††thanks: Corresponding Author

# Strange partners of the doubly charmed tetraquark $T^{+}_{cc}$

S. S. Agaev  Institute for Physical Problems, Baku State University, Az–1148
Baku, Azerbaijan  K. Azizi  Department of Physics, University of Tehran, North
Karegar Avenue, Tehran 14395-547, Iran  Department of Physics, Doǧuş
University, Dudullu-Ümraniye, 34775 Istanbul, Turkey  H. Sundu  Department of
Physics, Kocaeli University, 41380 Izmit, Turkey  Department of Physics
Engineering, Istanbul Medeniyet University, 34700 Istanbul, Turkey

###### Abstract

The spectroscopic parameters and widths of the axial-vector
$T_{cc:s}^{\mathrm{AV}}$ and scalar $T_{cc:s}^{\mathrm{S}}$,
$\widetilde{T}_{cc:s}^{\mathrm{S}}$ strange partners of the doubly charmed
exotic meson $T_{cc}^{+}$ with the content $cc\overline{u}\overline{s}$, are
calculated in the framework of the QCD sum rule method. We model
$T_{cc:s}^{\mathrm{AV}}$ as the diquark-antidiquark state composed of axial-
vector and scalar components, whereas scalar particles $T_{cc:s}^{\mathrm{S}}$
and $\widetilde{T}_{cc:s}^{\mathrm{S}}$ are built of axial-vector and scalar
diquarks, respectively. The masses and current couplings of these tetraquarks
are calculated in the context of the two-point sum rule approach by taking
into account the quark, gluon and mixed condensates up to dimension $10$. The
full width of the state $T_{cc:s}^{\mathrm{AV}}$ is found from analysis of the
processes $T_{cc:s}^{\mathrm{AV}}\to D^{0}D_{s}^{\ast+}$ and
$T_{cc:s}^{\mathrm{AV}}\to D^{\ast}(2007)^{0}D_{s}^{+}$. Decays to
$D^{0}D_{s}^{+}$, $D^{\ast}(2007)^{0}D_{s}^{\ast+}$ and $D^{0}D_{s}^{+}$
mesons are utilized in the case of the scalar tetraquarks
$T_{cc:s}^{\mathrm{S}}$ and $\widetilde{T}_{cc:s}^{\mathrm{S}}$, respectively.
The partial widths of the aforementioned decays are determined via the strong
couplings $g_{1}$, $g_{2}$, $G_{1}$, $G_{2}$ and $\widetilde{G}$, which
describe the strong interactions of the particles at the relevant tetraquark-
meson-meson vertices. These couplings are computed using the QCD three-point
sum rule method, most appropriate for the strong decays under study. The
predictions $m=(3995\pm 143)~{}\mathrm{MeV}$ and $\Gamma_{\mathrm{AV}}=(72\pm
13)~{}\mathrm{MeV}$, as well as $m_{\mathrm{S}}=(4128\pm 142)~{}\mathrm{MeV}$,
$\Gamma_{\mathrm{S}}=(213\pm 47)~{}\mathrm{MeV}$ and
$\widetilde{m}_{\mathrm{S}}=(4035\pm 145)~{}\mathrm{MeV}$ and
$\widetilde{\Gamma}_{\mathrm{S}}=(123\pm 32)~{}\mathrm{MeV}$ obtained for the
masses and widths of these tetraquarks in the present work can be useful in
future experimental investigations of the doubly charmed four-quark
resonances.

##  I Introduction

Recently, the LHCb Collaboration discovered the doubly charmed axial-vector
four-quark meson $T_{cc}^{+}$ with content $cc\overline{u}\overline{d}$
Aaij:2021vvq ; LHCb:2021auc . It was observed as a narrow peak in
$D^{0}D^{0}\pi^{+}$ invariant mass distribution, and is the first doubly
charmed state seen in the experiment. The state $T_{cc}^{+}$ has interesting
features: its mass $m_{\exp}=3875.1~{}\mathrm{MeV}+\delta m_{\exp}$, where
$\delta m_{\exp}=-273\pm 61\pm 5_{-14}^{+11}~{}\mathrm{KeV}$, is less than
two-meson $D^{0}D^{\ast}(2010)^{+}$ threshold , but is very close to it. It is
also the longest living exotic meson observed till now, because it has a very
small full width $\Gamma=410\pm 165\pm 43_{-38}^{+18}~{}\mathrm{KeV}$.

It is worth noting that the four-quark states
$QQ\overline{q}\overline{q}^{\prime}$ containing two heavy quarks $Q$ were
already in the center of theoretical investigations Esposito:2013fma ;
Navarra:2007yw ; Karliner:2017qjm ; Eichten:2017ffp . The reason is that some
of these tetraquarks may be stable against the strong and electromagnetic
decays, and therefore can transform to conventional mesons only through weak
interaction. Since widths of such particles would be very small, they may be
discovered in various exclusive and inclusive processes relatively easily. In
the case of tetraquarks $bb\overline{q}\overline{q}^{\prime}$ with different
spin-parities and light antidiquark contents
$\overline{q}\overline{q}^{\prime}$, this assumption seems is confirmed by
numerous studies performed in the context of various models of high energy
physics. Thus, stability of the axial-vector tetraquark
$T_{bb}^{-}=bb\overline{u}\overline{d}$ was proved in Refs. Navarra:2007yw ;
Karliner:2017qjm ; Eichten:2017ffp ; Agaev:2018khe . In Ref. Agaev:2018khe  it
was demonstrated that the mass of such particle, $m=(10035\pm
260)~{}\mathrm{MeV}$, is below the $B^{-}\overline{B}^{\ast 0}$ and
$B^{-}\overline{B}^{0}\gamma$ thresholds. This means that the double-beauty
tetraquark $T_{bb}^{-}$ is stable against the strong and radiative decays and
dissociates to ordinary mesons only weakly. The full width and mean lifetime
of $T_{bb}^{-}$ were evaluated in Refs. Agaev:2018khe ; Agaev:2020mqq  using
various semileptonic and nonleptonic decay channels of this particle. Results
obtained for these parameters, $\Gamma_{\mathrm{full}}=(7.72\pm 1.23)\times
10^{-8}~{}\mathrm{MeV}$ and $\tau=8.53_{-1.18}^{+1.57}~{}\mathrm{fs,}$ are
useful for experimental investigations of the tetraquark $T_{bb}^{-}$. Some of
other $bb\overline{q}\overline{q}^{\prime}$ states were also identified as
stable particles, and their full widths were calculated via allowed weak
transformations Xing:2018bqt ; Agaev:2020zag ; Agaev:2020dba ; Agaev:2019lwh .

The doubly charmed tetraquarks $cc\overline{q}\overline{q}^{\prime}$, as
interesting objects, were undergone of intensive studies as well
Navarra:2007yw ; Karliner:2017qjm ; Eichten:2017ffp ; Du:2012wp ; Wang:2017uld
; Wang:2017dtg ; Braaten:2020nwp ; Cheng:2020wxa ; Meng:2020knc ;
Junnarkar:2018twb ; Guo:2021yws ; Padmanath:2022cvl . Thus, the axial-vector
tetraquark $cc\overline{u}\overline{d}$ was explored in Ref. Navarra:2007yw
in the framework of QCD sum rule method. Properties of the four-quark mesons
$cc\overline{q}\overline{q}^{\prime}$ with spin-parities
$J^{\mathrm{P}}=0^{-},~{}0^{+},~{}1^{-}$ and $1^{+}$ were analyzed in Ref.
Du:2012wp . New investigations of these tetraquarks proved once more the
unstable nature of the axial-vector tetraquark $cc\overline{u}\overline{d}$
Karliner:2017qjm ; Eichten:2017ffp ; Wang:2017uld ; Wang:2017dtg ;
Braaten:2020nwp ; Cheng:2020wxa . In contrast to these conclusions, studies
carried out in the context of the constituent quark model and lattice
simulations demonstrated existence of the stable axial-vector tetraquark
$cc\overline{u}\overline{d}$ lying $\approx 23~{}\mathrm{MeV}$ below the two-
meson threshold Meng:2020knc ; Junnarkar:2018twb .

Detailed investigations of doubly charmed tetraquarks were carried out also in
our articles Agaev:2019qqn ; Agaev:2018vag . In the first of these
publications, we calculated various parameters of pseudoscalar and scalar
states $cc\overline{u}\overline{d}$. Our analyses demonstrated that these
four-quark systems are unstable particles and break up to conventional mesons
trough strong interactions. Full widths of these particles were evaluated
using their kinematically allowed decays to $D^{+}D^{\ast}(2007)^{0}$,
$D^{0}D^{\ast}(2010)^{+}$ and $D^{0}D^{+}$ mesons, respectively. Interesting
hypothetical pseudoscalar tetraquarks $cc\overline{s}\overline{s}$ and
$cc\overline{d}\overline{s}$ carrying two units of electric charge were object
of the second paper.

Experimental observation of the resonance $T_{cc}^{+}$ triggered new analyses
aimed to explain the measured parameters of this particle Agaev:2021vur ;
Feijoo:2021ppq ; Yan:2021wdl ; Fleming:2021wmk ; Agaev:2022ast . In our papers
Agaev:2021vur ; Agaev:2022ast , we addressed the relevant problems in the
framework of QCD sum rule method. In Ref. Agaev:2021vur , the doubly charmed
axial-vector resonance $T_{cc}^{+}$ was treated as the diquark-antidiquark
state $cc\overline{u}\overline{d}$. The mass and coupling of this tetraquark
were calculated by means of the QCD two-point sum rule approach.

To estimate the full width of $T_{cc}^{+}$, we specified its decay channels
and used QCD three-point sum rule method to find the strong couplings at
relevant vertices. Because $T_{cc}^{+}$ is a very narrow state, its possible
decay modes were objects in numerous analyses. Indeed, $T_{cc}^{+}$ was
discovered in the $D^{0}D^{0}\pi^{+}$ mass distribution. One of popular ways
to explain such final state is the chain of transformations
$T_{cc}^{+}\rightarrow D^{0}D^{\ast+}\rightarrow D^{0}D^{0}\pi^{+}$. But the
mass of $T_{cc}^{+}$ is below the $D^{0}D^{\ast+}$ threshold, therefore the
process $T_{cc}^{+}\rightarrow D^{0}D^{\ast+}$ is kinematically forbidden, and
may proceed only virtually. Alternatively, production of $D^{0}D^{0}\pi^{+}$
may run through a process $T_{cc}^{+}\rightarrow
T_{cc;\overline{u}\overline{u}}^{0}\pi^{+}$ where
$T_{cc;\overline{u}\overline{u}}^{0}$ is a scalar tetraquark. Then three-meson
final state may appear due to the decay
$T_{cc;\overline{u}\overline{u}}^{0}\rightarrow D^{0}D^{0}$. To calculate the
full width of $T_{cc}^{+}$, we explored also another mode
$T_{cc}^{+}\rightarrow\widetilde{T}_{cc;\overline{u}\overline{d}}^{+}\pi^{0}\rightarrow
D^{0}D^{+}\pi^{0}$ with $\widetilde{T}_{cc;\overline{u}\overline{d}}^{+}$
being a scalar tetraquark. Results found for the mass and width of the
diquark-antidiquark state $T_{cc}^{+}$ are in nice agreement with the LHCb
data.

