# Predictions of the Strange partner of $T_{cc}$ in the quark delocalization
color screening model

Xuejie Liu1<EMAIL_ADDRESS>Dianyong Chen2,4111Corresponding author
<EMAIL_ADDRESS>Hongxia Huang3<EMAIL_ADDRESS>Jialun Ping3
<EMAIL_ADDRESS>1School of Physics, Henan Normal University, Xinxiang
453007, P. R. China  2School of Physics, Southeast University, Nanjing 210094,
P. R. China  3Department of Physics, Nanjing Normal University, Nanjing
210023, P.R. China  4Lanzhou Center for Theoretical Physics, Lanzhou
University, Lanzhou 730000, P. R. China

###### Abstract

Inspired by the detection of $T_{cc}$ tetraquark state by LHCb Collaboration,
we preform a systemical investigation of the low-lying doubly heavy charm
tetraquark states with strangeness in the quark delocalization color screening
model in the present work. Two kinds of configurations, the meson-meson
configuration and diquark-antidiquark configuration, are considered in the
calculation. Our estimations indicate that the coupled channel effects play
important role in the multiquark system, and a bound state with $J^{P}=1^{+}$
and a resonance state with $J^{P}=0^{+}$ have been predicted. The mass of the
bound state is evaluated to be $(3971\sim 3975)$ MeV, while the mass and width
of the resonance are determined to be $(4113\sim 4114)$ MeV and $(14.3\sim
16.1)$ MeV, respectively.

###### pacs:

13.75.Cs, 12.39.Pn, 12.39.Jh

##  I Introduction

In the recent two decades, an increasing number of charmonium-like states have
been observed experimentally, which provide a good opportunity of searching
for multiquark states. As the first confirmed charmonium-like state,
$Z_{c}(3900)$ was first observed in the year of 2013 by the
BESIIIBESIII:2013ris  and Belle Belle:2013yex  Collaborations in the
$\pi^{+}J/\psi$ invariant mass spectrum of the process
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ at a center of mass energy of 4.26 GeV,
and then the authors of Ref. Xiao:2013iha  further confirmed the existence of
$Z_{c}(3900)$ by using the data sample collected by CLEO-c detector in the
same process but at $\sqrt{s}=4.170$ GeV. The partial wave analysis of the
process $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ with the data sample accumulated
at $\sqrt{s}=4.23$ and $4.26$ GeV indicated that the spin and parity of the
$Z_{c}(3900)^{\pm}$ state are $1^{+}$ BESIII:2017bua . The observations
indicate that such a new particle can not be simply interpreted in the
conventional quark-antiquark and three-quark schemes. Thus, some exotic
interpretations, such as tetraquark state Agaev:2016dev ; Dias:2013xfa ;
Wang:2013vex ; Deng:2014gqa , hadronic molecular state Wang:2013daa ;
Wilbring:2013cha ; Dong:2013iqa ; Ke:2013gia ; Gutsche:2014zda ;
Esposito:2014hsa ; Chen:2015igx ; Gong:2016hlt ; Ke:2016owt , have been
proposed. Besides the resonance interpretations, $Z_{c}(3900)$ has also been
considered as the kinematic effects Swanson:2014tra ; HALQCD:2016ofq ;
Szczepaniak:2015eza ; Chen:2011xk ; Chen:2013coa ; Swanson:2015bsa , which
indicated that $Z_{c}(3900)$ was not a genuine resonance.

Figure 1: (Color online). The similarity of the hidden charm and doubly
charmed states. Hereinafter, $T_{cc\bar{s}}$ is used to refer the doubly
charmed state with strangeness.

In the resonance frame, the quark component of $Z_{c}(3900)$ is
$c\bar{c}q\bar{q}$. The flavor independence of the strong interactions
naturally indicates the possible existence of the strange partner of
$Z_{c}(3900)$, whose quark components are $c\bar{c}s\bar{q}$. Such kind of
charmonium-like states with strangeness have been predicted theoretically in
various model, such as tetraquark scenarios Ebert:2008kb ; Ferretti:2020ewe ,
hadronic molecular model Lee:2008uy ; Dias:2013qga , the hadro-quarkonium
model Ferretti:2020ewe  and initial single chiral particle emission mechanism
Chen:2013wca . In the year of 2020, the BES III Collaboration observed a new
states named $Z_{cs}(3985)$ in the $K^{+}$ recoil mass distributions of the
process $e^{+}e^{-}\rightarrow K^{+}D_{s}^{-}D^{*0}/K^{+}D_{s}^{*-}D^{0}$
BESIII:2020qkh . Later on, the LHCb Collaboration reported their observation
of two exotic structures, $Z_{cs}(4000)$ and $Z_{cs}(4220)$, in the $J/\psi
K^{+}$ invariant mass spectrum of the $B^{+}\rightarrow J/\psi\phi K^{+}$
decay in 2021 LHCb:2021uow . Since the observed masses of $Z_{cs}(3985)$ and
$Z_{cs}(4000)$ were similar, these two states may be considered as the same
one (hereinafter, we use $Z_{cs}(3985)$ to refer to this state). It’s
interesting to notice that $Z_{c}(3900)$ is located in the vicinity of the
$D^{\ast}\bar{D}$ threshold, while $Z_{cs}(3985)$ is close to
$D_{s}^{\ast}\bar{D}$ threshold, thus one can consider $Z_{cs}(3985)$ as a
strange partner of $Z_{c}(3900)$. Consequently, the hadronic molecular
Chen:2021erj ; Wan:2020oxt ; Xu:2020evn ; Albuquerque:2021tqd ; Ozdem:2021yvo
; Ozdem:2021hka ; Wang:2020rcx ; Meng:2020ihj ; Yang:2020nrt ; Du:2020vwb ,
compact tetraquark Maiani:2021tri ; Shi:2021jyr ; Giron:2021sla  and hadro-
quarkonium Ferretti:2020ewe  scenarios have been proposed to decode the nature
of $Z_{cs}(3985)$.

In the naive multiquark scenario, if there are multiquark states composed of
$c\bar{c}q\bar{q}$, the states composed of $cc\bar{q}\bar{q}$ are also
expected to exist and have been considered to be the molecular $D^{*+}D^{0}$
states Chen:2021vhg ; Santowsky:2021bhy ; Chen:2021tnn ; Feijoo:2021ppq ;
Wang:2021yld ; Wang:2021ajy ; Ren:2021dsi ; Deng:2021gnb ; Xin:2021wcr ;
Azizi:2021aib ; Ozdem:2021hmk ; Albaladejo:2021vln ; Dai:2021vgf ; Dai:2022ulk
; Du:2021zzh ; Zhao:2021cvg ; Ke:2021rxd , and compact states Guo:2021yws ;
Weng:2021hje ; Agaev:2021vur . Recently, the LHCb Collaboration reported the
observation of the first doubly charmed tetraquark state $T_{cc}^{+}(3875)$ in
the $D^{0}D^{0}\pi^{+}$ mass spectrum just below the $D^{*+}D^{0}$ mass
threshold LHCb:2021vvq ; LHCb:2021auc  with $I(J^{P})=1(1^{+})$. As indicated
in Fig. 1, the quark components of $T_{cc}(3875)$ are $cc\bar{q}\bar{q}$,
which indicate that $T_{cc}(3875)$ could be a good candidate of compact
tetraquark state. In Refs. Guo:2021yws ; Weng:2021hje , the authors
investigated the mass spectrum of the $S-$wave doubly heavy tetraquark states
$QQ\bar{q}\bar{q}$ based on the improved chromomagnetic interaction model and
found a stable $cc\bar{u}\bar{d}$ tetraquark state with $I(J^{P})=0(1^{+})$
below the $D^{*+}D^{0}$ threshold, which is well consistent with the observed
$T_{cc}^{+}(3875)$. Moreover, the QCD sum rule estimation in Ref.
Agaev:2021vur  also supported the compact tetraquark interpretation. In
addition, the observed mass of $T_{cc}^{+}(3875)$ is only several hundred keV
below the threshold of $D^{0}D^{\ast+}$, which imply that $T_{cc}^{+}(3875)$
could be interpreted as a shallow molecular state composed of
$D^{0}D^{\ast+}+h.c.$. Further estimations by using the quark models
Chen:2021vhg ; Santowsky:2021bhy ; Chen:2021tnn ; Feijoo:2021ppq ;
Wang:2021yld ; Wang:2021ajy ; Ren:2021dsi ; Deng:2021gnb , QCD sum rules
Xin:2021wcr ; Azizi:2021aib ; Ozdem:2021hmk , heavy quark symmetry
Albaladejo:2021vln ; Dai:2021vgf ; Dai:2022ulk ; Du:2021zzh  and Bethe-
Salpeter equations Zhao:2021cvg ; Ke:2021rxd  indicated that
$T_{cc}^{+}(3875)$ could be a good candidate of $D^{0}D^{\ast+}+h.c.$
molecular state.

Similar to the relation between $Z_{cs}(3985)$ and $Z_{c}(3900)$, one can
expect the existence of the strange partner of $T_{cc}(3875)$, i.e., the
tetraquark states composed of $cc\bar{q}\bar{s}$. Actually, before the
observation of $T_{cc}^{+}(3875)$, the Lattice QCD estimations in Ref.
Junnarkar:2018twb  predicted that the $T_{cc\bar{s}}$ state with $J^{P}=1^{+}$
was about 10 MeV below the threshold of $D^{+}D_{s}^{\ast-}$, while the
estimations by using the heavy quark symmetry in Ref. Eichten:2017ffp  found
its mass to be about 180 MeV above the corresponding threshold. In Ref.
Luo:2017eub , the predicted $T_{cc\bar{s}}$ tetraquark state with
$J^{P}=1^{+}$ was below the threshold of $D^{+}D_{s}^{\ast-}$, while those
with $J^{P}=0^{+}$ and $2^{+}$ were both above the corresponding thresholds.
After the observation of $T_{cc}^{+}$, the authors in Ref. Dai:2021vgf  took
advantage of the experimental information on the binding energy of
$T_{cc}^{+}$ to fix the cutoff regulator of the loops in the Bethe-Salapeter
equation and a $D_{s}^{*}D^{*}$ bound state with $J^{P}=1^{+}$ was predicted.
Besides, the color-magnetic model estimations in Ref. Karliner:2021wju
implied that both $T_{cc}^{+}$ and $T_{cc\bar{s}}^{+}$ system could be stable
against the strong interactions. However, the state $T_{cc\bar{s}}^{+}$ was
not found in the quark model but if the mixing of S$-$D wave was taken into
account, this state may be obtained Deng:2021gnb . As mentioned above,
theorists have not reach an agreement on the existence of $T_{cc\bar{s}}$
tetraquark states. In the present work, we perform a system estimations of
$T_{cc\bar{s}}$ system by using the quark delocalization color screening model
(QDCSM) in an attempt to further explore the existence of the possible bounded
and resonant states in the $T_{cc\bar{s}}$ system.

This paper is organized as follows. After the introduction, the details of the
QDCSM and resonating group method (RGM) are presented in Section II. Our
numerical results and the related discussions for $T_{cc\bar{s}}$ system are
given in Section III, and the last section is devoted to a short summary.

