# Impact of a thermal medium on newly observed $Z_{cs}(3985)$ resonance and
its $b$-partner

J.Y. Süngü  Department of Physics, Kocaeli University, 41001 Izmit, Turkey  A.
Türkan  Özyeğin University, Department of Natural and Mathematical Sciences,
Çekmeköy,Istanbul,Turkey  H. Sundu  Department of Physics, Kocaeli University,
41001 Izmit, Turkey  E. Veli Veliev  Department of Physics, Kocaeli
University, 41001 Izmit, Turkey

###### Abstract

Motivated by the very recent discovery of the strange hidden-charm exotic
state $Z_{cs}(3985)$ by the BESIII Collaboration, we study the possible
interpretation of this exotic state both at $T=0$ and $T\neq 0$. We
analytically compute the mass and meson-current coupling constant of this
resonance with spin-parity $J^{P}=1^{+}$ at finite temperature approximation
up to the sixth order of the thermal operator dimension including non-
perturbative contributions. Extracting thermal mass and meson-current coupling
constant sum rules, the modifications on properties of $Z_{cs}(3985)$ state in
hot medium is determined. As a by-product, the hadronic parameters of the
bottom partner of $Z_{cs}(3985)$ is estimated as well. The search of
temperature effects on the hadronic parameters of hidden-charm meson
$Z_{cs}(3985)$ and the bottom partner make us understand the phase
transitions, chiral symmetry breaking, and the properties of hot-dense matter
in QCD. Moreover, the full width of the resonance $Z_{cs}(3985)$ is calculated
as $(12.0\pm 0.8)~{}\mathrm{MeV}$ using the strong decay in the tetraquark
picture. Results for width and mass are in reasonable agreement with existing
experimental data, and results of other theoretical works. The obtained
information about the parameters of considered states is useful for
experimental investigations of exotic mesons.

##  I Introduction

Though many new exotic states named as XYZ states above $D\overline{D}$
threshold have recently been observed in different experiments, their
substructures can not be explained yet clearly. There are a lot of candidates
for exotic hadrons in the charmonium sector of Quantum Chromodynamics (QCD)
studied at vacuum, hot medium and also in nuclear medium, such as
$X(3872),Z_{c}(3900),Z_{c}(4430),Y(4260),Z_{b}(10610),$ and
$Z_{b}^{\prime}(10650)$ Brambilla:2019esw ; Liu:2019zoy ; VeliVeliev:2018eaw ;
Azizi:2020itk ; Isik:2020fwl ; Chen:2016qju ; Azizi:2020yhs ; Agaev:2016dev ;
Agaev:2017tzv ; Agaev:2017lmc ; Agaev:TJP ; Ozdem:2017jqh ; Hay:TJP  which
yields a new horizon for understanding the inner structure of strongly
interacting matter. Investigating for charged charmonium-like states is one of
the most promising ways of searching exotic mesons since they must contain at
least four quarks and thus cannot be a conventional hadron.

Recently, for the first time, the BESIII Collaboration reported the charged
strange hidden charmonium-like structure $Z_{cs}(3985)$ near the
$D^{-}_{s}D^{*0}$ and $D^{*-}_{s}D^{0}$ mass thresholds in the $K^{+}$ recoil-
mass spectrum for events collected at $\sqrt{s}=4.681$ GeV in the processes of
$e^{+}e^{-}\rightarrow K^{+}(D^{-}_{s}D^{*0}+D^{*-}_{s}D^{0})$ Ablikim:2020hsk
. The significance was estimated to be $5.3\sigma$. This discovery could
maintain some unique hints to uncover the secrets of charged exotic $Z$
structures. This new hadronic structure is assigned in the class of exotic
state as the strange partner of $Z_{c}(3900)$ and studied in many different
models in the literature in the molecular and tetraquark scenarios 1831062 ;
1831033 ; 1831047 ; Wang:2020kej ; Meng:2020ihj ; Liu:2020nge ; Wan:2020oxt ;
Sun:2020hjw ; Chen:2020yvq ; Wang:2020rcx ; Azizi:2020zyq ; Wang:2020iqt ;
1832695 . Its mass and width are defined in experiment as:

$\displaystyle M_{Z_{cs}}$ $\displaystyle=$ $\displaystyle
3982.5^{+1.8}_{-2.6}\pm 2.1~{}\mathrm{MeV},$ $\displaystyle\Gamma_{Z_{cs}}$
$\displaystyle=$ $\displaystyle 12.8^{+5.3}_{-4.4}\pm 3.0~{}\mathrm{MeV}.$ (1)

Meanwhile, the features of matter under extreme conditions of high
temperatures and/or densities have attracted the curiosity of high energy
physicists Ayala:2020rmb ; Ayala:2016vnt ; Fu:2019hdw ; Zhao:2020nwy ;
Irikura:2020 . QCD, the theory of strong interactions, expects that nuclear
matter undergoes a phase transition from a state of deconfined quarks and
gluons forming a new state of matter, named as the quark-gluon plasma (QGP),
at a critical temperature $T_{c}\cong 155$ MeV $(\sim 10^{12}K)$ Aoki:2006br ;
Andronic:2017pug ; Steinbrecher:2018phh ; Fischer:2018sdj  which is in
excellent agreement with the freeze-out temperature for hadrons measured by
the ALICE collaboration at LHC producing ${}^{4}\textrm{He}$ and
${}^{4}\overline{\textrm{He}}$ nuclei in Pb–Pb collisions at
$\sqrt{s_{NN}}=2.76$ TeV in the rapidity range $|y|<1$ Acharya:2017bso ;
Floris:2014pta . Nevertheless its short life time ($10fm/c\approx 3\times
10^{-23}s$) and thermalization time ($0.2fm/c\approx 7\times 10^{-25}s$),
which makes measurements harder are a big challenge for experimentalist.

The phases of QCD are characterized by a variety of condensates in which
numerous particles interact with each other through strong force. The
materialization of condensates lessens the energy of a system, and also
condensates break symmetries in QCD. Chiral symmetry breaking (CSB) is
identified by a non-vanishing chiral condensate $\langle\bar{q}q\rangle$, here
$q$ is the quark field. Luckily, at extreme temperatures, it is predicted that
quark masses are decreased from their effective mass values in hot medium to
their bare ones and CSB is almost restored. Namely, the condensates depending
on temperature and baryon density play a key role in the structure of hadrons
Hatsuda:1992bv ; Foka:2016vta ; Bratkovskaya:2017gxq .

Recreating longer-lived QGP as well as a large number and assortment of
particles in laboratory conditions can provide us to explore the properties of
QGP and also understand the QCD vacuum, confinement, and hadronization phase
of the QGP. For a more precise interpretation of heavy-ion collision
experiments, deviations of the hadronic parameters depending on temperature
are vital and worth calculating. From the theoretical point of perspective,
these results have delivered some surprising and stimulating new theoretical
studies of hot matter Lerambert-Potin:2021ohy ; Grefa:2021qvt ;
Pinkanjanarod:2020mgi .

Charm and bottom quarks are excellent probes of the hot and dense state of
deconfined quarks and gluons. Heavy quarks are created at the initial stages
of the hard-scattering collisions and interact with the constituents of the
newly produced QGP through both elastic and inelastic processes. These quarks,
which can be studied through their decays into leptons, lose energy while
propagating through the QGP medium. The formation of a QGP phase also lets
them move freely and recombine to produce exotic states. They can coalesce to
create standard and possibly exotic bound states at the end of the QGP phase.

On the other hand, although a coherent picture of collision dynamics is
emerging, finding signatures of QGP remains unclear. Probably verification of
QGP formation will not come from a unique signal, and evidence based on well-
focused observations will have to be collected. Some signatures supporting the
creation of the QGP have been reported; suppression (and regeneration) of
heavy quarkonia, jet-quenching, the non-viscous flow, radiation of photons and
dileptons.

Depending on these ideas, the purpose of this article is to evaluate the mass,
meson-current coupling constant and decay width of $Z_{cs}(3985)$ assuming it
has four-quark content $[\bar{c}cu\bar{s}]$ including quark, gluon and quark-
gluon mixed condensates up to dimension six using the QCD Sum Rule (QCDSR)
approach at finite temperature. In this case, the vacuum condensate
expressions are replaced with their thermal condensates. This analysis can
give us some hints on the nature of the $Z_{cs}(3985)$ and also provide
insights into the nature of the produced hot and dense matter which is
predicted to exist in the initial stages of the universe and also in the core
of neutron stars Baym:2017whm .

The paper is organized as follows. The Thermal QCDSR (TQCDSR) approach is
introduced employed in our calculations in Section II. Numerical analysis of
mass and meson-current coupling constant of $Z_{cs}(3985)$ and its b-partner
(after that we will use $Z_{cs}$ and $Z_{bs}$ for shortness) is discussed in
Section III. In the next section, the decay width of $Z_{cs}(3985)$ is
evaluated. After summarizing in Section V, we present the explicit form of the
two-point thermal spectral densities $\rho^{\mathrm{QCD}}(s,T)$ which are
obtained from the TQCDSR theory in Appendix A.