The hadronic molecule model $D^{0}D^{\ast+}$ for the resonance $T_{cc}^{+}$
was examined in Ref. Agaev:2022ast . It turned out that the mass and width of
this system exceed experimental data of the LHCb Collaboration. Our studies
showed that a preferable assignment for the LHCb resonance $T_{cc}^{+}$ is the
diquark-antidiquark model, but because parameters of the molecule
$D^{0}D^{\ast+}$ suffer from theoretical uncertainties, we did not exclude the
molecule picture for $T_{cc}^{+}$.

The four-quark states containing strange $s$-quark alongside the heavy diquark
$cc$ form another interesting class of the doubly charmed particles. These
exotic mesons may have diquark-antidiquark or molecular structures. In the
context of various methods, masses of the doubly charmed strange tetraquarks
with different spin-parities and structures were considered in Refs.
Karliner:2021wju ; Chen:2022ros ; Praszalowicz:2022sqx ; Xin:2021wcr ;
Ren:2021dsi .

In the present article, we are going to explore features of axial-vector
$T_{cc:s}^{\mathrm{AV}}$ and scalar $T_{cc:s}^{\mathrm{S}}$,
$\widetilde{T}_{cc:s}^{\mathrm{S}}$ diquark-antidiquark systems
$cc\overline{u}\overline{s}$ (in what follows, $T_{\mathrm{AV}}$,
$T_{\mathrm{S}}$ and $\widetilde{T}_{\mathrm{S}}$, respectively) by computing
their masses, current couplings and widths. We model the tetraquark
$T_{\mathrm{AV}}$ as a four-quark meson composed of a heavy axial-vector
diquark and light scalar antidiquark. The scalar tetraquarks $T_{\mathrm{S}}$
and $\widetilde{T}_{\mathrm{S}}$ are considered as particles built of the
axial-vector and scalar constituents, respectively. To calculate the masses
and couplings of these structures, we use QCD two-point sum rules by taking
into account various vacuum condensates up to dimension $10$. The width of
$T_{\mathrm{AV}}$ is evaluated by considering its $S$-wave decays
$T_{\mathrm{AV}}\rightarrow D^{0}D_{s}^{\ast+}$ and
$T_{\mathrm{AV}}\rightarrow D^{\ast}(2007)^{0}D_{s}^{+}$. The widths of scalar
tetraquarks $T_{\mathrm{S}}$ and $\widetilde{T}_{\mathrm{S}}$ are estimated
using their decays to $D^{0}D_{s}^{+}$, $D^{\ast}(2007)^{0}D_{s}^{\ast+}$ and
$D^{0}D_{s}^{+}$ mesons, respectively. Partial widths of these processes are
determined by the strong couplings $g_{1}$, $g_{2}$, $G_{1}$, $G_{2}$ and
$\widetilde{G}$ at tetraquark-meson-meson vertices
$T_{\mathrm{AV}}D^{0}D_{s}^{\ast+}$,
$T_{\mathrm{AV}}D^{\ast}(2007)^{0}D_{s}^{+}$, $T_{\mathrm{S}}D^{0}D_{s}^{+}$,
$T_{\mathrm{S}}D^{\ast}(2007)^{0}D_{s}^{\ast+}$and
$\widetilde{T}_{\mathrm{S}}D^{0}D_{s}^{+}$, respectively.

This article is structured in the following manner: In Sec. II, we calculate
the mass and current coupling of the tetraquarks $T_{\mathrm{AV}}$,
$T_{\mathrm{S}}$ and $\widetilde{T}_{\mathrm{S}}$ in the framework of QCD two-
point sum rule method. The strong couplings $g_{1}$, $g_{2}$ and partial
widths of the decays $T_{\mathrm{AV}}\rightarrow D^{0}D_{s}^{\ast+}$ and
$T_{\mathrm{AV}}\rightarrow D^{\ast}(2007)^{0}D_{s}^{+}$, as well as the full
width of $T_{\mathrm{AV}}$ are found in Sec. III. Section IV is devoted to
computation of the strong couplings$\ G_{1}$, $G_{2}$ and $\widetilde{G}$ and
widths of the tetraquarks $T_{\mathrm{S}}$ and $\widetilde{T}_{\mathrm{S}}$.
Section V is reserved for discussion ans concluding notes.

##  II Spectroscopic parameters of the tetraquarks $T_{\mathrm{AV}}$,
$T_{\mathrm{S}}$, and $\widetilde{T}_{\mathrm{S}}$

In this section, we calculate the masses and current couplings of the
tetraquarks $T_{\mathrm{AV}}$, $T_{\mathrm{S}}$, and
$\widetilde{T}_{\mathrm{S}}$ using the two-point sum rule method
Shifman:1978bx ; Shifman:1978by . The key component in the sum rule analysis
is an interpolating current for a hadron under consideration. In the diquark-
antidiquark model, the four-quark mesons $T_{\mathrm{AV}}$, $T_{\mathrm{S}}$,
and $\widetilde{T}_{\mathrm{S}}$ are built of the heavy diquark $cc$ and light
antidiquark $\overline{u}\overline{s}$. In the case of the axial-vector state
$T_{\mathrm{AV}}$, we suggest that it is composed of the axial-vector diquark
$c^{T}C\gamma_{\mu}c$ and scalar antidiquark
$\overline{u}\gamma_{5}C\overline{s}^{T}$ with $C$ being the charge
conjugation matrix. We take into account that the axial-vector diquark
$c_{a}^{T}C\gamma_{\mu}c_{b}$, where $a$ and $b$ are color indices, has a
symmetric flavor but an antisymmetric color organization, and its flavor-color
structure is $(\mathbf{6}_{f},\overline{\mathbf{3}}_{c})$ Du:2012wp . Then,
the color-singlet current $J_{\mu}(x)$ for $T_{\mathrm{AV}}$ can be
constructed using the color-triplet light antidiquark field
$\overline{u}_{a}\gamma_{5}C\overline{s}_{b}^{T}-\overline{u}_{b}\gamma_{5}C\overline{s}_{a}^{T}$.
Because both components of this field lead to identical terms in $J_{\mu}(x)$,
it is enough to preserve in calculations one of them. As a result, for the
current $J_{\mu}(x)$, we get the following expression

$J_{\mu}(x)=\left[c_{a}^{T}(x)C\gamma_{\mu}c_{b}(x)\right]\left[\overline{u}_{a}(x)\gamma_{5}C\overline{s}_{b}^{T}(x)\right],$
(1)

which belongs to the
$[\overline{\mathbf{3}}_{c}]_{cc}\otimes[\mathbf{3}_{c}]_{\overline{u}\overline{s}}$
representation of the color group $SU_{c}(3)$.

In the case of the scalar exotic meson $cc\overline{u}\overline{s}$, we
consider two structures $T_{\mathrm{S}}$ and $\widetilde{T}_{\mathrm{S}}$
which can describe this tetraquark. We assume that $T_{\mathrm{S}}$ is a
scalar tetraquark built of axial-vector components. The interpolating current
for such state is given by the expression

$J(x)=[c_{a}^{T}(x)C\gamma_{\mu}c_{b}(x)][\overline{u}_{a}(x)\gamma^{\mu}C\overline{s}_{b}^{T}(x)],$
(2)

which is from the antitriplet-triplet representation of the color group
$SU_{c}(3)$.

We model $\widetilde{T}_{\mathrm{S}}$ by supposing that it is made of the
scalar diquark $c_{a}^{T}C\gamma_{5}c_{b}$ and antidiquark
$\overline{u}_{a}\gamma_{5}C\overline{s}_{b}^{T}$. The heavy diquark has
symmetric flavor and color structure $(\mathbf{6}_{f},\mathbf{6}_{c})$, i.e.,
has a color-flavor sextet organization Du:2012wp . Then the light antidiquark
should be also symmetric in the color indices
$\overline{u}_{a}\gamma_{5}C\overline{s}_{b}^{T}+\overline{u}_{b}\gamma_{5}C\overline{s}_{a}^{T}$.
The interpolating current $\widetilde{J}(x)$ of such tetraquark contains
relevant diquark and antidiquark fields and has
$[\mathbf{6}_{c}]_{cc}\otimes[\overline{\mathbf{6}}_{c}]_{\overline{u}\overline{s}}$
color-structure. In investigations, we employ

$\widetilde{J}(x)=[c_{a}^{T}(x)C\gamma_{5}c_{b}(x)][\overline{u}_{a}(x)\gamma_{5}C\overline{s}_{b}^{T}(x)],$
(3)

because components of the antidiquark field generate in the current
$\widetilde{J}$ two equal terms.

###  II.1 The mass and coupling of $T_{\mathrm{AV}}$

To extract sum rules for the spectroscopic parameters of the tetraquark
$T_{\mathrm{AV}}$, we begin from analysis of the correlation function
$\Pi_{\mu\nu}(p)$,

$\Pi_{\mu\nu}(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{{\dagger}}(0)\\}|0\rangle.$ (4)

First, $\Pi_{\mu\nu}(p)$ should be expressed using the mass $m$ and coupling
$f$ of the tetraquark $T_{\mathrm{AV}}$: This expression is the hadronic
representation of the $\Pi_{\mu\nu}(p)$ and forms the physical side of the
desired sum rules. To this end, we saturate the correlation function
$\Pi_{\mu\nu}(p)$ with a complete set of states with quantum numbers
$J^{\mathrm{P}}=1^{+}$ and perform the integration over $x$ in Eq. (4),

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{\langle
0|J_{\mu}|T_{\mathrm{AV}}(p,\epsilon)\rangle\langle
T_{\mathrm{AV}}(p,\epsilon)|J_{\nu}^{\dagger}|0\rangle}{m^{2}-p^{2}}+\cdots.$
(5)

We write down the contribution arising from the ground-state particle
$T_{\mathrm{AV}}$ explicitly, and denote ones due to higher resonances and
continuum states by the dots.