##  II Quark delocalization color screening model and the resonanting group
method

###  II.1 Quark delocalization color screening model

The QDCSM is an extension of the native quark cluster model DeRujula:1975qlm ;
Isgur:1978xj ; Isgur:1978wd ; Isgur:1979be  and also developed with aim of
addressing multiquark systems. For the tetraquark system, the Hamiltonian
reads,

$H=\sum_{i=1}^{4}\left(m_{i}+\frac{\bm{p}_{i}^{2}}{2m_{i}}\right)-T_{CM}+\sum_{j>i=1}^{4}V(r_{ij}),\\\
$ (1)

where $T_{CM}$ is the center-of-mass kinetic energy, who is usually subtracted
without losing generality since we mainly focus on the internal relative
motions of the multiquark system. The interplay is two body potential, which
includes color-confining potential $V_{\mathrm{CON}}$, one-gluon exchange
potential $V_{\mathrm{OGE}}$, and the potential results from Goldstone-boson
exchange, $V_{\chi}$, i.e.,

$V(r_{ij})=V_{\mathrm{CON}}(r_{ij})+V_{\mathrm{OGE}}(r_{ij})+V_{\chi}(r_{ij}).$
(2)

In the present work, we focus on the $S-$wave low-lying positive
$T_{cc\bar{s}}$ tetraquark system with positive parity. In this case, the
spin-orbit and tensor interactions vanish and the potential
$V_{\mathrm{OGE}}(r_{ij})$ becomes,

$\displaystyle V_{\mathrm{OGE}}(r_{ij})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\alpha_{s}^{q_{i}q_{j}}\bm{\lambda}^{c}_{i}\cdot\bm{\lambda}^{c}_{j}$
(3)
$\displaystyle\left[\frac{1}{r_{ij}}-\frac{\pi}{2}\delta(\bm{r}_{ij})(\frac{1}{m^{2}_{i}}+\frac{1}{m^{2}_{j}}+\frac{4\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}}{3m_{i}m_{j}})\right],$

where $m_{i}$ is the quark mass, $\bm{\sigma}_{i}$ and $\bm{\lambda^{c}}_{i}$
are the Pauli matrices and SU(3) color matrices, respectively. The
$\alpha_{s}^{q_{i}q_{j}}$ is the quark-gluon coupling constant, which offers a
consistent description of mesons from light to heavy-quark sector. The values
of $\alpha_{ij}$ are associated with the quark flavors and in the present work
they are fixed by reproducing the mass difference of the low-lying mesons with
$S=0$ and $S=1$.

Table 1: Three sets of model parameters involved in the present estimations.  |  Parameters |  QDCSM1 |  QDCSM2 |  QDCSM3
---|---|---|---|---
|  $m_{u}$(MeV) |  313 |  313 |  313
Quark Mass |  $m_{s}$(MeV) |  536 |  536 |  536
|  $m_{c}$(MeV) |  1728 |  1728 |  1728
|  b(fm) |  0.29 |  0.3 |  0.315
|  $a_{c}$(MeV $fm^{-2}$) |  101 |  101 |  101
|  $V_{0_{uu}}$(MeV) |  -2.3928 |  -2.2543 |  -2.0689
Confinement |  $V_{0_{us}}$(MeV) |  -1.9137 |  -1.7984 |  -1.6429
|  $V_{0_{uc}}$(MeV) |  -1.4175 |  -1.3231 |  -1.2052
|  $V_{0_{ss}}$(MeV) |  -1.3448 |  -1.2826 |  -1.2745
|  $V_{0_{sc}}$(MeV) |  -0.7642 |  -0.6739 |  -0.5452
|  $V_{0_{cc}}$(MeV) |  0.6063 |  0.7555 |  0.9829
|  $\alpha_{s}^{uu}$ |  0.2292 |  0.2567 |  0.3019
|  $\alpha_{s}^{us}$ |  0.2655 |  0.2970 |  0.3484
|  $\alpha_{s}^{uc}$ |  0.3437 |  0.3805 |  0.4405
OGE |  $\alpha_{s}^{ss}$ |  0.3856 |  0.3604 |  0.3360
|  $\alpha_{s}^{sc}$ |  0.5969 |  0.6608 |  0.7649
|  $\alpha_{s}^{cc}$ |  1.5101 |  1.6717 |  1.9353

The confining potential $V_{\mathrm{CON}}(r_{ij})$ is,

$V_{\mathrm{CON}}(r_{ij})=-a_{c}\bm{\lambda^{c}_{i}\cdot\lambda^{c}_{j}}\left[f(r_{ij})+V_{0_{q_{i}q_{j}}}\right],$
(4)

where the $V_{0_{q_{i}q_{j}}}$ is determined by the mass differences of the
theoretical esmations and experimental measurement of each kind of meson,
which is also quark flavor related parameter. In the QDCSM, the function
$f(r_{ij})$ is defined as,

$f(r_{ij})=\left\\{\begin{array}[]{ll}r_{ij}^{2}&\qquad\mbox{if }i,j\mbox{
occur in the same cluster},\\\
\frac{1-e^{-\mu_{ij}r_{ij}^{2}}}{\mu_{ij}}&\qquad\mbox{if }i,j\mbox{ occur in
different cluster},\\\ \end{array}\right.$ (5)

where the color screening parameter $\mu_{ij}$ relevant to the light quarks
can be determined by fitting the deuteron properties, $NN$ and $NY$ scattering
phase shifts Chen:2011zzb ; Ping:1993me ; Wang:1998nk , which are
$\mu_{qq}=0.45$, $\mu_{qs}=0.19$ and $\mu_{ss}=0.08$. The parameter $\mu_{ij}$
satisfy the relation $\mu_{qs}^{2}=\mu_{qq}\mu_{ss}$, where $q$ represents $u$
or $d$ quark. When extending to the heavy-quark case, we found that the
dependence of the parameter $\mu_{cc}$ is rather weak in the calculation of
the spectrum of $P_{c}$ states by taking the value of $\mu_{cc}$ from
$10^{-4}$ to $10^{-2}\ \mathrm{fm}^{-2}$ Huang:2015uda . Moreover, when
$\mu_{ij}$ is rather small, the exponential function can be approximated to
be,

$\displaystyle e^{-\mu_{ij}r_{ij}^{2}}$ $\displaystyle=$ $\displaystyle
1-\mu_{ij}r_{ij}^{2}+\mathcal{O}(\mu_{ij}^{2}r_{ij}^{4}).$ (6)

in the small $r$ region. Accordingly, the confinement potential between two
clusters is approximated to be,

$\displaystyle V_{\mathrm{CON}}(r_{ij})$ $\displaystyle=$ $\displaystyle-
a_{c}\bm{\mathbf{\lambda}}^{c}_{i}\cdot\bm{\mathbf{\lambda}}^{c}_{j}~{}\left(\frac{1-e^{-\mu_{ij}\mathbf{r}_{ij}^{2}}}{\mu_{ij}}+V_{0_{ij}}\right)$
$\displaystyle\approx$ $\displaystyle-
a_{c}\bm{\mathbf{\lambda}}^{c}_{i}\cdot\bm{\mathbf{\lambda}}^{c}_{j}~{}(r_{ij}^{2}+V_{0_{ij}}),$
(7)

which is the same with the expression of two quarks in the same cluster. Thus,
when the value of the $\mu_{ij}$ is very small, the screened confinement will
return to the quadratic form, which is why the results are insensitive to the
value of $\mu_{cc}$. So in the present work, we take $\mu_{cc}=0.01\
\mathrm{fm}^{-2}$. Then $\mu_{sc}$ and $\mu_{uc}$ are obtained by the relation
$\mu^{2}_{sc}=\mu_{ss}\mu_{cc}$ and $\mu^{2}_{uc}=\mu_{uu}\mu_{cc}$,
respectively.

The Goldstone-boson exchange interactions between light quarks appear because
the dynamical breaking of chiral symmetry. For the $T_{cc\bar{s}}$ system, the
$\pi$ exchange interaction vanishes because there is no unflavor quark pair in
the tetraquark state, and then the concrete form of the Goldstone-boson
exchange potential becomes,

$\displaystyle V^{\chi}_{ij}$ $\displaystyle=$ $\displaystyle
V_{K}(\bm{r}_{ij})\sum_{a=4}^{7}\lambda_{i}^{a}\cdot\lambda_{j}^{a}+$ (8)
$\displaystyle
V_{\eta}(\bm{r}_{ij})\left[\left(\lambda_{i}^{8}\cdot\lambda_{j}^{8}\right)\cos\theta_{P}-(\lambda_{i}^{0}\cdot\lambda_{j}^{0})\sin\theta_{P}\right],$

with

$\displaystyle V_{\chi}(\bm{r}_{ij})$ $\displaystyle=$
$\displaystyle{\frac{g_{ch}^{2}}{{4\pi}}}{\frac{m_{\chi}^{2}}{{\
12m_{i}m_{j}}}}{\frac{\Lambda_{\chi}^{2}}{{\Lambda_{\chi}^{2}-m_{\chi}^{2}}}}m_{\chi}$
(9)
$\displaystyle\left\\{(\bm{\sigma}_{i}\cdot\bm{\sigma}_{j})\left[Y(m_{\chi}\,r_{ij})-{\frac{\Lambda_{\chi}^{3}}{m_{\chi}^{3}}}Y(\Lambda_{\chi}\,r_{ij})\right]\right\\},$
$\displaystyle~{}~{}~{}~{}~{}~{}\chi=\\{K,\eta\\},$

where $Y(x)=e^{-x}/x$ is the standard Yukawa function. The $\bm{\lambda^{a}}$
is the SU(3) flavor Gell-Mann matrix. The mass of the $K$ and $\eta$ meson is
taken from the experimental value ParticleDataGroup:2018ovx . The chiral
coupling constant, $g_{ch}$, is determined from the $\pi NN$ coupling constant
through,

$\frac{g_{ch}^{2}}{4\pi}=\left(\frac{3}{5}\right)^{2}\frac{g_{\pi
NN}^{2}}{4\pi}{\frac{m_{u,d}^{2}}{m_{N}^{2}}},$ (10)

where the SU(3) flavor symmetry only broken by the different masses of the
light quarks. All the other model parameters are the same as the ones in Ref.
Liu:2021xje , where three different sets of parameters were used to study the
$c\bar{c}s\bar{s}$ tetraquark system and some experimental discovered
charmonium-like state, such as $\chi_{c0}(3930)$, $X(4350)$, $X(4500)$,
$X(4700)$ and $X(4274)$, coule be well explained. For the sake of
completeness, we collect the relevant model parameters in Table 1.

In the QDCSM, the single-particle orbital wave functions in the ordinary quark
cluster model are the left and right centered single Gaussian functions, which
are,

$\displaystyle\phi_{\alpha}(\bm{S_{i}})=\left(\frac{1}{\pi
b^{2}}\right)^{\frac{3}{4}}e^{-\frac{(\bm{r_{\alpha}}-\frac{1}{2}\bm{S_{i}})^{2}}{2b^{2}}},$
$\displaystyle\phi_{\beta}(-\bm{S_{i}})=\left(\frac{1}{\pi
b^{2}}\right)^{\frac{3}{4}}e^{-\frac{(\bm{r_{\beta}}+\frac{1}{2}\bm{S_{i}})^{2}}{2b^{2}}}.\
$ (11)

The quark delocalization is realized by writing the single-particle orbital
wave function as a linear combination of the left and right Gaussians, which
are,

$\displaystyle{\psi}_{\alpha}(\bm{S_{i}},\epsilon)$ $\displaystyle=$
$\displaystyle\left({\phi}_{\alpha}(\bm{S_{i}})+\epsilon{\phi}_{\alpha}(-\bm{S_{i}})\right)/N(\epsilon),$
$\displaystyle{\psi}_{\beta}(-\bm{S_{i}},\epsilon)$ $\displaystyle=$
$\displaystyle\left({\phi}_{\beta}(-\bm{S_{i}})+\epsilon{\phi}_{\beta}(\bm{S_{i}})\right)/N(\epsilon),$
$\displaystyle N(\epsilon)$ $\displaystyle=$
$\displaystyle\sqrt{1+\epsilon^{2}+2\epsilon e^{{-S}_{i}^{2}/4b^{2}}},$ (12)

where $\epsilon(\bm{S}_{i})$ is the delocalization parameter determined by the
dynamics of the quark system rather than free parameters. In this way, the
system can choose its most favorable configuration through its dynamics in a
larger Hilbert space.

###  II.2 The resonating group method

Figure 2: The meson-meson configuration (diagram (a)) and diquark-antidiquark
configuration (diagram (b)) in the $T_{cc\bar{s}}$ tetraquark system.