##  II THEORETICAL FRAMEWORK for two-point correlator

The QCD sum rules technique is a successful and powerful non-perturbative
method Shifman ; Reinders:1984sr , which is widely applied to study the mass
spectra and decay properties of hadrons. To find the variations of mass and
meson-current coupling of $Z_{cs}$ with increasing temperature, we adopt the
QCDSR formalism to TQCDSR. We start the calculation by writing down the
correlation function Bochkarev:1985ex :

$\Pi_{\mu\nu}(q,T)=i\int d^{4}x~{}e^{iq\cdot
x}\langle\Omega|\mathcal{T}\\{\eta_{\mu}(x)\eta_{\nu}^{{\dagger}}(0)\\}|\Omega\rangle,$
(2)

where $\mathcal{T}$ represents the time ordering operator, $\Omega$ symbolizes
the thermal medium, $T$ is the temperature and $\eta_{\mu}(x)$ is the
interpolating current accompanying to resonance $Z_{cs}$.

To derive the TQCDSR we start to compute the correlation function in
connection with the physical degrees of freedom. The correlation function is
expressed by saturating via a complete set of states with the same quantum
number $J^{P}=1^{+}$ of $Z_{cs}$ state and then Eq. (2) is integrated for $x$:

$\Pi^{\mathrm{Phys}}_{\mu\nu}(q,T)=\frac{\langle\Omega|\eta_{\mu}|Z_{cs}(q)\rangle\langle
Z_{cs}(q)|\eta_{\nu}^{\dagger}|\Omega\rangle}{m_{Z_{cs}}^{2}(T)-q^{2}}+\ldots,$
(3)

where $m_{Z_{cs}}(T)$ is the temperature-dependent ground state mass of axial-
vector state $Z_{cs}$ and three dots indicate the higher states and continuum.
The definition of the matrix element of temperature-dependent meson-current
coupling constants is:

$\langle\Omega|\eta_{\mu}|Z_{cs}(q)\rangle=\lambda_{Z_{cs}}(T)m_{Z_{cs}}(T)\varepsilon_{\mu},$
(4)

here $\varepsilon_{\mu}$ is the polarization vector. So the correlation
function for the physical side can be written concerning the thermal ground
state mass and meson-current coupling constant in the form below:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(q,T)$ $\displaystyle=$
$\displaystyle\frac{m_{Z_{cs}}^{2}\left(T\right)\lambda_{Z_{cs}}^{2}\left(T\right)}{m_{Z_{cs}}^{2}\left(T\right)-q^{2}}\bigg{(}-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{Z_{cs}}^{2}\left(T\right)}\bigg{)}$
(5) $\displaystyle+$ $\displaystyle\ldots.$

In our computations, the chosen structure for both physical and QCD parts of
the correlator is $(g_{\mu\nu})$ to obtain the TQCDSR for the mass and meson-
current coupling constant. Then isolating ground state contributions from the
higher resonances and continuum states by taking derivative, namely using
Borel transformation, the physical side is determined as:

$\displaystyle\mathcal{\widehat{B}}(q^{2})\Pi^{\mathrm{Phys}}(q^{2},T)=m_{Z_{cs}}^{2}(T)\lambda_{Z_{cs}}^{2}(T)~{}e^{-m_{Z_{cs}}^{2}(T)/M^{2}},$
(6)

here $M$ is the Borel parameter in the QCDSR model.

The next step is to determine the QCD part in which the correlation function
is expressed with the quark and gluon degrees of freedom. First, we choose the
concerned current for the $Z_{cs}$ state with $J^{P}=1^{+}$ constructed in
tetraquark picture as:

$\displaystyle\eta_{\mu}(x)=i\epsilon_{abc}\epsilon_{dec}\big{[}\big{(}s_{a}^{T}(x)C\gamma_{5}c_{b}(x)\big{)}\big{(}\overline{u}_{d}(x)\gamma_{\mu}C\overline{c}^{T}_{e}(x)\big{)}\big{]},~{}$
(7)

here $\epsilon_{abc}$ and $\epsilon_{dec}$ are anti-symmetric Levi-Civita
symbols, $a,b,c,d,e$ are color indices, and $C$ is the charge conjugation
matrix.

The QCD part of the correlation function $\Pi_{\mu\nu}^{\mathrm{QCD}}(q,T)$
can be described, as usual with a dispersion integral:

$\Pi^{\mathrm{QCD}}(q^{2},T)=\int_{\mathcal{M}^{2}}^{\infty}\frac{\rho^{\mathrm{QCD}}(s,T)}{s-q^{2}}ds,$
(8)

where $\mathcal{M}^{2}=(m_{u}+m_{s}+2m_{c})^{2}$, and the spectral density
function $\rho^{\mathrm{QCD}}(s,T)$ is given by the imaginary part of the
correlation function:

$\rho^{\mathrm{QCD}}(s,T)=\frac{1}{\pi}Im[\Pi^{QCD}].$ (9)

Having completed lengthy calculations the QCD side of the correlation function
in terms of the heavy and light quark propagators reads:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(q,T)=-\frac{i}{2}\int
d^{4}xe^{iq\cdot
x}\epsilon_{abc}\tilde{\epsilon}_{dec}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}^{\prime}\tilde{\epsilon}_{d^{\prime}e^{\prime}c^{\prime}}^{\prime}$
$\displaystyle\times\Big{(}\mathrm{Tr}[\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{\nu}S_{u}^{d^{\prime}d}(-x)\gamma_{\mu}]\mathrm{Tr}[\widetilde{S}_{s}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\gamma_{5}]$
$\displaystyle-\mathrm{Tr}[\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{u}^{d^{\prime}d}(-x)\gamma_{\mu}]\mathrm{Tr}[\widetilde{S}_{s}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\gamma_{\nu}]$
$\displaystyle-\mathrm{Tr}[\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{\nu}S_{u}^{d^{\prime}d}(-x)\gamma_{5}]\mathrm{Tr}[\widetilde{S}_{s}^{aa^{\prime}}(x)\gamma_{\mu}S_{c}^{bb^{\prime}}(x)\gamma_{5}]$
$\displaystyle+\mathrm{Tr}[\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{u}^{d^{\prime}d}(-x)\gamma_{5}]\mathrm{Tr}[\widetilde{S}_{s}^{aa^{\prime}}(x)\gamma_{\mu}S_{c}^{bb^{\prime}}(x)\gamma_{\nu}]\Big{)}$

and for compactness we used the following notation in Eq. (II):

$\widetilde{S}^{aa^{\prime}}(x)=CS^{aa^{\prime}T}(x)~{}C.$

By the way, at finite temperatures, the additional operators arise in the
short distance expansion of the product of two quark bilinear operators since
the failure of Lorentz invariance with the preferred reference frame and
spilling of the residual $\mathcal{O}(3)$ symmetry, and accordingly the
thermal heavy and light quark propagators include new terms. So we modify the
vacuum condensates by their thermal averages.

In the calculations we use the following definition of the thermal light quark
propagator $S_{q}^{ij}(x)$ Azizi:2016ddw ; Azizi:2014maa ; Mallik:1997pq :

$\displaystyle
S_{q}^{ij}(x)=i\frac{\not{x}}{2\pi^{2}x^{4}}\delta_{ij}-\frac{m_{q}}{4\pi^{2}x^{2}}\delta_{ij}-\frac{\langle\bar{q}q\rangle_{T}}{12}\delta_{ij}$
$\displaystyle-\frac{x^{2}}{192}m_{0}^{2}\langle\bar{q}q\rangle_{T}\Big{[}1-i\frac{m_{q}}{6}\not{x}\Big{]}\delta_{ij}+\frac{i}{3}\Big{[}\not{x}\Big{(}\frac{m_{q}}{16}\langle\bar{q}q\rangle_{T}$
$\displaystyle-\frac{1}{12}\langle
u^{\mu}\theta_{\mu\nu}^{f}u^{\nu}\rangle\Big{)}+\frac{1}{3}(u\cdot
x)\not{u}\langle u^{\mu}\theta_{\mu\nu}^{f}u^{\nu}\rangle\Big{]}\delta_{ij}$
$\displaystyle-\frac{ig_{s}G_{ij}^{\alpha\beta}}{32\pi^{2}x^{2}}\Big{(}\not{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\not{x}\Big{)}-i\delta_{ij}\frac{x^{2}\not{x}\langle\bar{q}q\rangle^{2}_{T}}{7776}g_{s}^{2},$
(11)

where $m_{q}$ is the light quark mass, $\langle\bar{q}q\rangle_{T}$ denotes
the temperature-dependent light quark condensate, $u_{\mu}$ is the four-
velocity of hot matter, and $\theta_{\mu\nu}^{f}$ is the fermionic part of the
energy momentum tensor. Also the gluon condensate depending on the gluonic
part of the energy-momentum tensor $\theta_{\lambda\sigma}^{g}$ is
Mallik:1997pq :

