# On the structures of new scalar resonances $T_{cs0}^{a}(2900)^{++}$ and
$T_{cs0}^{a}(2900)^{0}$

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, Turkiye  H. Sundu  Department of
Physics, Kocaeli University, 41380 Izmit, Turkiye  Department of Physics
Engineering, Istanbul Medeniyet University, 34700 Istanbul, Turkiye

###### Abstract

We investigate properties of the new scalar resonances
$T_{cs0}^{a}(2900)^{++}$ and $T_{cs0}^{a}(2900)^{0}$, which were recently
reported by the LHCb collaboration. These states were observed as resonant
structures in $D_{s}^{+}\pi^{+}$ and $D_{s}^{+}\pi^{-}$ invariant mass
distributions in $B^{+}$ meson decays. We argue that $T_{cs0}^{a}(2900)^{++}$
and $T_{cs0}^{a}(2900)^{0}$ may be modeled as molecules
$\mathcal{M}^{++}=D_{s}^{\ast+}\rho^{+}$ and
$\mathcal{M}^{0}=D_{s}^{\ast+}\rho^{-}$ of conventional vector mesons,
respectively. The mass $m$ and current coupling $f$ of the molecule
$\mathcal{M}^{++}$ are calculated using two-point sum rule method. The sum
rule analysis is performed by taking into account vacuum condensates up to
dimension $8$. The obtained result for the mass, $m=(2917\pm
135)~{}\mathrm{MeV}$, permits us to consider the molecule $\mathcal{M}^{++}$
as one of possible models of the resonance $T_{cs0}^{a}(2900)^{++}$. Because
the second structure $T_{cs0}^{a}(2900)^{0}$ is isospin partner of the doubly
charged state, it should have the mass close to $m$.

##  I Introduction

A few months ago the LHCb collaboration reported about observation of three
new exotic hadrons labeled as $P^{\psi s}(4338)^{0}$, $T_{cs0}^{a}(2900)^{0}$
and $T_{cs0}^{a}(2900)^{++}$, respectively LHCb:2022xob ; LHCb:2022bkt . First
of them is presumably a pentaquark discovered with a high significance in the
$J/\psi\Lambda$ invariant mass distribution in the decay $B^{-}\rightarrow
J/\psi\Lambda p$. Remaining two resonances are four-quark mesons
$T_{cs0}^{a}(2900)^{0}$ and $T_{cs0}^{a}(2900)^{++}$ (hereafter $T_{cs0}^{a0}$
and $T_{cs0}^{a++}$, respectively) fixed in the processes $B^{0}\rightarrow
D^{0}D_{s}^{+}\pi^{-}$ and $B^{+}\rightarrow D^{-}D_{s}^{+}\pi^{+}$. They are
scalar particles and were seen in the mass distributions of the mesons
$D_{s}^{+}\pi^{-}$ and $D_{s}^{+}\pi^{+}$. The LHCb measured the masses and
widths of the exotic mesons $T_{cs0}^{a0}$

$\displaystyle m_{1\mathrm{exp}}$ $\displaystyle=$ $\displaystyle(2892\pm
14\pm 15)~{}\mathrm{MeV},$ $\displaystyle\Gamma_{1\mathrm{exp}}$
$\displaystyle=$ $\displaystyle(119\pm 26\pm 13)~{}\mathrm{MeV},$ (1)

and $T_{cs0}^{a++}$

$\displaystyle m_{2\mathrm{exp}}$ $\displaystyle=$ $\displaystyle(2923\pm
17\pm 20)~{}\mathrm{MeV},$ $\displaystyle\Gamma_{2\mathrm{exp}}$
$\displaystyle=$ $\displaystyle(137\pm 32\pm 17)~{}\mathrm{MeV}.$ (2)

The collaboration also provided the following data for these structures:

$\displaystyle m_{\mathrm{exp}}$ $\displaystyle=$ $\displaystyle(2908\pm 11\pm
20)~{}\mathrm{MeV},\ $ $\displaystyle\Gamma_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle(136\pm 23\pm 11)~{}\mathrm{MeV},$ (3)

supposing that they are isospin partners and share these parameters.