In the present work, the RGM is employed to carry out the dynamical
calculation. When dealing with the two-cluster system in this method, one can
only consider the relative motion between the clusters, while the two clusters
are frozen inside Kamimura:1981oxj . So the wave function of the
$T_{cc\bar{s}}$ system can be constructed as,

$\displaystyle\psi_{4q}$ $\displaystyle=$
$\displaystyle\mathcal{A}\left[\left[\psi_{A}(\bm{\rho}_{A})\psi_{B}(\bm{\rho}_{B})\right]^{[\sigma]IS}\otimes\chi_{L}(\textbf{R})\right]^{J},$
(13)

where the symbol $\mathcal{A}$ is the antisymmetry operator, which is defined
as

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle 1-P_{13}.$ (14)

where the $P_{13}$ indicates the exchange of the particle positions with
numbers $1$ and $3$ from the Fig. 2. $[\sigma]=[222]$ gives the total color
symmetry. The symbols $I$, $S$, $L$, and $J$ represent flavor, spin, orbit
angular momentum, and total angular momentum of $T_{cc\bar{s}}$ system,
respectively. $\psi_{A}$ and $\psi_{B}$ are the wave functions of the two-
quark cluster, which are,

$\displaystyle\psi_{A}$ $\displaystyle=$ $\displaystyle\left(\frac{1}{2\pi
b^{2}}\right)^{3/4}e^{-\bm{\rho_{A}}^{2}/(4b^{2})}\eta_{I_{A}}S_{A}\chi_{A}^{c},$
$\displaystyle\psi_{B}$ $\displaystyle=$ $\displaystyle\left(\frac{1}{2\pi
b^{2}}\right)^{3/4}e^{-\bm{\rho_{B}}^{2}/(4b^{2})}\eta_{I_{B}}S_{B}\chi_{B}^{c},$
(15)

where $\eta_{I}$, $S$, and $\chi$ represent the flavor, spin and internal
color terms of the cluster wave functions, respectively. According to Fig. 2,
we adopt different Jacobi coordinates for different diagrams. As for the
meson-meson configuration in Fig. 2-(a), the Jacobi coordinates are defined
as,

$\displaystyle\bm{\rho_{A}}$ $\displaystyle=$
$\displaystyle{\bm{r}_{q_{1}}-\bm{r}_{\bar{q}_{2}}},\ \ \ \
\bm{\rho_{B}}={\bm{r}_{q_{3}}-\bm{r}_{\bar{q}_{4}}},$
$\displaystyle\bm{R_{A}}$ $\displaystyle=$
$\displaystyle\frac{m_{1}{\bm{r}_{q_{1}}}+m_{2}{\bm{r}_{\bar{q}_{2}}}}{m_{1}+m_{2}},$
$\displaystyle\bm{R_{B}}$ $\displaystyle=$
$\displaystyle\frac{m_{3}{\bm{r}_{q_{3}}}+m_{4}{\bm{r}_{\bar{q}_{4}}}}{m_{3}+m_{4}},$
$\displaystyle\bm{R}$ $\displaystyle=$ $\displaystyle\bm{R_{A}-R_{B}},$
$\displaystyle\bm{R_{c}}$ $\displaystyle=$
$\displaystyle\frac{m_{1}{\bm{r}_{q_{1}}}+m_{2}{\bm{r}_{\bar{q}_{2}}}+m_{3}{\bm{r}_{q_{3}}}+m_{4}{\bm{r}_{\bar{q}_{4}}}}{m_{1}+m_{2}+m_{3}+m_{4}}.$
(16)

where the subscript $q/\bar{q}$ indicates the quark or antiquark particle,
while the number indicates the quark position in Fig. 2-(a). As for the
diquark-antidiquark configuration as shown in Fig. 2-(b), the relevant Jacobi
coordinates can be obtained by interchanging $\bm{r}_{q_{3}}$ with
$\bm{r}_{\bar{q}_{2}}$ in Eq. (II.2).

Form the variational principle, after variation with respect to the relative
motion wave function $\chi\bm{(}R)=\sum_{L}\chi_{L}\bm{(}R)$, one obtains the
RGM equation, which is,

$\displaystyle\int
H\left(\bm{R,R^{\prime}}\right)\chi\left(\bm{R^{\prime}}\right)d\bm{R}^{\prime}=E$
$\displaystyle\int
N\left(\bm{R,R^{\prime}}\right)\chi\left(\bm{R^{\prime}}\right)d\bm{R}^{\prime},$
(17)

where $H(\bm{R,R^{\prime}})$ and $N(\bm{R,R^{\prime}})$ are Hamiltonian and
norm kernels, respectively. The eigenenergy $E$ and the wave functions can be
obtained by solving the RGM equation. In the present estimation, the function
$\chi(\bm{R})$ can be expanded by gaussian bases, which is

$\displaystyle\chi\bm{(R)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi}}\sum_{L}\left(\frac{1}{\pi
b^{2}}\right)^{3/4}\sum_{i}^{n}C_{i,L}$ (18) $\displaystyle\times\int
e^{-\frac{1}{2}\bm{(}R-S_{i})^{2}/b^{2}}Y^{L}\left(\hat{\bm{S}_{i}}\right)d\hat{\bm{S}_{i}},$

where $C_{i,L}$ is the expansion coefficient, and $n$ is the number of
gaussian bases, which is determined by the stability of the results.
$\bm{S}_{i}$ is the separation of two reference centers. $\bm{R}$ is the
dynamic coordinate defined in Eq. (II.2). After including the motion of the
center of mass, i.e.,

$\phi_{C}(\bm{R_{c}})=\left(\frac{4}{\pi
b^{2}}\right)^{3/4}\mathrm{e}^{\frac{-2\bm{R_{c}}^{2}}{b^{2}}},$ (19)

one can rewrite Eq. (13) as,

$\displaystyle\psi_{4q}$ $\displaystyle=$
$\displaystyle\mathcal{A}\sum_{i,L}C_{i,L}\int\frac{d\hat{\bm{S_{i}}}}{\sqrt{4\pi}}\prod_{\alpha=1}^{2}\phi_{\alpha}\left(\bm{S_{i}}\right)\prod_{\alpha=3}^{4}\phi_{\beta}\left(\bm{-S_{i}}\right)$
(20)
$\displaystyle\times\left[\left[\eta_{I_{A}S_{A}}\eta_{I_{B}S_{B}}\right]^{IS}Y^{L}(\hat{\bm{S_{i}}})\right]^{J}\left[\chi_{A}^{c}\chi_{B}^{c}\right]^{[\sigma]},$

where $\phi_{\alpha}(\bm{S_{i}})$ and $\phi_{\beta}(\bm{-S_{i}})$ are the
single-particle orbital wave functions with different reference centers, whose
specific expressions have been presented in Eq. (II.1).

With the reformulated ansatz as shown in Eq. (20), the RGM equation becomes an
algebraic eigenvalue equation, which is,

$\displaystyle\sum_{j,L}C_{J,L}H_{i,j}^{L,L^{\prime}}$ $\displaystyle=$
$\displaystyle E\sum_{j}C_{j,L^{\prime}}N_{i,j}^{L^{\prime}},$ (21)

where $N_{i,j}^{L^{\prime}}$ and $H_{i,j}^{L,L^{\prime}}$ are the overlap of
the wave functions and the matrix elements of the Hamiltonian, respectively.
By solving the generalized eigenvalue problem, we can obtain the energies of
the tetraquark systems $E$ and the corresponding expansion coefficients
$C_{j,L}$. Finally, the relative motion wave function between two clusters can
be obtained by substituting the $C_{j,L}$ into Eq. (18). As for the flavor,
spin and color wave functions of the tetraquark system, they are constructed
in a two step way. One can first construct the wave functions for the two
clusters, and then coupling the wave functions of two clusters to form the
wave function of tetraquark system. The details of the flavor, spin and color
wave functions of tetraquark system are collected in the Appendix A.

Table 2:  The relevant channels for all possible $J^{P}$ quantum numbers. $J^{P}=0^{+}$ |  | $J^{P}=1^{+}$ |  | $J^{P}=2^{+}$
---|---|---|---|---
index |  $F^{i};S^{j}_{s};\chi^{c}_{k}$ |  channels |  |  index |  $F^{i};S^{j}_{s};\chi^{c}_{k}$ |  channels |  |  index |  $F^{i};S^{j}_{s};\chi^{c}_{k}$ |  channels
[i;j;k] |  |  [i;j;k] |  |  [i;j;k]
1 |  [1,1,1] |  $D^{0}D_{s}^{+}$ |  |  1 |  [1,3,1] |  $D^{0}D_{s}^{*+}$ |  |  1 |  [1,6,1] |  $D^{*}D_{s}^{*+}$
2 |  [1,2,1] |  $D^{*}D_{s}^{*+}$ |  |  2 |  [1,4,1] |  $D^{*}D_{s}^{+}$ |  |  2 |  [2,6,4] |  $(cc)(\bar{q}\bar{s})$
3 |  [2,1,3] |  $(cc)(\bar{q}\bar{s})$ |  |  3 |  [1,5,1] |  $D^{*}D_{s}^{*+}$ |  |
4 |  [2,2,4] |  $(cc)(\bar{q}\bar{s})$ |  |  4 |  [2,3,3] |  $(cc)(\bar{q}\bar{s})$ |  |
|  |  5 |  [2,4,4] |  $(cc)(\bar{q}\bar{s})$ |  |
|  |  6 |  [2,5,4] |  $(cc)(\bar{q}\bar{s})$ |  |

##  III Numerical RESULTS AND DISCUSSIONS

In this work, only the low-lying $S-$wave $T_{cc\bar{s}}$ tetraquark state are
considered and the spin of the tetraquark system can be 0, 1, and 2. Thus, the
spin parity of $T_{cc\bar{s}}$ tetraquark states can be $0^{+}$, $1^{+}$ and
$2^{+}$, respectively. Moreover, in the present estimations, both the meson-
meson and diquark-antidiquark configurations are considered. In general, there
are two types of color structures for the meson-meson configuration, which are
color singlet-singlet $(\textbf{1}_{c}\otimes\textbf{1}_{c})$ and the color
octet-octet $(\textbf{8}_{c}\otimes\textbf{8}_{c})$. The later color structure
have been taken into account by introducing the color screening effects in the
model , thus, we only consider the color singlet-singlet structures in the
present estimations. A for the diquark-antidiquark configuration, both the
antitriplet-triplet $(\bar{\textbf{3}}_{c}\otimes\textbf{3}_{c})$ and sextet-
antisextet $(\textbf{6}_{c}\otimes\bar{\textbf{6}}_{c})$ structure are taken
into account. All the relevant channels for all possible quantum numbers are
listed in Table 2, where $F^{i};S^{j}_{s};\chi^{c}_{k}$ shows the necessary
basis combinations in flavor ($F^{i}$), spin ($S^{j}_{s}$) and color
($\chi^{c}_{k}$) degrees of freedom.

Table 3:  The low-lying eigenenergies (in unit of MeV) of $T_{cc\bar{s}}$ tetraquark states with $J^{P}=0^{+}$.  Index |  Channel |  Threshold | QDCSM1 |  | QDCSM2 |  | QDCSM3
---|---|---|---|---|---|---|---
|  |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$ |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$ |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$
1 |  $D^{0}D_{s}^{+}$ |  3833 |  3836.3 |  3836.2 |  3836.2 |  |  3836.3 |  3836.3 |  3836.2 |  |  3836.2 |  3836.2 |  3836.2
2 |  $D^{*}D_{s}^{*+}$ |  4119 |  4119.7 |  |  |  |  4120.9 |  |  |  |  4121.2 |  |
3 |  $(cc)(\bar{q}\bar{s})$ |  |  4589.3 |  4299.8 |  |  |  4585.1 |  4291.8 |  |  |  4574.7 |  4277.9 |
4 |  $(cc)(\bar{q}\bar{s})$ |  |  4321.3 |  |  |  |  4316.5 |  |  |  |  4308.0 |  |
Table 4:  The same as Table 3 but for the tetraquark states with $J^{P}=1^{+}$.  Index |  Channel |  Threshold | QDCSM1 |  | QDCSM2 |  | QDCSM3
---|---|---|---|---|---|---|---
|  |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$ |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$ |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$
1 |  $D^{0}D_{s}^{*+}$ |  3977 |  3978.2 |  3977.1 |  3971.1 |  |  3978.2 |  3977.7 |  3973.8 |  |  3978.2 |  3978.1 |  3974.8
2 |  $D^{*}D_{s}^{+}$ |  3975 |  3978.0 |  |  |  |  3978.1 |  |  |  |  3978.2 |  |
3 |  $D^{*}D_{s}^{*+}$ |  4119 |  4110.8 |  |  |  |  4117.2 |  |  |  |  4118.1 |  |
4 |  $(cc)(\bar{q}\bar{s})$ |  |  4544.2 |  4128.2 |  |  |  4535.4 |  4127.2 |  |  |  4518.9 |  4124.1 |
5 |  $(cc)(\bar{q}\bar{s})$ |  |  4132.7 |  |  |  |  4132.5 |  |  |  |  4130.7 |  |
6 |  $(cc)(\bar{q}\bar{s})$ |  |  4337.5 |  |  |  |  4334.1 |  |  |  |  4327.8 |  |
Table 5:  The same as Table 3 but for the tetraquark states with $J^{P}=2^{+}$.  Index |  Channel |  Threshold | QDCSM1 |  | QDCSM2 |  | QDCSM3
---|---|---|---|---|---|---|---
|  |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$ |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$ |  |  $E_{sc}$ |  $E_{cc}$ |  $E_{mix}$
1 |  $D^{*}D_{s}^{*+}$ |  4119 |  4122.0 |  — |  4121.5 |  |  4122.2 |  — |  4122.1 |  |  4122.3 |  — |  4122.2
2 |  $(cc)(\bar{q}\bar{s})$ |  |  4367.1 |  — |  |  |  4366.3 |  — |  |  |  4364.1 |  — |

###  III.1 Bound State

With the above preparations, the low-lying $S-$wave $T_{cc\bar{s}}$ tetraquark
states are systematically explored herein. In Tables 3\- 5, we collect the
estimated eigenenergies of the $T_{cc\bar{s}}$ tetraquark states with
different $J^{P}$ quantum numbers. In those tables, the index of the first
column represents the symbols of each channel and in the second and third
columns we list all the involved channels and the corresponding theoretical
threshold, respectively. Moreover, $E_{sc}$ is the eigenenergy obtained in the
single channel estimations, $E_{cc}$ and $E_{mix}$ are the eigenenergies
estimated by considering the coupled channel effects in each kind of
configuration, and in both configurations, respectively.