$\displaystyle\langle
Tr^{c}G_{\alpha\beta}G_{\mu\nu}\rangle=\frac{1}{24}(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu})\langle
G_{\lambda\sigma}^{a}G^{a\lambda\sigma}\rangle$
$\displaystyle+\frac{1}{6}\Big{[}g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}-2(u_{\alpha}u_{\mu}g_{\beta\nu}-u_{\alpha}u_{\nu}g_{\beta\mu}$
$\displaystyle-
u_{\beta}u_{\mu}g_{\alpha\nu}+u_{\beta}u_{\nu}g_{\alpha\mu})\Big{]}\langle
u^{\lambda}{\theta}_{\lambda\sigma}^{g}u^{\sigma}\rangle.$ (12)

The heavy quark propagator $S_{Q}^{ij}(x)$ $(Q=c,b)$ is described as in Ref.
Mallik:1997pq :

$\displaystyle S_{Q}^{ij}(x)=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot
x}\Bigg{[}\frac{\delta_{ij}\Big{(}{\\!\not\\!{k}}+m_{Q}\Big{)}}{k^{2}-m_{Q}^{2}}$
$\displaystyle-\frac{gG_{ij}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}\Big{(}{\\!\not\\!{k}}+m_{Q}\Big{)}+\Big{(}{\\!\not\\!{k}}+m_{Q}\Big{)}\sigma_{\alpha\beta}}{(k^{2}-m_{Q}^{2})^{2}}$
$\displaystyle+\frac{g^{2}}{12}G_{\alpha\beta}^{A}G_{A}^{\alpha\beta}\delta_{ij}m_{Q}\frac{k^{2}+m_{Q}{\\!\not\\!{k}}}{(k^{2}-m_{Q}^{2})^{4}}+\ldots\Bigg{]},$
(13)

where for the external gluon field $G_{ij}^{\alpha\beta}$, the below short-
hand notation is employed:

$G_{ij}^{\alpha\beta}\equiv G_{A}^{\alpha\beta}\lambda_{ij}^{A}/2,$

here $\lambda_{A}^{ij}$ are Gell-Mann matrices, $i,\,j$ are color indices and
$A=1,\,2\,\ldots 8$ are the number of gluon flavours. The first term in Eq.
(II) denote the perturbative contribution to the heavy quark propagator and
the others are non-perturbative terms.

Then taking into account tensor structure of
$\Pi_{\mu\nu}^{\mathrm{QCD}}(q,T)$ we can write:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(q^{2},T)$ $\displaystyle=$
$\displaystyle\Pi_{0}^{\mathrm{QCD}}(q^{2},T)\frac{q_{\mu}q_{\nu}}{q^{2}}$
(14) $\displaystyle+$
$\displaystyle\Pi_{1}^{\mathrm{QCD}}(q^{2},T)(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}),$

where $\Pi_{0}^{\mathrm{QCD}}(q^{2},T)$ and $\Pi_{1}^{\mathrm{QCD}}(q^{2},T)$
are invariant functions. According to the idea of the QCDSR, we should choose
the same structures $(g_{\mu\nu})$ for the mass and the meson-current coupling
constant sum rules in both $\Pi_{\mu\nu}^{\mathrm{Phys}}(q^{2},T)$ and
$\Pi_{\mu\nu}^{\mathrm{QCD}}(q^{2},T)$.

Now using these definitions, transferring the continuum contribution to the
QCD part, applying Borel transformation to both parts of the sum rules and
equating them, thermal meson-current coupling constant sum rule for the axial-
vector meson $Z_{cs}$ up to the dimension-six condensates is written as
follows:

$\displaystyle
m_{Z_{cs}}^{2}\left(T\right)\lambda_{Z_{cs}}^{2}(T)e^{-m_{Z_{cs}}^{2}(T)/M^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\int_{\mathcal{M}^{2}}^{s_{0}(T)}ds\rho^{\mathrm{QCD}}(s,T)e^{-s/M^{2}}$
(15)

and then taking the derivative of Eq. (II) in terms of $(-1/M^{2})$ we reach
the thermal mass sum rule of $Z_{cs}$:

$m_{Z_{cs}}^{2}(T)=\frac{\int_{\mathcal{M}^{2}}^{s_{0}(T)}dss\rho^{\mathrm{QCD}}(s,T)e^{-s/M^{2}}}{\int_{\mathcal{M}^{2}}^{s_{0}(T)}ds\rho^{\mathrm{QCD}}(s,T)e^{-s/M^{2}}},$
(16)

where $s_{0}(T)$ symbolize the thermal continuum threshold parameter which
separates the ground state from higher states. The next step is to carry out
the numerical analysis to determine the values of hadronic parameters of
resonance $Z_{cs}$ and also replace $c$ quark with $b$ quark to obtain the
$b$-partner $Z_{bs}$ of $Z_{cs}$ in tetraquark assumption.

##  III Analysis of the mass and meson-current coupling constant of the
$Z_{cs}$ and $Z_{bs}$

To get the values of mass and meson-current coupling constant of the hidden-
charm system $Z_{cs}$ in the TQCDSR approach, we require some parameters e.g.
quark masses, quark, gluon, and mixed vacuum and thermal condensates. The
vacuum values of these input parameters are listed in Table 1.

Table 1: Input parameters. Parameters | Values
---|---
$m_{u}$ |  $2.16^{+0.49}_{-0.26}\mathrm{MeV}$ Zyla
$m_{d}$ |  $4.67{}^{+0.48}_{-0.17}\mathrm{MeV}$ Zyla
$m_{s}$ |  $93^{+11}_{-5}\mathrm{MeV}$ Zyla
$m_{c}$ |  $1.23\pm 0.09~{}\mathrm{GeV}$ Eidemuller:2000rc ; Zyla
$m_{b}$ |  $4.18^{+0.03}_{-0.02}\mathrm{MeV}$ Zyla
$\langle 0|\bar{q}q|0\rangle$ |  $(-275(5))^{3}~{}\mathrm{MeV}^{3}$ Gubler:2018ctz
$\langle 0|\bar{s}s|0\rangle$ |  $(-296(11))^{3}~{}\mathrm{MeV}^{3}$ Gubler:2018ctz
$\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle$ |  $0.028(3)~{}\mathrm{GeV}^{4}$Horsley:2012ra
$m_{0}^{2}$ |  $(0.8\pm 0.1)~{}\mathrm{GeV}^{2}$ Shifman ; Reinders:1984sr

Further, we need temperature-dependent quark, gluon condensates, and also
energy density as well. Thermal quark condensates are obtained fitting data
from Ref. Gubler:2018ctz , which is consistent with the Lattice QCD data:

$\displaystyle\frac{\langle\bar{q}q\rangle_{T}}{\langle
0|\bar{q}q|0\rangle}=\mu_{1}e^{c_{1}T}+\mu_{2},$ (17)

where $q$ denotes the $u$ or $d$ quarks while for the $s$ quark

$\displaystyle\frac{\langle\bar{s}s\rangle_{T}}{\langle
0|\bar{s}s|0\rangle}=\mu_{3}e^{c_{2}T}+\mu_{4},$ (18)

here $c_{1}=\mathrm{0.040~{}MeV^{-1}}$, $c_{2}=\mathrm{0.516~{}MeV^{-1}}$,
$\mu_{1}$$=-6.534\times 10^{-4}$, $\mu_{2}=1.015$, $\mu_{3}=-2.169\times
10^{-5}$, $\mu_{4}$$=1.002$ are coefficients Azizi:2019cmj  and are
trustworthy up to a temperature $T=180~{}\mathrm{MeV}$ and $\langle
0|\bar{q}q|0\rangle$ denotes the condensate of the light quarks at $T=0$.

The gluonic and fermionic parts of the energy density can be parametrized as
in Ref. Azizi:2016ddw  taking into account the Lattice QCD data given in Ref.
Cheng:2007jq :

$\displaystyle\langle u^{\mu}\mathrm{\theta}^{f}_{\mu\nu}u^{\nu}\rangle_{T}$
$\displaystyle=$ $\displaystyle(\tau_{1}e^{c_{3}T}+\mu_{5})~{}T^{4},$ (19)
$\displaystyle\langle u^{\mu}\mathrm{\theta}^{g}_{\mu\nu}u^{\nu}\rangle_{T}$
$\displaystyle=$ $\displaystyle(\tau_{2}e^{c_{4}T}-\mu_{6})~{}T^{4},$ (20)

where $\tau_{1}=0.009$, $c_{3}=24.876~{}\mathrm{GeV^{-1}}$, $\mu_{5}=0.024$,
$\tau_{2}=0.091$, $c_{4}=21.277~{}\mathrm{GeV^{-1}}$ and $\mu_{6}=0.731$
Azizi:2019cmj .

