both the antitriplet-triplet type
and singlet-singlet type components. We have to explore the two-body strong
decays to diagnose their nature.

In Ref.[26], we assign the $Z^{\pm}_{c}(3900)$ to be the diquark-antidiquark
type tetraquark state with the quantum numbers $J^{PC}=1^{+-}$, calculate the
hadronic coupling constants $G_{ZJ/\psi\pi}$, $G_{Z\eta_{c}\rho}$,
$G_{ZD\bar{D}^{*}}$ with the three-point QCD sum rules based on the rigorous
duality for the first time, then obtain the partial decay widths to diagnose
the nature of the $Z_{c}^{\pm}(3900)$. Thereafter, the rigorous duality has
been successfully applied to study the strong decays of the exotic states
$X(4140)$, $X(4274)$, $X(4660)$, $Z_{c}(4600)$ and $P_{c}(4312)$ [27, 28, 29,
30, 31, 32].

In this work, we extend our previous work on the masses of the hidden-charm
tetraquark states and molecular states with the strangeness [19, 25], to
explore the two-body strong decays of the $Z_{cs}(3985/4000)$ with the
possible assignments $J^{PC}=1^{+-}$, so as to diagnose their nature.

The article is arranged as follows: we acquire the QCD sum rules for the
hadronic coupling constants $G_{ZJ/\psi K}$, $G_{Z\eta_{c}K^{*}}$,
$G_{ZD^{*}\bar{D}_{s}}$ in section 2; in Sect.3, we present the numerical
results and discussions; and Sect.4 is reserved for our conclusion.

##  2 The hadronic coupling constants of the $Z_{cs}(3985/4000)$

We investigate the two-body strong decays $Z_{cs}^{-}(3985/4000)\to J/\psi
K^{-}$, $\eta_{c}K^{*-}$, $D^{*0}\bar{D}_{s}^{-}$, $D^{0}\bar{D}_{s}^{*-}$
using the three-point correlation functions $\Pi_{\mu\nu}^{1}(p,q)$,
$\Pi_{\mu\nu}^{2}(p,q)$ and $\Pi_{\mu\nu}^{3}(p,q)$, respectively,

$\displaystyle\Pi_{\mu\nu}^{1}(p,q)$ $\displaystyle=$ $\displaystyle i^{2}\int
d^{4}xd^{4}y\,e^{ipx}e^{iqy}\,\langle
0|T\left\\{J_{\mu}^{J/\psi}(x)J_{5}^{K}(y)J^{\dagger}_{\nu}(0)\right\\}|0\rangle\,,$
$\displaystyle\Pi_{\mu\nu}^{2}(p,q)$ $\displaystyle=$ $\displaystyle i^{2}\int
d^{4}xd^{4}y\,e^{ipx}e^{iqy}\,\langle
0|T\left\\{J_{5}^{\eta_{c}}(x)J_{\mu}^{K^{*}}(y)J^{\dagger}_{\nu}(0)\right\\}|0\rangle\,,$
$\displaystyle\Pi_{\mu\nu}^{3}(p,q)$ $\displaystyle=$ $\displaystyle i^{2}\int
d^{4}xd^{4}y\,e^{ipx}e^{iqy}\,\langle
0|T\left\\{J_{\mu}^{D^{*}}(x)J_{5}^{D_{s}}(y)J^{\dagger}_{\nu}(0)\right\\}|0\rangle\,,$
(4)

where the currents

$\displaystyle J_{\mu}^{J/\psi}(x)$ $\displaystyle=$
$\displaystyle\bar{c}(x)\gamma_{\mu}c(x)\,,$ $\displaystyle J_{5}^{K}(y)$
$\displaystyle=$ $\displaystyle\bar{u}(y)i\gamma_{5}s(y)\,,$ $\displaystyle
J_{5}^{\eta_{c}}(x)$ $\displaystyle=$
$\displaystyle\bar{c}(x)i\gamma_{5}c(x)\,,$ $\displaystyle J_{\mu}^{K^{*}}(y)$
$\displaystyle=$ $\displaystyle\bar{u}(y)\gamma_{\mu}s(y)\,,$ $\displaystyle
J_{\mu}^{D^{*}}(x)$ $\displaystyle=$
$\displaystyle\bar{u}(x)\gamma_{\mu}c(x)\,,$ $\displaystyle J_{5}^{D_{s}}(y)$
$\displaystyle=$ $\displaystyle\bar{c}(y)i\gamma_{5}s(y)\,,$ $\displaystyle
J^{\dagger}_{\nu}(0)$ $\displaystyle=$ $\displaystyle
J_{\nu}^{T\dagger}(0)\,,\,J_{\nu}^{M\dagger}(0)\,,$ (5) $\displaystyle
J_{\nu}^{T\dagger}(0)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{\\{}c^{Tn}(0)C\gamma_{\nu}u^{m}(0)\,\bar{c}^{k}(0)\gamma_{5}C\bar{s}^{Tj}(0)-c^{Tn}(0)C\gamma_{5}u^{m}(0)\,\bar{c}^{k}(0)\gamma_{\nu}C\bar{s}^{Tj}(0)\Big{\\}}\,,$
$\displaystyle J_{\nu}^{M\dagger}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{\\{}\bar{s}(0)\gamma_{\nu}c(0)\,\bar{c}(0)i\gamma_{5}u(0)+\bar{s}(0)i\gamma_{5}c(0)\,\bar{c}(0)\gamma_{\nu}u(0)\Big{\\}}\,,$
(6)

interpolate the mesons $J/\psi$, $K$, $\eta_{c}$, $K^{*}$, $D^{*}$, $D_{s}$,
$Z^{T}_{cs}$ and $Z^{M}_{cs}$, respectively, the superscripts $T$ and $M$
denote the tetraquark-type and molecule-type four-quark currents/tetraquarks,
respectively [19, 25].

We insert a complete set of intermediate hadronic states with the same quantum
numbers as the currents into the three-point correlation functions, and
isolate the ground state contributions,

$\displaystyle\Pi_{\mu\nu}^{T,1}(p,q)$ $\displaystyle=$
$\displaystyle\left\\{\frac{f_{K}M_{K}^{2}f_{J/\psi}M_{J/\psi}\lambda_{Z}^{T}G^{T}_{ZJ/\psi
K}}{m_{u}+m_{s}}\frac{i}{(M_{Z}^{2}-p^{\prime
2})(M_{J/\psi}^{2}-p^{2})(M_{K}^{2}-q^{2})}\right.$ (7)
$\displaystyle+\frac{i}{(M_{Z}^{2}-p^{\prime
2})(M_{J/\psi}^{2}-p^{2})}\int_{s^{0}_{K}}^{\infty}du\frac{\rho_{ZK^{\prime}}(p^{\prime
2},p^{2},u)}{u-q^{2}}$ $\displaystyle+\frac{i}{(M_{Z}^{2}-p^{\prime
2})(M_{K}^{2}-q^{2})}\int_{s^{0}_{J/\psi}}^{\infty}ds\frac{\rho_{Z\psi^{\prime}}(p^{\prime
2},s,q^{2})}{s-p^{2}}$
$\displaystyle\left.+\frac{i}{(M_{J/\psi}^{2}-p^{2})(M_{K}^{2}-q^{2})}\int_{s^{0}_{Z}}^{\infty}ds^{\prime}\frac{\rho_{Z^{\prime}J/\psi}(s^{\prime},p^{2},q^{2})+\rho_{Z^{\prime}K}(s^{\prime},p^{2},q^{2})}{s^{\prime}-p^{\prime
2}}+\cdots\right\\}g_{\mu\nu}+\cdots$ $\displaystyle=$ $\displaystyle
i\,\Pi_{T}^{1}(p^{\prime 2},p^{2},q^{2})\,g_{\mu\nu}+\cdots\,,$