Additionally, we define the binding energy $E_{b}$ of the $T_{cc\bar{s}}$
tetraquark states as $E_{bi}=E_{i}-E_{4}(\infty)$ to identify whether or not
the tetraquark states are stable against the strong interactions, where
$E_{4}(\infty)$ is the lowest possible threshold of the two meson structure
estimated in the QDCSM. and $i$ represents the different situation of channel
coupling. Such a subtraction procedure can greatly reduce the influence of the
model parameters on the binding energies. If $E_{b}>0$, the tetraqaurk systems
can fall apart into two mesons via the strong interactions. If $E_{b}<0$, the
strong decay into two mesons is forbidden kinemetically and therefore the
decay can only occur via either the weak or electromagnetic interaction.

For the $T_{cc\bar{s}}$ tetraquark system with $J^{P}=0^{+}$, there are two
channels in the meson-configuration and two channels in the diquark-
antidiquark configuration. The estimated eigenenergies of $T_{cc\bar{s}}$
state with $J^{P}=0^{+}$ are listed in Table 3. The theoretical thresholds of
the meson-meson channels are also presented for comparison. With the
parameters in QDCSM1, the single channel estimations in the meson-meson
configuration find that the eigenenergies are all above the corresponding
threshold, which indicate that the single channel estimations do not support
the existence of the bound states. In addition, when considering the coupled
channels effects in the meson-meson configurations, we find the estimated
eigenenergy is 3836.2 MeV, which is above the threshold of $D^{0}D_{s}^{+}$.
The lowest eigenenergy obtained by coupled channel estimations in the meson-
meson configuration is very close to the one of the single channel estimations
in the $D^{0}D_{s}^{+}$ channel, which indicates that the coupled channel
effect in the meson-meson configuration is rather weak. As for the diquark-
antidiquark configuration, both the single channel estimations and the coupled
channel estimations indicate that the eigenenergies are above the threshold of
$D^{0}D_{s}^{+}$. Different from the meson-meson configuration, we find the
eigenenergy obtained from the coupled channel estimation is at least 20 MeV
below the lowest one of the single channel estimation, which indicate the
coupled channels effect in the diquark-antidiquark configuration is much
strong. Moreover, we extend the coupled channel effect in both configurations,
and the eigenenergy is estimated to be 3836.2 MeV, which is still above the
threshold of $D^{0}D_{s}^{+}$. The results estimated with the parameters in
QDCSM2 and QDCSM3 are very similar with those obtained with the parameter in
QDCSM1 and no stable tetraquark state is found.

For the $T_{cc\bar{s}}$ tetraquark system with $J^{P}=1^{+}$, there are six
channels, including three channels in the meson-meson configuration and three
channels in the diquark-antidiquark configuration. From Table 4, the estimated
results of three sets of model parameters are almost identical. When
considering the single channel estimations in the meson-meson configuration,
we find that the estimated eigenenergy of $D^{0}D_{s}^{*+}$ and
$D^{*}D_{s}^{+}$ channels are above the theoretical threshold of the
corresponding physical channels, which indicates that these channels are
scattering channels in single channel calculations. However, a bound state in
the $D^{*}D_{s}^{*+}$ channel with the bound energy about $1\sim 10$ MeV is
obtained with all three sets of model parameters. Besides, by the coupling
channels with the meson-meson configuration, the estimated eigenenergy is
slightly above the lowest theoretical threshold of the $D^{*}D_{s}^{+}$, which
show that the effect of couple channels in the meson-meson configuration is
rather weak. For the diquark-antidiquark configuration, the estimated
eigenenergies obtained for the single-channel and channel-coupled estimations
are above the theoretical threshold of the lowest channel $D^{*}D_{s}^{+}$.
Nevertheless, when the channel coupling between the two configuration are
taken into account, a shallow bound state is detected, although the magnitude
of the bound energy is slightly different with different sets of the model
parameters.

Table 6:  Contributions of each terms in Hamiltonian to the energy of the $D^{0}D_{s}^{*+}$ bound state with $J^{P}=1^{+}$ in unit of MeV. $E_{M}$ stands for the sum of two mesons threshold. Our estimations indicate the contributions of $\eta$ meson exchange potential are all less than $0.05$ MeV in different sets of model parameters. Thus, the contributions from $\eta$ meson exchange are not presented. | QDCSM1 |  | QDCSM2 |  | QDCSM3
---|---|---|---|---|---
|  $H_{T}$ |  $V_{\mathrm{CON}}$ |  $V_{\mathrm{OGE}}$ |  $V_{K}$ |  |  $H_{T}$ |  $V_{\mathrm{CON}}$ |  $V_{\mathrm{OGE}}$ |  $V_{K}$ |  |  $H_{T}$ |  $V_{\mathrm{CON}}$ |  $V_{\mathrm{OGE}}$ |  $V_{K}$
$E_{sc}$ |  1081.3 |  -901.7 |  -506.6 |  $\sim 0.0$ |  |  1011.2 |  -783.9 |  -554.2 |  $\sim 0.0$ |  |  917.9 |  -615.9 |  -628.8 |  $\sim 0.0$
$E_{cc}$ |  1073.9 |  -895.9 |  -505.8 |  -0.1 |  |  1008.8 |  -782.5 |  -553.5 |  -0.1 |  |  917.1 |  -615.5 |  -628.5 |  $\sim 0.0$
$E_{mix}$ |  1049.0 |  -820.4 |  -558.1 |  -4.4 |  |  998.4 |  -752.4 |  -573.7 |  -3.5 |  |  915.3 |  -609.8 |  -635.4 |  -0.3
$E_{M}$ |  1079.6 |  -903.3 |  -506.1 |  $\sim 0.0$ |  |  1008.7 |  -784.7 |  -553.8 |  $\sim 0.0$ |  |  915.0 |  -616.3 |  -628.5 |  $\sim 0.0$
$\Delta E_{sc}$ |  1.7 |  1.6 |  -0.5 |  $\sim 0.0$ |  |  2.5 |  0.8 |  0.4 |  $\sim 0.0$ |  |  2.9 |  0.4 |  -0.3 |  $\sim 0.0$
$\Delta E_{cc}$ |  -5.7 |  7.4 |  0.3 |  -0.1 |  |  0.1 |  2.2 |  -0.3 |  -0.1 |  |  2.1 |  0.8 |  0.0 |  $\sim 0.0$
$\Delta E_{mix}$ |  -30.6 |  82.9 |  -52.0 |  -4.4 |  |  -10.3 |  32.3 |  -19.9 |  -3.5 |  |  0.3 |  5.5 |  -7.2 |  -0.3

In view of the above conclusions, we estimate the average values of each terms
in the Hamiltonian to examine how a shallow $D^{*}D_{s}^{+}$ bound state with
$J^{P}=1^{+}$ is created. In Table 6, we present the contributions of each
interaction by considering the single channel and coupled channel
calculations. In addition, the average values of each interaction of two
conventional $D^{*}$and $D_{s}^{+}$ mesons without interactions, i.e., the
distance between the two mesons are large enough, are also listed in the table
for comparison. From the Table, one finds that the magnitude of the average
values of each terms for different sets of model parameter are very similar.
Here, we define $\Delta E_{sc}=E_{sc}-E_{M}$, $\Delta E_{cc}=E_{cc}-E_{M}$ and
$\Delta E_{mix}=E_{mix}-E_{M}$. From our estimations, we find the
contributions of the confinement potential to $\Delta E_{sc}$, $\Delta E_{cc}$
and $\Delta{E}_{mix}$ are positive, which indicate the confinement potential
hinders the $D^{*}$ and $D_{s}^{+}$ subclusters from forming a bound states.
For the kinetic energy term, with more physical channels taking into
consideration, the properties of kinetic energy basically transforms gradually
from repulsion towards very strong attraction, whereas the similar conclusions
can be drawn for the one-gluon-exchange interaction. In addition, in the meson
exchange interactions, the meson exchange potential contributions to $\Delta
E_{sc}$, $\Delta E_{cc}$ and $\Delta{E}_{mix}$ are negligible, in
particularly, the contributions from $\eta$ meson exchange potential are all
less than 0.05 MeV, which are not listed in the table. According to the above
analysis, one can find that the kinetic energy term and the one-gluon-exchange
potential have deep attractions in the channel coupling calculations with both
the meson-meson and diquark-antidiquark configurations, However, the
confinement potential displays as repulsive interaction, which weaken the
overall attraction. Such a phenomenon illustrates the very delicate
competition between the kinetic energy and the interaction potential from
various sources in the Hamiltonian.

For the $T_{cc\bar{s}}$ tetraquark system with $J^{P}=2^{+}$, only one physics
channel in the meson-meson configuration and one channel in the diquark-
antidiquark configuration exists. From Table 5, one can find the eigenenergies
obtained from the single channel estimation is higher than the physical meson-
meson channel. After considering the coupled channel effect between the meson-
meson and diquark-antidiquark configurations, the estimated eigenenergy is
still above the threshold of $D^{\ast}D_{s}^{\ast+}$, which indicates that
there is no bound state in the $T_{cc\bar{s}}$ tetraquark system with
$J^{P}=2^{+}$.

Figure 3: A sketch diagram of the resonance shape in the real-scaling method.

###  III.2 Resonance States

In the bound state estimations, we find one bound state with $J^{P}=1^{+}$
while there is no bound state in the $J^{P}=0^{+}$ and $J^{P}=2^{+}$ systems.
In the following, we will employ the real scaling method to explore the
possible resonance states in the $T_{cc\bar{s}}$ tetraquark system. To
determine whether these resonance states could be detected by the open
channels, we perform a channel coupling estimation by including all the meson-
meson and diquark-antidiquark channels in the estimations.

The real scaling method is developed to identify the genuine resonances from
the states with discrete energies with finite volume real_Method  . In this
method, a factor $\textbf{S}_{m}$, which is the distance between two clusters,
is adopted to scale the finite volume. So with the increase of the distance
between two clusters, the continuum state will fall off toward its threshold,
the energy of the bound state remains unchanged, while a resonance state will
tend to be stable. If the energy of a scattering state is far away from the
one of the resonance, the coupling between the resonance and the scattering
states is rather weak, and the energy of the resonance is almost stable. When
the energy of the scattering state approaches the one of the resonance due to
the increasing of $\textbf{S}_{m}$, the coupling will become strong, and if
$\textbf{S}_{m}$ increases further, the energy gap between the resonance and
scattering states will increase and the coupling will become weak again. In
this way, an avoided crossing structure appears. This is a general feature of
two interacting energy levels. Because of the continuum nature of the
scattering states, the avoided crossing structure will show up repeatedly with
the increasing of $\textbf{S}_{m}$ as shown in Fig. 3 and the resonance line
corresponds to the energy of the resonance state. In addition, from the slopes
of resonance and scattering states, the decay width can be estimated by,

$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle
4|V_{min}(S)|\frac{\sqrt{|k_{r}||k_{c}|}}{|k_{r}-k_{c}|}$ (22)

where $k_{r}$ and $k_{c}$ are the slopes of the resonance and scattering
states, respectively. While, $V_{min}(S)$ is the minimal energy difference
between the resonance and the scattering state at avoided crossing point. This
method has been successfully applied to investigate the pentaquark
Hiyama:2005cf ; Hiyama:2018ukv , the dibaryon Xia:2021tof , and the tetraquark
systems Jin:2020jfc ; Liu:2021xje ; Liu:2022vyy .