Also the temperature-dependent gluon condensate $\langle G^{2}\rangle_{T}$ is
defined as in Ref. Gubler:2018ctz :

$\displaystyle\delta\Big{\langle}\frac{\alpha_{s}G^{2}}{\pi}\Big{\rangle}_{T}$
$\displaystyle=$ $\displaystyle-\frac{8}{9}[\delta
T^{\mu}_{\mu}(T)-m_{u}\delta\langle\bar{u}u\rangle_{T}$ (21) $\displaystyle-$
$\displaystyle
m_{d}\delta\langle\bar{d}d\rangle_{T}-m_{s}\delta\langle\bar{s}s\rangle_{T}],$

where the vacuum subtracted values of the related quantities are employed as

$\displaystyle\delta f(T)\equiv f(T)-f(0)$ (22)

and

$\displaystyle\delta T^{\mu}_{\mu}(T)=\varepsilon(T)-3p(T),$ (23)

here $p(T)$ is the pressure and $\varepsilon(T)$ is the energy density.
Considering recent Lattice evaluations Bazavov:2014pvz ; Borsanyi:2013bia  the
fit function of $\delta T^{\mu}_{\mu}(T)$ as obtained in Ref. Azizi:2019cmj :

$\displaystyle\frac{\delta T^{\mu}_{\mu}(T)}{T^{4}}$ $\displaystyle=$
$\displaystyle(\mu_{7}e^{c_{5}T}+\mu_{8})$ (24)

with $\mu_{7}=0.020$, $c_{5}=29.412~{}\mathrm{GeV^{-1}}$, $\mu_{8}=0.115$.
Moreover the following expression for the temperature-dependent strong
coupling Kaczmarek:2004gv ; Morita:2007hv  is taken into account in the
calculations being $\Lambda_{\overline{MS}}\simeq T_{c}/1.14$:

$\displaystyle g_{s}^{-2}(T)=\frac{11}{8\pi^{2}}\ln\Big{(}\frac{2\pi
T}{\Lambda_{\overline{MS}}}\Big{)}+\frac{51}{88\pi^{2}}\ln\Big{[}2\ln\Big{(}\frac{2\pi
T}{\Lambda_{\overline{MS}}}\Big{)}\Big{]}.$ (25)

As for the temperature-dependent continuum threshold $s_{0}(T)$ belonging to
$Z_{cs}$ state is another auxiliary parameter that can be defined as in the
following form Dominguez:2016roi ; Borsanyi:2010bp ; Bhattacharya:2014ara :

$\displaystyle\frac{s_{0}(T)}{s_{0}(0)}=\bigg{(}\frac{\langle\bar{q}q\rangle_{T}}{\langle
0|\bar{q}q|0\rangle}\bigg{)}^{2/3}.$ (26)

Continuum threshold parameter $s_{0}(0)$ is not completely independent of the
mass of the first excited state of $Z_{cs}$. According to the QCDSR formalism,
the physical quantities shouldn’t be connected with the auxiliary parameters
$M^{2}$ and $s_{0}$. However $M^{2}$ and $s_{0}$ are susceptible to the
selection of the parameters of the theory.

In the QCDSR method, OPE convergence points us the lower bound on $M^{2}$, and
the pole contribution (PC) yields the upper bound, i.e the highest-dimensional
condensates should contribute no more than $\sim 20\%$ to the QCD side while
the continuum is less than $50\%$ of the total terms.

In this context, the maximum allowed $M^{2}$ needs be fixed to obey the
restriction dictated on $\mathrm{PC}$. At the maximum value of $M^{2}$ the
constraint $\mathrm{PC}>0.2$ is typical for multiquark systems and we get:

$\mathrm{PC}=\frac{\Pi(M^{2},\ s_{0})}{\Pi(M^{2},\ \infty)}=0.24,$ (27)

here $\Pi(M^{2},\ s_{0})$ is the Borel-transformed and subtracted invariant
amplitude $\Pi^{\mathrm{OPE}}(p^{2})$.

To ensure the convergence of the $\mathrm{OPE}$, at the minimum limit of
$M^{2}$ we use the limitation $R\leq 0.15$. The lower bound of Borel window is
defined from the convergence of the $\mathrm{OPE}$ by the ratio below:

$R(M^{2})=\frac{\Pi^{\mathrm{(Dim5+Dim6)}}(M^{2},\ s_{0})}{\Pi(M^{2},\
s_{0})}=0.15,$ (28)

here $\mathrm{Dim5}$ and $\mathrm{Dim6}$ show the contributions to the
correlation function of the sum of the last two terms in the operator product
expansion.

Considering all these constraints, according to our analyses, the continuum
threshold, and Borel parameters are fixed as follows for the $Z_{cs}$
resonance, respectively:

$\displaystyle 4~{}\mathrm{GeV^{2}}$ $\displaystyle\leq$ $\displaystyle
M^{2}\leq 5~{}\mathrm{GeV^{2}},$ $\displaystyle 19.5~{}\mathrm{GeV^{2}}$
$\displaystyle\leq$ $\displaystyle s_{0}~{}~{}\leq 20.5~{}\mathrm{GeV^{2}},$
(29)

and for the $Z_{bs}$ state as well:

$\displaystyle 13~{}\mathrm{GeV^{2}}$ $\displaystyle\leq$ $\displaystyle
M^{2}\leq 15~{}\mathrm{GeV^{2}},$ $\displaystyle 122~{}\mathrm{GeV^{2}}$
$\displaystyle\leq$ $\displaystyle s_{0}~{}~{}\leq 126~{}\mathrm{GeV^{2}}.$
(30)

The philosophy of the QCDSR method dictates that the dependence of hadronic
quantities on Borel parameter $M^{2}$ and continuum threshold $s_{0}$ should
stay steady in the selected working region. This means that we can obtain
reliable results from the extracted sum rules. We see the stability of sum
rules according to the model parameters drawing graphs. Here we only present a
plot for the $Z_{cs}$ state in Figure 1.

Figure 1: The vacuum mass of the $Z_{cs}$ state versus Borel parameter $M^{2}$
for fixed continuum threshold values in tetraquark picture.

In the end, the below results for $Z_{cs}$ resonance in $T=0$ limit of the
TQCDSR model is obtained:

$\displaystyle m_{Z_{cs}}$ $\displaystyle=$ $\displaystyle
3.996^{+0.068}_{-0.067}~{}\mathrm{GeV},$ $\displaystyle\lambda_{Z_{cs}}$
$\displaystyle=$ $\displaystyle 0.72^{+0.06}_{-0.05}\times
10^{-2}~{}\mathrm{GeV^{4}},$

whereas we set a range for the state $Z_{bs}$ as

$\displaystyle m_{Z_{bs}}$ $\displaystyle=$ $\displaystyle(10.379\sim
10.557)~{}\mathrm{GeV},$ $\displaystyle\lambda_{Z_{bs}}$ $\displaystyle=$
$\displaystyle(2.73\sim 3.36)\times 10^{-2}~{}\mathrm{GeV^{4}},$

which are consistent with the experimental and theoretical estimations in Ref.
1831062 ; 1831033 ; 1831047 ; Wang:2020kej ; Meng:2020ihj ; Liu:2020nge ;
1832695 ; Azizi:2020zyq ; Wang:2020iqt  within the limits of uncertainties
Ablikim:2020hsk .

Next, we define the modifications of mass and meson-current coupling constant
of the $Z_{cs}$ in terms of temperature. In this manner, the ratio of changing
the mass and meson-current constant graphs are drawn as a function of the
temperature for the tetraquark assumption in Figure 2 and 3, respectively.

Figure 2: The ratio of the temperature-dependent mass to vacuum mass of the
$Z_{cs}$ state, respectively in the tetraquark picture for fixed values of
$s_{0}(0)$. Figure 3: The ratio of the temperature-dependent meson-current
coupling constant to vacuum meson-current coupling constant of the $Z_{cs}$
state, respectively in the tetraquark picture for fixed values of $s_{0}(0)$.

##  IV Strong decays of the tetraquark $Z_{cs}(3985)$

The quark component of the newly observed resonance $Z_{cs}$ should be
$[c\bar{c}s\bar{u}]$ rather than the pure $c\bar{c}$ since it is a charged
particle with strangeness and mass of this tetraquark is large enough to make
kinematically allowed the strong decay modes
$D_{s}^{-}D^{*0}/D_{s}^{*-}D^{0}$. In this part of the paper, we have
discussed the details of the decays $Z_{cs}\rightarrow
D_{s}^{-}D^{*0}/D_{s}^{*-}D^{0}$.