The function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ can be simplified using the
matrix element,

$\langle 0|J_{\mu}|T_{\mathrm{AV}}(p,\epsilon)\rangle=fm\epsilon_{\mu},$ (6)

where $\epsilon_{\mu}$ is the polarization vector of $T_{\mathrm{AV}}$. It is
not difficult to demonstrate that in terms of $m$ and $f$ the function
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ takes the following form:

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{m^{2}f^{2}}{m^{2}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m^{2}}\right)+\cdots.$
(7)

The QCD side of the sum rules $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ has to be
computed in the operator product expansion ($\mathrm{OPE}$) with some fixed
accuracy. The function $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ can be obtained by
inserting the explicit form of the interpolating current $J_{\mu}(x)$ into Eq.
(4), and contracting corresponding heavy and light quark fields. After these
operations, we get

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p)=i\int
d^{4}xe^{ipx}\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{s}^{b^{\prime}b}(-x)\right.$
$\displaystyle\left.\times\gamma_{5}S_{u}^{a^{\prime}a}(-x)\right]\left\\{\mathrm{Tr}\left[\gamma_{\nu}\widetilde{S}_{c}^{ba^{\prime}}(x)\gamma_{\mu}S_{c}^{ab^{\prime}}(x)\right]\right.$
$\displaystyle\left.-\mathrm{Tr}\left[\gamma_{\nu}\widetilde{S}_{c}^{aa^{\prime}}(x)\gamma_{\mu}S_{c}^{bb^{\prime}}(x)\right]\right\\}.$
(8)

In Eq. (8), $S_{c}^{ab}(x)$ and $S_{q}^{ab}(x)$ are the $c$ and $q(u,s)$-quark
propagators: their analytic expressions were presented in Ref. Agaev:2020zad .
We have also introduced the notation

$\widetilde{S}_{c(q)}(x)=CS_{c(q)}^{T}(x)C.$ (9)

The QCD sum rules are obtained by equating the invariant amplitudes which
correspond to the same Lorentz structures both in
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ and $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. The
amplitudes corresponding to the structures $\sim g_{\mu\nu}$ do not receive
contributions from the spin-$0$ particles, therefore, they are suitable for
our purposes. Having denoted these invariant amplitudes by
$\Pi^{\mathrm{Phys}}(p^{2})$ and $\Pi^{\mathrm{OPE}}(p^{2})$, respectively,
and equating them, we find an expression which is convenient for further
processing. First, we encounter here problems connected with necessity to
suppress the contributions of the higher resonances and continuum states. For
these purposes, we make use of the Borel transformation in the sum rule
equality. Afterwards, employing the assumption about quark-hadron duality, it
is possible to subtract these effects from the obtained expression. After
these manipulations, the sum rule equality acquires dependence on the Borel
$M^{2}$ and continuum threshold $s_{0}$ parameters.

It is evident that the Borel transformation of $\Pi^{\mathrm{Phys}}(p^{2})$ is
trivial. The Borel transformation and continuum subtraction convert the
invariant amplitude $\Pi^{\mathrm{OPE}}(p^{2})$ to the form,

$\Pi(M^{2},s_{0})=\int_{\mathcal{M}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}),$
(10)

where $\mathcal{M}=2m_{c}+m_{s}$. In numerical computations, we set $m_{u}=0$
and $m_{s}^{2}=0$, but take into account terms proportional to $m_{s}$. In Eq.
(10), $\rho^{\mathrm{OPE}}(s)$ is the two-point spectral density computed as
the imaginary part of the correlation function. The second component of the
invariant amplitude $\Pi(M^{2})$ includes nonperturbative contributions
extracted directly from $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. We calculate
$\Pi(M^{2},s_{0})$ by taking into account the nonperturbative terms up to
dimension $10$. Explicit expression of $\Pi(M^{2},s_{0})$ is rather
cumbersome, therefore we do not provide it here.

The sum rules for $m$ and $f$ read

$m^{2}=\frac{\Pi^{\prime}(M^{2},s_{0})}{\Pi(M^{2},s_{0})},$ (11)

and

$f^{2}=\frac{e^{m^{2}/M^{2}}}{m^{2}}\Pi(M^{2},s_{0}),$ (12)

where $\Pi^{\prime}(M^{2},s_{0})=d\Pi(M^{2},s_{0})/d(-1/M^{2})$.

The formulas in Eqs. (11) and (12) depend on the various quark, gluon and
mixed condensates. They are universal quantities extracted from numerous
analyses:

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\ \langle\overline{s}s\rangle=(0.8\pm
0.1)\langle\overline{q}q\rangle,$ $\displaystyle\langle\overline{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle,\
\langle\overline{s}g_{s}\sigma
Gs\rangle=m_{0}^{2}\langle\overline{s}s\rangle,$ $\displaystyle
m_{0}^{2}=(0.8\pm 0.2)~{}\mathrm{GeV}^{2}$
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle\langle
g_{s}^{3}G^{3}\rangle=(0.57\pm 0.29)~{}\mathrm{GeV}^{6},$ $\displaystyle
m_{s}=93_{-5}^{+11}~{}\mathrm{MeV},\ m_{c}=(1.27\pm 0.02)~{}\mathrm{GeV}.$
(13)

We also added the masses of $c$ and $s$ quarks to this list.

It is known that the Borel and continuum threshold parameters, $M^{2}$ and
$s_{0}$, are auxiliary quantities of calculations, which have to satisfy some
constraints imposed on $\Pi(M^{2},s_{0})$ by the dominance of the pole
contribution ($\mathrm{PC}$) and convergence of the $\mathrm{OPE}$. We use the
following definition for $\mathrm{PC}$:

$\mathrm{PC}=\frac{\Pi(M^{2},s_{0})}{\Pi(M^{2},\infty)}.$ (14)

The restriction on $\mathrm{PC}$ is necessary to fix the maximum value of the
Borel parameter. The low boundary of the window for the Borel parameter is
obtained from the convergence of the $\mathrm{OPE}$. To this end, we employ
the expression

$R(M^{2})=\frac{\Pi^{\mathrm{DimN}}(M^{2},s_{0})}{\Pi(M^{2},s_{0})},$ (15)

where $\Pi^{\mathrm{DimN}}(M^{2},s_{0})$ is the contribution of the last three
terms in $\mathrm{OPE}$, i.e., $\mathrm{DimN=Dim(8+9+10)}$.

Apart from these constraints, the region for $M^{2}$ should lead to stable
predictions for extracted physical quantities. Our computations prove that the
regions for $M^{2}$ and $s_{0}$,

$M^{2}\in[3,4.5]~{}\mathrm{GeV}^{2},\ s_{0}\in[20.5,21.5]~{}\mathrm{GeV}^{2},$
(16)

satisfy all restrictions mentioned above. In fact, in these regions, on
average in $s_{0}$, the pole contribution changes within the limits:

$0.79\geq\mathrm{PC}\geq 0.49.$ (17)

At $M^{2}=3~{}\mathrm{GeV}^{2}$, the contributions to $\Pi(M^{2},s_{0})$ of
the last three terms in $\mathrm{OPE}$ is less than $0.01$.

In Fig. 1, we plot the dependence of the mass $m$ of the tetraquark
$T_{\mathrm{AV}}$ on $M^{2}$ and $s_{0}$. It is clear, that the window for
$M^{2}$, where parameters of $T_{\mathrm{AV}}$ are extracted, can be
considered as a relatively stable plateau. Nevertheless, one sees a residual
dependence of $m$ on the Borel parameter $M^{2}$. This effect allows one to
find ambiguities of the sum rule calculations. Another source of the
theoretical uncertainties is the continuum threshold parameter $s_{0}$. The
region for $s_{0}$ has to meet the constraints coming from the dominance of
$\mathrm{PC}$ and convergence of the $\mathrm{OPE}$. The parameter
$\sqrt{s_{0}}$ bears also information on the mass of the $T_{\mathrm{AV}}$
tetraquark’s first radial excitation.

Figure 1: Mass $m$ of the tetraquark $T_{\mathrm{AV}}$ as a function of the
Borel parameter $M^{2}$ (left), and the continuum threshold parameter $s_{0}$
(right).

Our results for the spectral parameters of the tetraquark $T_{\mathrm{AV}}$
read

$\displaystyle m$ $\displaystyle=$ $\displaystyle(3995\pm
143)~{}\mathrm{MeV},$ $\displaystyle f$ $\displaystyle=$ $\displaystyle(2.4\pm
0.5)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (18)

The predictions for the mass $m$ and coupling $f$ are obtained as average of
results for these quantities in the working windows (16). These values
correspond to the sum rules’ predictions approximately at middle of these
regions, i.e., to results at $M^{2}=3.5~{}\mathrm{GeV}^{2}$ and
$s_{0}=21~{}\mathrm{GeV}^{2}$, where $\mathrm{PC}\approx 0.65$ guarantees the
ground-state nature of $T_{\mathrm{AV}}$. As it has been noted above,
$\sqrt{s_{0}}-m\geq 480~{}\mathrm{MeV}$ allows us to estimate the mass of the
excited tetraquark as $m^{\ast}\approx(m+480)~{}\mathrm{MeV}$, which is
reasonable for double-heavy tetraquarks.

###  II.2 Parameters of the tetraquarks $T_{\mathrm{S}}$ and
$\widetilde{T}_{\mathrm{S}}$

To find the mass $m_{\mathrm{S}}$ and current coupling $f_{\mathrm{S}}$ of the
scalar tetraquark $T_{\mathrm{S}}$, we start from the correlation function
$\Pi(p)$

$\Pi(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{J(x)J^{{\dagger}}(0)\\}|0\rangle,$ (19)

where the current $J(x)$ is given by Eq. (2). In terms of the tetraquark’s
physical parameters, $\Pi(p)$ is determined by the expression,

$\Pi^{\mathrm{Phys}}(p)=\frac{\langle 0|J|T_{\mathrm{S}}(p)\rangle\langle
T_{\mathrm{S}}(p)|J^{\dagger}|0\rangle}{m_{\mathrm{S}}^{2}-p^{2}}+\cdots.$
(20)

The function $\Pi^{\mathrm{Phys}}(p)$ can be rewritten by employing the matrix
element,

$\langle 0|J|T_{\mathrm{S}}(p)\rangle=m_{\mathrm{S}}f_{\mathrm{S}}.$ (21)

Then, it is easy to see that in terms of the parameters $m_{\mathrm{S}}$ and
$f_{\mathrm{S}}$ the $\Pi^{\mathrm{Phys}}(p)$ has the form,

$\Pi^{\mathrm{Phys}}(p)=\frac{m_{\mathrm{S}}^{2}f_{\mathrm{S}}^{2}}{m_{\mathrm{S}}^{2}-p^{2}}+\cdots.$
(22)

To obtain the QCD side of the sum rules, $\Pi^{\mathrm{OPE}}(p)$, we insert
the interpolating current $J(x)$ into Eq. (19), and contract the relevant
quark fields. After these manipulations, we get

$\displaystyle\Pi^{\mathrm{OPE}}(p)=i\int
d^{4}xe^{ipx}\mathrm{Tr}\left[\gamma_{\mu}\widetilde{S}_{s}^{b^{\prime}b}(-x)\right.$
$\displaystyle\left.\times\gamma_{\nu}S_{u}^{a^{\prime}a}(-x)\right]\left\\{\mathrm{Tr}\left[\gamma^{\nu}\widetilde{S}_{c}^{aa^{\prime}}(x)\gamma^{\mu}S_{c}^{bb^{\prime}}(x)\right]\right.$
$\displaystyle\left.-\mathrm{Tr}\left[\gamma^{\nu}\widetilde{S}_{c}^{ba^{\prime}}(x)\gamma^{\mu}S_{c}^{ab^{\prime}}(x)\right]\right\\}.$
(23)

Figure 2: Dependence of the $T_{\mathrm{S}}$ tetraquark’s mass
$m_{\mathrm{S}}$ on the Borel parameter $M^{2}$ (left), and on the continuum
threshold parameter $s_{0}$ (right).