Because the new scalar resonances were fixed in the $D_{s}^{+}\pi^{-}$ and
$D_{s}^{+}\pi^{+}$ mass distributions, the processes $T_{cs0}^{a0}\rightarrow
D_{s}^{+}\pi^{-}$ and $T_{cs0}^{a++}\rightarrow D_{s}^{+}\pi^{+}$ are their
main decay channels. Then, it is clear that they are built of quarks
$cd\overline{s}\overline{u}$ and $cu\overline{s}\overline{d}$, provided the
four-quark picture can be applied to model these states. They are also fully
open flavor tetraquarks, and hence belong to a family of exotic mesons
established by the structures $X_{0}(2900)$ and $X_{1}(2900)$. In fact, in
Refs. LHCb:2020A ; LHCb:2020  the LHCb informed on scalar $X_{0}(2900)$ and
vector $X_{1}(2900)$ resonances (in what follows $X_{0}$ and $X_{1}$) in the
invariant mass distribution of $D^{-}K^{+}$ mesons in the channel
$B^{+}\rightarrow D^{+}D^{-}K^{+}$. This means that the resonances $X_{0}$ and
$X_{1}$ decay to mesons $D^{-}K^{+}$, and in the four-quark model are composed
of four different valence quarks $ud\overline{s}\overline{c}$. This fact
placed $X_{0}$ and $X_{1}$ to special position in $XYZ$ family of exotic
mesons, because they were first evidences for the fully open flavor
tetraquarks.

The discovery of the resonances $X_{0}$ and $X_{1}$ triggered interesting
theoretical investigations aimed to understand their nature, calculate their
masses, and estimate widths of these states Karliner:2020vsi ; Wang:2020xyc ;
Chen:2020aos ; Liu:2020nil ; Molina:2020hde ; Hu:2020mxp ; He:2020jna ;
Lu:2020qmp ; Zhang:2020oze ; Huang:2020ptc ; Xue:2020vtq ; Yang:2021izl ;
Wu:2020job ; Abreu:2020ony ; Wang:2020prk ; Xiao:2020ltm ; Dong:2020rgs ;
Burns:2020xne ; Bondar:2020eoa ; Chen:2020eyu ; Albuquerque:2020ugi .
Naturally, authors of numerous publications made different assumptions about
internal structures of $X_{0}$ and $X_{1}$, and invoked various models and
calculational schemes to evaluate their parameters. For instance, in Refs.
Karliner:2020vsi ; Wang:2020xyc  $X_{0}$ was treated as a scalar diquark-
antidiquark state $[sc][\overline{u}\overline{d}]$. Contrary, in Ref.
Chen:2020aos  $X_{0}$ was assigned as the $S$-wave hadronic molecule
$D^{\ast-}K^{\ast+}$, whereas for $X_{1}$ the authors adopted a diquark-
antidiquark model.

We studied the resonance $X_{0}$ by considering it as a molecule
$\overline{D}^{\ast 0}K^{\ast 0}$ and evaluated its mass and width
Agaev:2020nrc . Obtained results allowed us to confirm the molecule nature of
$X_{0}$. We explored in Ref. Agaev:2021knl  also the resonance $X_{1}$ and
interpreted it as a vector diquark-antidiqurk state
$X_{\mathrm{V}}=[ud][\overline{c}\overline{s}]$. The discovery of $X_{0}$ and
$X_{1}$ by LHCb made the diquark-antidiquarks $[ud][\overline{c}\overline{s}]$
objects of intensive investigations. Indeed, the vector state $X_{\mathrm{V}}$
from this group of particles presumably was seen in experiment as the
resonance $X_{1}$. The masses and full widths of the ground-state and radially
excited scalar particles $X_{0}^{(\prime)}=[ud][\overline{c}\overline{s}]$
were computed in Ref. Agaev:2022eeh . Spectroscopic parameters and widths of
the axial-vector and pseudoscalar tetraquarks $X_{\mathrm{AV}}$ and
$X_{\mathrm{PS}}$ with the same content were calculated Sundu:2022kyd , as
well.