$\bullet$ We start with the first process $Z_{cs}\rightarrow D_{s}^{-}D^{*0}$.
Initially, we need to compute the strong coupling corresponding to the vertex
$Z_{cs}D_{s}^{-}D^{*0}$ which quantitatively defines strong interactions
between the tetraquark and two conventional mesons. To obtain the QCD three-
point sum rules for the related coupling, we begin the calculation by writing
the correlation function:

$\displaystyle\Pi_{\mu\nu}(p,p^{\prime},T)$ $\displaystyle=$ $\displaystyle
i^{2}\int d^{4}xd^{4}ye^{i(p^{\prime}\cdot y-p\cdot
x)}\langle\Omega|\mathcal{T}\\{\eta_{\nu}^{D^{\ast 0}}(y)$ (31)
$\displaystyle\times\eta^{D_{s}^{-}}(0)\eta^{Z_{cs}}_{\mu}(x)\\}|\Omega\rangle,$

where $\eta^{Z_{cs}}_{\mu}(x)$, $\eta^{D_{s}^{-}}(0)$ and $\eta_{\nu}^{D^{\ast
0}}(y)$ symbolize the interpolating currents for the tetraquark $Z_{cs}$ and
mesons $D_{s}^{-}$ and $D^{\ast 0}$, respectively. The four-momenta of the
tetraquark $Z_{cs}$ and meson $D^{\ast 0}$ are $p$ and $p^{\prime}$,
respectively; and so the momentum of the meson $D_{s}^{-}$ is
$q=p-p^{\prime}$. The current $\eta^{Z_{cs}}_{\mu}(x)$ is given by Eq. (7) in
Section II, while the remaining two currents, we employ:

$\eta_{\nu}^{D^{\ast 0}}(y)=\overline{u}^{g}(y)i\gamma_{\nu}c^{g}(y),\ \
\eta^{D_{s}^{-}}(0)=i\overline{c}^{f}(0)\gamma_{5}s^{f}(0),$ (32)

where $g$ and $f$ are the color indices. Then, we apply the standard
prescription of the TQCDSR technique and determine the correlation function
$\Pi_{\mu\nu}(p,p^{\prime},T)$ using both physical parameters of the hadrons
involved in the process and quark-gluon degrees of freedom. Isolating the
ground-state contribution to the correlation function in Eq. (31) from
contributions of higher resonances and continuum states for the physical side
of the TQCDSR $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime},T)$, we get:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime},T)=\frac{\langle\Omega|\eta_{\nu}^{D^{\ast}}|D^{\ast}(p^{\prime})\rangle\langle\Omega|\eta^{D_{s}}|D_{s}(q)\rangle}{(m_{Z_{cs}}^{2}-p^{2})(m_{D^{\ast}}^{2}-p^{\prime
2})}$ $\displaystyle\times\frac{\langle
D_{s}(q)D^{\ast}(p^{\prime})|Z_{cs}(p)\rangle\langle
Z_{cs}(p)|\eta_{\mu}^{\dagger
Z_{cs}}|\Omega\rangle}{(m_{D_{s}}^{2}-q^{2})}+\ldots~{}~{}~{}~{}~{}$ (33)

To simplify this expression, we introduce the following matrix elements in
terms of the meson’s physical parameters:

$\displaystyle\langle\Omega|\eta^{D^{\ast}}_{\nu}|D^{\ast}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
f_{D^{\ast}}(T)m_{D^{\ast}}(T)\varepsilon^{\prime}_{\nu},\ $
$\displaystyle\langle Z_{cs}(p)|\eta_{\mu}^{Z_{cs}}|\Omega\rangle$
$\displaystyle=$
$\displaystyle\lambda_{Z_{cs}}(T)m_{Z_{cs}}(T)\varepsilon^{\ast}_{\mu},$
$\displaystyle\langle\Omega|\eta^{D_{s}}|D_{s}(q)\rangle$ $\displaystyle=$
$\displaystyle f_{D_{s}}(T)\frac{m^{2}_{D_{s}}(T)}{m_{c}+m_{s}},$ (34)

here $m_{D^{\ast}}(T)$, $m_{Z_{cs}}(T)$, $m_{D_{s}}(T)$ and $f_{D^{\ast}}(T)$,
$\lambda_{Z_{cs}}(T)$, $f_{D_{s}}(T)$ are the temperature-dependent masses and
decay constants of the mesons $D^{\ast}(2007)^{0}$, $Z_{cs}(3985)$ and
$D_{s}(1968)^{-}$, respectively. $\varepsilon^{\prime}_{\nu}$ and
$\varepsilon_{\mu}^{*}$ are the polarization vectors of the
$D^{\ast}(2007)^{0}$ and $Z_{cs}(3985)$ states, respectively. Then, we model
$\langle D_{s}(q)D^{\ast}(p^{\prime},s^{\prime})|Z_{cs}(p,s)\rangle$ in Eq.
(34) as follows:

$\displaystyle\langle
D_{s}(q)D^{\ast}(p^{\prime})|Z_{cs}(p)\rangle=g_{1}(T)[(p\cdot p^{\prime})$
$\displaystyle\times(\varepsilon^{\prime*}\cdot\varepsilon)-(p\cdot\varepsilon^{\prime*})(p^{\prime}\cdot\varepsilon)]$
(35)

denoting the strong coupling of the vertex
$Z_{cs}(p)D_{s}(q)D^{\ast}(p^{\prime})$ with $g_{1}(T)$. Then, it is easy to
get the physical part of the correlation function in Eq. (33):

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime},T)=\frac{g_{1}(T)f_{D^{*}}(T)m_{D^{*}}(T)\lambda_{Z_{cs}}(T)m_{Z_{cs}}(T)}{(m_{c}+m_{s})}~{}$
$\displaystyle\times\frac{f_{D_{s}}(T)m_{D_{s}}^{2}(T)}{(m_{Z_{cs}}^{2}(T)-p^{2})(m^{2}_{D^{*}}(T)-p^{\prime
2})(m_{D_{s}}^{2}(T)-q^{2})}$
$\displaystyle\times\bigg{[}\frac{m_{Z_{cs}}^{2}(T)+m_{D^{\ast}}^{2}(T)-m_{D_{s}}^{2}(T)}{2}g_{\mu\nu}-p^{\prime}_{\mu}p_{\nu}\bigg{]}+\ldots$
(36)

The correlation function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime},T)$ )
contains the two different Lorentz structures proportional to
$p_{\mu}^{\prime}p_{\nu}$, $g_{\mu\nu}$ and one of which should be chosen to
get the sum rules. We select the structure $g_{\mu\nu}$ to work with the
invariant amplitude $\Pi^{\mathrm{Phys}}(p^{2},p^{\prime 2},T)$. Afterwards,
we carry out the double Borel transformation of this amplitude over variables
$p^{2}$ and $p^{\prime 2}$. This operation allows us to arrive the physical
side of the sum rule.

In order to get the other side, i.e. QCD side, of the three-point sum rule, we
derive $\Pi_{\mu\nu}(p,p^{\prime},T)$ in terms of the quark propagators:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,p^{\prime},T)=i^{4}\int
d^{4}xd^{4}ye^{i(p^{\prime}\cdot y-p\cdot x)}\epsilon_{abc}\epsilon_{dec}$
$\displaystyle\times\mathrm{Tr}[S_{c}^{ef}(x)\gamma_{5}S_{s}^{fa}(-x)\gamma_{5}\widetilde{S}_{c}^{gb}(y-x)\gamma_{\nu}\widetilde{S}_{u}^{dg}(x-y)\gamma_{\mu}].~{}$
(37)

The correlation function $\Pi_{\mu\nu}^{\mathrm{QCD}}(p,p^{\prime},T)$ is
computed with dimension-6 accuracy, and has the same Lorentz structures as
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime},T)$. The double Borel
transformation $\mathcal{B}\Pi^{\mathrm{QCD}}(p^{2},p^{\prime 2},T)$ unveils
the second side of the sum rule. The Borel transformed and subtracted
amplitude $\Pi^{\mathrm{QCD}}(p^{2},p^{\prime 2},T)$ can be written depending
on the spectral density $\widetilde{\rho}(s,s^{\prime},T)$ which is
proportional to the imaginary part of $\Pi^{\mathrm{QCD}}(p,p^{\prime},T)$,

$\displaystyle\Pi(\mathbf{M}^{2},\mathbf{\
s}_{0},T)=\int_{\mathcal{M}^{2}}^{s_{0}}ds\int_{m_{c}^{2}}^{s_{0}^{\prime}}ds^{\prime}\widetilde{\rho}(s,s^{\prime},T)$
$\displaystyle\times e^{-s/M_{1}^{2}}e^{-s^{\prime}/M_{2}^{2}},$ (38)

where $\mathbf{M}^{2}=(M_{1}^{2},\ M_{2}^{2})$ and $\mathbf{s}_{0}=(s_{0},\
s_{0}^{\prime})$ represent the Borel mass and continuum threshold parameters,
respectively. The pair of parameters $(M_{1},s_{0})$ corresponds to the
initial tetraquark’s channels, whereas $(M_{2},s_{0}^{\prime})$ depicts the
final-state meson. $\mathcal{M}=(2m_{c}+m_{s}+m_{u})$ and
$\widetilde{\rho}(s,s^{\prime},T)$ are spectral densities computed as the
imaginary parts of the corresponding terms in
$\Pi^{\mathrm{QCD}}_{\mu\nu}(p,p^{\prime},T)$. Next, by equating
$\mathcal{B}\Pi^{\mathrm{QCD}}(p^{2},p^{\prime 2},T)$ and Borel transformation
of $\Pi^{\mathrm{Phys}}(p^{2},p^{\prime 2},T)$, and doing continuum
subtraction, the sum rule for the coupling $g_{1}(T)$ is determined as:

$\displaystyle
g_{1}(T)=\frac{2(m_{D_{s}}^{2}(T)-q^{2})}{f_{D^{\ast}}(T)m_{D^{\ast}}(T)\lambda_{Z_{cs}}(T)m_{Z_{cs}}(T)f_{D_{s}}(T)m_{D_{s}}^{2}(T)}$
$\displaystyle\times\frac{(m_{c}+m_{s}){\Pi}(\mathbf{M}^{2},\mathbf{\
s}_{0},q^{2})}{\big{(}m_{Z_{cs}}^{2}(T)+m_{D^{\ast}}^{2}(T)-m_{D_{s}}^{2}(T)\big{)}e^{-m_{Z_{cs}}^{2}/M_{1}^{2}}e^{-m_{D^{*}}^{2}/M_{2}^{2}}}$
(39)

Note that, the $g_{1}(T)$ is a function of $T$, and also rely on the Borel and
continuum threshold parameters, but are not explicitly specified in Eq. (39)
as arguments of $g_{1}$. After that, we introduce a new variable
$Q^{2}=-q^{2}$ and denote the obtained function as $g_{1}(Q^{2})$.