$\displaystyle\Pi_{\mu\nu}^{M,1}(p,q)$ $\displaystyle=$
$\displaystyle\left\\{\frac{f_{K}M_{K}^{2}f_{J/\psi}M_{J/\psi}\lambda_{Z}^{M}G^{M}_{ZJ/\psi
K}}{m_{u}+m_{s}}\frac{1}{(M_{Z}^{2}-p^{\prime
2})(M_{J/\psi}^{2}-p^{2})(M_{K}^{2}-q^{2})}+\cdots\right\\}g_{\mu\nu}+\cdots$
(8) $\displaystyle=$ $\displaystyle\Pi^{1}_{M}(p^{\prime
2},p^{2},q^{2})\,g_{\mu\nu}+\cdots\,,$

$\displaystyle\Pi_{\mu\nu}^{T,2}(p,q)$ $\displaystyle=$
$\displaystyle\left\\{\frac{f_{\eta_{c}}M_{\eta_{c}}^{2}f_{K^{*}}M_{K^{*}}\lambda_{Z}^{T}G_{Z\eta_{c}K^{*}}^{T}}{2m_{c}}\frac{i}{(M_{Z}^{2}-p^{\prime
2})(M_{\eta_{c}}^{2}-p^{2})(M_{K^{*}}^{2}-q^{2})}+\cdots\right\\}g_{\mu\nu}+\cdots$
(9) $\displaystyle=$ $\displaystyle i\,\Pi_{T}^{2}(p^{\prime
2},p^{2},q^{2})\,g_{\mu\nu}+\cdots\,,$

$\displaystyle\Pi_{\mu\nu}^{M,2}(p,q)$ $\displaystyle=$
$\displaystyle\left\\{\frac{f_{\eta_{c}}M_{\eta_{c}}^{2}f_{K^{*}}M_{K^{*}}\lambda_{Z}^{M}G_{Z\eta_{c}K^{*}}^{M}}{2m_{c}}\frac{1}{(M_{Z}^{2}-p^{\prime
2})(M_{\eta_{c}}^{2}-p^{2})(M_{K^{*}}^{2}-q^{2})}+\cdots\right\\}g_{\mu\nu}+\cdots$
(10) $\displaystyle=$ $\displaystyle\Pi_{M}^{2}(p^{\prime
2},p^{2},q^{2})\,g_{\mu\nu}+\cdots\,,$

$\displaystyle\Pi_{\mu\nu}^{T,3}(p,q)$ $\displaystyle=$
$\displaystyle\left\\{\frac{f_{D_{s}}M_{D_{s}}^{2}f_{D^{*}}M_{D^{*}}\lambda_{Z}^{T}G^{T}_{ZD^{*}\bar{D}_{s}}}{m_{c}+m_{s}}\frac{i}{(M_{Z}^{2}-p^{\prime
2})(M_{D^{*}}^{2}-p^{2})(M_{D_{s}}^{2}-q^{2})}+\cdots\right\\}g_{\mu\nu}+\cdots$
(11) $\displaystyle=$ $\displaystyle i\,\Pi_{T}^{3}(p^{\prime
2},p^{2},q^{2})\,g_{\mu\nu}+\cdots\,,$

$\displaystyle\Pi_{\mu\nu}^{M,3}(p,q)$ $\displaystyle=$
$\displaystyle\left\\{\frac{f_{D_{s}}M_{D_{s}}^{2}f_{D^{*}}M_{D^{*}}\lambda_{Z}^{M}G^{M}_{ZD^{*}\bar{D}_{s}}}{m_{c}+m_{s}}\frac{1}{(M_{Z}^{2}-p^{\prime
2})(M_{D^{*}}^{2}-p^{2})(M_{D_{s}}^{2}-q^{2})}+\cdots\right\\}g_{\mu\nu}+\cdots$
(12) $\displaystyle=$ $\displaystyle\Pi_{M}^{3}(p^{\prime
2},p^{2},q^{2})\,g_{\mu\nu}+\cdots\,,$

where $p^{\prime}=p+q$, the $f_{J/\psi}$, $f_{K}$, $f_{\eta_{c}}$,
$f_{K^{*}}$, $f_{D^{*}}$, $f_{D_{s}}$, $\lambda_{Z}^{T}$ and $\lambda_{Z}^{M}$
are the decay constants of the mesons $J/\psi$, $K$, $\eta_{c}$, $K^{*}$,
$D^{*}$, $D_{s}$, $Z_{cs}^{T}$ and $Z_{cs}^{M}$, respectively,

$\displaystyle\langle 0|J_{\mu}^{J/\psi}(0)|J/\psi(p)\rangle$ $\displaystyle=$
$\displaystyle f_{J/\psi}M_{J/\psi}\,\xi_{\mu}\,\,,$ $\displaystyle\langle
0|J_{\mu}^{K^{*}}(0)|K^{*}(q)\rangle$ $\displaystyle=$ $\displaystyle
f_{K^{*}}M_{K^{*}}\,\varepsilon_{\mu}\,\,,$ $\displaystyle\langle
0|J_{\mu}^{D^{*}}(0)|D^{*}(p)\rangle$ $\displaystyle=$ $\displaystyle
f_{D^{*}}M_{D^{*}}\,\varsigma_{\mu}\,\,,$ $\displaystyle\langle
0|J_{5}^{K}(0)|K(q)\rangle$ $\displaystyle=$
$\displaystyle\frac{f_{K}M_{K}^{2}}{m_{u}+m_{s}}\,\,,$ $\displaystyle\langle
0|J_{5}^{\eta_{c}}(0)|\eta_{c}(p)\rangle$ $\displaystyle=$
$\displaystyle\frac{f_{\eta_{c}}M_{\eta_{c}}^{2}}{2m_{c}}\,\,,$
$\displaystyle\langle 0|J_{5}^{D_{s}}(0)|D_{s}(q)\rangle$ $\displaystyle=$
$\displaystyle\frac{f_{D_{s}}M_{D_{s}}^{2}}{m_{c}+m_{s}}\,\,,$
$\displaystyle\langle 0|J^{T/M}_{\nu}(0)|Z_{cs}^{T/M}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\lambda_{Z}^{T/M}\,\zeta_{\nu}\,\,,$ (13)

and the $\xi$, $\varepsilon$, $\varsigma$ and $\zeta$ are polarization vectors
of the $J/\psi$, $K^{*}$, $D^{*}$ and $Z_{cs}^{T/M}$, respectively, the
hadronic coupling constants $G^{T}_{ZJ/\psi K}$, $G^{M}_{ZJ/\psi K}$,
$G^{T}_{Z\eta_{c}K^{*}}$, $G^{M}_{Z\eta_{c}K^{*}}$,
$G_{ZD^{*}\bar{D}_{s}}^{T}$ and $G_{ZD^{*}\bar{D}_{s}}^{M}$ are defined by