It is worth to emphasize that the fully open flavor exotic mesons were already
in agenda of researchers. Thus, the scalar diquark-antidiquark state
$X_{c}=[su][\overline{c}\overline{d}]$ was investigated in Ref. Agaev:2016lkl
, in which its mass and full width were computed in the framework of QCD sum
rule method using the $C\gamma_{5}\otimes\gamma_{5}C$ and
$C\gamma_{\mu}\otimes\gamma^{\mu}C$ structures and, accordingly, the scalar-
scalar and axial-axial interpolating currents. The scalar, pseudoscalar and
axial-vector diquark-antidiquark states $[sd][\overline{u}\overline{c}]$,
which carry two units of electric charge $-2|e|$ were studied in Ref.
Agaev:2017oay . The particles $Z^{++}$ with the content
$[cu][\overline{s}\overline{d}]$ are positively charged counterparts of these
states and have the same masses and decay widths. Parameters of the vector
tetraquark $Z_{\mathrm{V}}^{++}$ became available recently Agaev:2021jsz .
Knowledge gained during these investigations is very useful to examine the
resonances $T_{cs0}^{a0/++}$.

In our view, the resonance $T_{cs0}^{a++}$ is a more interesting object for
exploration, because it has additional attractive feature as a first doubly
charged tetraquark observed in the experiment. As is seen, the $T_{cs0}^{a++}$
has a content which is identical to one of the tetraquarks $Z^{++}$. The
scalar tetraquark $Z_{\mathrm{S}}^{++}$ has the mass and width Agaev:2017oay

$m_{\mathrm{Z}_{\mathrm{S}}}=2628_{-153}^{+166}~{}\mathrm{MeV},\
\Gamma_{\mathrm{Z}_{\mathrm{S}}}=(66.89\pm 15.11)~{}\mathrm{MeV,}$ (4)

which are far from the parameters of $T_{cs0}^{a++}$.

As we have noted above, the neutral scalar resonance $T_{cs0}^{a0}$ is
composed of quarks $cd\overline{s}\overline{u}$. The quark content and spin-
parity of $T_{cs0}^{a0}$ coincide with parameters of the scalar tetraquark
$\overline{X}_{c}=[cd][\overline{s}\overline{u}]$. The latter is an
antiparticle of $X_{c}$ and should have the same parameters as $X_{c}$

$m_{\mathrm{S}}=(2634\pm 62)~{}\mathrm{MeV}~{}\mbox{and}~{}\
\Gamma_{\mathrm{S}}=(57.7\pm 11.6)~{}\mathrm{MeV},$ (5)

and

$m_{\mathrm{A}}=(2590\pm 60)~{}\mathrm{MeV}~{}\mbox{and}~{}\
\Gamma_{\mathrm{A}}=(63.4\pm 14.2)~{}\mathrm{MeV},$ (6)

which were found using for $X_{c}$ scalar-scalar or axial-axial currents . Let
us note that the prediction $(2.55\pm 0.09)~{}\mathrm{GeV}$ for the mass of
the $X_{c}$ was obtained in Ref. Chen:2016mqt , as well. In other words,
neither $T_{cs0}^{a++}$ nor $T_{cs0}^{a0}$ can be interpreted as the ground-
level scalar diquark-antidiquark systems.

In this situation, we can study the $T_{cs0}^{a++}$ and $T_{cs0}^{a0}$ within
a hadronic molecule model, i.e., as a bound state of conventional mesons. Let
us analyze in details the resonance $T_{cs0}^{a++}$. Interpretation of a
molecule $D_{s}^{+}\pi^{+}$ as the resonance $T_{cs0}^{a++}$ seems is
difficult, because the mass of such system is considerably smaller than
$m_{2\mathrm{exp}}$. The hadronic molecules $D_{s}^{\ast+}\rho^{+}$ and
$D^{\ast+}K^{\ast+}$ or their superposition provide alternative choices for
$T_{cs0}^{a++}$. Two-particle thresholds for these molecules are equal to
$2887~{}\mathrm{MeV}$ and $2902\ \mathrm{MeV}$, respectively. They cannot
decay to $D_{s}^{\ast+}+\rho^{+}$ and $D^{\ast+}+K^{\ast+}$ meson pairs if
their masses are less than these thresholds. Otherwise, the molecules
$D_{s}^{\ast+}\rho^{+}$ and $D^{\ast+}K^{\ast+}$ with masses above these
limits dissociate to these mesons. In both cases, decay to a pair of the
pseudoscalar $D_{s}^{+}$ and $\pi^{+}$ mesons are kinematically allowed
channels for the molecules $D_{s}^{\ast+}\rho^{+}$ and $D^{\ast+}K^{\ast+}$.