The sum rule in Eq. (39 ) includes mass and decay constant’s temperature-
dependence of the final mesons, so we need numerical values of these
parameters. Therefore they are computed with the standard sum rule method and
findings in vacuum are given in Table 2.

Parameters | Numeric Values ($\mathrm{GeV}$)
---|---
$m_{D^{0}}$ | $1.861^{+0.063}_{-0.062}$
$m_{D_{s}^{-}}$ | $1.966^{+0.060}_{-0.059}$
$m_{D^{*0}}$ | $2.005^{+0.051}_{-0.050}$
$m_{D_{s}^{*-}}$ | $2.071^{+0.025}_{-0.024}$
$f_{D^{0}}$ | $0.22^{+0.01}_{-0.01}$
$f_{D_{s}^{-}}$ | $0.27^{+0.01}_{-0.01}$
$f_{D^{*0}}$ | $0.23^{+0.01}_{-0.01}$
$f_{D_{s}^{*-}}$ | $0.27^{+0.01}_{-0.01}$
Table 2: Obtained mass and coupling constant values of $D$ mesons at $T=0$
produced in the decays of tetraquark $Z_{cs}$.

The following is the function that best fits the graphs we have drawn for the
temperature dependencies of the mass and couplings:

$F_{n}(T)=A_{n}\mathrm{e}^{\frac{T}{B_{n}}}+C_{n},$ (40)

here $F_{n}$, $A_{n}$, $B_{n}$ and $C_{n}$ are fitting parameters. Numerical
analysis lets us fix these parameters as in Table 3.

$F_{n}$ | $A_{n}(\mathrm{GeV})$ | $B_{n}(\mathrm{GeV})$ | $C_{n}(\mathrm{GeV})$
---|---|---|---
$m_{D_{s}^{-}}$ | $-1.375\times 10^{-4}$ | $0.021$ | $1.970$
$m_{D^{*0}}$ | $-1.135\times 10^{-4}$ | $0.020$ | $2.008$
$m_{D_{s}^{*-}}$ | $-1.407\times 10^{-4}$ | $0.020$ | $2.077$
$m_{Z_{cs}}$ | $-2.206\times 10^{-4}$ | $0.020$ | $4.001$
$m_{D^{0}}$ | $-1.198\times 10^{-4}$ | $0.021$ | $1.865$
| $A_{n}(\mathrm{GeV})$ | $B_{n}(\mathrm{GeV})$ | $C_{n}(\mathrm{GeV})$
$f_{D_{s}^{-}}$ | $-4.370\times 10^{-5}$ | $0.021$ | $0.271$
$f_{D^{*0}}$ | $-3.068\times 10^{-5}$ | $0.020$ | $0.190$
$f_{D_{s}^{*-}}$ | $-3.690\times 10^{-5}$ | $0.021$ | $0.271$
$f_{D^{0}}$ | $-4.170\times 10^{-5}$ | $0.021$ | $0.224$
| $A_{n}(\mathrm{GeV^{4}})$ | $B_{n}(\mathrm{GeV})$ | $C_{n}(\mathrm{GeV^{4}})$
$\lambda_{Z_{cs}}$ | $-5.769\times 10^{-4}$ | $0.023$ | $0.007$
Table 3: Fit parameters for the mass and meson-current coupling constant of
$D_{s}^{-},D^{*0},D_{s}^{*-},D^{0}$ and $Z_{cs}$ states.

Then to continue the evaluation of $g_{1}(Q^{2})$, we need to determine
$\mathbf{M}^{2}$ and $\mathbf{s}_{0}$. The limitations imposed on these
auxiliary parameters have been mentioned before. The working region for Borel
mass and continuum threshold is the same range used in the mass and meson-
current coupling constant calculation. The decay width of the considered
process should be computed using the strong coupling at the $D_{s}^{-}$
meson’s mass shell $Q^{2}=-m_{D_{s}}^{2}$, which is not accessible to the sum
rule calculations. We avoid this problem by adopting a fitting procedure.
Using the fit function below;

$g_{fit}(Q^{2})=\frac{g_{0}}{1-a\frac{Q^{2}}{m^{2}_{Z_{cs}}}+b\big{(}\frac{Q^{2}}{m^{2}_{Z_{cs}}}\big{)}^{2}}$
(41)

where $g_{0}=0.22,~{}a=-1.37,~{}b=-0.84$ are the fit coefficients for the
coupling $g_{1}$ which gives at the mass shell $Q^{2}=-m_{D_{s}}^{2}$ in
vacuum:

$g_{1}(-m_{D_{s}}^{2})=0.36\pm 0.02~{}\mathrm{GeV^{-1}}.$ (42)

The decay width of $Z_{cs}\rightarrow D_{s}^{-}D^{*0}$ is extracted by the
following expression

$\displaystyle\Gamma_{1}\left[Z_{cs}\rightarrow
D_{s}^{-}D^{\ast}(2007)^{0}\right]=\frac{g_{1}^{2}(T)m_{D^{\ast}}^{2}(T)}{24\pi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\xi(m_{Z_{cs}}(T),m_{D^{\ast}}(T),m_{D_{s}^{-}}(T))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\Big{[}3+\frac{2\xi^{2}(m_{Z_{cs}}(T),m_{D^{\ast}}(T),m_{D_{s}^{-}}(T))}{m_{D^{\ast}}^{2}(T)}\Big{]},$
(43)

where

$\xi(a,b,c)=\frac{1}{2a}\sqrt{a^{4}+b^{4}+c^{4}-2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})}.$
(44)

Employing the vacuum value of strong coupling from Eq. (42) and the
$m_{D^{*}}$ from Table 2, the decay width of $Z_{cs}\rightarrow
D_{s}^{-}D^{\ast}(2007)^{0}$

$\Gamma_{1}\left[Z_{cs}\rightarrow D_{s}^{-}D^{\ast}(2007)^{0}\right]=(4.35\pm
0.44)\ \mathrm{MeV}\text{.}$ (45)

$\bullet$ The second process $\widetilde{Z}_{cs}\rightarrow D_{s}^{*-}D^{0}$
can be handled in the same way as the first process. However here, we utilize
the following current expressions for the $D^{0}$ and $D_{s}^{*-}$ mesons;

$\eta_{\nu}^{D^{0}}(y)=i\overline{u}^{g}(y)\gamma_{5}c^{g}(y),\eta^{D_{s}^{*-}}(0)=\overline{c}^{f}(0)\gamma_{\nu}s^{f}(0),$
(46)

and introduce the new matrix elements:

$\displaystyle\langle\Omega|\eta^{D^{0}}|D^{0}(p^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\frac{f_{D^{0}}(T)m_{D^{0}}^{2}(T)}{m_{c}},$
$\displaystyle\langle Z_{cs}(p)|\eta_{\mu}^{Z_{cs}}|\Omega\rangle$
$\displaystyle=$
$\displaystyle\lambda_{Z_{cs}}(T)~{}m_{Z_{cs}}(T)~{}\varepsilon^{\ast}_{\mu},$
$\displaystyle\langle\Omega|\eta^{D_{s}^{*}}_{\nu}|D_{s}^{*}(q,\varepsilon^{{}^{\prime}})\rangle$
$\displaystyle=$ $\displaystyle
f_{D_{s}^{*}}(T)~{}m_{D_{s}^{*}}(T)~{}\varepsilon^{{}^{\prime}}_{\nu}.$ (47)

By applying the standard procedures mentioned above for the first process,
$\widetilde{\Pi}^{\mathrm{Phys}}(p,p^{\prime},T)$ and
$\widetilde{\Pi}^{\mathrm{OPE}}(p,p^{\prime},T)$ yield the sum rule

$\displaystyle
g_{2}(q^{2})=\frac{(m_{D_{s}^{*}}^{2}(T)-q^{2})}{f_{D^{0}}(T)m_{D^{0}}^{2}(T)\lambda_{Z_{cs}}(T)m_{Z_{cs}}(T)f_{D_{s}^{\ast}}(T)m_{D_{s}^{\ast}(T)}}$
$\displaystyle\times\frac{2m_{c}\widetilde{\Pi}(\mathbf{M}^{2},\mathbf{s}_{0},q^{2})}{\big{(}m_{Z_{cs}}^{2}(T)+m_{D_{s}^{\ast}}^{2}(T)-m_{D^{0}}^{2}(T)\big{)}e^{-m_{Z_{cs}}^{2}/M_{1}^{2}}e^{-m_{D^{0}}^{2}/M_{2}^{2}}}.$
(48)