$\displaystyle\langle J/\psi(p)K(q)|Z^{T}_{cs}(p^{\prime})\rangle$
$\displaystyle=$
$\displaystyle-\xi^{*}(p)\cdot\zeta(p^{\prime})\,G^{T}_{ZJ/\psi K}\,,$
$\displaystyle\langle\eta_{c}(p)K^{*}(q)|Z_{cs}^{T}(p^{\prime})\rangle$
$\displaystyle=$
$\displaystyle-\varepsilon^{*}(q)\cdot\zeta(p^{\prime})\,G^{T}_{Z\eta_{c}K^{*}}\,,$
$\displaystyle\langle D^{*}(p)D_{s}(q)|Z^{T}_{cs}(p^{\prime})\rangle$
$\displaystyle=$
$\displaystyle-\varsigma^{*}(p)\cdot\zeta(p^{\prime})\,G^{T}_{ZD^{*}\bar{D}_{s}}\,,$
$\displaystyle\langle J/\psi(p)K(q)|Z^{M}_{cs}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
i\xi^{*}(p)\cdot\zeta(p^{\prime})\,G^{M}_{ZJ/\psi K}\,,$
$\displaystyle\langle\eta_{c}(p)K^{*}(q)|Z_{cs}^{M}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
i\varepsilon^{*}(q)\cdot\zeta(p^{\prime})\,G^{M}_{Z\eta_{c}K^{*}}\,,$
$\displaystyle\langle D^{*}(p)D_{s}(q)|Z^{M}_{cs}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
i\varsigma^{*}(p)\cdot\zeta(p^{\prime})\,G^{M}_{ZD^{*}\bar{D}_{s}}\,.$ (14)

The hadronic spectral densities $\rho_{ZK^{\prime}}(p^{\prime 2},p^{2},u)$,
$\rho_{Z\psi^{\prime}}(p^{\prime 2},s,q^{2})$,
$\rho_{Z^{\prime}J/\psi}(s^{\prime},p^{2},q^{2})$,
$\rho_{Z^{\prime}K}(s^{\prime},p^{2},q^{2})$ in the component
$\Pi_{T}^{1}(p^{\prime 2},p^{2},q^{2})$ stand for the transitions between the
ground states and continuum states (including the first radial excited
states). In fact, such hadronic spectral densities also exist in the big
$\\{\\}$ in the components $\Pi_{T}^{2}(p^{\prime 2},p^{2},q^{2})$,
$\Pi_{T}^{3}(p^{\prime 2},p^{2},q^{2})$, $\Pi_{M}^{1}(p^{\prime
2},p^{2},q^{2})$, $\Pi_{M}^{2}(p^{\prime 2},p^{2},q^{2})$,
$\Pi_{M}^{3}(p^{\prime 2},p^{2},q^{2})$, we neglect them for simplicity. In
this work, we choose the tensor structure $g_{\mu\nu}$ to explore the hadronic
coupling constants $G^{T/M}_{ZJ/\psi K}$, $G^{T/M}_{Z\eta_{c}K^{*}}$ and
$G^{T/M}_{ZD^{*}\bar{D}_{s}}$, and neglect other tensor structures for
simplicity.

We accomplish the operator product expansion up to the vacuum condensates of
dimension 5 and neglect the tiny gluon condensate contributions, just like in
our previous works [26, 27, 28, 29, 30], then obtain the QCD spectral
densities $\rho_{QCD}(p^{\prime 2},s,u)$ through double dispersion relation,
and write the correlation functions at the QCD side in the form,

$\displaystyle\Pi_{QCD}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\int_{\Delta_{s}^{2}}^{\infty}ds\int_{\Delta_{u}^{2}}^{\infty}du\frac{\rho_{QCD}(p^{\prime
2},s,u)}{(s-p^{2})(u-q^{2})}\,,$ (15)

where the $\Delta_{s}^{2}$ and $\Delta_{u}^{2}$ are the thresholds. While at
the hadron side, we obtain the hadron spectral densities
$\rho_{H}(s^{\prime},s,u)$ through triple dispersion relation, and write the
correlation functions in the form,

$\displaystyle\Pi_{H}(p^{\prime 2},p^{2},q^{2})$ $\displaystyle=$
$\displaystyle\int_{\Delta_{s}^{\prime
2}}^{\infty}ds^{\prime}\int_{\Delta_{s}^{2}}^{\infty}ds\int_{\Delta_{u}^{2}}^{\infty}du\frac{\rho_{H}(s^{\prime},s,u)}{(s^{\prime}-p^{\prime
2})(s-p^{2})(u-q^{2})}\,,$ (16)

according to Eqs.(7)-(12), where the $\Delta_{s}^{\prime 2}$ are the
thresholds. We match the hadron side with the QCD side of the correlation
functions bellow the continuum thresholds according to the dispersion relation
at the QCD side to warrant rigorous quark-hadron duality,

$\displaystyle\int_{\Delta_{s}^{2}}^{s_{0}}ds\int_{\Delta_{u}^{2}}^{u_{0}}du\frac{\rho_{QCD}(p^{\prime
2},s,u)}{(s-p^{2})(u-q^{2})}$ $\displaystyle=$
$\displaystyle\int_{\Delta_{s}^{2}}^{s_{0}}ds\int_{\Delta_{u}^{2}}^{u_{0}}du\left[\int_{\Delta_{s}^{\prime
2}}^{\infty}ds^{\prime}\frac{\rho_{H}(s^{\prime},s,u)}{(s^{\prime}-p^{\prime
2})(s-p^{2})(u-q^{2})}\right]\,,$ (17)

where the $s_{0}$ and $u_{0}$ are the continuum thresholds, we accomplish the
integral over $ds^{\prime}$ firstly, and introduce some unknown parameters to
parameterize the contributions involving the higher resonances and continuum
states in the $s^{\prime}$ channel [26, 27], we take account of the higher
resonances and continuum states rather than neglecting them. For example,
according to Eq.(7), we write down the correlation function at the hadron side
explicitly,

$\displaystyle\int_{\Delta_{s}^{2}}^{s_{0}}ds\int_{\Delta_{u}^{2}}^{u_{0}}du\frac{\rho_{QCD}(p^{\prime
2},s,u)}{(s-p^{2})(u-q^{2})}$ $\displaystyle=$ $\displaystyle\frac{A_{ZJ/\psi
K}}{(M_{Z}^{2}-p^{\prime
2})(M_{J/\psi}^{2}-p^{2})(M_{K}^{2}-q^{2})}+\frac{C^{T}_{J/\psi
K}}{(M_{J/\psi}^{2}-p^{2})(M_{K}^{2}-q^{2})}\,,$

where

$\displaystyle A_{ZJ/\psi K}$ $\displaystyle=$
$\displaystyle\frac{f_{K}M_{K}^{2}f_{J/\psi}M_{J/\psi}\lambda_{Z}^{T}G^{T}_{ZJ/\psi
K}}{m_{u}+m_{s}}\,,$ $\displaystyle C^{T}_{J/\psi K}$ $\displaystyle=$
$\displaystyle\int_{s^{0}_{Z}}^{\infty}ds^{\prime}\frac{\rho_{Z^{\prime}J/\psi}(s^{\prime},p^{2},q^{2})+\rho_{Z^{\prime}K}(s^{\prime},p^{2},q^{2})}{s^{\prime}-p^{\prime
2}}\,,$ (19)

where the unknown parameter $C_{J/\psi K}^{T}$ parameterizes the contributions
involving the higher resonances and continuum states with the same quantum
numbers as the $Z_{cs}$ (in the $s^{\prime}$ channel), the unknown parameters
$C_{J/\psi K}^{M}$, $C_{\eta_{c}K^{*}}^{T}$, $C_{\eta_{c}K^{*}}^{M}$,
$C_{D^{*}\bar{D}_{s}}^{T}$, and $C_{D^{*}\bar{D}_{s}}^{M}$ in Eqs.(2)-(2) are
implied in the same way.