Because the remaining operations with the correlation function
$\Pi^{\mathrm{OPE}}(p)$ are standard ones, we omit the further details and
provide only the final results for the mass $m_{\mathrm{S}}$ and coupling
$f_{\mathrm{S}}$:

$\displaystyle m_{\mathrm{S}}$ $\displaystyle=$ $\displaystyle(4128\pm
142)~{}\mathrm{MeV},$ $\displaystyle f_{\mathrm{S}}$ $\displaystyle=$
$\displaystyle(3.84\pm 0.34)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (24)

Let us note that the values of $m_{\mathrm{S}}$ and $f_{\mathrm{S}}$ are found
utilizing for the parameters $M^{2}$ and $s_{0}$ working regions

$M^{2}\in[3,4.5]~{}\mathrm{GeV}^{2},\ s_{0}\in[21.5,22.5]~{}\mathrm{GeV}^{2}.$
(25)

In these regions the pole contribution varies within boundaries

$0.69\geq\mathrm{PC}\geq 0.51,$ (26)

and $R(3~{}\mathrm{GeV}^{2})<0.01$. The behavior of $m_{\mathrm{S}}$ as a
function of $M^{2}$ and $s_{0}$ is depicted in Fig. 2.

Similar analysis for the tetraquark $\widetilde{T}_{\mathrm{S}}$ gives

$\displaystyle\widetilde{\Pi}^{\mathrm{OPE}}(p)=i\int
d^{4}xe^{ipx}\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{s}^{b^{\prime}b}(-x)\right.$
$\displaystyle\left.\times\gamma_{5}S_{u}^{a^{\prime}a}(-x)\right]\left\\{\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{c}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\right]\right.$
(27)
$\displaystyle\left.+\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{c}^{ba^{\prime}}(x)\gamma_{5}S_{c}^{ab^{\prime}}(x)\right]\right\\}.$
(28)

The spectroscopic parameters of $\widetilde{T}_{\mathrm{S}}$ are equal to

$\displaystyle\widetilde{m}_{\mathrm{S}}$ $\displaystyle=$
$\displaystyle(4035\pm 145)~{}\mathrm{MeV},$
$\displaystyle\widetilde{f}_{\mathrm{S}}$ $\displaystyle=$
$\displaystyle(7.7\pm 1.2)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (29)

To extract $\widetilde{m}_{\mathrm{S}}$ and $\widetilde{f}_{\mathrm{S}}$ , we
used the windows Eq. (16) for the parameters $M^{2}$ and $s_{0}$, which
satisfy all of the sum rule constraints in this case as well.

##  III Width of the tetraquark $T_{\mathrm{AV}}$

In the previous section, we have calculated the masses and couplings of the
tetraquarks $T_{\mathrm{AV}}$, $T_{\mathrm{S}}$ and
$\widetilde{T}_{\mathrm{S}}$. This information forms a basis to determine the
kinematically allowed decay channels of these particles. In the case of the
tetraquark $T_{\mathrm{AV}}$ the channels $T_{\mathrm{AV}}\rightarrow
D^{0}D_{s}^{\ast+}$ and $T_{\mathrm{AV}}\rightarrow
D^{\ast}(2007)^{0}D_{s}^{+}$ are its $S$-wave modes, thresholds for which
equal to $\approx 3975~{}\mathrm{MeV}$ and $3977~{}\mathrm{MeV}$,
respectively. We calculate the full width of $T_{\mathrm{AV}}$ by taking into
account these two channels.

It is worth noting, that we use central value of the mass $m$. But sum rule
computations depend on auxiliary parameters $M^{2}$ and $s_{0}$, which
restrict precision of the method by generating uncertainties in extracted
physical observables. Thus, the maximal predicted value for $m$ is
$4138~{}\mathrm{MeV}$ which, however does not affect considerably our result
for its full widths. The reason is that even for such $m$ other strong decays
of $T_{\mathrm{AV}}$, for instance, $P$-wave modes $T_{\mathrm{AV}}\rightarrow
D^{0}D_{s0}^{\ast}(2317)^{+}$ or $T_{\mathrm{AV}}\rightarrow
D_{0}^{\ast}(2400)^{0}D_{s}^{+}$ are kinematically forbidden processes because
$4138~{}\mathrm{MeV}$ is below relevant two-meson thresholds.

At the $m$ less than the limit $3975~{}\mathrm{MeV}$ the axial-vector state
$T_{\mathrm{AV}}$ cannot decay to $D^{0}D_{s}^{\ast+}$ and
$D^{\ast}(2007)^{0}D_{s}^{+}$. The process $T_{\mathrm{AV}}\rightarrow
D^{0}D^{0}K^{+}$, analog of $T_{cc}^{+}\rightarrow D^{0}D^{0}\pi^{+}$, could
not also take place: A threshold for this decay $4224~{}\mathrm{MeV}$ is
significantly larger than $3975~{}\mathrm{MeV}$. Therefore, at
$m<3975~{}\mathrm{MeV}$ the tetraquark $T_{\mathrm{AV}}$ becomes a strong-
interaction stable particle, which is excluded by the experimental data of the
LHCb Collaboration and theoretical analyses of diquark masses. In fact, the
axial-vector states $T_{cc}^{+}=cc\overline{u}\overline{d}$ and
$T_{\mathrm{AV}}=cc\overline{u}\overline{s}$ contain the same heavy diquark
$c^{T}C\gamma_{\mu}c$ and ”good” light scalar antidiquarks
$\overline{u}\gamma_{5}C\overline{d}^{T}$ and
$\overline{u}\gamma_{5}C\overline{s}^{T}$ in $(\mathbf{3}_{f},\mathbf{3}_{c})$
flavor-color representations, respectively. The mass splitting $\delta$
between ”good” $J^{\mathrm{P}}=0^{+}$ diquarks $us$ and $ud$ was analyzed in
Ref. Burden:1996nh  and found equal to $145~{}\mathrm{MeV}$. This gap leads to
the mass of $T_{\mathrm{AV}}$ approximately around of $4020~{}\mathrm{MeV}$.
Our prediction for $m$, as well as for the $T_{\mathrm{AV}}-T_{cc}^{+}$ mass
splitting $\Delta=120~{}\mathrm{MeV}$ are close to these estimates. Thus, the
experimental-theoretical analyses rule out the small mass region for
$T_{\mathrm{AV}}$, and confirm its unstability against strong decays to
conventional open-charmed mesons.

###  III.1 Decay $T_{\mathrm{AV}}\rightarrow D^{0}D_{s}^{\ast+}$

Here, we consider in a detailed form the process $T_{\mathrm{AV}}\rightarrow
D^{0}D_{s}^{\ast+}$. The partial width of this decay contains various physical
parameters of the initial and final particles, such as their masses and decay
constants. These parameters are known from other sources or have been computed
in the present article.

The partial width depends also on the strong interaction’s coupling $g_{1}$ of
the corresponding tetraquark and mesons at the vertex
$T_{\mathrm{AV}}D^{0}D_{s}^{\ast+}$. It is convenient to determine $g_{1}$ by
means of the QCD three-point sum rule method. To this end, we analyze the
correlation function

$\displaystyle\Pi_{\mu\nu}(p,p^{\prime})=i^{2}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\langle
0|\mathcal{T}\\{J_{\nu}^{D_{s}^{\ast}}(y)$ $\displaystyle\times
J^{D^{0}}(0)J_{\mu}^{\dagger}(x)\\}|0\rangle,$ (30)

where $J_{\mu}(x)$,$\ J_{\nu}^{D_{s}^{\ast}}(y)$ and $J^{D^{0}}(0)$ are the
interpolating currents for the tetraquark $T_{\mathrm{AV}}$, the vector
$D_{s}^{\ast+}$ and pseudoscalar $D^{0}$ mesons, respectively. The four-
momenta of $T_{\mathrm{AV}}$ and $D_{s}^{\ast+}$ are denoted by $p$ and
$p^{\prime}$, then momentum of the meson $D^{0}$ is equal to $q=p-p^{\prime}$.

The $J_{\mu}(x)$ is determined by Eq. (1), whereas for currents of two final-
state mesons, we use

$\displaystyle\ J_{\nu}^{D_{s}^{\ast}}(x)$ $\displaystyle=$
$\displaystyle\overline{s}_{j}(x)\gamma_{\nu}c_{j}(x),$ $\displaystyle
J^{D^{0}}(x)$ $\displaystyle=$
$\displaystyle\overline{u}_{i}(x)i\gamma_{5}c_{i}(x),$ (31)

where $i$ and $j$ are color indices.

To continue our study in the framework of the sum rule method, we calculate
the correlation function $\Pi_{\mu\nu}(p,p^{\prime})$ by employing the
physical parameters of the particles involved into this process. For the
physical side of the sum rule, $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$,
we obtain

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=\frac{\langle
0|J_{\nu}^{D^{\ast}}|D_{s}^{\ast+}(p^{\prime},\varepsilon^{\prime})\rangle\langle
0|J^{D^{0}}|D^{0}(q)\rangle}{(p^{\prime
2}-m_{D_{s}^{\ast}}^{2})(q^{2}-m_{D}^{2})}$ $\displaystyle\times\frac{\langle
D^{0}(q)D_{s}^{\ast+}(p^{\prime},\varepsilon^{\prime})|T_{\mathrm{AV}}(p,\epsilon)\rangle\langle
T_{\mathrm{AV}}(p,\epsilon)|J_{\mu}^{\dagger}|0\rangle}{(p^{2}-m^{2})}+\cdots,$
(32)

where $m_{D_{s}^{\ast}}$ and $m_{D}$ are the masses of the mesons
$D_{s}^{\ast+}$ and $D^{0}$, respectively. To derive this expression, we
separate the contributions of the ground-state particles from the effects of
the higher resonances and continuum states in Eq. (30). Hence, in Eq. (32),
the ground-state term is written down explicitly, whereas ellipses stand for
the other contributions.

The function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$ can be simplified by
introducing the $D_{s}^{\ast+}$ and $D^{0}$ mesons’ matrix elements

$\displaystyle\langle
0|J_{\nu}^{D^{\ast}}|D_{s}^{\ast+}(p^{\prime},\varepsilon^{\prime})\rangle$
$\displaystyle=$ $\displaystyle
m_{D_{s}^{\ast}}f_{D_{s}^{\ast}}\varepsilon_{\nu}^{\prime},\ $
$\displaystyle\langle 0|J^{D^{0}}|D^{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{m_{D}^{2}f_{D}}{m_{c}},$ (33)

with $f_{D_{s}^{\ast}}$ and $f_{D}$ being their decay constants. Here,
$\varepsilon_{\nu}^{\prime}$ is the polarization vector of the meson
$D_{s}^{\ast+}$.