In our paper Agaev:2022eyk , we investigated the resonance $T_{cs0}^{a++}$ by
modeling it as a the hadronic molecule $D^{\ast+}K^{\ast+}$. The results for
the mass and width of such compound $\ (2924\pm 107)~{}\mathrm{MeV},~{}$and
$(123\pm 25)~{}\mathrm{MeV}$ are consistent with parameters of $T_{cs0}^{a++}$
given by Eq. (2). The resonances $T_{cs0}^{a0/++}$ were investigated using
different models and approaches in Refs. Chen:2022svh ; Ge:2022dsp ;
Wei:2022wtr ; Liu:2022hbk . It is interesting that conclusions made about
nature of these structures also differ from each other. Thus, in Ref.
Chen:2022svh  the one-boson exchange model was used to explore the
interactions in $D^{(\ast)}K^{(\ast)}$ systems. Analysis performed in this
article allowed the authors to assign $T_{cs0}^{a++}$ to be an isovector
$D^{\ast+}K^{\ast+}$ molecule state with the spin-parity
$J^{\mathrm{P}}=0^{+}$ and mass $2891~{}\mathrm{MeV}$. Interpretation of the
new tetraquark candidate $T_{cs0}^{a}$ as the resonance-like structure induced
by threshold effects was suggested in Ref. Ge:2022dsp . Here, it was argued
that the triangle singularity generated by the $\chi_{c1}K^{\ast}D^{\ast}$
loop peaks around the threshold $D^{\ast}K^{\ast}$ and may simulate
$T_{cs0}^{a}$.

A multiquark color flux-tube model was used to investigate the resonances
$T_{cs0}^{a0/++}$ in the framework of the diqark-antidiquark model Wei:2022wtr
. The authors found that a system $[cu][\overline{s}\overline{d}]$ built of
the color antitriplet diquark and triplet antidiquark with the mass
$2923~{}\mathrm{MeV}$ is very nice candidate to the resonance $T_{cs0}^{a++}$.
The properties of the charmed-strange tetraquarks were studied in Ref.
Liu:2022hbk  by employing a nonrelativistic potential quark model.

In present work, we explore the spectroscopic parameters of the molecule
$\mathcal{M}^{++}=D_{s}^{\ast+}\rho^{+}$. The molecular structure
$\mathcal{M}^{0}=D_{s}^{\ast+}\rho^{-}$ may be considered as a model for
$T_{cs0}^{a0}$. We calculate the mass and current coupling of
$\mathcal{M}^{++}$ using QCD two-point sum rule method, and confront our
predictions with the experimental data of the LHCb Collaboration.

This article is structured in the following manner: In Sec. II, we derive the
sum rules for the mass $m$ and current coupling $f$ of the molecule
$\mathcal{M}^{++}$ in the context of QCD sum rule method. Numerical analysis
of the quantities $m$ and $f$ is performed in Sec. III, where we determine
working windows for the Borel and continuum subtraction parameters, and
evaluate $m$ and $f$. The section IV contains our concluding remarks.

##  II Spectroscopic parameters of the hadronic molecule
$\mathcal{M}^{++}=D_{s}^{\ast+}\rho^{+}$

We compute the mass $m$ and current coupling $f$ of the hadronic molecule
$\mathcal{M}^{++}$ using the QCD two-point sum rule method Shifman:1978bx ;
Shifman:1978by . To obtain sum rules for $m$ and $f$, we start to the analysis
by considering the following correlation function:

$\Pi(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{J(x)J^{{\dagger}}(0)\\}|0\rangle,$ (7)

where $\mathcal{T}$ is the time-ordering operator, and $J(x)$ stands for the
interpolating current of the molecule $\mathcal{M}^{++}$.