We accomplish the integral over the variable $ds^{\prime}$ firstly, and do not
need the continuum threshold parameters $s^{\prime}_{0}$, and the duality is
rigorous in the sense that we do not set $s_{0}=s^{\prime}_{0}=s^{0}_{Z}$
approximately [35]. If we set $s_{0}=s_{0}^{\prime}=s^{0}_{Z}$ in the present
case, then
$\sqrt{s_{0}}>M_{Z}>\psi^{\prime}>\eta_{c}^{\prime}>M_{D_{s}^{*\prime}}>M_{D_{s}^{\prime}}$,
the contaminations from the excited states are out of control. In fact,
without accomplishing the integral over the variable $ds^{\prime}$ firstly,
the representations at the QCD side and at the hadron side cannot match with
each other, up to now, no one can approve that they match with each other
without merging the $s$ and $s^{\prime}$ channels
$s_{0}=s^{\prime}_{0}=s^{0}_{Z}$ at the hadron side by hand, to be serious,
such approximations $s_{0}=s^{\prime}_{0}$ are wrong, as the $s$ and
$s^{\prime}$ channels are quite different.

In the $s$ and $u$ channels, we deal with the conventional mesons, and choose
the standard vector and pseudoscalar currents to interpolate them, direct
calculations indicate that we can reproduce their masses satisfactorily with
the two-point QCD sum rules below the continuum thresholds $s_{0}$ and
$u_{0}$, respectively, the quark-hadron duality is reliable.

We set $p^{\prime 2}=p^{2}$ and $p^{\prime 2}=4p^{2}$ in the correlation
functions $\Pi^{1/2}_{T/M}(p^{\prime 2},p^{2},q^{2})$ and
$\Pi^{3}_{T/M}(p^{\prime 2},p^{2},q^{2})$, respectively, and accomplish the
double Borel transform in regard to the variables $P^{2}=-p^{2}$ and
$Q^{2}=-q^{2}$ respectively, then set the Borel parameters
$T_{1}^{2}=T_{2}^{2}=T^{2}$ to get the six QCD sum rules,

$\displaystyle\frac{f_{K}M_{K}^{2}f_{J/\psi}M_{J/\psi}\lambda_{Z}^{T}G^{T}_{ZJ/\psi
K}}{m_{u}+m_{s}}\frac{1}{M_{Z}^{2}-M_{J/\psi}^{2}}\left[\exp\left(-\frac{M_{J/\psi}^{2}}{T^{2}}\right)-\exp\left(-\frac{M_{Z}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{M_{K}^{2}}{T^{2}}\right)$
$\displaystyle+C^{T}_{J/\psi
K}\exp\left(-\frac{M_{J/\psi}^{2}+M_{K}^{2}}{T^{2}}\right)=\frac{1}{64\sqrt{2}\pi^{4}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\int_{0}^{u^{0}_{K}}du\sqrt{1-\frac{4m_{c}^{2}}{s}}u\left(2s+4m_{c}^{2}-3m_{s}m_{c}\right)$
$\displaystyle\exp\left(-\frac{s+u}{T^{2}}\right)-\frac{m_{s}\left[2\langle\bar{q}q\rangle+\langle\bar{s}s\rangle\right]}{24\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\left(s+2m_{c}^{2}\right)\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{288\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\frac{s+20m_{c}^{2}}{\sqrt{s\left(s-4m_{c}^{2}\right)}}\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{24\sqrt{2}\pi^{2}T^{2}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\left(s+2m_{c}^{2}\right)\exp\left(-\frac{s}{T^{2}}\right)\,,$
(20)

$\displaystyle\frac{f_{K}M_{K}^{2}f_{J/\psi}M_{J/\psi}\lambda_{Z}^{M}G^{M}_{ZJ/\psi
K}}{m_{u}+m_{s}}\frac{1}{M_{Z}^{2}-M_{J/\psi}^{2}}\left[\exp\left(-\frac{M_{J/\psi}^{2}}{T^{2}}\right)-\exp\left(-\frac{M_{Z}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{M_{K}^{2}}{T^{2}}\right)$
$\displaystyle+C^{M}_{J/\psi
K}\exp\left(-\frac{M_{J/\psi}^{2}+M_{K}^{2}}{T^{2}}\right)=\frac{1}{128\sqrt{2}\pi^{4}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\int_{0}^{u^{0}_{K}}du\sqrt{1-\frac{4m_{c}^{2}}{s}}u\left(2s+4m_{c}^{2}-3m_{s}m_{c}\right)$
$\displaystyle\exp\left(-\frac{s+u}{T^{2}}\right)-\frac{m_{s}\left[2\langle\bar{q}q\rangle+\langle\bar{s}s\rangle\right]}{48\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\left(s+2m_{c}^{2}\right)\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{288\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\frac{s+20m_{c}^{2}}{\sqrt{s\left(s-4m_{c}^{2}\right)}}\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{48\sqrt{2}\pi^{2}T^{2}}\int_{4m_{c}^{2}}^{s^{0}_{J/\psi}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\left(s+2m_{c}^{2}\right)\exp\left(-\frac{s}{T^{2}}\right)\,,$
(21)

$\displaystyle\frac{f_{\eta_{c}}M_{\eta_{c}}^{2}f_{K^{*}}M_{K^{*}}\lambda_{Z}^{T}G^{T}_{Z\eta_{c}K^{*}}}{2m_{c}}\frac{1}{M_{Z}^{2}-M_{\eta_{c}}^{2}}\left[\exp\left(-\frac{M_{\eta_{c}}^{2}}{T^{2}}\right)-\exp\left(-\frac{M_{Z}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{M_{K^{*}}^{2}}{T^{2}}\right)$
$\displaystyle+C^{T}_{\eta_{c}K^{*}}\exp\left(-\frac{M_{\eta_{c}}^{2}+M_{K^{*}}^{2}}{T^{2}}\right)=-\frac{1}{64\sqrt{2}\pi^{4}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\int_{0}^{u^{0}_{K^{*}}}du\sqrt{1-\frac{4m_{c}^{2}}{s}}u\left(2s-3m_{s}m_{c}\right)$
$\displaystyle\exp\left(-\frac{s+u}{T^{2}}\right)+\frac{m_{s}\langle\bar{q}q\rangle}{8\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\,s\,\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle+\frac{m_{s}\langle\bar{s}g_{s}\sigma
Gs\rangle}{96\sqrt{2}\pi^{2}T^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\,s\,\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle+\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{72\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\frac{s+2m_{c}^{2}}{\sqrt{s\left(s-4m_{c}^{2}\right)}}\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle-\frac{m_{c}\left[\langle\bar{q}g_{s}\sigma
Gq\rangle+\langle\bar{s}g_{s}\sigma
Gs\rangle\right]}{24\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\exp\left(-\frac{s}{T^{2}}\right)\,,$
(22)