The matrix element of the vertex $T_{\mathrm{AV}}D^{0}D_{s}^{\ast+}$ can be
modeled in the form

$\displaystyle\langle
D^{0}(q)D_{s}^{\ast+}(p^{\prime},\varepsilon^{\prime})|T_{\mathrm{AV}}(p,\epsilon)\rangle=g_{1}(q^{2})\left[\left(p\cdot
p^{\prime}\right)\left(\epsilon\cdot\varepsilon^{\prime\ast}\right)\right.$
$\displaystyle\left.-\left(p^{\prime}\cdot\epsilon\right)\left(p\cdot\varepsilon^{\prime\ast}\right)\right],$
(34)

where we denote the strong coupling at $T_{\mathrm{AV}}D^{0}D_{s}^{\ast+}$ by
$g_{1}(q^{2})$. Then, one can easily find that

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=g_{1}(q^{2})\frac{m_{D_{s}^{\ast}}f_{D_{s}^{\ast}}m_{D}^{2}f_{D}fm}{m_{c}(p^{2}-m^{2})(q^{2}-m_{D}^{2})}$
$\displaystyle\times\frac{1}{(p^{\prime
2}-m_{D_{s}^{\ast}}^{2})}\left(\frac{m^{2}+m_{D_{s}^{\ast}}^{2}-q^{2}}{2}g_{\mu\nu}-p_{\nu}p_{\mu}^{\prime}\right)+\cdots.$
(35)

The double Borel transformation of the correlation function over the variables
$-p^{2}$ and $-p^{\prime 2}$ is determined by the expression

$\displaystyle\mathcal{B}\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=g_{1}(q^{2})\frac{m_{D_{s}^{\ast}}f_{D_{s}^{\ast}}m_{D}^{2}f_{D}fm}{m_{c}(q^{2}-m_{D}^{2})}e^{-m^{2}/M_{1}^{2}}$
$\displaystyle\times
e^{-m_{D_{s}^{\ast}}^{2}/M_{2}^{2}}\left(\frac{m^{2}+m_{D_{s}^{\ast}}^{2}-q^{2}}{2}g_{\mu\nu}-p_{\mu}p_{\nu}^{\prime}\right)+\cdots.$

The function $\mathcal{B}\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$ has two
Lorentz structures which are proportional to $g_{\mu\nu}$ and
$p_{\nu}p_{\mu}^{\prime}$. In our analysis, we work with the Borel
transformation of the invariant amplitude $\Pi^{\mathrm{Phys}}(p^{2},p^{\prime
2},q^{2})$ which corresponds to the structure $\sim g_{\mu\nu}$.

To derive the QCD side of the three-point sum rule, we calculate
$\Pi_{\mu\nu}(p,p^{\prime})$ using the quark propagators, and obtain

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})=i^{3}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}$
$\displaystyle\times\left\\{\mathrm{Tr}\left[\gamma_{\nu}S_{c}^{ja}(y-x)\gamma_{\mu}\widetilde{S}_{c}^{ib}(-x)\gamma_{5}\widetilde{S}_{u}^{ai}(x)\gamma_{5}\right.\right.$
$\displaystyle\left.\times
S_{s}^{bj}(x-y)\right]-\mathrm{Tr}\left[\gamma_{\nu}S_{c}^{jb}(y-x)\gamma_{\mu}\widetilde{S}_{c}^{ia}(-x)\right.$
$\displaystyle\left.\left.\times\gamma_{5}\widetilde{S}_{u}^{ai}(x)\gamma_{5}S_{s}^{bj}(x-y)\right]\right\\}.$
(37)

The correlation function $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})$ is
computed by taking into account the nonperturbative contributions up to
dimension $6$. It contains the same Lorentz structures as
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$. Let us denote by
$\Pi^{\mathrm{OPE}}(p^{2},p^{\prime 2},q^{2})$ the invariant amplitude that
corresponds to the term proportional to $g_{\mu\nu}$ in
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})$. The double Borel transformation,
$\mathcal{B}\Pi^{\mathrm{OPE}}(p^{2},p^{\prime 2},q^{2})$, establishes the
second component of the sum rule. Having equated
$\mathcal{B}\Pi^{\mathrm{OPE}}(p^{2},p^{\prime 2},q^{2})$ and
$\mathcal{B}\Pi^{\mathrm{Phys}}(p^{2},p^{\prime 2},q^{2})$, and carried out
the continuum subtraction, we derive the sum rule for the strong coupling
$g_{1}(q^{2})$.

The Borel transformed and subtracted amplitude
$\Pi^{\mathrm{OPE}}(p^{2},p^{\prime 2},q^{2})$ can be expressed by means of
the spectral density $\widetilde{\rho}(s,s^{\prime},q^{2})$ which is
proportional to the relevant imaginary part of the
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})$

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

The Borel and continuum threshold parameters are denoted in Eq. (38) by
$\mathbf{M}^{2}=(M_{1}^{2},\ M_{2}^{2})$ and $\mathbf{s}_{0}=(s_{0},\
s_{0}^{\prime})$, respectively. Then, the sum rule for $g_{1}(q^{2})$ reads

$\displaystyle
g_{1}(q^{2})=\frac{2m_{c}}{m_{D_{s}^{\ast}}f_{D_{s}^{\ast}}m_{D}^{2}f_{D}fm}\frac{q^{2}-m_{D}^{2}}{m^{2}+m_{D_{s}^{\ast}}^{2}-q^{2}}$
$\displaystyle\times
e^{m^{2}/M_{1}^{2}}e^{m_{D_{s}^{\ast}}^{2}/M_{2}^{2}}\Pi(\mathbf{M}^{2},\mathbf{s}_{0},q^{2}).$
(39)

The coupling $g_{1}(q^{2})$ is also a function of the Borel and continuum
threshold parameters which for simplicity are omitted in Eq. (39). We also
introduce a new variable $Q^{2}=-q^{2}$ and denote the obtained function
$g_{1}(Q^{2})$.

It is seen that Eq. (39) contains the mass and coupling of the tetraquark
$T_{\mathrm{AV}}$ as well as the masses and decay constants of the mesons
$D^{0}$ and $D_{s}^{\ast+}$. The relevant input parameters are collected in
Table 1, which contains also the masses and decay constants of the mesons
appearing at final stages of other processes. The masses of all mesons and
decay constants $f_{D}$ and $f_{D_{s}}$ are borrowed from Ref. PDG:2022 . For
decay constants of the mesons $D_{s}^{\ast+}$ and $D^{\ast}(2007)^{0}$, we use
predictions obtained in the context of the QCD lattice method Lubicz:2016bbi .

Besides, for numerical analysis, we should determine working regions for
parameters $\mathbf{M}^{2}$ and $\mathbf{s}_{0}$. The restrictions imposed on
$\mathbf{M}^{2}$ and $\mathbf{s}_{0}$ are standard for sum rule computations
and have been discussed above. The windows for $M_{1}^{2}$ and $s_{0}$
correspond to the $T_{\mathrm{AV}}$ channel and coincide with regions from Eq.
(16). The second pair of parameters $(M_{2}^{2},\ s_{0}^{\prime})$ for the
$D_{s}^{\ast+}$ channel are fixed within boundaries:

$M_{2}^{2}\in[2.5,3.5]~{}\mathrm{GeV}^{2},\
s_{0}^{\prime}\in[6,8]~{}\mathrm{GeV}^{2}.$ (40)

The regions for the Borel and continuum subtraction parameters are chosen in
such a way that to minimize also dependence of $g_{1}(Q^{2})$ on them.

The width of the decay $T_{\mathrm{AV}}\rightarrow D^{0}D_{s}^{\ast+}$ has to
be calculated by means of the strong coupling at the $D^{0}$ meson’s mass
shell $q^{2}=m_{D}^{2}$, which is not calculable by the sum rule method. To
avoid this obstacle, we use a fit function $\mathcal{F}_{i}(Q^{2})$ which at
the momenta $Q^{2}>0$ coincides with results of the sum rule analyses, but can
be extended to the $Q^{2}<0$ domain to find $g_{1}(-m_{D}^{2})$. For these
purposes, we employ the function $\mathcal{F}_{i}(Q^{2})$

$\mathcal{F}_{i}(Q^{2})=\mathcal{F}_{i}^{0}\mathrm{\exp}\left[c_{i}^{1}\frac{Q^{2}}{m^{2}}+c_{i}^{2}\left(\frac{Q^{2}}{m^{2}}\right)^{2}\right],$
(41)

where $\mathcal{F}_{i}^{0}$, $c_{i}^{1}$ and $c_{i}^{2}$ are fitting
parameters. Numerical computations show that
$\mathcal{F}_{1}^{0}=7.86~{}\mathrm{GeV}^{-1}$, $c_{1}^{1}=7.33$, and
$c_{1}^{2}=-4.94$ lead to nice agreement with the sum rule’s data (see, Fig.
3).

At the mass shell, $q^{2}=m_{D}^{2}$, this function predicts

$g_{1}\equiv\mathcal{F}_{1}(-m_{D}^{2})=(1.26\pm 0.14)~{}\mathrm{GeV}^{-1}.$
(42)

The width of the process $T_{\mathrm{AV}}\rightarrow D^{0}D_{s}^{\ast+}$ is
determined by the following formula:

$\Gamma\left[T_{\mathrm{AV}}\rightarrow
D^{0}D_{s}^{\ast+}\right]=\frac{g_{1}^{2}m_{D_{s}^{\ast}}^{2}\lambda}{24\pi}\left(3+\frac{2\lambda^{2}}{m_{D_{s}^{\ast}}^{2}}\right),$
(43)

where $\lambda=\lambda\left(m,m_{D_{s}^{\ast}},m_{D}\right)$ and

$\displaystyle\lambda\left(a,b,c\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2a}\left[a^{4}+b^{4}+c^{4}\right.$ (44)
$\displaystyle\left.-2\left(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2}\right)\right]^{1/2}.$

Employing the coupling $g_{1}$ from Eq. (42), it is easy to find the partial
width of the process $T_{\mathrm{AV}}\rightarrow D^{0}D_{s}^{\ast+}$

$\Gamma\left[T_{\mathrm{AV}}\rightarrow D^{0}D_{s}^{\ast+}\right]=(53\pm
12)~{}\mathrm{MeV}.$ (45)

Figure 3: The sum rule predictions and fit function for the strong coupling
$g_{1}(Q^{2})$. The red diamond shows the point $Q^{2}=-m_{D}^{2}$.