In the molecule model colorless four-quark structures come from the singlet-
singlet $\mathbf{[1}_{c}\mathbf{]\otimes[1}_{c}\mathbf{]}$ and octet-octet
$\mathbf{[8}_{c}\mathbf{]\otimes[8}_{c}\mathbf{]}$ terms of the color group
$SU_{c}(3)$. In the case of $\mathcal{M}^{++}$, we suppose that the hadronic
molecule $\mathcal{M}^{++}$ is made of two vector mesons $D_{s}^{\ast+}$ and
$\rho^{+}$, and consider the singlet-singlet type current. Then, in the
$\mathbf{[1}_{c}\mathbf{]}_{\overline{s}c}\mathbf{\otimes[1}_{c}\mathbf{]}_{\overline{d}u}$
representation $J(x)$ has the following form

$J(x)=[\overline{s}_{a}(x)\gamma^{\mu}c_{a}(x)][\overline{d}_{b}(x)\gamma_{\mu}u_{b}(x)],$
(8)

with $a$ and $b$ being color indices.

It is worth to note that $J(x)$ couples not only to the molecule
$\mathcal{M}^{++}$ but also to diquark-antidiquark states. The reason is that
a molecule current by means of Fierz transformation can be expressed as the
sum of different diquark-antidiquark currents with some numerical factors
Wang:2020rcx . In other words, the molecule current is a special weighted sum
of diquark-antidiquark currents. Contrary, a diquark-antidiquark current can
be rewritten via molecule structures Refs. Chen:2022sbf ; Xin:2021wcr . Only
comparison with experimental data can justify a choice of molecule or diquark-
antidiquark type structures to model the resonance $\mathcal{M}^{++}$.

In the sum rule approach, the correlator $\Pi(p)$ has to be presented using
physical parameters of $\mathcal{M}^{++}$, and also written down in terms of
different quark-gluon condensates of QCD. For the physical side of the sum
rule, we get

$\Pi^{\mathrm{Phys}}(p)=\frac{\langle
0|J|\mathcal{M}^{++}(p)\rangle\langle\mathcal{M}^{++}(p)|J^{\dagger}|0\rangle}{m^{2}-p^{2}}+\cdots,$
(9)

where $p$ is the four momentum of $\mathcal{M}^{++}$. In Eq. (9) the term
shown explicitly is contribution of ground-state particle $\mathcal{M}^{++}$,
whereas ellipses stand for effects of higher resonances and continuum states
in the $\mathcal{M}^{++}$ channel. To derive the physical side
$\Pi^{\mathrm{Phys}}(p)$ of the sum rule from Eq. (7), we insert a complete
set of intermediate states with content and quantum numbers of the
$\mathcal{M}^{++}$ state, and carry out integration over $x$.

Introducing the physical parameters of $\mathcal{M}^{++}$ through the matrix
element

$\langle 0|J|\mathcal{M}^{++}\rangle=fm,$ (10)

we recast $\Pi^{\mathrm{Phys}}(p)$ into the final form

$\Pi^{\mathrm{Phys}}(p)=\frac{f^{2}m^{2}}{m^{2}-p^{2}}+\cdots.$ (11)

The function $\Pi^{\mathrm{Phys}}(p)$ includes only one Lorentz structure,
namely the unit matrix $\mathrm{I}$, and term in rhs of Eq. (11) is the
invariant amplitude $\Pi^{\mathrm{Phys}}(p^{2})$ corresponding to this
structure.