$\displaystyle\frac{f_{\eta_{c}}M_{\eta_{c}}^{2}f_{K^{*}}M_{K^{*}}\lambda_{Z}^{M}G^{M}_{Z\eta_{c}K^{*}}}{2m_{c}}\frac{1}{M_{Z}^{2}-M_{\eta_{c}}^{2}}\left[\exp\left(-\frac{M_{\eta_{c}}^{2}}{T^{2}}\right)-\exp\left(-\frac{M_{Z}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{M_{K^{*}}^{2}}{T^{2}}\right)$
$\displaystyle+C^{M}_{\eta_{c}K^{*}}\exp\left(-\frac{M_{\eta_{c}}^{2}+M_{K^{*}}^{2}}{T^{2}}\right)=\frac{1}{128\sqrt{2}\pi^{4}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\int_{0}^{u^{0}_{K^{*}}}du\sqrt{1-\frac{4m_{c}^{2}}{s}}u\left(2s-3m_{s}m_{c}\right)$
$\displaystyle\exp\left(-\frac{s+u}{T^{2}}\right)-\frac{m_{s}\langle\bar{q}q\rangle}{16\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}s\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}\langle\bar{s}g_{s}\sigma
Gs\rangle}{192\sqrt{2}\pi^{2}T^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\,s\,\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{72\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\frac{s+2m_{c}^{2}}{\sqrt{s\left(s-4m_{c}^{2}\right)}}\exp\left(-\frac{s}{T^{2}}\right)$
$\displaystyle+\frac{m_{c}\left[\langle\bar{q}g_{s}\sigma
Gq\rangle+\langle\bar{s}g_{s}\sigma
Gs\rangle\right]}{48\sqrt{2}\pi^{2}}\int_{4m_{c}^{2}}^{s^{0}_{\eta_{c}}}ds\sqrt{1-\frac{4m_{c}^{2}}{s}}\exp\left(-\frac{s}{T^{2}}\right)\,,$
(23)

$\displaystyle\frac{f_{D_{s}}M_{D_{s}}^{2}f_{D^{*}}M_{D^{*}}\lambda_{Z}^{T}G^{T}_{ZD^{*}\bar{D}_{s}}}{4(m_{c}+m_{s})}\frac{1}{\widetilde{M}_{Z}^{2}-M_{D^{*}}^{2}}\left[\exp\left(-\frac{M_{D^{*}}^{2}}{T^{2}}\right)-\exp\left(-\frac{\widetilde{M}_{Z}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{M_{D_{s}}^{2}}{T^{2}}\right)$
$\displaystyle+C^{T}_{D^{*}\bar{D}_{s}}\exp\left(-\frac{M_{D^{*}}^{2}+M_{D_{s}}^{2}}{T^{2}}\right)=-\frac{m_{c}\langle\bar{s}g_{s}\sigma
Gs\rangle}{96\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s^{0}_{D^{*}}}ds\left(\frac{9}{2}-\frac{10m_{c}^{2}}{s}+\frac{3m_{c}^{4}}{2s^{2}}\right)\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{96\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{u^{0}_{D_{s}}}du\left(2+\frac{3m_{c}^{2}}{u}\right)\frac{m_{c}^{2}}{u}\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\frac{m_{c}\langle\bar{q}g_{s}\sigma
Gq\rangle}{96\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{u^{0}_{D_{s}}}du\left(\frac{3}{2}+\frac{4m_{c}^{2}}{u}-\frac{3m_{c}^{4}}{2u^{2}}\right)\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma
Gq\rangle}{192\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{u^{0}_{D_{s}}}du\frac{1}{u-m_{c}^{2}}\left(u-9m_{c}^{2}-\frac{5m_{c}^{4}}{u}-\frac{3m_{c}^{6}}{u^{2}}\right)\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)\,,$
(24)

$\displaystyle\frac{f_{D_{s}}M_{D_{s}}^{2}f_{D^{*}}M_{D^{*}}\lambda_{Z}^{M}G^{M}_{ZD^{*}\bar{D}_{s}}}{4(m_{c}+m_{s})}\frac{1}{\widetilde{M}_{Z}^{2}-M_{D^{*}}^{2}}\left[\exp\left(-\frac{M_{D^{*}}^{2}}{T^{2}}\right)-\exp\left(-\frac{\widetilde{M}_{Z}^{2}}{T^{2}}\right)\right]\exp\left(-\frac{M_{D_{s}}^{2}}{T^{2}}\right)$
$\displaystyle+C^{M}_{D^{*}\bar{D}_{s}}\exp\left(-\frac{M_{D^{*}}^{2}+M_{D_{s}}^{2}}{T^{2}}\right)=-\frac{3}{64\sqrt{2}\pi^{4}}\int_{m_{c}^{2}}^{s^{0}_{D^{*}}}ds\int_{m_{c}^{2}}^{u^{0}_{D_{s}}}du\left(1-\frac{m_{c}^{2}}{s}\right)\left(1-\frac{m_{c}^{2}}{u}\right)$
$\displaystyle\left(2s-m_{c}^{2}-\frac{m_{c}^{4}}{s}\right)\left(u-m_{c}^{2}+2m_{s}m_{c}\right)\exp\left(-\frac{s+u}{T^{2}}\right)$
$\displaystyle+\frac{m_{c}\langle\bar{s}s\rangle}{8\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{s^{0}_{D^{*}}}ds\left(1-\frac{m_{c}^{2}}{s}\right)\left(2s-m_{c}^{2}-\frac{m_{c}^{4}}{s}\right)\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\frac{3m_{c}\langle\bar{q}q\rangle}{8\sqrt{2}\pi^{2}}\int_{m_{c}^{2}}^{u^{0}_{D_{s}}}du\left(1-\frac{m_{c}^{2}}{u}\right)\left(u-m_{c}^{2}+2m_{s}m_{c}\right)\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\frac{m_{s}\langle\bar{s}s\rangle}{16\sqrt{2}\pi^{2}}\left(1+\frac{m_{c}^{2}}{T^{2}}\right)\int_{m_{c}^{2}}^{s^{0}_{D^{*}}}ds\left(1-\frac{m_{c}^{2}}{s}\right)\left(2s-m_{c}^{2}-\frac{m_{c}^{4}}{s}\right)\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle-\frac{3m_{c}^{3}\langle\bar{q}g_{s}\sigma
Gq\rangle}{32\sqrt{2}\pi^{2}T^{4}}\int_{m_{c}^{2}}^{u^{0}_{D_{s}}}du\left(1-\frac{m_{c}^{2}}{u}\right)\left(u-m_{c}^{2}+2m_{s}m_{c}\right)\exp\left(-\frac{u+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle-\frac{m_{s}m_{c}^{4}\langle\bar{s}g_{s}\sigma
Gs\rangle}{96\sqrt{2}\pi^{2}T^{6}}\int_{m_{c}^{2}}^{s^{0}_{D^{*}}}ds\left(1-\frac{m_{c}^{2}}{s}\right)\left(2s-m_{c}^{2}-\frac{m_{c}^{4}}{s}\right)\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)$
$\displaystyle+\frac{m_{c}\langle\bar{s}g_{s}\sigma
Gs\rangle}{16\sqrt{2}\pi^{2}T^{2}}\left(1-\frac{m_{c}^{2}}{2T^{2}}\right)\int_{m_{c}^{2}}^{s^{0}_{D^{*}}}ds\left(1-\frac{m_{c}^{2}}{s}\right)\left(2s-m_{c}^{2}-\frac{m_{c}^{4}}{s}\right)\exp\left(-\frac{s+m_{c}^{2}}{T^{2}}\right)\,,$
(25)

where $\widetilde{M}_{Z}^{2}=\frac{M_{Z}^{2}}{4}$, the $s^{0}_{J/\psi}$,
$u^{0}_{K}$, $s^{0}_{\eta_{c}}$, $u^{0}_{K^{*}}$, $s^{0}_{D^{*}}$ and
$u^{0}_{D_{s}}$ are the continuum threshold parameters. There exist end-
divergences at the endpoints $s=4m_{c}^{2}$ and $u=m_{c}^{2}$, we regularize
the endpoint divergences with the simple replacements
$\frac{1}{\sqrt{s-4m_{c}^{2}}}\to\frac{1}{\sqrt{s-4m_{c}^{2}+4m_{s}^{2}}}$ and
$\frac{1}{u-m_{c}^{2}}\to\frac{1}{u-m_{c}^{2}+4m_{s}^{2}}$ by adding a small
squared $s$-quark mass $4m_{s}^{2}$ [26, 27, 28, 29, 30].