###  III.2 Process $T_{\mathrm{AV}}\rightarrow D^{\ast}(2007)^{0}D_{s}^{+}$

The second decay $T_{\mathrm{AV}}\rightarrow D^{\ast}(2007)^{0}D_{s}^{+}$ can
be explored by the same manner. The correlation function which should be
considered in this case is

$\displaystyle\widetilde{\Pi}_{\mu\nu}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle i^{2}\int d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\langle
0|\mathcal{T}\\{J^{D_{s}}(y)$ (46) $\displaystyle\times J_{\nu}^{D^{\ast
0}}(0)J_{\mu}^{\dagger}(x)\\}|0\rangle,$

where $\ J_{\nu}^{D^{\ast 0}}(0)$ and $J^{D_{s}}(y)$ are the interpolating
currents of the mesons $D^{\ast}(2007)^{0}$ and $D_{s}^{+}$, respectively.
These currents are determined by the formulas

$\displaystyle J_{\nu}^{D^{\ast 0}}(x)$ $\displaystyle=$
$\displaystyle\overline{u}_{i}(x)\gamma_{\nu}c_{i}(x),$ $\displaystyle
J^{D_{s}}(x)$ $\displaystyle=$
$\displaystyle\overline{s}_{j}(x)i\gamma_{5}c_{j}(x).$ (47)

To find the physical side of the sum rule, we use recipes described above and
get

$\displaystyle\widetilde{\Pi}_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=g_{2}(q^{2})\frac{m_{D^{\ast
0}}f_{D^{\ast
0}}m_{D_{s}}^{2}f_{D_{s}}fm}{(m_{c}+m_{s})(p^{2}-m^{2})(q^{2}-m_{D^{\ast
0}}^{2})}$ $\displaystyle\times\frac{1}{(p^{\prime
2}-m_{D_{s}}^{2})}\left(\frac{m^{2}-m_{D_{s}}^{2}+q^{2}}{2}g_{\mu\nu}-q_{\mu}p_{\nu}\right)+\cdots,$
(48)

where $m_{D^{\ast 0}}$, $m_{D_{s}}$ and $f_{D^{\ast 0}}$, $f_{D_{s}}$ are the
masses and decay constants of the mesons $D^{\ast}(2007)^{0}$ and $D_{s}^{+}$,
respectively.

In deriving of Eq. (48), we have used the following matrix elements:

$\displaystyle\langle 0|J_{\nu}^{D^{\ast 0}}|D^{\ast
0}(q,\varepsilon^{\prime})\rangle=m_{D^{\ast 0}}f_{D^{\ast
0}}\varepsilon_{\nu}^{\prime},\ $ $\displaystyle\langle
0|J^{D_{s}}|D_{s}^{+}\rangle=\frac{m_{D_{s}}^{2}f_{D_{s}}}{m_{c}+m_{s}},$
$\displaystyle\langle D^{\ast
0}(q,\varepsilon^{\prime})D_{s}^{+}(p^{\prime})|T_{\mathrm{AV}}(p,\epsilon)\rangle=g_{2}(q^{2})\left[\left(p\cdot
q\right)\left(\epsilon\cdot\varepsilon^{\prime\ast}\right)\right.$
$\displaystyle\left.-\left(q\cdot\epsilon\right)\left(p\cdot\varepsilon^{\prime\ast}\right)\right].$
(49)

Here $g_{2}$ is the strong coupling, which corresponds to the vertex
$T_{\mathrm{AV}}D^{\ast}(2007)^{0}D_{s}^{+}$ and is defined at the mass shell
of the $D^{\ast}(2007)^{0}$ meson,

$g_{2}=\mathcal{F}_{2}(-m_{D^{\ast 0}}^{2}).$ (50)

The fit function $\mathcal{F}_{2}(Q^{2})$ is given by Eq. (41) with the
parameters: $\mathcal{F}_{2}^{0}=5.36~{}\mathrm{GeV}^{-1}$, $c_{2}^{1}=3.68$,
and $c_{2}^{2}=-15.73$. The partial width of this decay can be calculated by
means of the formula in Eq. (43) with evident replacements $g_{1}\rightarrow
g_{2}$, $m_{D_{s}^{\ast}}^{2}\rightarrow m_{D^{\ast 0}}^{2}$ and
$\lambda\left(m,m_{D_{s}^{\ast}},m_{D}\right)\rightarrow\lambda\left(m,m_{D^{\ast
0}},m_{D_{s}}\right)$.

In the sum rule computations, we use Eq. (25) and
$M^{2}_{2}\in[2.5,3.5]~{}\mathrm{GeV}^{2}$,
$s^{\prime}_{0}\in[5,7]~{}\mathrm{GeV}^{2}$. As a result, we find

$g_{2}\equiv\mathcal{F}_{2}(-m_{D^{\ast 0}}^{2})=(7.8\pm 0.9)\cdot 10^{-1}\
\mathrm{GeV}^{-1}.$ (51)

Then, the width of the process $T_{\mathrm{AV}}\rightarrow
D^{\ast}(2007)^{0}D_{s}^{+}$ is equal to

$\Gamma\left[T_{\mathrm{AV}}\rightarrow
D^{\ast}(2007)^{0}D_{s}^{+}\right]=(19\pm 5)~{}\mathrm{MeV}.$ (52)

For the full width of the exotic axial-vector meson $T_{\mathrm{AV}}$, we get

$\Gamma_{\mathrm{AV}}=(72\pm 13)~{}\mathrm{MeV}.$ (53)

Quantity | Value (in $\mathrm{MeV}$ units)
---|---
$m_{D_{s}^{\ast}}$ | $2112.2\pm 0.4$
$m_{D}$ | $1864.84\pm 0.05$
$m_{D_{s}}$ | $1969.0\pm 1.4$
$m_{D^{\ast 0}}$ | $2006.85\pm 0.05$
$f_{D_{s}^{\ast}}$ | $268.8\pm 6.5$
$f_{D}$ | $212.0\pm 0.7$
$f_{D_{s}}$ | $249.9\pm 0.5$
$f_{D^{\ast 0}}$ | $223.5\pm 8.4$
Table 1: Masses and decay constants of the mesons $D_{s}^{\ast+}$, $D^{0}$,
$D_{s}^{+}$, and $D^{\ast}(2007)^{0}$, which have been used in numerical
computations.

This result is rather stable prediction for $\Gamma_{\mathrm{AV}}$, because in
the region $3995~{}\mathrm{MeV}\leq m\leq 4138~{}\mathrm{MeV}$ there are not
other allowed decays of $T_{\mathrm{AV}}$, and its full width may undergone
only small variations due to changes of $m$ and $\lambda$ in Eq. (43).

##  IV Full widths of $T_{\mathrm{S}}$ and $\widetilde{T}_{\mathrm{S}}$

In the case of the scalar tetraquark $T_{\mathrm{S}}$, the $S$-wave decay
channels which contribute to its full width are the processes
$T_{\mathrm{S}}\rightarrow D^{0}D_{s}^{+}$ and $T_{\mathrm{S}}\rightarrow
D^{\ast}(2007)^{0}D_{s}^{\ast+}$. The two-meson thresholds for these decays
$3833~{}\mathrm{MeV}$ and $4119~{}\mathrm{MeV}$ make them dominant channels of
the tetraquark $T_{\mathrm{S}}$. There are other channels via of which the
scalar four-quark state $cc\overline{u}\overline{s}$ may transform to
conventional charmed mesons. For example, decays to meson pairs
$D_{0}^{\ast}(2400)^{0}D_{s0}^{\ast}(2317)^{+}$ and
$D^{\ast}(2007)^{0}D_{s0}^{\ast}(2317)^{+}$ are possible $S$\- and $P$-wave
modes of $cc\overline{u}\overline{s}$, respectively. But thresholds for
production of these and other pairs exceed considerably the maximal value
$m_{\mathrm{S}}=4270~{}\mathrm{MeV}$ predicted for $T_{\mathrm{S}}$. In the
case of $\widetilde{T}_{\mathrm{S}}$, we explore the decay
$\widetilde{T}_{\mathrm{S}}\rightarrow D^{0}D_{s}^{+}$ allowed by its mass
$\widetilde{m}_{\mathrm{S}}=4035~{}\mathrm{MeV}$.

###  IV.1 Decay modes $T_{\mathrm{S}}\rightarrow D^{0}D_{s}^{+}$ and
$T_{\mathrm{S}}\rightarrow D^{\ast}(2007)^{0}D_{s}^{\ast+}$

The treatment of the decays $T_{\mathrm{S}}\rightarrow D^{0}D_{s}^{+}$ and
$T_{\mathrm{S}}\rightarrow D^{\ast}(2007)^{0}D_{s}^{\ast+}$ in the context of
the three-point sum rule approach does differ from our analysis made in the
previous section. Here, one should find the strong couplings $G_{1}$ and
$G_{2}$ at the relevant vertices. For the coupling $G_{1}$ the correlation
function of interest is

$\displaystyle\Pi(p,p^{\prime})=i^{2}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\langle 0|\mathcal{T}\\{J^{D_{s}}(y)$
$\displaystyle\times J^{D^{0}}(0)J^{\dagger}(x)\\}|0\rangle,$ (54)

where all interpolating currents have been defined above. Thus, the current
$J(x)$ of the scalar tetraquark $T_{\mathrm{S}}$ has been introduced in Eq.
(2), whereas $J^{D^{0}}(x)$ and $J^{D_{s}}(x)$ have been determined by Eqs.
(31) and (47), respectively.

The physical side of the sum rule has the form

$\displaystyle\Pi^{\mathrm{Phys}}(p,p^{\prime})=\frac{\langle
0|J^{D_{s}}|D_{s}^{+}(p^{\prime})\rangle\langle
0|J^{D^{0}}|D^{0}(q)\rangle}{(p^{\prime 2}-m_{D_{s}}^{2})(q^{2}-m_{D}^{2})}$
$\displaystyle\times\frac{\langle
D^{0}(q)D_{s}^{+}(p^{\prime})|T_{\mathrm{S}}(p)\rangle\langle
T_{\mathrm{S}}(p)|J^{\dagger}|0\rangle}{(p^{2}-m_{\mathrm{S}}^{2})}+\cdots.$
(55)

The matrix elements of the mesons $D^{0}$ and $D_{s}^{+}$ have been defined by
Eqs. (33) and (49), respectively. We model the matrix element $\langle
D^{0}(q)D_{s}^{+}(p^{\prime})|T_{\mathrm{S}}(p)\rangle$ as

$\langle
D^{0}(q)D_{s}^{+}(p^{\prime})|T_{\mathrm{S}}(p)\rangle=G_{1}(q^{2})p\cdot
p^{\prime}.$ (56)

These matrix elements allow us to rewrite $\Pi^{\mathrm{Phys}}(p,p^{\prime})$
in a simplified form,

$\displaystyle\Pi^{\mathrm{Phys}}(p,p^{\prime})=G_{1}(q^{2})\frac{m_{D}^{2}f_{D}m_{D_{s}}^{2}f_{D_{s}}f_{\mathrm{S}}m_{\mathrm{S}}}{m_{c}(m_{c}+m_{s})(p^{2}-m_{\mathrm{S}}^{2})}$
$\displaystyle\times\frac{m_{\mathrm{S}}^{2}+m_{D_{s}}^{2}-q^{2}}{2(q^{2}-m_{D}^{2})(p^{\prime
2}-m_{D_{s}}^{2})}+\cdots.$ (57)

As is seen, the correlation function $\Pi^{\mathrm{Phys}}(p,p^{\prime})$ has a
simple Lorentz structure proportional to $\mathrm{I}$, therefore we employ the
whole expression in Eq. (57) as an invariant amplitude to derive the sum rule
for $G_{1}(q^{2})$.