Selecting the auxiliary parameters according to the same criteria as stated
above and to determine the coupling constant $g_{2}$ at $T=0$, employing the
fit function in Eq. (41 ), with coefficients $g_{0}=0.24,~{}a=-1.52$, and
$b=-0.92$ at the mass shell $Q^{2}=-m^{2}_{D_{s}^{*}}$ reads:

$g_{2}(-m_{D_{s}^{*}}^{2})=0.47\pm 0.03~{}\mathrm{GeV^{-1}}.$ (49)

The decay width of the second process is defined by the formula;

$\displaystyle\Gamma_{2}[\widetilde{Z}_{cs}\rightarrow
D_{s}^{*-}D^{0}]=\frac{g_{2}^{2}(T)m_{D_{s}^{*-}}^{2}(T)}{24\pi}$
$\displaystyle~{}~{}~{}~{}~{}~{}\times\xi\big{(}m_{\widetilde{Z}_{cs}}(T),m_{D_{s}^{*-}}(T),m_{D^{0}}(T)\big{)}$
$\displaystyle\times\bigg{(}3+\frac{2\big{[}\xi(m_{\widetilde{Z}_{cs}}(T),m_{D_{s}^{*-}}(T),m_{D^{0}}(T))\big{]}^{2}}{m_{D_{s}^{*-}}^{2}(T)}\bigg{)},$
(50)

and our prediction for this decay channel is:

$\Gamma_{2}[\widetilde{Z}_{cs}\rightarrow D_{s}^{*-}D^{0}]=(7.65\pm 0.67)\
\mathrm{MeV}\text{.}$ (51)

As a result, we get the partial widths of these decays in the present section
and using Eqs. (45) and (51), the full width and mean lifetime of $Z_{cs}$ are
foreseen as:

$\displaystyle\Gamma^{\mathrm{full}}$ $\displaystyle=$ $\displaystyle(12.0\pm
0.8)~{}\mathrm{MeV},$ $\displaystyle\tau$ $\displaystyle=$
$\displaystyle(5.5\pm 0.5)\times 10^{-19}s.$ (52)

Our result for the $\Gamma^{\mathrm{{full}}}$ is consistent with the measured
value in BESIII Ablikim:2020hsk .

The last step is to analyse the variation of partial decay widths in terms of
temperature. We draw the $\Gamma_{Z_{cs}}(T)/\Gamma_{Z_{cs}}(0)$ versus
$T/T_{c}$ in Figure (4). As is seen from this figure decay width is
dramatically increased with growing temperature.

Figure 4: Variation of the ratio of temperature-dependent decay width of
$Z_{cs}\rightarrow D_{s}^{-}D^{*0}/D_{s}^{*-}D^{0}$ to its value in vacuum
according to $T/T_{c}$ for $T_{c}=155~{}\mathrm{MeV}$.

##  V Summary and Results

Collisions of heavy-ions in laboratory conditions allow us to create and
investigate the strongly interacting matter in hot medium. Many facilities
contribute to probing the chiral as well as the deconfinement phase transition
from hadronic matter to the QGP and mapping out different domains of the QCD
phase diagram. Among these experiments, Relativistic Heavy Ion Collider (RHIC)
and Large Hadron Collider (LHC) energies form deconfined matter characterized
by vanishing baryon densities and high temperatures consistent with lattice
data. Mapping out the region of a first-order transition at large chemical
potential is a primary aim of current and upcoming experimental programs.
Recently, RHIC and LHC have radically increased the energy levels that can be
attained by heavy nuclei collisions at near-light speeds bringing them in line
with those of the conditions in the early universe. In addition to these
improvements, future experiments at the Facility for Antiproton and Ion
Research (FAIR) and at the Nuclotron-based Ion Collider (NICA) will generate a
wealth of data.

Also, the ALICE experiment is at the CERN LHC finalizing a major upgrade and
will restart its operations with a new computing system in order to handle a
data volume roughly $100$ times larger than during the previous operational
period in 2022. Further constraints can be set by future higher precision
measurements during Run 3. The LHCC review of the ALICE $3$ plans has begun
and is expected to be concluded in March 2022 and deliver highly impactful
physics results.

However, in heavy-ion collisions, the QGP lifetime is $(\sim 10fm/c)$, and the
hadrons would be erased from the spectrum in an extremely short time as seen
from Eq. (52). Hence it is almost impossible to work with a literally external
probe in these collisions, whereas a hadron would continue to exist although
with changing mass in QGP medium. Alternatively, such an imaging method can be
performed using particles produced during the parton-parton scatterings. So,
searching the signals of QGP provide knowledge to quantitatively understand
the charmonium/bottomonium suppression of conventional and also exotic states
Chen:2021akx ; Abreu:2016qci ; Abreu:2017pos  in heavy-ion collisions at high
temperatures. This issue may be one of the focus areas of research in the near
future regarding QGP signals.

Meanwhile, the newly observed resonance $Z_{cs}(4000)$ with strangeness by
LHCb collaboration got people to think whether they are the same state as the
$Z_{cs}(3985)$. But, according to our lifetime calculation of $Z_{cs}(3985)$
detected in BESIII, has a narrower state from the $Z_{cs}(4000)$ in LHCb and
they should be different states. This can be tested in future experiments and
distinguish the two-state interpretation from the one-state scheme.

In this work, we propose a novel picture, i.e Thermal QCDSR, to understand the
nature of $Z_{cs}(3985)$ with $J^{P}=1^{+}$ and also we reduce our results
$T=0$ to compare with experimental and theoretical data in the literature. We
also estimate the hadronic parameters of b-partner of $Z_{cs}(3985)$ which we
hope to be detected in the near future experiments. According to our numerical
evaluations, changes in mass and meson-current coupling constant values are
fixed up to $T\cong 100~{}\mathrm{MeV}$, but start to decrease after this
point for both of them. At critical transition temperature the values of mass
and meson-current coupling constant of $Z_{cs}(3985)$ change up to
$10\%,66\%$, respectively. The thermal width of the $Z_{cs}(3985)$ meson (see
Figure 4) exhibits an increase of roughly a factor 4.6 near $T_{c}$.

As a result, by looking at the numerical analysis we find that the resonance
$Z_{cs}(3985)$ can be well defined as a diquark-antidiquark candidate with
quark content $[\bar{c}cu\bar{s}]$. The variations on decay width of
$Z_{cs}(3985)$ will provide valuable input to our understanding of the heavy
quark system in heavy-ion collisions. However, XYZ exotic states should be
tested in more precise experimental data in the future and we need more
experimental studies on the dominant decay channels of $Z_{cs}(3985)$ to pin
down its inner configuration.

##  Appendix A Thermal spectral density Functions

In this part, the results of our evaluations for the spectral density is
presented for the mass and meson-current coupling constant as a function of
the temperature belonging to the $Z_{cs}(3985)$ resonance in the tetraquark
picture using the following abbreviations as $\Theta$ is the step function
(for brevity, we don’t give the spectral density belonging to the decay width
here):

$\displaystyle L(s,x)$ $\displaystyle=$ $\displaystyle sx(1-x)-m_{c}^{2},$
$\displaystyle L^{{}^{\prime}}(s,x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\frac{(-1+x_{2})\Big{[}-sx_{1}x_{2}(-1+x_{1}+x_{2})+m_{c}^{2}(x_{1}+x_{2})(x_{1}^{2}+x_{1}(-1+x_{2})+(-1+x_{2})x_{2})\Big{]}}{(x_{1}^{2}+x_{1}(-1+x_{2})+(-1+x_{2})x_{2})^{2}},$
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
x_{1}^{2}+x_{1}(x_{2}-1)+x_{2}(x_{2}-1),$ $\displaystyle\beta$
$\displaystyle=$ $\displaystyle x_{1}+x_{2}-1,$ $\displaystyle\zeta$
$\displaystyle=$ $\displaystyle x_{1}(x_{1}-1)+x_{2}(x_{1}-1)+x_{2}^{2},$
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle x_{1}+x_{2},$

and also we separate the thermal spectral density functions in terms of
dimensions:

$\displaystyle\rho^{\mathrm{QCD}}(s,T)$ $\displaystyle=$
$\displaystyle\rho^{\mathrm{pert.}}(s)+\rho^{\langle\bar{q}q\rangle}(s,T)+\rho^{G^{2}+\langle\theta_{00}\rangle}(s,T)+\rho^{\langle
qGq\rangle}(s,T)+\rho^{\langle\bar{q}q\rangle^{2}}(s,T).$ (53)

The explicit form of spectral densities is performed with the integrals over
the Feynman parameters $x,x_{1},x_{2},$ and $x_{3}$ as follows:

$\displaystyle\rho^{pert}(s)$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}\frac{dx_{2}}{3072\alpha^{8}\beta\pi^{6}}\bigg{\\{}\Big{(}-\alpha\eta
m_{c}^{2}+\beta sx_{1}x_{2}\Big{)}^{2}\Big{(}12\alpha^{3}\eta
m_{c}^{3}m_{s}x_{2}+3\alpha^{2}\eta^{2}m_{c}^{4}x_{1}x_{2}-48\alpha^{2}\beta
m_{c}m_{s}sx_{1}x_{2}^{2}$ (54) $\displaystyle-$ $\displaystyle
26\alpha\beta\eta
m_{c}^{2}sx_{1}^{2}x_{2}^{2}+35\beta^{2}s^{2}x_{1}^{3}x_{2}^{3}\Big{)}\bigg{\\}}\Theta[L^{{}^{\prime}}(s,x_{1},x_{2})],$
$\displaystyle\rho^{\langle\bar{q}q\rangle}(s,T)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx_{1}\int_{0}^{1-dx_{1}}\frac{dx_{2}}{64\alpha^{6}\pi^{4}}\bigg{\\{}15\beta^{3}m_{s}s^{2}\langle\bar{s}s\rangle
x_{1}^{3}x_{2}^{3}-2\alpha^{3}\eta^{2}m_{c}^{5}\Big{(}2\langle\bar{u}u\rangle
x_{1}+\langle\bar{s}s\rangle x_{2}\Big{)}+4\alpha^{2}\beta\eta
m_{c}^{3}sx_{1}x_{2}\Big{(}3\langle\bar{u}u\rangle x_{1}$ (55)
$\displaystyle+$ $\displaystyle 2\langle\bar{s}s\rangle
x_{2}\Big{)}-2\alpha\beta^{2}m_{c}s^{2}x_{1}^{2}x_{2}^{2}\Big{(}4\langle\bar{u}u\rangle
x_{1}+3\langle\bar{s}s\rangle x_{2}\Big{)}+4\alpha\beta
m_{c}^{2}m_{s}sx_{1}x_{2}\Big{(}-4\beta\eta\langle\bar{s}s\rangle
x_{1}x_{2}+\langle\bar{u}u\rangle\zeta^{2}\Big{)}-\alpha^{2}\eta
m_{c}^{4}m_{s}$ $\displaystyle\times$
$\displaystyle\Big{(}-3\beta\eta\langle\bar{s}s\rangle
x_{1}x_{2}+4\langle\bar{u}u\rangle\zeta^{2}\Big{)}\bigg{\\}}\Theta[L^{{}^{\prime}}(s,x_{1},x_{2})],$
$\displaystyle\rho^{G^{2}+\langle\theta_{00}\rangle}(s,T)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}dx_{2}\Bigg{\\{}\frac{1}{2304\alpha^{6}\beta\pi^{6}}\Bigg{[}3\beta
x_{2}\Big{[}4\alpha^{3}\eta m_{c}^{3}m_{s}\Big{(}8\beta\pi^{2}\langle
u^{\mu}\theta^{f}_{\mu\nu}u^{\nu}\rangle+g_{s}^{2}\langle
u^{\mu}\theta^{g}_{\mu\nu}u^{\nu}\rangle
x_{1}\Big{)}+\alpha^{2}\eta^{2}m_{c}^{4}x_{1}$ (56) $\displaystyle\times$
$\displaystyle\Big{(}64\beta\pi^{2}\langle
u^{\mu}\theta^{f}_{\mu\nu}u^{\nu}\rangle+g_{s}^{2}\langle
u^{\mu}\theta^{g}_{\mu\nu}u^{\nu}\rangle(3x_{1}-x_{2})\Big{)}-8\alpha^{2}\beta
m_{c}m_{s}sx_{1}\Big{(}16\beta\pi^{2}\langle
u^{\mu}\theta^{f}_{\mu\nu}u^{\nu}\rangle+g_{s}^{2}\langle
u^{\mu}\theta^{g}_{\mu\nu}u^{\nu}\rangle x_{1}\Big{)}x_{2}$ $\displaystyle-$
$\displaystyle 4\alpha\beta\eta
m_{c}^{2}sx_{1}^{2}\Big{(}208\beta\pi^{2}\langle
u^{\mu}\theta^{f}_{\mu\nu}u^{\nu}\rangle+g_{s}^{2}\langle
u^{\mu}\theta^{g}_{\mu\nu}u^{\nu}\rangle(4x_{1}-5x_{2})\Big{)}x_{2}+5\beta^{2}s^{2}x_{1}^{3}\Big{(}192\beta\pi^{2}\langle
u^{\mu}\theta^{f}_{\mu\nu}u^{\nu}\rangle+g_{s}^{2}$ $\displaystyle\times$
$\displaystyle\langle
u^{\mu}\theta^{g}_{\mu\nu}u^{\nu}\rangle(3x_{1}-5x_{2})\Big{)}x_{2}^{2}\Big{]}+\langle\frac{\alpha_{s}GG}{\pi}\Big{\rangle}\pi^{2}\Big{[}15\beta^{3}s^{2}x_{1}^{3}(5x_{1}-9x_{2})x_{2}^{3}-12\alpha\beta
m_{c}m_{s}sx_{1}x_{2}^{2}\Big{(}4(-1+x_{1})^{2}x_{1}^{2}$ $\displaystyle+$
$\displaystyle
4(-1+x_{1})x_{1}(-1+2x_{1})x_{2}+7(-1+x_{1})x_{1}x_{2}^{2}+(-1+3x_{1})x_{2}^{3}+x_{2}^{4}\Big{)}+3\alpha^{2}m_{c}^{3}\Big{(}m_{s}(-1+x_{1})x_{1}^{3}(-12$
$\displaystyle+$ $\displaystyle
13x_{1})x_{2}+m_{s}(-1+x_{1})x_{1}^{2}(-24+37x_{1})x_{2}^{2}+m_{s}x_{1}\Big{(}12+x_{1}(-59+48x_{1})\Big{)}x_{2}^{3}+m_{s}x_{1}(-25+34x_{1})x_{2}^{4}$
$\displaystyle+$ $\displaystyle
m_{s}(-2+13x_{1})x_{2}^{5}+2m_{s}x_{2}^{6}\Big{)}+3\alpha^{2}\eta^{2}m_{c}^{4}x_{1}x_{2}\Big{(}10x_{1}^{2}+(9-8x_{2})x_{2}-x_{1}(9+x_{2})\Big{)}-8\alpha\beta\eta
m_{c}^{2}sx_{1}^{2}x_{2}^{2}$ $\displaystyle\times$
$\displaystyle\Big{(}13x_{1}^{2}+(18-17x_{2})x_{2}-x_{1}(12+7x_{2})\Big{)}\Big{]}\Bigg{]}\Bigg{\\}}\Theta[L^{{}^{\prime}}(s,x_{1},x_{2})],$
$\displaystyle\rho^{\langle qGq\rangle}(s,T)$ $\displaystyle=$
$\displaystyle\frac{m_{c}^{2}m_{o}^{2}m_{s}\langle\bar{u}u\rangle}{64\pi^{4}}+\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}dx_{2}\Bigg{\\{}-\frac{\beta
m_{o}^{2}}{192\alpha^{5}\pi^{4}}\Big{[}-3\alpha\beta\eta
m_{c}^{2}m_{s}\langle\bar{s}s\rangle
x_{1}x_{2}+8\beta^{2}m_{s}s\langle\bar{s}s\rangle x_{1}^{2}x_{2}^{2}$ (57)
$\displaystyle+$ $\displaystyle 3\alpha^{2}\eta
m_{c}^{3}\Big{(}2\langle\bar{u}u\rangle x_{1}+\langle\bar{s}s\rangle
x_{2}\Big{)}-3\alpha\beta m_{c}sx_{1}x_{2}\Big{(}3\langle\bar{u}u\rangle
x_{1}+2\langle\bar{s}s\rangle
x_{2}\Big{)}\Big{]}\Bigg{\\}}\Theta[L^{{}^{\prime}}(s,x_{1},x_{2})],$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}(s,T)$ $\displaystyle=$
$\displaystyle\frac{1}{1296\pi^{4}}\Bigg{[}\int_{0}^{1}dx\Bigg{\\{}-108m_{c}^{2}\pi^{2}\langle\bar{s}s\rangle\langle\bar{u}u\rangle+m_{c}m_{s}\langle\bar{u}u\rangle\Big{(}g_{s}^{2}\langle\bar{u}u\rangle(-1+x)+108\pi^{2}\langle\bar{s}s\rangle
x\Big{)}\Bigg{\\}}\Theta[L(s,x)]$ (58) $\displaystyle+$
$\displaystyle\int_{0}^{1-x_{1}}dx_{1}\int_{0}^{1-x_{1}-x_{2}}dx_{2}\Bigg{\\{}\beta^{2}g_{s}^{2}\big{[}\langle\bar{s}s\rangle^{2}+\langle\bar{u}u\rangle^{2}\big{]}x_{1}x_{2}(3\alpha\eta
m_{c}^{2}-8\beta
sx_{1}x_{2})\Bigg{\\}}\Bigg{]}\Theta[L^{{}^{\prime}}(s,x_{1},x_{2})].~{}~{}~{}~{}$

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