Now we take a short digression to illustrate the end-point divergences, we
often encounter the typical integral,

$\displaystyle I_{21}$ $\displaystyle=$ $\displaystyle\int
d^{4}k_{1}\frac{1}{(k_{1}^{2}-m_{c}^{2})^{2}}\frac{1}{((p-k_{1})^{2}-m_{c}^{2})}\,,$
(26)

and calculate it by using the Cutkosky’s rules,

$\displaystyle I_{21}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
A}\int
d^{4}k_{1}\frac{1}{k_{1}^{2}-A}\frac{1}{(p-k_{1})^{2}-m_{c}^{2}}\mid_{A\to
m_{c}^{2}}$ (27) $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
A}\frac{(-2\pi i)^{2}}{2\pi
i}\int_{(\sqrt{A}+m_{c})^{2}}^{\infty}dt\frac{1}{t-p^{2}}\int
d^{4}k_{1}k^{4}k_{2}\delta^{4}(k_{1}+k_{2}-p)\delta(k_{1}^{2}-A)\delta(k_{2}^{2}-m_{c}^{2})$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial A}\frac{(-2\pi
i)^{2}}{2\pi
i}\int_{(\sqrt{A}+m_{c})^{2}}^{\infty}dt\frac{1}{t-p^{2}}\frac{\pi}{2}\frac{\sqrt{\lambda(t,A,m_{c}^{2})}}{t}\mid_{A\to
m_{c}^{2}}$ $\displaystyle=$ $\displaystyle\frac{(-2\pi i)^{2}}{2\pi
i}\int_{4m_{c}^{2}}^{\infty}dt\frac{1}{t-p^{2}}\frac{\pi}{2}\frac{-1}{\sqrt{t(t-4m_{c}^{2})}}\,,$

divergence at the end-point $t=4m_{c}^{2}$ appears. Such terms are companied
with the small $s$-quark mass $m_{s}$ and play a minor important role, we
regularize the endpoint divergences by adding a small squared $s$-quark mass
$4m_{s}^{2}$ empirically, just like in previous works [26, 27, 28, 29, 30]. If
we vary the $4m_{s}^{2}$ by adding a small uncertainty $4m_{s}^{2}\pm\delta$,
and we can obtain almost invariant numerical results.

Furthermore, we smear the dependencies of the parameters $C_{J/\psi K}^{T}$,
$C_{J/\psi K}^{M}$, $C_{\eta_{c}K^{*}}^{T}$, $C_{\eta_{c}K^{*}}^{M}$,
$C_{D^{*}\bar{D}_{s}}^{T}$, and $C_{D^{*}\bar{D}_{s}}^{M}$ on the Lorentz
invariants $p^{\prime 2}$, $p^{2}$, $q^{2}$, and take them as free parameters,
and search for the best values to delete the contaminations from the high
resonances and continuum states to acquire stable QCD sum rules with
variations of the Borel parameters $T^{2}$.

##  3 Numerical results and discussions

At the QCD side, we take the standard values $\langle\bar{q}q\rangle=-(0.24\pm
0.01\,\rm{GeV})^{3}$, $\langle\bar{s}s\rangle=(0.8\pm
0.1)\langle\bar{q}q\rangle$, $\langle\bar{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$, $\langle\bar{s}g_{s}\sigma
Gs\rangle=m_{0}^{2}\langle\bar{s}s\rangle$, $m_{0}^{2}=(0.8\pm
0.1)\,\rm{GeV}^{2}$ at the energy scale $\mu=1\,\rm{GeV}$ [33, 34, 36], and
take the $\overline{MS}$ quark 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 [7]. We set $m_{u}=m_{d}=0$ and take account of the
energy-scale dependence of the input parameters,

$\displaystyle\langle\bar{q}q\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}q\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{s}s\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{s}s\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$ $\displaystyle
m_{c}(\mu)$ $\displaystyle=$ $\displaystyle
m_{c}(m_{c})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle m_{s}(\mu)$ $\displaystyle=$ $\displaystyle m_{s}({\rm
2GeV})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}({\rm
2GeV})}\right]^{\frac{12}{33-2n_{f}}}\,,$ $\displaystyle\alpha_{s}(\mu)$
$\displaystyle=$
$\displaystyle\frac{1}{b_{0}t}\left[1-\frac{b_{1}}{b_{0}^{2}}\frac{\log
t}{t}+\frac{b_{1}^{2}(\log^{2}{t}-\log{t}-1)+b_{0}b_{2}}{b_{0}^{4}t^{2}}\right]\,,$
(28)

from the renormalization group equation,
$t=\log\frac{\mu^{2}}{\Lambda_{QCD}^{2}}$, $b_{0}=\frac{33-2n_{f}}{12\pi}$,
$b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$,
$b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$,
$\Lambda_{QCD}=210\,\rm{MeV}$, $292\,\rm{MeV}$ and $332\,\rm{MeV}$ for the
flavors $n_{f}=5$, $4$ and $3$, respectively [7, 37], and evolve all the input
parameters to the typical energy scale $\mu=m_{c}(m_{c})\approx 1.3\,\rm{GeV}$
to extract hadronic coupling constants [19, 25, 38].

At the hadron side, we take the parameters as $M_{K}=0.4937\,\rm{GeV}$,
$M_{K^{*}}=0.8917\,\rm{GeV}$, $M_{J/\psi}=3.0969\,\rm{GeV}$,
$M_{\eta_{c}}=2.9834\,\rm{GeV}$, $M_{D^{*}}=2.007\,\rm{GeV}$ and
$D_{s}=1.969\,\rm{GeV}$ [7], $f_{K}=0.156\,\rm{GeV}$ [7],
$f_{K^{*}}=0.220\,\rm{GeV}$, $\sqrt{u^{0}_{K}}=1.0\,\rm{GeV}$,
$\sqrt{u^{0}_{K^{*}}}=1.3\,\rm{GeV}$ [39], $f_{D_{s}}=240\,\rm{MeV}$,
$\sqrt{u^{0}_{D_{s}}}=2.6\,\rm{GeV}$, $f_{D^{*}}=263\,\rm{MeV}$,
$\sqrt{s^{0}_{D^{*}}}=2.5\,\rm{GeV}$ [40], $f_{J/\psi}=0.418\,\rm{GeV}$,
$f_{\eta_{c}}=0.387\,\rm{GeV}$ [41], $\sqrt{s^{0}_{J/\psi}}=3.6\,\rm{GeV}$,
$\sqrt{s^{0}_{\eta_{c}}}=3.5\,\rm{GeV}$ [7], $M_{Z}=3.99\,\rm{GeV}$,
$\lambda_{Z}^{T}=2.85\times 10^{-2}\,\rm{GeV}^{5}$ [19],
$\lambda_{Z}^{M}=1.96\times 10^{-2}\,\rm{GeV}^{5}$ [25],
$\frac{f_{K}M^{2}_{K}}{m_{u}+m_{s}}=-\frac{\langle\bar{q}q\rangle+\langle\bar{s}s\rangle}{f_{K}(1-\delta_{K})}$
from the Gell-Mann-Oakes-Renner relation, $\delta_{K}=0.50$ [42].