After some calculations, we find also the QCD side of the required sum rule

$\displaystyle\Pi^{\mathrm{OPE}}(p,p^{\prime})=\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}$
$\displaystyle\times\left\\{\mathrm{Tr}\left[\gamma_{5}S_{c}^{ja}(y-x)\gamma_{\mu}\widetilde{S}_{c}^{ib}(-x)\gamma_{5}\widetilde{S}_{u}^{ai}(x)\gamma^{\mu}\right.\right.$
$\displaystyle\left.\times
S_{s}^{bj}(x-y)\right]-\mathrm{Tr}\left[\gamma_{5}S_{c}^{jb}(y-x)\gamma_{\mu}\widetilde{S}_{c}^{ia}(-x)\right.$
$\displaystyle\left.\left.\times\gamma_{5}\widetilde{S}_{u}^{ai}(x)\gamma^{\mu}S_{s}^{bj}(x-y)\right]\right\\}.$
(58)

Then the sum rule for the coupling $G_{1}(q^{2})$ reads

$\displaystyle
G_{1}(q^{2})=\frac{2m_{c}(m_{c}+m_{s})}{m_{D}^{2}f_{D}m_{D_{s}}^{2}f_{D_{s}}f_{\mathrm{S}}m_{\mathrm{S}}}\frac{q^{2}-m_{D}^{2}}{m_{\mathrm{S}}^{2}+m_{D_{s}}^{2}-q^{2}}$
$\displaystyle\times
e^{m_{\mathrm{S}}^{2}/M_{1}^{2}}e^{m_{D_{s}}^{2}/M_{2}^{2}}\Pi_{\mathrm{scal}}(\mathbf{M}^{2},\mathbf{s}_{0},q^{2}),$
(59)

where $\Pi_{\mathrm{scal}}(\mathbf{M}^{2},\mathbf{s}_{0},q^{2})$ is the
correlation function $\Pi^{\mathrm{OPE}}(p,p^{\prime})$ after the Borel
transformation and subtraction operations. It is calculated by taking into
account the nonperturbative terms up to dimension $6$, as two similar
functions in the previous section.

Computations carried out in accordance with a scheme described above, give the
following predictions

$G_{1}=\mathcal{F}_{3}(-m_{D}^{2})=(1.01\pm 0.21)\ \mathrm{GeV}^{-1},$ (60)

where parameters of the function $\mathcal{F}_{3}(Q^{2})$ are:
$\mathcal{F}_{3}^{0}=4.10~{}\mathrm{GeV}^{-1}$, $c_{3}^{1}=4.86$, and
$c_{3}^{2}=-9.90$. It is worth nothing that the fit function is given by Eq.
(41) with substitution $m\rightarrow m_{S}$. In the sum rule computations for
the $T_{S}$ channel, we employ the parameters from Eq. (25).

The partial width of this decay is determined by the expression

$\Gamma\left[T_{\mathrm{S}}\rightarrow
D^{0}D_{s}^{+}\right]=\frac{G_{1}^{2}m_{D_{s}}^{2}\lambda}{8\pi}\left(1+\frac{\lambda^{2}}{m_{D_{s}}^{2}}\right),$
(61)

with $\lambda$ being equal to
$\lambda\left(m_{\mathrm{S}},m_{D_{s}},m_{D}\right)$. Numerical computations
yield

$\Gamma\left[T_{\mathrm{S}}\rightarrow D^{0}D_{s}^{+}\right]=(138\pm
41)~{}\mathrm{MeV}.$ (62)

To determine the coupling $G_{2}$ and study the decay
$T_{\mathrm{S}}\rightarrow D^{\ast}(2007)^{0}D_{s}^{\ast+}$, we analyze the
correlation function

$\displaystyle\widehat{\Pi}_{\mu\nu}(p,p^{\prime})=i^{2}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\langle
0|\mathcal{T}\\{J_{\mu}^{D_{s}^{\ast}}(y)$ $\displaystyle\times
J_{\nu}^{D^{\ast 0}}(0)J^{\dagger}(x)\\}|0\rangle,$ (63)

with $J_{\mu}^{D_{s}^{\ast}}(y)$ and $J_{\nu}^{D^{\ast 0}}(0)$ being the
interpolating currents of the vector mesons $D_{s}^{\ast+}$ and
$D^{\ast}(2007)^{0}$ given by Eqs. (31) and (47), respectively.

The function $\widehat{\Pi}_{\mu\nu}(p,p^{\prime})$ in terms of the physical
parameters of the tetraquark $T_{\mathrm{S}}$ and mesons $D_{s}^{\ast+}$ and
$D^{\ast}(2007)^{0}$ is equal to expression

$\displaystyle\widehat{\Pi}_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=G_{2}(q^{2})\frac{m_{D_{s}^{\ast}}f_{D_{s}^{\ast}}m_{D^{\ast
0}}f_{D^{\ast
0}}f_{\mathrm{S}}m_{\mathrm{S}}}{(p^{2}-m_{\mathrm{S}}^{2})(p^{\prime
2}-m_{D_{s}^{\ast}}^{2})}$ $\displaystyle\times\frac{1}{(q^{2}-m_{D^{\ast
0}}^{2})}\left(\frac{m_{\mathrm{S}}^{2}-m_{D_{s}^{\ast}}^{2}-q^{2}}{2}g_{\mu\nu}-q_{\mu}p_{\nu}^{\prime}\right)+\cdots,$

where the vertex $\langle
D^{\ast}(2007)^{0}(q,\varepsilon)D_{s}^{\ast+}(p^{\prime},\varepsilon^{\prime})|T_{\mathrm{S}}(p)\rangle$
is modeled in the form

$\displaystyle\langle
D^{\ast}(2007)^{0}(q,\varepsilon)D_{s}^{\ast+}(p^{\prime},\varepsilon^{\prime})|T_{\mathrm{S}}(p)\rangle=G_{2}(q^{2})$
$\displaystyle\times\left[\left(p^{\prime}\cdot
q\right)\left(\varepsilon^{\ast}\cdot\varepsilon^{\prime\ast}\right)-\left(q\cdot\varepsilon^{\prime\ast}\right)\left(p^{\prime}\cdot\varepsilon^{\ast}\right)\right].$
(65)

In terms of the quark-gluon degrees of freedom,
$\widehat{\Pi}_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})$ is given by the formula

$\displaystyle\widehat{\Pi}_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})=\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\left\\{\mathrm{Tr}\left[\gamma_{\mu}S_{c}^{ia}(y-x)\right.\right.$
$\displaystyle\left.\times\gamma_{\alpha}\widetilde{S}_{c}^{jb}(-x)\gamma_{\nu}\widetilde{S}_{s}^{bj}(x)\gamma^{\alpha}S_{u}^{ai}(x-y)\right]$
$\displaystyle\left.-\mathrm{Tr}\left[\gamma_{\mu}S_{c}^{ib}(y-x)\gamma_{\alpha}\widetilde{S}_{c}^{ja}(-x)\gamma_{\nu}\widetilde{S}_{s}^{bj}(x)\gamma^{\alpha}S_{u}^{ai}(x-y)\right]\right\\}.$
(66)

Omitting details of rather standard computations, let us write down the final
results:

$G_{2}=\mathcal{F}_{4}(-m_{D^{\ast 0}}^{2})=(2.11\pm 0.43)\
\mathrm{GeV}^{-1},$ (67)

where parameters of the function $\mathcal{F}_{4}(Q^{2})$ are:
$\mathcal{F}_{4}^{0}=3.17~{}\mathrm{GeV}^{-1}$, $c_{4}^{1}=1.14$, and
$c_{4}^{2}=-2.51$.

The partial width of this decay is equal to

$\Gamma\left[T_{\mathrm{S}}\rightarrow
D^{\ast}(2007)^{0}D_{s}^{\ast+}\right]=(75\pm 22)~{}\mathrm{MeV}.$ (68)

Using Eqs. (62) and (68), we find

$\Gamma_{\mathrm{S}}=(213\pm 47)~{}\mathrm{MeV}.$ (69)

###  IV.2  Decay $\widetilde{T}_{\mathrm{S}}\rightarrow D^{0}D_{s}^{+}$

For the decay $\widetilde{T}_{\mathrm{S}}\rightarrow D^{0}D_{s}^{+}$ one
should employ the following correlation function

$\displaystyle\widetilde{\Pi}^{\mathrm{OPE}}(p,p^{\prime})=-\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}$
$\displaystyle\times\left\\{\mathrm{Tr}\left[\gamma_{5}S_{c}^{ja}(y-x)\gamma_{5}\widetilde{S}_{c}^{ib}(-x)\gamma_{5}\widetilde{S}_{u}^{ai}(x)\gamma_{5}\right.\right.$
$\displaystyle\left.\times
S_{s}^{bj}(x-y)\right]+\mathrm{Tr}\left[\gamma_{5}S_{c}^{jb}(y-x)\gamma_{5}\widetilde{S}_{c}^{ia}(-x)\right.$
$\displaystyle\left.\left.\times\gamma_{5}\widetilde{S}_{u}^{ai}(x)\gamma_{5}S_{s}^{bj}(x-y)\right]\right\\}.$
(70)

Then $\widetilde{G}(q^{2})$ can be extracted from the sum rule

$\displaystyle\widetilde{G}(q^{2})=\frac{2m_{c}(m_{c}+m_{s})}{m_{D}^{2}f_{D}m_{D_{s}}^{2}f_{D_{s}}\widetilde{f}_{\mathrm{S}}\widetilde{m}_{\mathrm{S}}}\frac{q^{2}-m_{D}^{2}}{\widetilde{m}_{\mathrm{S}}^{2}+m_{D_{s}}^{2}-q^{2}}$
$\displaystyle\times
e^{\widetilde{m}_{\mathrm{S}}^{2}/M_{1}^{2}}e^{m_{D_{s}}^{2}/M_{2}^{2}}\widetilde{\Pi}_{\mathrm{scal}}(\mathbf{M}^{2},\mathbf{s}_{0},q^{2}),$
(71)

where $\widetilde{\Pi}_{\mathrm{scal}}(\mathbf{M}^{2},\mathbf{s}_{0},q^{2})$
is the Borel transformed and subtracted correlation function
$\widetilde{\Pi}^{\mathrm{OPE}}(p,p^{\prime})$.

Computations give the following predictions:

$\widetilde{G}=\mathcal{F}_{5}(-m_{D}^{2})=(1.07\pm 0.21)\ \mathrm{GeV}^{-1},$
(72)

where parameters of the function $\mathcal{F}_{5}(Q^{2})$, obtained from Eq.
(41) after the replacement $m\rightarrow\tilde{m}_{S}$, are:
$\mathcal{F}_{5}^{0}=4.06~{}\mathrm{GeV}^{-1}$, $c_{5}^{1}=4.56$, and
$c_{5}^{2}=-7.82$.

The partial width of this decay is determined by the expression Eq. (61) after
substitutions $G_{1}\rightarrow\widetilde{G}$, $m_{\mathrm{S}}\rightarrow$
$\widetilde{m}_{\mathrm{S}}$ and $\lambda\rightarrow$ $\widetilde{\lambda}=$
$\lambda(\widetilde{m}_{\mathrm{S}},m_{D_{s}},m_{D})$.