In the present work, we expand the spacial wave function with a set of
gaussians with differences $\textbf{S}_{m}$, $(m=1,2,3,\ldots,n)$ and the
distance with the relative motion of two clusters can be scaled. So we
calculate the energy eigenvalues of the $T_{cc\bar{s}}$ tetraquark system by
taking the value of the largest distance $(S_{m})$ between two clusters from
4.0 to 9.0 fm to check if there is any resonance state. Here, we take the
results of the QDCSM1 as examples, which are shown in Fig. 4 with different
$J^{P}$ quantum numbers. For the $T_{cc\bar{s}}$ tetraquark system with
$J^{P}=0^{+}$ as shown in Fig. 4-(a), one can note that the lower black
horizontal line corresponds to the physical threshold of $D_{s}^{+}D^{0}$,
while the upper blue horizontal line with the energy to be about 4114 MeV,
locates below the threshold of $D^{*}D_{s}^{+*}$, which corresponds to a
resonance state since the resonance behavior appearing in the Fig. 4-(a) as
the finite space is constantly expanding. Moreover, the resonance state is
estimated by considering the full channel coupling, and the present result
indicates that its main ingredient is $D^{*}D_{s}^{+*}$. In other words, the
effect of the channel coupling push the energy of the physical channel
$D^{*}D_{s}^{+*}$ a bit below its threshold. In addition, the width of this
resonance state is estimated to be about $14.3$ MeV according to Eq. (22).

For the $T_{cc\bar{s}}$ tetraquark system with $J^{P}=1^{+}$ as shown in Fig.
4-(b), it is obvious that the lowest red horizontal line locates at the energy
of $3971$ MeV, which is below the threshold of the $D^{0}D_{s}^{+*}$, and this
represents the bound states of $T_{cc\bar{s}}$ tetraquark system with
$J^{P}=1^{+}$. This conclusion is consistent with the estimations in the last
subsection. Moreover, two additional horizontal lines are also presented,
which stand for the threshold of $D^{*}D_{s}^{+}$ and $D^{*}D_{s}^{*+}$,
respectively. The present estimations indicate that there is no resonance
state in the $T_{cc\bar{s}}$ tetraquark system with $J^{P}=1^{+}$, and the
bound state in the $D^{*}D_{s}^{*+}$ channel becomes the scattering state by
the effect of the channel coupling. For the $T_{cc\bar{s}}$ tetraquark system
with $J^{P}=2^{+}$ as shown in Fig. 4-(c), there is one horizontal line, which
represents the threshold of $D^{*}D_{s}^{*+}$. It is clearly to conclude that
there are no bound or resonant states in the $T_{cc\bar{s}}$ tetraquark system
with $J^{P}=2^{+}$.

In addition, we perform the same estimations for the $T_{cc\bar{s}}$
tetraquark system in the QDCSM2 and QDCSM3. The results are similar to those
of QDCSM1. We summarize the results obtained from three sets of model
parameters in Table 7. By taking the coupled channel effects into
consideration, we find one resonance state with a mass $4113\sim 4114$ MeV for
the $T_{cc\bar{s}}$ tetraquark system with $J^{P}=0^{+}$. The dominant
component of the resonance state is $D^{*}D_{s}^{*+}$ with the percentage of
this component to be about $80\%$. Moreover, the decay width of this resonance
state is predicted to be $14.3\sim 16.1$ MeV. For the $J^{P}=1^{+}$ system,
there is a bound state with energy range $(3971.1\sim 3974.8)$ MeV and no
resonance state is obtained. For the $T_{cc\bar{s}}$ tetraquark system with
$J^{P}=2^{+}$, no resonance or bound state is obtained by the channel coupling
estimations.

Figure 4: The stabilization plots of the energies of the $T_{cc\bar{s}}$ tetraquark systems.  Table 7: The energies and widths of the $T_{cc\bar{s}}$ tertraquark states.  State | Parameter Sets
---|---
|  $J^{P}$ |  QDCSM1 |  QDCSM2 |  QDCSM3
Bound |  $1^{+}$ |  3971.1 |  3973.8 |  3974.8
Resonance |  $0^{+}$ |  $4114/14.3$ |  $4144/15.8$ |  $4143/16.1$

##  IV Summary

In the present work, the $T_{cc\bar{s}}$ tetraquark system with the quantum
number $J^{P}=0^{+},1^{+},2^{+}$ are systemically investigated to search for
the possible bound state and resonance state by using the RGM in the QDCSM
framework. In the model, both meson-meson and diquark-antidiquark
configurations are taken into account, and the single-channel and the coupled
channel calculations are preformed to obtain the energy of the $T_{cc\bar{s}}$
tetraquark system. In addition, a stabilization calculation is carried out to
seek for possible resonance states. Furthermore, to check whether the
estimated results are parameter dependent, three different sets of model
parameters are employed in the calculation and we find the qualitative results
of three sets of model parameters for the $T_{cc\bar{s}}$ tetraquark system
are very similar.

From the present estimations, we find that the coupled channel effects plays
important role in the $T_{cc\bar{s}}$ tetraquark system. After taking the
coupled channel effects into consideration, we predict one bound state with
the energy to be $3971.1\sim 3974.8$ MeV and $J^{P}=1^{+}$. Moreover, one
resonance state with $J^{P}=0^{+}$ is also obtained, the resonance mass and
width are estimated to be $4113\sim 4114$ MeV and $14.3\sim 16.1$ MeV,
respectively. The predictions in the present work could be experimentally
detected in the future by LHCb and Belle II. Additionally the theoretical and
further experimental investigations for properties of the $T_{cc\bar{s}}$
tetraquark could pave the way for possible doubly and triply tetraquark
states.

###### Acknowledgements.

This work is supported partly by the National Natural Science Foundation of
China under Contract No. 12175037, No. 12335001, No. 11775118 and No.
11535005. This work is also supported by china Postdoctoral Science Foundation
funded project No. 2021M690626, and No. 1107020201.

##  Appendix A The wave function of the open heavy charm tetraquark with
strangeness

###  A.1 The color wave function

Plenty of color structures in multiquark systems will be available with
respect to those of conventional hadrons such as $q\bar{q}$ mesons and $qqq$
baryons. In this appendix, we present how to construct the colorless wave
function for a tetraquark system.

For the meson-meson configurations, the color wave functions of a $q\bar{q}$
cluster would be,

$\displaystyle C^{1}_{[111]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}\Big{(}r\bar{r}+g\bar{g}+b\bar{b}\Big{)},$
$\displaystyle C^{2}_{[21]}$ $\displaystyle=$ $\displaystyle r\bar{b},\qquad
C^{3}_{[21]}=-r\bar{g},$ $\displaystyle C^{4}_{[21]}$ $\displaystyle=$
$\displaystyle g\bar{b},\qquad C^{5}_{[21]}=-b\bar{g},$ $\displaystyle
C^{6}_{[21]}$ $\displaystyle=$ $\displaystyle g\bar{r},\qquad
C^{7}_{[21]}=b\bar{r},$ $\displaystyle C^{8}_{[21]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}r\bar{r}-g\bar{g}\Big{)},$
$\displaystyle C^{9}_{[21]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{6}}\Big{(}-r\bar{r}-g\bar{g}+2b\bar{b}\Big{)},$
(23)

where the subscript [111] and [21] stand for color-singlet ($\textbf{1}_{c}$)
and color-octet ($\textbf{8}_{c}$), respectively. So, the
$\mathrm{SU}(3)_{\mathrm{color}}$ wave functions of color-singlet (two color-
singlet cluters, $\textbf{1}_{c}\otimes\textbf{1}_{c}$) and hidden-color (two
color-octet clusters, $\textbf{8}_{c}\otimes\textbf{8}_{c}$) channels are
given, respectively,

$\displaystyle\chi^{c}_{1}$ $\displaystyle=$ $\displaystyle
C^{1}_{[111]}C^{1}_{[111]},$ $\displaystyle\chi^{c}_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{8}}\Big{(}C^{2}_{[21]}C^{7}_{[21]}-C^{4}_{[21]}C^{5}_{[21]}-C^{3}_{[21]}C^{6}_{[21]}$
(24)
$\displaystyle+C^{8}_{[21]}C^{8}_{[21]}-C^{6}_{[21]}C^{3}_{[21]}+C^{9}_{[21]}C^{9}_{[21]}$
$\displaystyle-C^{5}_{[21]}C^{4}_{[21]}+C^{7}_{[21]}C^{2}_{[21]}\Big{)}.$

For the diquark-antidiquark structure, the color wave functions of the diquark
clusters are,

$\displaystyle C^{1}_{[2]}$ $\displaystyle=$ $\displaystyle rr,\qquad
C^{2}_{[2]}=\sqrt{\frac{1}{2}}\Big{(}rg+gr\Big{)},$ $\displaystyle
C^{3}_{[2]}$ $\displaystyle=$ $\displaystyle gg,\qquad
C^{4}_{[2]}=\sqrt{\frac{1}{2}}\Big{(}rb+br\Big{)},$ $\displaystyle
C^{5}_{[2]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}gb+bg\Big{)},\qquad C^{6}_{[2]}=bb,$
$\displaystyle C^{7}_{[11]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}rg-gr\Big{)},\qquad
C^{8}_{[11]}=\sqrt{\frac{1}{2}}\Big{(}rb-br\Big{)},$ $\displaystyle
C^{9}_{[11]}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{1}{2}}\Big{(}gb-
bg\Big{)}.$ (25)

While the color wave functions of the antidiquark clusters can be writen as,

$\displaystyle C^{1}_{[22]}$ $\displaystyle=$
$\displaystyle\bar{r}\bar{r},\qquad
C^{2}_{[22]}=-\sqrt{\frac{1}{2}}\Big{(}\bar{r}\bar{g}+\bar{g}\bar{r}\Big{)},$
$\displaystyle C^{3}_{[22]}$ $\displaystyle=$
$\displaystyle\bar{g}\bar{g},\qquad
C^{4}_{[22]}=\sqrt{\frac{1}{2}}\Big{(}\bar{r}\bar{b}+\bar{b}\bar{r}\Big{)},$
$\displaystyle C^{5}_{[22]}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}\Big{(}\bar{g}\bar{b}+\bar{b}\bar{g}\Big{)},\qquad
C^{6}_{[22]}=\bar{b}\bar{b},$ $\displaystyle C^{7}_{[211]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}\bar{r}\bar{g}-\bar{g}\bar{r}\Big{)},\qquad
C^{8}_{[211]}=-\sqrt{\frac{1}{2}}\Big{(}\bar{r}\bar{b}-\bar{b}\bar{r}\Big{)},$
$\displaystyle C^{9}_{[211]}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}\bar{g}\bar{b}-\bar{b}\bar{g}\Big{)}.$
(26)

The color-singlet wave functions of the diquark-antidiquark configuration can
be the product of color sextet and antisextet clusters
($\textbf{6}_{c}\otimes\bar{\textbf{6}}_{c}$) or the product of color-triplet
and antitriplet cluster ($\textbf{3}_{c}\otimes\bar{\textbf{3}}_{c}$), which
read,

$\displaystyle\chi^{c}_{3}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{6}}\Big{(}C^{1}_{[2]}C^{1}_{[22]}-C^{2}_{[2]}C^{[2]}_{[22]}+C^{3}_{[2]}C^{3}_{[22]}$
$\displaystyle+C^{4}_{[2]}C^{4}_{[22]}-C^{5}_{[2]}C^{5}_{[22]}+C^{6}_{2}C^{6}_{22}\Big{)},$
$\displaystyle\chi^{c}_{4}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}\Big{(}C^{7}_{[11]}C^{7}_{[211]}-C^{8}_{[11]}C^{8}_{[211]}+C^{9}_{[11]}C^{9}_{[211]}\Big{)}.$
(27)