In calculations, we fit the unknown parameters to be $C^{T}_{J/\psi
K}=0.00045+0.00038\times T^{2}\,\rm{GeV}^{8}$,
$C_{\eta_{c}K^{*}}^{T}=-0.00184-0.00068\times T^{2}\,\rm{GeV}^{8}$,
$C^{T}_{D^{*}\bar{D}_{s}}=0+0\times T^{2}\,\rm{GeV}^{8}$, $C^{M}_{J/\psi
K}=0.0004+0.00017\times T^{2}\,\rm{GeV}^{8}$,
$C^{M}_{\eta_{c}K^{*}}=0.00094+0.00033\times T^{2}\,\rm{GeV}^{8}$, and
$C^{M}_{D^{*}\bar{D}_{s}}=-0.023-0.00405\times T^{2}\,\rm{GeV}^{8}$ to acquire
the flat Borel platforms $T^{2}_{max}-T^{2}_{min}=1\,\rm{GeV}^{2}$, where the
max and min represent the maximum and minimum values, respectively, see Fig.1
for example. In the picture of tetraquark states, the Borel parameters are
$T^{2}_{J/\psi K}=(2.7-3.7)\,\rm{GeV}^{2}$,
$T^{2}_{\eta_{c}K^{*}}=(2.5-3.5)\,\rm{GeV}^{2}$ and
$T^{2}_{D^{*}\bar{D}_{s}}=(3.1-4.1)\,\rm{GeV}^{2}$ for the hadronic coupling
constants $G_{ZJ/\psi K}^{T}$, $G^{T}_{Z\eta_{c}K^{*}}$ and
$G^{T}_{ZD^{*}\bar{D}_{s}}$, respectively, while in the picture of molecular
states, the Borel parameters $T^{2}_{J/\psi K}=(2.7-3.7)\,\rm{GeV}^{2}$,
$T^{2}_{\eta_{c}K^{*}}=(2.3-3.3)\,\rm{GeV}^{2}$ and
$T^{2}_{D^{*}\bar{D}_{s}}=(2.3-3.3)\,\rm{GeV}^{2}$ for the hadronic coupling
constants $G_{ZJ/\psi K}^{M}$, $G^{M}_{Z\eta_{c}K^{*}}$ and
$G^{M}_{ZD^{*}\bar{D}_{s}}$, respectively, where we add the subscripts $J/\psi
K$, $\eta_{c}K^{*}$ and $D^{*}\bar{D}_{s}$ to denote the corresponding
channels. The uncertainties originate from the uncertainties of the input
parameters at the QCD side, in general, can be absorbed into the decay
constants and coupling constants together, for example, the $f_{J/\psi}$,
$f_{K}$, $\lambda_{Z}^{T}$ and $G^{T}_{ZJ/\psi K}$ in the QCD sum rules in
Eq.(2).

If we use the symbol $\xi$ to stand for the input parameters at the QCD side,
the uncertainties $\xi\to\xi+\delta\xi$ lead to the uncertainties
$f_{J/\psi}f_{K}\lambda^{T}_{Z}G^{T}_{ZJ/\psi K}\to
f_{J/\psi}f_{K}\lambda^{T}_{Z}G^{T}_{ZJ/\psi
K}+\delta\,f_{J/\psi}f_{K}\lambda^{T}_{Z}G^{T}_{ZJ/\psi K}$, $C^{T}_{J/\psi
K}\to C^{T}_{J/\psi K}+\delta C^{T}_{J/\psi K}$, where

$\displaystyle\delta\,f_{J/\psi}f_{K}\lambda^{T}_{Z}G^{T}_{ZJ/\psi K}$
$\displaystyle=$ $\displaystyle f_{J/\psi}f_{K}\lambda^{T}_{Z}G^{T}_{ZJ/\psi
K}\left(\frac{\delta f_{J/\psi}}{f_{J/\psi}}+\frac{\delta
f_{K}}{f_{K}}+\frac{\delta\lambda^{T}_{Z}}{\lambda^{T}_{Z}}+\frac{\delta
G^{T}_{ZJ/\psi K}}{G^{T}_{ZJ/\psi K}}\right)\,,$ (29)

we can set $\frac{\delta f_{J/\psi}}{f_{J/\psi}}=\frac{\delta
f_{K}}{f_{K}}=\frac{\delta\lambda^{T}_{Z}}{\lambda^{T}_{Z}}=\frac{\delta
G^{T}_{ZJ/\psi K}}{G^{T}_{ZJ/\psi K}}$ approximately, then

$\displaystyle\delta\,f_{J/\psi}f_{K}\lambda^{T}_{Z}G^{T}_{ZJ/\psi K}$
$\displaystyle=$ $\displaystyle f_{J/\psi}f_{K}\lambda^{T}_{Z}G^{T}_{ZJ/\psi
K}\frac{4\delta G^{T}_{ZJ/\psi K}}{G^{T}_{ZJ/\psi K}}\,,$ (30)

to avoid overestimating the uncertainty of the hadronic coupling constant. In
calculations, we estimate the uncertainties according to Eq.(30) analogously
and neglect the uncertainties of the unknown parameters $C^{T}_{J/\psi K}$,
$C_{\eta_{c}K^{*}}^{T}$, $C^{T}_{D^{*}\bar{D}_{s}}$, $C^{M}_{J/\psi K}$,
$C^{M}_{\eta_{c}K^{*}}$, and $C^{M}_{D^{*}\bar{D}_{s}}$, except in the case of
the $\delta m_{c}(m_{c})$ for the $C^{M}_{D^{*}\bar{D}_{s}}$, where we have to
take account of the uncertainty to acquire flat Borel platform.

Now let us obtain the values of the hadronic coupling constants routinely,

$\displaystyle G^{T}_{ZJ/\psi K}$ $\displaystyle=$ $\displaystyle 1.79\pm
0.10\,\rm{GeV}\,,$ $\displaystyle|G^{T}_{Z\eta_{c}K^{*}}|$ $\displaystyle=$
$\displaystyle 2.97\pm 0.22\,\rm{GeV}\,,$
$\displaystyle|G^{T}_{ZD^{*}\bar{D}_{s}}|$ $\displaystyle=$ $\displaystyle
0.71\pm 0.04\,\rm{GeV}\,,$ $\displaystyle G^{M}_{ZJ/\psi K}$ $\displaystyle=$
$\displaystyle 1.23\pm 0.07\,\rm{GeV}\,,$ $\displaystyle
G^{M}_{Z\eta_{c}K^{*}}$ $\displaystyle=$ $\displaystyle 2.20\pm
0.16\,\rm{GeV}\,,$ $\displaystyle|G^{M}_{ZD^{*}\bar{D}_{s}}|$ $\displaystyle=$
$\displaystyle 9.60\pm 1.27\,\rm{GeV}\,.$ (31)

Figure 1:  The hadronic coupling constants $G^{T}_{ZJ/\psi K}$ (I) and
$G^{M}_{ZJ/\psi K}$ (II) with variations of the Borel parameter $T^{2}$, where
the regions between the two vertical lines are the Borel windows.