The QCD side of the sum rules, $\Pi^{\mathrm{OPE}}(p)$, should be calculated
in the operator product expansion ($\mathrm{OPE}$) with certain accuracy. To
find $\Pi^{\mathrm{OPE}}(p)$, we use in Eq. (7) the interpolating current
$J(x)$, and contract the corresponding heavy and light quark fields using the
Wick’s theorem. By performing these operations for $\Pi^{\mathrm{OPE}}(p)$, we
obtain

$\displaystyle\Pi^{\mathrm{OPE}}(p)=i\int
d^{4}xe^{ipx}\mathrm{Tr}\left[\gamma_{\mu}S_{c}^{aa^{\prime}}(x)\gamma_{\nu}S_{s}^{a^{\prime}a}(-x)\right]$
$\displaystyle\times\mathrm{Tr}\left[\gamma^{\mu}S_{u}^{bb^{\prime}}(x)\gamma^{\nu}S_{d}^{b^{\prime}b}(-x)\right],$
(12)

with $S_{c}(x)$ and $S_{u(s,d)}(x)$ being the quark propagators. The explicit
expressions for the heavy and light quarks propagators are collected in Ref.
Agaev:2020zad .

The $\Pi^{\mathrm{OPE}}(p)$ has also a trivial structure $\sim\mathrm{I}$ and
is characterized by an amplitude $\Pi^{\mathrm{OPE}}(p^{2})$. After equating
the invariant amplitudes $\Pi^{\mathrm{Phys}}(p^{2})$ and
$\Pi^{\mathrm{OPE}}(p^{2})$, one obtains the QCD sum rule equality. In order
to suppress contributions of higher resonances and continuum states one has to
apply the Borel transformation to both sides of this expression. We apply the
continuum subtraction supported by the quark-hadron duality assumption as
well. These operations generate dependence of the sum rule equality on the
Borel $M^{2}$ and continuum threshold $s_{0}$ parameters. Final expression and
its derivative over $d/d(-1/M^{2})$ can be used to derive sum rules for the
mass $m$ and coupling $f$ of the molecule $\mathcal{M}^{++}$

$m^{2}=\frac{\Pi^{\prime}(M^{2},s_{0})}{\Pi(M^{2},s_{0})},$ (13)

and

$f^{2}=\frac{e^{m^{2}/M^{2}}}{m^{2}}\Pi(M^{2},s_{0}).$ (14)

Here, $\Pi(M^{2},s_{0})$ is the invariant amplitude
$\Pi^{\mathrm{OPE}}(p^{2})$ after Borel transformation and subtraction
procedures, and $\Pi^{\prime}(M^{2},s_{0})=d\Pi(M^{2},s_{0})/d(-1/M^{2})$.

The Borel transformation of the amplitude $\Pi^{\mathrm{Phys}}(p^{2})$ has the
simple form

$\mathcal{B}\Pi^{\mathrm{Phys}}(p^{2})=fme^{-m^{2}/M^{2}},$ (15)

whereas the correlator $\Pi(M^{2},s_{0})$ is given by the expression

$\Pi(M^{2},s_{0})=\int_{(m_{c}+m_{s})^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}).$
(16)

The spectral density $\rho^{\mathrm{OPE}}(s)$ is calculated as an imaginary
part of the amplitude $\Pi^{\mathrm{OPE}}(p^{2})$. The term $\Pi(M^{2})$ in
Eq. (16) is the Borel transformations of some of terms evaluated directly from
their expressions in $\Pi^{\mathrm{OPE}}(p)$. In this article, we neglect the
masses of the $u$ and $d$ quarks, but take into account terms $\sim m_{s}$
setting, at the same time, $m_{s}^{2}=0$. Computations are performed by
including into analysis the vacuum expectation values of the nonperturbative
operators up to dimension $8$. The higher dimensional contributions to
$\Pi(M^{2},s_{0})$ are obtained as products of basic vacuum condensates using
the factorization procedure. First few terms in $\mathrm{OPE}$ do not contains
such condensates. They appear at higher dimensions and are numerically small.
Therefore, we neglect the ambiguities caused by the factorization and their
impact on the sum rules’ results.

Analytical expressions of $\rho^{\mathrm{OPE}}(s)$ and $\Pi(M^{2})$ are rather
lengthy and not presented here explicitly.