###  A.2 The flavor wave function

For the flavor degree of freedom, the different coupling methods generate
different flavor wave function. From the Table 2, the $T_{cc\bar{s}}$
tetraquark flavor wave function can be categorized as $F^{i}_{m}$ and
$F_{d}^{i}$, where the subscript $m$ and $d$ refer to meson-meson and the
diquark-antidiquark configurations, respectively. Distinctive structures are
gotten the quark coupling arrange. For the meson-meson structure, the coupling
orders can be accessed as,

$\displaystyle F^{1}_{m}$ $\displaystyle=$
$\displaystyle(c\bar{q})-(c\bar{s}),$ (28)

while for the diquark-antidiquark structure, the flavor wave function should
be written as

$\displaystyle F^{2}_{d}$ $\displaystyle=$
$\displaystyle(cc)-(\bar{q}\bar{s})$ (29)

###  A.3 The spin wave function

The total spin $S$ of tetraquark states ranges from 0 to 2. All of them are
considered. The wave functions of two body clusters are,

$\displaystyle\chi_{11}$ $\displaystyle=$ $\displaystyle\alpha\alpha,$
$\displaystyle\chi_{10}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}\alpha\beta+\beta\alpha\Big{)},$
$\displaystyle\chi_{1-1}$ $\displaystyle=$ $\displaystyle\beta\beta,$
$\displaystyle\chi_{00}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}\alpha\beta-\beta\alpha\Big{)}.$ (30)

Then, the total spin wave functions $S^{i}_{s}$ are obtained by considering
the coupling of two subcluster spin wave functions with SU(2) algebra, and the
total spin wave functions of four-quark states can be read as,

$\displaystyle S^{1}_{0}$ $\displaystyle=$ $\displaystyle\chi_{00}\chi_{00},$
$\displaystyle S^{2}_{0}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{3}}\Big{(}\chi_{11}\chi_{1-1}-\chi_{10}\chi_{10}+\chi_{1-1}\chi_{11}\Big{)},$
$\displaystyle S^{3}_{1}$ $\displaystyle=$ $\displaystyle\chi_{00}\chi_{11},$
$\displaystyle S^{4}_{1}$ $\displaystyle=$ $\displaystyle\chi_{11}\chi_{00},$
$\displaystyle S^{5}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}}\Big{(}\chi_{11}\chi_{10}-\chi_{10}\chi_{11}\Big{)},$
$\displaystyle S^{6}_{2}$ $\displaystyle=$ $\displaystyle\chi_{11}\chi_{11}.$
(31)

## References

  * (1) M. Ablikim et al. [BESIII], Phys. Rev. Lett. 110 (2013), 252001 doi:10.1103/PhysRevLett.110.252001 [arXiv:1303.5949 [hep-ex]].
  * (2) Z. Q. Liu et al. [Belle], Phys. Rev. Lett. 110 (2013), 252002 [erratum: Phys. Rev. Lett. 111 (2013), 019901] doi:10.1103/PhysRevLett.110.252002 [arXiv:1304.0121 [hep-ex]].
  * (3) T. Xiao, S. Dobbs, A. Tomaradze and K. K. Seth, Phys. Lett. B 727 (2013), 366-370 doi:10.1016/j.physletb.2013.10.041 [arXiv:1304.3036 [hep-ex]].
  * (4) M. Ablikim et al. [BESIII], Phys. Rev. Lett. 119, no.7, 072001 (2017) doi:10.1103/PhysRevLett.119.072001 [arXiv:1706.04100 [hep-ex]].
  * (5) S. S. Agaev, K. Azizi and H. Sundu, Phys. Rev. D 93 (2016) no.7, 074002 doi:10.1103/PhysRevD.93.074002 [arXiv:1601.03847 [hep-ph]].
  * (6) J. M. Dias, F. S. Navarra, M. Nielsen and C. M. Zanetti, Phys. Rev. D 88 (2013) no.1, 016004 doi:10.1103/PhysRevD.88.016004 [arXiv:1304.6433 [hep-ph]].
  * (7) Z. G. Wang and T. Huang, Phys. Rev. D 89 (2014) no.5, 054019 doi:10.1103/PhysRevD.89.054019 [arXiv:1310.2422 [hep-ph]].
  * (8) C. Deng, J. Ping and F. Wang, Phys. Rev. D 90 (2014), 054009 doi:10.1103/PhysRevD.90.054009 [arXiv:1402.0777 [hep-ph]].
  * (9) Z. G. Wang and T. Huang, Eur. Phys. J. C 74 (2014) no.5, 2891 doi:10.1140/epjc/s10052-014-2891-6 [arXiv:1312.7489 [hep-ph]].
  * (10) E. Wilbring, H. W. Hammer and U. G. Meißner, Phys. Lett. B 726 (2013), 326-329 doi:10.1016/j.physletb.2013.08.059 [arXiv:1304.2882 [hep-ph]].
  * (11) Y. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 88 (2013) no.1, 014030 doi:10.1103/PhysRevD.88.014030 [arXiv:1306.0824 [hep-ph]].
  * (12) H. W. Ke, Z. T. Wei and X. Q. Li, Eur. Phys. J. C 73 (2013) no.10, 2561 doi:10.1140/epjc/s10052-013-2561-0 [arXiv:1307.2414 [hep-ph]].
  * (13) T. Gutsche, M. Kesenheimer and V. E. Lyubovitskij, Phys. Rev. D 90 (2014) no.9, 094013 doi:10.1103/PhysRevD.90.094013 [arXiv:1410.0259 [hep-ph]].
  * (14) A. Esposito, A. L. Guerrieri and A. Pilloni, Phys. Lett. B 746 (2015), 194-201 doi:10.1016/j.physletb.2015.04.057 [arXiv:1409.3551 [hep-ph]].
  * (15) D. Y. Chen and Y. B. Dong, Phys. Rev. D 93 (2016) no.1, 014003 doi:10.1103/PhysRevD.93.014003 [arXiv:1510.00829 [hep-ph]].
  * (16) Q. R. Gong, Z. H. Guo, C. Meng, G. Y. Tang, Y. F. Wang and H. Q. Zheng, Phys. Rev. D 94 (2016) no.11, 114019 doi:10.1103/PhysRevD.94.114019 [arXiv:1604.08836 [hep-ph]].
  * (17) H. W. Ke and X. Q. Li, Eur. Phys. J. C 76 (2016) no.6, 334 doi:10.1140/epjc/s10052-016-4183-9 [arXiv:1601.03575 [hep-ph]].
  * (18) E. S. Swanson, Phys. Rev. D 91 (2015) no.3, 034009 doi:10.1103/PhysRevD.91.034009 [arXiv:1409.3291 [hep-ph]].
  * (19) Y. Ikeda et al. [HAL QCD], Phys. Rev. Lett. 117 (2016) no.24, 242001 doi:10.1103/PhysRevLett.117.242001 [arXiv:1602.03465 [hep-lat]].
  * (20) A. P. Szczepaniak, Phys. Lett. B 747 (2015), 410-416 doi:10.1016/j.physletb.2015.06.029 [arXiv:1501.01691 [hep-ph]].
  * (21) D. Y. Chen and X. Liu, Phys. Rev. D 84 (2011), 034032 doi:10.1103/PhysRevD.84.034032 [arXiv:1106.5290 [hep-ph]].
  * (22) D. Y. Chen, X. Liu and T. Matsuki, Phys. Rev. D 88 (2013) no.3, 036008 doi:10.1103/PhysRevD.88.036008 [arXiv:1304.5845 [hep-ph]].
  * (23) E. S. Swanson, Int. J. Mod. Phys. E 25 (2016) no.07, 1642010 doi:10.1142/S0218301316420106 [arXiv:1504.07952 [hep-ph]].
  * (24) D. Ebert, R. N. Faustov and V. O. Galkin, Eur. Phys. J. C 58, 399-405 (2008) doi:10.1140/epjc/s10052-008-0754-8 [arXiv:0808.3912 [hep-ph]].
  * (25) J. Ferretti and E. Santopinto, JHEP 04 (2020), 119 doi:10.1007/JHEP04(2020)119 [arXiv:2001.01067 [hep-ph]].
  * (26) S. H. Lee, M. Nielsen and U. Wiedner, J. Korean Phys. Soc. 55 (2009), 424 doi:10.3938/jkps.55.424 [arXiv:0803.1168 [hep-ph]].
  * (27) J. M. Dias, X. Liu and M. Nielsen, Phys. Rev. D 88 (2013) no.9, 096014 doi:10.1103/PhysRevD.88.096014 [arXiv:1307.7100 [hep-ph]].
  * (28) D. Y. Chen, X. Liu and T. Matsuki, Phys. Rev. Lett. 110, no.23, 232001 (2013) doi:10.1103/PhysRevLett.110.232001 [arXiv:1303.6842 [hep-ph]].
  * (29) M. Ablikim et al. [BESIII], Phys. Rev. Lett. 126 (2021) no.10, 102001 doi:10.1103/PhysRevLett.126.102001 [arXiv:2011.07855 [hep-ex]].
  * (30) R. Aaij et al. [LHCb], Phys. Rev. Lett. 127 (2021) no.8, 082001 doi:10.1103/PhysRevLett.127.082001 [arXiv:2103.01803 [hep-ex]].
  * (31) H. X. Chen, Phys. Rev. D 105 (2022) no.9, 094003 doi:10.1103/PhysRevD.105.094003 [arXiv:2103.08586 [hep-ph]].
  * (32) B. D. Wan and C. F. Qiao, Nucl. Phys. B 968 (2021), 115450 doi:10.1016/j.nuclphysb.2021.115450 [arXiv:2011.08747 [hep-ph]].
  * (33) Y. J. Xu, Y. L. Liu, C. Y. Cui and M. Q. Huang, Phys. Rev. D 104 (2021) no.9, 094028 doi:10.1103/PhysRevD.104.094028 [arXiv:2011.14313 [hep-ph]].
  * (34) R. M. Albuquerque, S. Narison and D. Rabetiarivony, Phys. Rev. D 103 (2021) no.7, 074015 doi:10.1103/PhysRevD.103.074015 [arXiv:2101.07281 [hep-ph]].
  * (35) U. Özdem and K. Azizi, Eur. Phys. J. Plus 136 (2021) no.9, 968 doi:10.1140/epjp/s13360-021-01977-w [arXiv:2102.09231 [hep-ph]].
  * (36) U. Özdem and A. K. Yıldırım, Phys. Rev. D 104 (2021) no.5, 054017 doi:10.1103/PhysRevD.104.054017 [arXiv:2104.13074 [hep-ph]].
  * (37) Q. N. Wang, W. Chen and H. X. Chen, Chin. Phys. C 45 (2021) no.9, 093102 doi:10.1088/1674-1137/ac0b3b [arXiv:2011.10495 [hep-ph]].
  * (38) L. Meng, B. Wang and S. L. Zhu, Phys. Rev. D 102 (2020) no.11, 111502 doi:10.1103/PhysRevD.102.111502 [arXiv:2011.08656 [hep-ph]].
  * (39) Z. Yang, X. Cao, F. K. Guo, J. Nieves and M. P. Valderrama, Phys. Rev. D 103 (2021) no.7, 074029 doi:10.1103/PhysRevD.103.074029 [arXiv:2011.08725 [hep-ph]].
  * (40) M. C. Du, Q. Wang and Q. Zhao, [arXiv:2011.09225 [hep-ph]].
  * (41) L. Maiani, A. D. Polosa and V. Riquer, Sci. Bull. 66 (2021), 1616-1619 doi:10.1016/j.scib.2021.04.040 [arXiv:2103.08331 [hep-ph]].
  * (42) P. P. Shi, F. Huang and W. L. Wang, Phys. Rev. D 103 (2021) no.9, 094038 doi:10.1103/PhysRevD.103.094038 [arXiv:2105.02397 [hep-ph]].
  * (43) J. F. Giron, R. F. Lebed and S. R. Martinez, Phys. Rev. D 104 (2021) no.5, 054001 doi:10.1103/PhysRevD.104.054001 [arXiv:2106.05883 [hep-ph]].
  * (44) R. Chen, Q. Huang, X. Liu and S. L. Zhu, Phys. Rev. D 104 (2021) no.11, 114042 doi:10.1103/PhysRevD.104.114042 [arXiv:2108.01911 [hep-ph]].
  * (45) A. Feijoo, W. H. Liang and E. Oset, Phys. Rev. D 104 (2021) no.11, 114015 doi:10.1103/PhysRevD.104.114015 [arXiv:2108.02730 [hep-ph]].
  * (46) F. L. Wang and X. Liu, Phys. Rev. D 104 (2021) no.9, 094030 doi:10.1103/PhysRevD.104.094030 [arXiv:2108.09925 [hep-ph]].
  * (47) F. L. Wang, R. Chen and X. Liu, [arXiv:2111.00208 [hep-ph]].
  * (48) H. Ren, F. Wu and R. Zhu, Adv. High Energy Phys. 2022 (2022), 9103031 doi:10.1155/2022/9103031 [arXiv:2109.02531 [hep-ph]].
  * (49) Q. Xin and Z. G. Wang, Eur. Phys. J. A 58 (2022) no.6, 110 doi:10.1140/epja/s10050-022-00752-4 [arXiv:2108.12597 [hep-ph]].
  * (50) K. Azizi and U. Özdem, Phys. Rev. D 104 (2021) no.11, 114002 doi:10.1103/PhysRevD.104.114002 [arXiv:2109.02390 [hep-ph]].
  * (51) U. Özdem, Phys. Rev. D 105 (2022) no.5, 054019 doi:10.1103/PhysRevD.105.054019 [arXiv:2112.10402 [hep-ph]].
  * (52) M. Albaladejo, Phys. Lett. B 829 (2022), 137052 doi:10.1016/j.physletb.2022.137052 [arXiv:2110.02944 [hep-ph]].
  * (53) L. R. Dai, E. Oset, A. Feijoo, R. Molina, L. Roca, A. M. Torres and K. P. Khemchandani, Phys. Rev. D 105 (2022) no.7, 074017 doi:10.1103/PhysRevD.105.074017 [arXiv:2201.04840 [hep-ph]].
  * (54) M. L. Du, V. Baru, X. K. Dong, A. Filin, F. K. Guo, C. Hanhart, A. Nefediev, J. Nieves and Q. Wang, Phys. Rev. D 105 (2022) no.1, 014024 doi:10.1103/PhysRevD.105.014024 [arXiv:2110.13765 [hep-ph]].
  * (55) M. J. Zhao, Z. Y. Wang, C. Wang and X. H. Guo, Phys. Rev. D 105 (2022) no.9, 096016 doi:10.1103/PhysRevD.105.096016 [arXiv:2112.12633 [hep-ph]].
  * (56) H. W. Ke, X. H. Liu and X. Q. Li, Eur. Phys. J. C 82 (2022) no.2, 144 doi:10.1140/epjc/s10052-022-10092-8 [arXiv:2112.14142 [hep-ph]].
  * (57) X. Chen and Y. Yang, Chin. Phys. C 46 (2022) no.5, 054103 doi:10.1088/1674-1137/ac4ee8 [arXiv:2109.02828 [hep-ph]].
  * (58) N. Santowsky and C. S. Fischer, Eur. Phys. J. C 82 (2022) no.4, 313 doi:10.1140/epjc/s10052-022-10272-6 [arXiv:2111.15310 [hep-ph]].
  * (59) C. Deng and S. L. Zhu, Phys. Rev. D 105 (2022) no.5, 054015 doi:10.1103/PhysRevD.105.054015 [arXiv:2112.12472 [hep-ph]].
  * (60) L. R. Dai, R. Molina and E. Oset, Phys. Rev. D 105 (2022) no.1, 016029 doi:10.1103/PhysRevD.105.016029 [arXiv:2110.15270 [hep-ph]].
  * (61) T. Guo, J. Li, J. Zhao and L. He, Phys. Rev. D 105 (2022) no.1, 014021 doi:10.1103/PhysRevD.105.014021 [arXiv:2108.10462 [hep-ph]].
  * (62) X. Z. Weng, W. Z. Deng and S. L. Zhu, Chin. Phys. C 46 (2022) no.1, 013102 doi:10.1088/1674-1137/ac2ed0 [arXiv:2108.07242 [hep-ph]].
  * (63) S. S. Agaev, K. Azizi and H. Sundu, Nucl. Phys. B 975 (2022), 115650 doi:10.1016/j.nuclphysb.2022.115650 [arXiv:2108.00188 [hep-ph]].
  * (64) R. Aaij et al. [LHCb], Nature Phys. 18 (2022) no.7, 751-754 doi:10.1038/s41567-022-01614-y [arXiv:2109.01038 [hep-ex]].
  * (65) R. Aaij et al. [LHCb], Nature Commun. 13 (2022) no.1, 3351 doi:10.1038/s41467-022-30206-w [arXiv:2109.01056 [hep-ex]].
  * (66) P. Junnarkar, N. Mathur and M. Padmanath, Phys. Rev. D 99 (2019) no.3, 034507 doi:10.1103/PhysRevD.99.034507 [arXiv:1810.12285 [hep-lat]].
  * (67) E. J. Eichten and C. Quigg, Phys. Rev. Lett. 119, no.20, 202002 (2017) doi:10.1103/PhysRevLett.119.202002 [arXiv:1707.09575 [hep-ph]].
  * (68) S. Q. Luo, K. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Eur. Phys. J. C 77, no.10, 709 (2017) doi:10.1140/epjc/s10052-017-5297-4 [arXiv:1707.01180 [hep-ph]].
  * (69) M. Karliner and J. L. Rosner, Phys. Rev. D 105 (2022) no.3, 034020 doi:10.1103/PhysRevD.105.034020 [arXiv:2110.12054 [hep-ph]].
  * (70) A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D 12 (1975), 147-162 doi:10.1103/PhysRevD.12.147
  * (71) N. Isgur and G. Karl, Phys. Rev. D 18 (1978), 4187 doi:10.1103/PhysRevD.18.4187
  * (72) N. Isgur and G. Karl, Phys. Rev. D 19 (1979), 2653 [erratum: Phys. Rev. D 23 (1981), 817] doi:10.1103/PhysRevD.19.2653
  * (73) N. Isgur and G. Karl, Phys. Rev. D 20 (1979), 1191-1194 doi:10.1103/PhysRevD.20.1191
  * (74) M. Chen, H. Huang, J. Ping and F. Wang, Phys. Rev. C 83 (2011), 015202 doi:10.1103/PhysRevC.83.015202
  * (75) J. L. Ping, F. Wang, G. H. Wu, L. J. Teng and J. T. Goldman,
  * (76) F. Wang, D. Qing, P. Xu and J. L. Ping, Nucl. Phys. A 631 (1998), 462C-466C doi:10.1016/S0375-9474(98)00048-7
  * (77) H. Huang, C. Deng, J. Ping and F. Wang, Eur. Phys. J. C 76 (2016) no.11, 624 doi:10.1140/epjc/s10052-016-4476-z [arXiv:1510.04648 [hep-ph]].
  * (78) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018) no.3, 030001 doi:10.1103/PhysRevD.98.030001
  * (79) X. Liu, H. Huang, J. Ping, D. Chen and X. Zhu, Eur. Phys. J. C 81 (2021) no.10, 950 doi:10.1140/epjc/s10052-021-09752-y [arXiv:2103.12425 [hep-ph]].
  * (80) M. Kamimura, Nucl. Phys. A 351 (1981), 456-480 doi:10.1016/0375-9474(81)90182-2
  * (81) J. Simon, J. Chem, Phys. 75, 2465 (1981).
  * (82) E. Hiyama, M. Kamimura, A. Hosaka, H. Toki and M. Yahiro, Phys. Lett. B 633 (2006), 237-244 doi:10.1016/j.physletb.2005.11.086 [arXiv:hep-ph/0507105 [hep-ph]].
  * (83) E. Hiyama, A. Hosaka, M. Oka and J. M. Richard, Phys. Rev. C 98 (2018) no.4, 045208 doi:10.1103/PhysRevC.98.045208 [arXiv:1803.11369 [nucl-th]].
  * (84) Z. Xia, S. Fan, X. Zhu, H. Huang and J. Ping, Phys. Rev. C 105 (2022) no.2, 025201 doi:10.1103/PhysRevC.105.025201 [arXiv:2105.14723 [hep-ph]].
  * (85) X. Liu, D. Chen, H. Huang, J. Ping, X. Chen and Y. Yang, Sci. China Phys. Mech. Astron. 66, no.2, 221012 (2023) doi:10.1007/s11433-022-1987-5 [arXiv:2204.08104 [hep-ph]].
  * (86) X. Jin, Y. Y. Xue, H. X. Huang, and J. L. Ping, Eur. Phys. J. C 80:1083 (2020).