Then we choose the masses $M_{K^{-}}=0.4937\,\rm{GeV}$,
$M_{K^{*+}}=0.8917\,\rm{GeV}$, $M_{J/\psi}=3.0969\,\rm{GeV}$,
$M_{\eta_{c}}=2.9834\,\rm{GeV}$, $M_{D^{*0}}=2.0069\,\rm{GeV}$,
$M_{D^{-}_{s}}=1.9690\,\rm{GeV}$, $M_{D^{0}}=1.8648\,\rm{GeV}$,
$M_{D^{*-}_{s}}=2.1122\,\rm{GeV}$ [7], $M_{Z_{c}}=3.99\,\rm{GeV}$ [19, 25],
and obtain the partial decay widths,

$\displaystyle\Gamma(Z_{cs}^{T}\to J/\psi K^{-})$ $\displaystyle=$
$\displaystyle 5.36\pm 0.60\,\rm{MeV}\,,$
$\displaystyle\Gamma(Z_{cs}^{T}\to\eta_{c}K^{*-})$ $\displaystyle=$
$\displaystyle 9.54\pm 1.41\,\rm{MeV}\,,$ $\displaystyle\Gamma(Z_{cs}^{T}\to
D^{*0}\bar{D}^{-}_{s})$ $\displaystyle=$ $\displaystyle 0.21\pm
0.02\,\rm{MeV}\,,$ $\displaystyle\Gamma(Z_{cs}^{T}\to D^{0}\bar{D}^{*-}_{s})$
$\displaystyle=$ $\displaystyle 0.20\pm 0.02\,\rm{MeV}\,,$ (32)
$\displaystyle\Gamma(Z_{cs}^{M}\to J/\psi K^{-})$ $\displaystyle=$
$\displaystyle 2.53\pm 0.29\,\rm{MeV}\,,$
$\displaystyle\Gamma(Z_{cs}^{M}\to\eta_{c}K^{*-})$ $\displaystyle=$
$\displaystyle 5.23\pm 0.76\,\rm{MeV}\,,$ $\displaystyle\Gamma(Z_{cs}^{M}\to
D^{*0}\bar{D}^{-}_{s})$ $\displaystyle=$ $\displaystyle 38.69\pm
10.24\,\rm{MeV}\,,$ $\displaystyle\Gamma(Z_{cs}^{M}\to D^{0}\bar{D}^{*-}_{s})$
$\displaystyle=$ $\displaystyle 37.06\pm 9.81\,\rm{MeV}\,,$ (33)

and the total widths,

$\displaystyle\Gamma_{Z_{cs}}^{T}$ $\displaystyle=$ $\displaystyle 15.31\pm
2.06\,\rm{MeV}\,,$ $\displaystyle\Gamma_{Z_{cs}}^{M}$ $\displaystyle=$
$\displaystyle 83.51\pm 21.09\,\rm{MeV}\,,$ (34)

which are consistent with the experimental data $13.8^{+8.1}_{-5.2}\pm
4.9\,\rm{MeV}$ from the BESIII collaboration [5] and $131\pm 15\pm
26\,\rm{MeV}$ from the LHCb collaboration [6], respectively. There are two
$Z_{cs}$ states with analogous masses, just like in the case of the
$Z_{c}^{\pm}$ states, there exist a tetraquark candidate $Z_{c}(3900)$ and a
molecule candidate $Z_{c}(3885)$ [26].

The physical $Z_{cs}$ states maybe have both the diquark-antidiquark type and
color-singlet-color-singlet type tetraquark Fock components, it is better to
choose the mixing currents or more physical currents to interpolate them,

$\displaystyle J_{\nu}(x)$ $\displaystyle=$ $\displaystyle
J^{T}_{\nu}(x)\,\cos\theta+J^{M}_{\nu}(x)\,\sin\theta\,,$ (35)

with the mixing angle $\theta$. As both the currents $J^{T}_{\nu}(x)$ and
$J^{M}_{\nu}(x)$ can lead to a mass about $3.99\pm 0.09\,\rm{GeV}$ in the case
of choosing the same constraints [19, 25]. The additional parameter $\theta$
cannot affect the predicted mass remarkably, but can affect the predicted
width remarkably. We can estimate the hadronic coupling constants $G$ with the
simple replacement,

$\displaystyle G$ $\displaystyle\to$ $\displaystyle
G^{T}\,\cos\theta+G^{M}\,\sin\theta\,,$ (36)

and compare the predicted total widths and branching fractions to the precise
experimental data in the future to estimate the mixing angle $\theta$, the
branching fractions have not been measured yet. The significance of the
$Z^{-}_{cs}(3985)$ is about $5.3\,\sigma$ [5], while the significance of the
$Z^{+}_{cs}(4000)$ is about $15\,\sigma$ [6], such large significances only
indicate that there maybe exist those resonance structures indeed, more
experimental data are still needed even to distinguish (and confirm) the
$Z_{cs}(3985)$ and $Z_{cs}(4000)$ unambiguously. In the present work, at
least, we can obtain the conclusion tentatively that larger color-singlet-
color-singlet component in the $Z_{cs}$ state leads to larger decay width, the
$Z_{cs}(3985)$ maybe have large diquark-antidiquark type Fock component, while
the $Z_{cs}(4000)$ maybe have large color-singlet-color-singlet type Fock
component.

##  4 Conclusion

In this work, we explore the hadronic coupling constants $G_{ZJ/\psi K}$,
$G_{Z\eta_{c}K^{*}}$, $G_{ZD^{*}\bar{D}_{s}}$ of the exotic states
$Z_{cs}(3985/4000)$ both in the pictures of the tetraquark states and
molecular states with the tentative assignments $J^{PC}=1^{+-}$ based on the
rigorous quark-hadron duality. We write down the three-point correlation
functions, and accomplish the operator product expansion up to the vacuum
condensates of dimension-5, and neglect the tiny gluon condensate
contributions, just like in our previous works, and obtain the spectral
densities through dispersion relation, then we acquire the rigorous quark-
hadron duality below the continuum thresholds $s_{0}$ and $u_{0}$ by
accomplishing the integral over the variable $ds^{\prime}$ firstly, and obtain
six QCD sum rules for the hadronic coupling constants. We investigate the two-
body strong decays of the axialvector tetraquark state and molecular state,
respectively, and obtain the total widths $\Gamma_{Z_{cs}}^{T}=15.31\pm
2.06\,\rm{MeV}$ and $\Gamma_{Z_{cs}}^{M}=83.51\pm 21.09\,\rm{MeV}$, which are
consistent with the experimental data $13.8^{+8.1}_{-5.2}\pm 4.9\,\rm{MeV}$
from the BESIII collaboration and $131\pm 15\pm 26\,\rm{MeV}$ from the LHCb
collaboration, respectively. The present calculations support assigning the
$Z_{cs}(3985)$ and $Z_{cs}(4000)$ to be the hidden-charm tetraquark state and
molecular state with the $J^{PC}=1^{+-}$, respectively. Or at least, the
$Z_{cs}(3985)$ maybe have large diquark-antidiquark type Fock component, while
the $Z_{cs}(4000)$ maybe have large color-singlet-color-singlet type Fock
component.

## Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number
12175068.

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