Numerical computations yield

$\widetilde{\Gamma}_{\mathrm{S}}=(123\pm 32)~{}\mathrm{MeV},$ (73)

which characterizes $\widetilde{T}_{\mathrm{S}}$ as a wide resonance.

##  V Discussion and concluding notes

The main motivation for present investigation is the LHCb discovery of the
very narrow doubly charmed axial-vector state $T_{cc}^{+}$. It is special in
two respects: First, $T_{cc}^{+}$ is the only doubly charmed resonance
observed experimentally. Secondly, it is narrowest candidate to a four-quark
meson. These circumstances made $T_{cc}^{+}$ an object of intensive
theoretical studies: Its structure and parameters were investigated in
numerous publications using various methods and models.

This discovery generated interest to possible counterparts of $T_{cc}^{+}$,
which may be four-quark systems with the same content but different spin-
parities. The doubly charmed tetraquarks containing strange $s$-quark(s) are
another class of particles closely related to $T_{cc}^{+}$. In present
article, we have concentrated namely on these particles and explored the
axial-vector and scalar doubly charmed strange tetraquarks $T_{\mathrm{AV}}$,
$T_{\mathrm{S}}$, and $\widetilde{T}_{\mathrm{S}}$ by calculating their masses
and widths.

It is worth to compare predictions obtained for the mass $m=(3995\pm
143)~{}\mathrm{MeV}$ and width $\Gamma_{\mathrm{AV}}=(72\pm
13)~{}\mathrm{MeV}$ of $T_{\mathrm{AV}}$ with parameters of $T_{cc}^{+}$
measured by the LHCb Collaboration. The mass gap between these two states, in
accordance with our findings, amounts to $120~{}\mathrm{MeV}$, which can be
considered as a reasonable estimate for particles with a $s$-quark difference
in their contents. The two-meson states $D^{0}D_{s}^{\ast+}$ and
$D^{\ast}(2007)^{0}D_{s}^{+}$ play for the tetraquark $T_{\mathrm{AV}}$ the
same role as $D^{0}D^{\ast}{}^{+}$ for the resonance $T_{cc}^{+}$. The mass of
$T_{cc}^{+}$ is very close but less than $D^{0}D^{\ast}{}^{+}$ threshold,
whereas $T_{\mathrm{AV}}$ lies $\approx 20~{}\mathrm{MeV}$ above corresponding
thresholds, which makes its decays to mesons $D^{0}D_{s}^{\ast+}$ and
$D^{\ast}(2007)^{0}D_{s}^{+}$ kinematically allowed processes. These $S$-wave
channels form width of the tetraquark $T_{\mathrm{AV}}$, and our prediction
for $\Gamma_{\mathrm{AV}}$ means that it is relatively wide resonance. In this
aspect, $T_{\mathrm{AV}}$ differs from the very narrow state $T_{cc}^{+}$,
because $T_{\mathrm{AV}}$ and $T_{cc}^{+}$ decay to ordinary mesons through
different mechanisms. Thus, the decay mode $T_{cc}^{+}\rightarrow
D^{0}D^{0}\pi^{+}$ in which $T_{cc}^{+}$ was discovered, may run due to the
transformation $T_{cc}^{+}\rightarrow D^{0}D^{\ast+}$. But this process is
forbidden, therefore the final state $D^{0}D^{0}\pi^{+}$ appears through
production of intermediate scalar tetraquarks, partial widths of which are
very small.

Another interesting question to be addressed here is the mass splitting
between the axial-vector $T_{\mathrm{AV}}$ and scalar $T_{\mathrm{S}}$
tetraquarks. The $T_{\mathrm{AV}}$ contains $J^{\mathrm{P}}=0^{+}$ ”good”
$(\mathbf{3}_{f},\mathbf{3}_{c})$ antidiquark $\overline{u}\overline{s}$,
whereas $T_{\mathrm{S}}$ is made of $J^{\mathrm{P}}=1^{+}$ ”bad”
$(\overline{\mathbf{6}}_{f},\mathbf{3}_{c})$ one. For the nonstrange diquark
the ”bad”-”good” mass difference $\delta(1^{+}-0^{+})_{ud}$ was found from the
QCD lattice calculations equal to $198~{}\mathrm{MeV}$ Francis:2021vrr , which
transforms to $\delta(1^{+}-0^{+})_{us}=135~{}\mathrm{MeV}$ in the case of
$\overline{u}\overline{s}$ diquark Karliner:2021wju . Then, in accordance with
this scheme, $m_{\mathrm{S}}$ should be equal approximately to
$4130~{}\mathrm{MeV}$. As is seen, our sum rule prediction for
$m_{\mathrm{S}}=(4128\pm 142)~{}\mathrm{MeV}$ agrees well with this estimate.
The tetraquark $\widetilde{T}_{\mathrm{S}}$ built of color-sextet scalar
diquarks resides between $T_{\mathrm{AV}}$ and $T_{\mathrm{S}}$ states. In a
situation when an existence of a scalar state $cc\overline{u}\overline{s}$
with ”good” components is forbidden, the axial-vector tetraquark
$T_{\mathrm{AV}}$ becomes the lightest particle in the spectrum.

The four-quark structure $cc\overline{u}\overline{s}$ with
$J^{\mathrm{P}}=1^{+}$ was studied in other publications in the framework of
alternative approaches. In the diquark-antidiquark and molecule models its
mass was estimated as $4106~{}\mathrm{MeV}$ and
$3974.8_{-0.5}^{+0.4}~{}\mathrm{MeV}$ in Refs. Karliner:2021wju  and
Ren:2021dsi , respectively. As it has been emphasized above, the prediction
around of $3975~{}\mathrm{MeV}$ contradicts to experimental-theoretical
constraints on the mass of $T_{\mathrm{AV}}$. Our result is less than the
prediction made in Ref. Karliner:2021wju  though $m$ in upper limit overlaps
with it.

One sees, that there are interesting open problems in physics of doubly
charmed four-quark mesons, which require additional theoretical and
experimental studies. The information gained in this article on the masses and
widths of the strange doubly-charmed tetraquarks gives new perspectives on
these states and can be used in future experimental investigations of
multiquark hadrons.

Data Availability Statement: No Data associated in the manuscript.

## References

  * (1) R. Aaij et al. [LHCb Collaboration], Nature Phys. 18, 751 (2022).
  * (2) R. Aaij et al. [LHCb Collaboration], Nature Commun. 13, 1 (2022).
  * (3) A. Esposito, M. Papinutto, A. Pilloni, A. D. Polosa, and N. Tantalo, Phys. Rev. D 88, 054029 (2013).
  * (4) F. S. Navarra, M. Nielsen, and S. H. Lee, Phys. Lett. B 649, 166 (2007).
  * (5) M. Karliner and J. L. Rosner, Phys. Rev. Lett. 119, 202001 (2017).
  * (6) E. J. Eichten and C. Quigg, Phys. Rev. Lett. 119, 202002 (2017).
  * (7) S. S. Agaev, K. Azizi, B. Barsbay and H. Sundu, Phys. Rev. D 99, 033002 (2019).
  * (8) S. S. Agaev, K. Azizi, B. Barsbay, and H. Sundu, Eur. Phys. J. A 57, 106 (2021).
  * (9) Y. Xing and R. Zhu, Phys. Rev. D 98, 053005 (2018).
  * (10) S. S. Agaev, K. Azizi, B. Barsbay, and H. Sundu, Chin. Phys. C 45, 013105 (2021).
  * (11) S. S. Agaev, K. Azizi, B. Barsbay, and H. Sundu, Eur. Phys. J. A 56, 177 (2020).
  * (12) S. S. Agaev, K. Azizi, B. Barsbay, and H. Sundu, Phys. Rev. D 101, 094026 (2020).
  * (13) M. L. Du, W. Chen, X. L. Chen, and S. L. Zhu, Phys. Rev. D 87, 014003 (2013).
  * (14) Z. G. Wang, Acta Phys. Polon. B 49, 1781 (2018).
  * (15) Z. G. Wang and Z. H. Yan, Eur. Phys. J. C 78, 19 (2018).
  * (16) E. Braaten, L. P. He, and A. Mohapatra, Phys. Rev. D 103, 016001 (2021).
  * (17) J. B. Cheng, S. Y. Li, Y. R. Liu, Z. G. Si, and T. Yao, Chin. Phys. C 45, 043102 (2021).
  * (18) Q. Meng, E. Hiyama, A. Hosaka, M. Oka, P. Gubler, K. U. Can, T. T. Takahashi, and H. S. Zong, Phys. Lett. B 814, 136095 (2021).
  * (19) P. Junnarkar, N. Mathur, and M. Padmanath, Phys. Rev. D 99, 034507 (2019).
  * (20) T. Guo, J. Li, J. Zhao, and L. He, Phys. Rev. D 105, 014021 (2022).
  * (21) M. Padmanath, and S. Prelovsek, Phys. Rev. Lett. 129, 032002 (2022).
  * (22) S. S. Agaev, K. Azizi, and H. Sundu, Phys. Rev. D 99, 114016 (2019).
  * (23) S. S. Agaev, K. Azizi, B. Barsbay, and H. Sundu, Nucl. Phys. B 939, 130 (2019).
  * (24) A. Feijoo, W. H. Liang, and E. Oset, Phys. Rev. D 104, 114015 (2021).
  * (25) M. J. Yan and M. P. Valderrama, Phys. Rev. D 105, 014007 (2022).
  * (26) S. Fleming, R. Hodges, and T. Mehen, Phys. Rev. D 104, 116010 (2021).
  * (27) S. S. Agaev, K. Azizi, and H. Sundu, Nucl. Phys. B 975, 115650 (2022).
  * (28) S. S. Agaev, K. Azizi, and H. Sundu, JHEP 06, 057 (2022).
  * (29) M. Karliner and J. L. Rosner, Phys. Rev. D 105, 034020 (2022).
  * (30) X. Chen, F. L. Wang, Y. Tan and Y. Yang arXiv:2206.10917 [hep-ph].
  * (31) M. Praszalowicz, Phys. Rev. D 106, 114005 (2022).
  * (32) Q. Xin and Z. G. Wang Eur. Phys. J. A 58, 110 (2022).
  * (33) H. Ren, F. G. Wu, and R. Zhu Adv. High Energy Phys. 2022, 9103031 (2022).
  * (34) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979).
  * (35) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 448 (1979).
  * (36) S. S. Agaev, K. Azizi and H. Sundu, Turk. J. Phys. 44, 95 (2020).
  * (37) C. J. Burden, L. Qian, C. D. Roberts, P. C. Tandy, and M. J. Thomson, Phys. Rev. C 55, 2649 (1997).
  * (38) R. L. Workman et al. [Particle Data Group], Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  * (39) V. Lubicz, A. Melis and S. Simula, PoS LATTICE 2016, 291 (2017).
  * (40) A. Francis, P. de Forcrand, R. Lewis, and K. Maltman, JHEP 05, 062 (2022).