any}\;\alpha\geq 0;$ (5.23)
$\|R\big{(}\widehat{A}(0)+E_{\pm}(\alpha)\big{)}\|\leq\frac{M_{\mathrm{b}}}{\mathrm{Re}\lambda+(\eta_{0}-M_{\mathrm{b}}g_{\Delta}+\overline{g})\alpha}\;\mbox{whenever}\;\mathrm{Re}\lambda>-(\eta_{0}-M_{\mathrm{b}}g_{\Delta}+\overline{g})\alpha.$

Let $q:[0,\infty)\to[0,\infty)$ be the function defined by
$q(\alpha)=(\eta_{0}-M_{\mathrm{b}}g_{\Delta}+\overline{g})\alpha$. From
(5.22) and (5.23) we see that conditions (i) and (ii) of Hypothesis (H3) are
satisfied. Since $g\in L^{\infty}({\mathbb{R}})$, by (1.12), from (5.13) and
(5.20) we obtain

(5.24) $\|E_{\pm}(\alpha)\|\leq\alpha(\|g\|_{\infty}+\eta_{0})\;\mbox{for
any}\;\alpha\geq 0,$

which implies that
$\limsup\limits_{\alpha\to\infty}\frac{\|E_{\pm}(\alpha)\|}{q(\alpha)}\leq\|g\|_{\infty}+\eta_{0}<\infty$.
Hence, condition (iii) of Hypothesis (H3) is satisfied. Finally, it follows
from (5.24) that Hypothesis (H4) is satisfied, proving the lemma. ∎

We are now ready to prove the main result of this section, the Lyapunov linear
stability of the planar front $\overline{w}$.

###### Proof of Proposition 1.4.

From Remark 5.2 and Lemma 5.4 the family of operators
${\widetilde{A}}_{\pm}(\alpha)=\widehat{A}(0)+E_{\pm}(\alpha)$, $\alpha\geq
0$, satisfies Hypotheses (H1), (H2’), (H3) and (H4). Hence, from Theorem 1.2
we conclude that the family of analytic semigroups generated by
${\widetilde{A}}_{\pm}(\alpha)$ is uniformly exponentially stable for
$\alpha\geq 0$. Setting $\varkappa=\frac{1}{2}$ in Theorem 1.2 we see that
there exists a constant ${\widetilde{M}}_{\mathrm{b}}>0$ such that

(5.25)
$\|e^{t{\widetilde{A}}_{\pm}(\alpha)}\|\leq{\widetilde{M}}_{\mathrm{b}}e^{-\frac{(\eta_{0}-M_{\mathrm{b}}g_{\Delta}+\overline{g})\alpha}{2}t}\;\mbox{for
any}\;t\geq 0,\,\alpha\geq 0.$

From (5.20) we have

(5.26)
$H(\xi_{2})=E_{\sigma}(\xi_{2}^{2})+c{\mathrm{i}}\eta_{1}\xi_{2}I_{L^{2}({\mathbb{R}})}\;\mbox{for
any}\;\xi_{2}\in{\mathbb{R}},\;\mbox{where}\;\sigma=\left\\{\begin{array}[]{ll}+,&\xi_{2}\geq
0,\\\ -,&\xi_{2}<0.\end{array}\right.$

From (5.12) and (5.26) one can readily check that

(5.27)
$e^{t\widehat{A}(\xi_{2})}={\mathscr{M}}_{e^{{\mathrm{i}}\eta_{1}\xi_{2}\cdot}}\,e^{t\big{(}\widehat{A}(0)+H(\xi_{2})\big{)}}{\mathscr{M}}_{e^{-{\mathrm{i}}\eta_{1}\xi_{2}\cdot}}=e^{c{\mathrm{i}}\eta_{1}\xi_{2}t}{\mathscr{M}}_{e^{{\mathrm{i}}\eta_{1}\xi_{2}\cdot}}\,e^{t{\widetilde{A}}_{\sigma}(\xi_{2}^{2})}{\mathscr{M}}_{e^{-{\mathrm{i}}\eta_{1}\xi_{2}\cdot}}$

for any $t\geq 0$ and $\xi_{2}\in{\mathbb{R}}$. From (5.25) and (5.27) we
conclude that

(5.28)
$\|e^{t\widehat{A}(\xi_{2})}\|\leq{\widetilde{M}}_{\mathrm{b}}e^{-\frac{(\eta_{0}-M_{\mathrm{b}}g_{\Delta}+\overline{g})\xi_{2}^{2}}{2}t}\;\mbox{for
any}\;t\geq 0,\,\xi_{2}\in{\mathbb{R}}.$

Since ${\mathscr{A}}$, the linearization along the front $\overline{w}$, is
unitary equivalent to ${\mathscr{M}}_{\widehat{A}}$, from (5.28) we see that
the semigroup generated by ${\mathscr{A}}$ is bounded, proving that the front
$\overline{w}$ is Lyapunov linearly stable. ∎

##  Appendix A Some Examples

In this section we give two examples to show that by perturbing an operator
satisfying Hypotheses (H1) and (H2) we might have unstable spectrum, even if
the space is finite dimensional and the perturbation is a self-adjoint,
bounded, uniformly negative definite operator.