##  III Numerical analysis

The sum rules in Eqs. (13) and (14) contain different quark, gluon and mixed
condensates. They are universal parameters, and were extracted from the
analysis of numerous processes. The mass $m$ and coupling $f$ depend also on
the masses of $c$ and $s$ quarks. To carry out numerical computations, one has
to fix values all of these parameters. Below, we list the values of these
condensates

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\ \langle\overline{s}s\rangle=(0.8\pm
0.1)\langle\overline{q}q\rangle,$ $\displaystyle\langle\overline{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle,\
\langle\overline{s}g_{s}\sigma
Gs\rangle=m_{0}^{2}\langle\overline{s}s\rangle,$ $\displaystyle
m_{0}^{2}=(0.8\pm 0.2)~{}\mathrm{GeV}^{2},$
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle\langle
g_{s}^{3}G^{3}\rangle=(0.57\pm 0.29)~{}\mathrm{GeV}^{6},$ $\displaystyle
m_{s}=93_{-5}^{+11}~{}\mathrm{MeV},\ m_{c}=1.27\pm 0.02~{}\mathrm{GeV}.$ (17)

Figure 1: The pole contribution to the correlator $\Pi(M^{2},s_{0})$ as a
function of the Borel parameter $M^{2}$ at different $s_{0}$. The horizontal
black line limits the border $\mathrm{PC}=0.5$. The red diamond marks the
point at which the mass $m$ of the molecule
$\mathcal{M}^{++}=D_{s}^{\ast+}\rho^{+}$ has effectively been extracted.

We included into Eq. (17) the masses of $c$ and $s$ quarks, as well.

The working regions for the parameters $M^{2}$ and $s_{0}$ should meet the
standard constraints imposed on the pole contribution ($\mathrm{PC}$) and
convergence of the operator product expansion. To quantify these restrictions,
it is convenient to introduce the formulas

$\mathrm{PC}=\frac{\Pi(M^{2},s_{0})}{\Pi(M^{2},\infty)},$ (18)

and

$R(M^{2})=\frac{\Pi^{\mathrm{Dim8}}(M^{2},s_{0})}{\Pi(M^{2},s_{0})}.$ (19)

First of them is a measure of the pole contribution and is necessary to find
the upper limit $M_{\mathrm{max}}^{2}$ of the Borel region. In sum rule
analyses of the conventional hadrons a constraint $\mathrm{PC}\geq 0.5$ is a
standard requirement. In the case of multiquark hadrons this constraint may be
satisfied, although it shrinks a region for $M^{2}$. The function $R(M^{2})$
is employed to find the lower limit, $M_{\mathrm{min}}^{2}$, of the window for
the Borel parameter. Here, $\Pi^{\mathrm{Dim8}}(M^{2},s_{0})$ indicates the
last term in the operator product expansion of $\ \Pi(M^{2},s_{0})$. For
computations performed up to dimension $8$ the constraint
$R(M_{\mathrm{min}}^{2})\leq 0.05$ is a reasonable requirement.

Analysis demonstrates that the regions for the parameters $M^{2}$ and $s_{0}$

$M^{2}\in[2,3]~{}\mathrm{GeV}^{2},\ s_{0}\in[11,12.2]~{}\mathrm{GeV}^{2},$
(20)

obey all the necessary constraints. Indeed, at $M^{2}=3~{}\mathrm{GeV}^{2}$
the pole contribution on average in $s_{0}$ is $0.50$, whereas at
$M^{2}=2~{}\mathrm{GeV}^{2}$ it becomes equal to $0.72$. In Fig. 1 the pole
contribution is depicted as a function of $M^{2}$ at various fixed $s_{0}$.
Only at $s_{0}=11~{}\mathrm{GeV}^{2}$ in a small region, $M^{2}\geq
2.8~{}\mathrm{GeV}^{2}$, the pole contribution is less than $0.5$. But on
average in $s_{0}$ the constraint $\mathrm{PC}\geq 0.5$ is satisfied in the
entire working region for the Borel parameter. At the minimum point,
$M^{2}=2~{}\mathrm{GeV}^{2}$, we get $R(2~{}\mathrm{GeV}^{2})\approx 0.022$
and the contribution of dimension-$8$ term does not exceed $2.2\%$ of the
whole result.