######  Example A.1.

Let $J_{0}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$ and
$Z_{0}=\begin{bmatrix}1&b_{0}\\\ b_{0}&\frac{1}{4}\end{bmatrix}$ with
$b_{0}\in\big{(}-\frac{1}{2},\frac{1-\sqrt{2}}{2}\big{)}$. One can readily
check that

(A.1) $\Big{\langle}Z_{0}\left(\begin{matrix}u\\\
v\end{matrix}\right),\left(\begin{matrix}u\\\
v\end{matrix}\right)\Big{\rangle}_{{\mathbb{C}}^{2}}=|u|^{2}+2b_{0}\mathrm{Re}(u\overline{v})+\frac{1}{4}|v|^{2}\geq(|u|-b_{0}|v|)^{2}+\Big{(}\frac{1}{4}-b_{0}^{2}\Big{)}|v|^{2}\;\mbox{for
any}\;u,v\in{\mathbb{C}}.$

From (A.1) we infer that $Z_{0}$ is a symmetric, positive definite matrix.
Since $b_{0}\in\big{(}-\frac{1}{2},\frac{1-\sqrt{2}}{2}\big{)}$, a simple
computation shows that
$\sigma(J_{0}-Z_{0})=\\{\mu_{0}^{*},\lambda_{0}^{*}\\}$, where
$\mu_{0}^{*}=-\frac{5}{8}-\sqrt{\frac{25}{64}-(1+4b_{0}-4b_{0}^{2})}<0$ and
$\lambda_{0}^{*}=-\frac{5}{8}+\sqrt{\frac{25}{64}-(1+4b_{0}-4b_{0}^{2})}>0$.
We obtain

(A.2)
$b_{1}:=\frac{1}{4}\min\\{\lambda_{0}^{*},m_{0}\\}>0,\;\mbox{where}\;m_{0}=\min\Big{\\{}\Big{|}Z_{0}^{\frac{1}{2}}\left(\begin{matrix}u\\\
v\end{matrix}\right)\Big{|}^{2}:\Big{|}\left(\begin{matrix}u\\\
v\end{matrix}\right)\Big{|}=1\Big{\\}}.$

It follows that $\sigma(J_{0}-b_{1}I_{2})=\\{-b_{1}\\}$ is stable,
$Y_{0}:=b_{1}I_{2}-Z_{0}$ is a symmetric, uniformly negative definite matrix
and $\lambda_{0}^{*}\in\sigma(J_{0}-b_{1}I_{2}+Y_{0})=\sigma(J_{0}-Z_{0})$. We
introduce the matrices

(A.3) $A_{0}=\begin{bmatrix}-b_{1}&1&0\\\ 0&-b_{1}&0\\\
0&0&0\end{bmatrix}\;\mbox{and}\;W_{0}=\begin{bmatrix}b_{1}-1&-b_{0}&0\\\
-b_{0}&b_{1}-\frac{1}{4}&0\\\ 0&0&-1\end{bmatrix}.$

One can readily check that $0$ is a simple eigenvalue of $A_{0}$ and that
$\sigma(A_{0})=\\{0,-b_{1}\\}$. Since any bounded operator is sectorial, we
immediately infer that $A_{0}$ defined in (A.3) satisfies Hypotheses (H1) and
(H2). Since $W_{0}=Y_{0}\oplus-1$ we have $W_{0}$ is a symmetric, uniform
negative definite matrix. However, $A_{0}+W_{0}=(J_{0}-Z_{0})\oplus-1$, which
shows that $\sigma(A_{0}+W_{0})=(J_{0}-Z_{0})\cup\\{-1\\}$, hence
$\lambda_{0}^{*}\in\sigma(A_{0}+W_{0})\cap(0,\infty)$.

This example shows that by perturbing an operator that satisfies Hypotheses
(H1) and (H2) we might generate unstable point spectrum, even if the
perturbation is symmetric, negative definite. Using a similar argument, one
can see that the essential spectrum can become unstable under the same type of
perturbation.

######  Example A.2.

We set ${\mathbb{H}}=L^{2}([1,2],{\mathbb{C}}^{2})\oplus{\mathbb{C}}$ and let
${\widetilde{A}}_{0},{\widetilde{W}}_{0}:{\mathbb{H}}\to{\mathbb{H}}$ be the
bounded linear operators defined by

(A.4) ${\widetilde{A}}_{0}={\mathscr{M}}_{J_{1}}\oplus
0,\quad{\widetilde{W}}_{0}={\mathscr{M}}_{Y_{1}}\oplus-1,$

where ${\mathscr{M}}_{J_{1}}$ and ${\mathscr{M}}_{Y_{1}}$ are the
multiplication operators on $L^{2}([1,2],{\mathbb{C}}^{2})$ by the matrix
valued functions $J_{1},Y_{1}:[1,2]\to{\mathbb{C}}^{2\times 2}$ defined by
$J_{1}(s)=s(J_{0}-b_{1}I_{2})$ and $Y_{1}(s)=sY_{0}=s(b_{1}I_{2}-Z_{0})$. Here
the matrices $J_{0}$, $Z_{0}$ and $Y_{0}$ were introduced in Example A.1
above. We recall that $\sigma(J_{0}-b_{1}I_{2})=\\{-b_{1}\\}$ and
$\sigma(J_{0}-Z_{0})=\\{\mu_{0}^{*},\lambda_{0}^{*}\\}$, with
$\mu_{0}^{*}<0<\lambda_{0}^{*}$.

Since the matrix $Y_{0}$ is symmetric, uniform negative definite, we
immediately infer that ${\mathscr{M}}_{Y_{1}}$ is self-adjoint, uniform
negative definite on $L^{2}([1,2],{\mathbb{C}}^{2})$. Hence,
${\widetilde{W}}_{0}$ is self-adjoint, uniformly negative definite on
${\mathbb{H}}$. Moreover, since the functions $J_{1}$ and $Y_{1}$ are
continuous by [8], it follows that

$\displaystyle\sigma({\mathscr{M}}_{J_{1}})$
$\displaystyle=\sigma_{\mathrm{ess}}({\mathscr{M}}_{J_{1}})=\overline{\bigcup_{s\in[1,2]}\sigma(J_{1}(s))}=[-2b_{1},-b_{1}],$
(A.5) $\displaystyle\sigma({\mathscr{M}}_{J_{1}+Y_{1}})$
$\displaystyle=\sigma_{\mathrm{ess}}({\mathscr{M}}_{J_{1}+Y_{1}})=\overline{\bigcup_{s\in[1,2]}s\sigma(J_{0}-Z_{0})}=[-2\mu_{0}^{*},-\mu_{0}^{*}]\,\cup[\lambda_{0}^{*},2\lambda_{0}^{*}].$

Since the linear operator $A_{0}$ is bounded on ${\mathbb{H}}$ we have it is
sectorial. Moreover, from (A.4) and (A.2) we infer that $0$ is a simple
eigenvalue of ${\widetilde{A}}_{0}$ and
$\sigma({\widetilde{A}}_{0})\setminus\\{0\\}=\sigma({\mathscr{M}}_{J_{1}})=[-2b_{1},-b_{1}]$,
thus
$\sup\,\mathrm{Re}\big{(}\sigma({\widetilde{A}}_{0})\setminus\\{0\\}\big{)}=-b_{1}<0$.
Hence, ${\widetilde{A}}_{0}$ satisfies Hypotheses (H1) and (H2). Even if
${\widetilde{W}}_{0}$ is self-adjoint, uniform negative definite on
${\mathbb{H}}$, from (A.2) we have
$\sigma_{\mathrm{ess}}({\widetilde{A}}_{0}+{\widetilde{W}}_{0})=\sigma_{\mathrm{ess}}({\mathscr{M}}_{J_{1}+Y_{1}})=[-2\mu_{0}^{*},-\mu_{0}^{*}]\,\cup[\lambda_{0}^{*},2\lambda_{0}^{*}]$,
which shows that
$\sigma_{\mathrm{ess}}({\widetilde{A}}_{0}+{\widetilde{W}}_{0})\cap(0,\infty)=[\lambda_{0}^{*},2\lambda_{0}^{*}]$
is nonempty.

## References

  * [1] S. Agmon, On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), 119–147.
  * [2] W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Second edition. Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011.
  * [3] W. Auzinger, Sectorial operators and normalized numerical range, Appl. Numer. Math. 45 (2003), no. 4, 367–388.
  * [4] C. Chicone, Y. Latushkin, Evolution semigroups in dynamical systems and differential equations, Math. Surv. Monogr. 70, AMS, Providence, 1999.
  * [5] Y.L. Daletskii, M.G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, Amer. Math. Soc. Transl., vol. 43, Amer. Math. Soc., Providence, RI 1974.
  * [6] K. Engel, R. Nagel, One-parameter semigroups for linear evolution equations. Springer-Verlag, New York, 2000.
  * [7] L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc. 236(1978), 385–394.
  * [8] V. Hardt, E. Wagenfuhrer, Spectral Properties of a multiplication operator, Math. Nachrichten, 178(1996), 135–156.
  * [9] B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds, (2010). arXiv:1001.4171v1, math. FA.
  * [10] B. Helffer and J. Sjöstrand, Improving semigroup bounds with resolvent estimate, Integral Equations Operator Theory 93 (2021), no. 3, Paper No. 36, 41 pp.
  * [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840, Springer, New York (1981).
  * [12] T. Kapitula, K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, 2013.
  * [13] T. Kato Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.
  * [14] Y. Latushkin, V. Yurov, Stability estimates for semigroups on Banach spaces, Discrete Contin. Dyn. Syst. 33 (2013), no. 11-12, 5203-5216.
  * [15] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications 16 (1995), Birkhäuser Verlag, Basel.
  * [16] Y. Mori, H. Matano, Stability of Front Solutions of the Bidomain Equation, Comm. Pure Appl. Math., Vol. LXIX (2016), 2364–2426.
  * [17] A. Pazy, Semigroups of linear operators and applications to partial differential operators. Appl. Math. Sci. vol. 44. Springer (1983).
  * [18] J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc. 284(1984), 847–857.
  * [19] M. Reed, B. Simon, Methods of Modern Mathematical Physics.II: Fourier analysis, Self-Adjointness, Academic Press, New York, 1975.
  * [20] K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space. Graduate Texts in Mathematics, 265. Springer, Dordrecht, 2012.
  * [21] H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc. 199 (1974), 141–162.
  * [22] H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980), 299–310.
  * [23] A. I. Volpert, V. A. Volpert, V. A. Volpert, Traveling wave solutions of parabolic systems. Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, R.I., 1994.