Another important question which should be taken into account in fixing the
regions in Eq. (20) is the prevalence of the perturbative contribution to
$\Pi(M^{2},s_{0})$ over the nonperturbative one, as well as the convergence of
the operator product expansion in these regions. From Fig. 2 it becomes clear
that even at $M^{2}=2~{}\mathrm{GeV}^{2}$ the perturbative contribution
constitutes more than $65\%$ of $\Pi(M^{2},s_{0})$, whereas the
nonperturbative contribution is less than $35\%$ and gradually falls with
increasing of $M^{2}$. The convergence of $\mathrm{OPE}$ implies reducing
contributions of the nonperturbative terms by increasing dimensions of the
corresponding operators. Here, one should take into account that due to
smallness of gluon condensates this hierarchy may be destroyed for few terms.
In the right panel of Fig. 2, we plot different nonperturbative contributions
to $\Pi(M^{2},s_{0})$. The contributions of terms $\mathrm{Dim3}$ and
$\mathrm{Dim6}$ are positive: contributions of other operators are negative
and their absolute values do not exceed the positive terms. The last two terms
in this expansion at $M^{2}=2~{}\mathrm{GeV}^{2}$ form only $0.22$ and $0.07$
parts of $\mathrm{Dim3}$ contribution, respectively. In other words, the
operator product expansion converges quite well.

Results for $m$ and $f$ are obtained by taking their mean values at different
choices of the parameters $M^{2}$ and $s_{0}$

$\displaystyle m$ $\displaystyle=$ $\displaystyle(2917~{}\pm
135)~{}\mathrm{MeV},$ $\displaystyle f$ $\displaystyle=$
$\displaystyle(4.65\pm 0.95)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (21)

The $m$ and $f$ from Eq. (21) effectively correspond to sum rules’ predictions
at $M^{2}=2.4~{}\mathrm{GeV}^{2}$ and $s_{0}=11.8~{}\mathrm{GeV}^{2}$ marked
in Fig. 1 by the red diamond, where the pole contribution is
$\mathrm{PC}\approx 0.64$. This fact ensures the ground-state nature of
$\mathcal{M}^{++}$ and reliability of obtained results.

The mass $m$ as functions of the parameters $M^{2}$ and $s_{0}$ is shown in
Fig. 3. Here, one can see a dependence of $m$ on the parameter $M^{2}$, though
the physical quantity should not depend on it. Nevertheless, such residual
dependence of $m$ and $f$ on the Borel parameter exists and generates
essential part of theoretical uncertainties shown in Eq. (21). There is also
dependence on the choice of $s_{0}$, but this effect may be used to extract
information on the mass of the first excited particle in the
$D_{s}^{\ast+}\rho^{+}$ channel. Because $\sqrt{s_{0}}$ should be less than
the mass $m^{\ast}$ of the first excited molecule $\mathcal{M}^{++\ast}$, we
find an estimate $m^{\ast}\geq m+500~{}\mathrm{MeV}$, which may be expected
for $\mathcal{M}^{++}$ molecule containing a $c$ quark.

Figure 2: Left: Perturbative and nonperturbative contributions to
$\Pi(M^{2},s_{0})$ normalized to $1$ as functions of the Borel parameter
$M^{2}$, Right: Normalized contributions of different operators to the
correlator as functions of $M^{2}$. All curves in the figure have been
calculated at $s_{0}=11.6~{}\mathrm{MeV}^{2}$.

Figure 3: The dependence of the mass $m$ of the molecule $\mathcal{M}^{++}$ on
the Borel parameter $M^{2}$ (left panel), and on the continuum threshold
parameter $s_{0}$ (right panel).

##  IV Concluding remarks

As is seen, the result $m=(2917~{}\pm 135)~{}\mathrm{MeV}$ for the mass of the
molecule $\mathcal{M}^{++}=D_{s}^{\ast+}\rho^{+}$ obtained in the current
paper agrees nicely with the LHCb datum $m_{2\mathrm{exp}}$. In our article
Agaev:2022eyk , we modeled the resonance $T_{cs0}^{a++}$ as the hadronic
molecule $D^{\ast+}K^{\ast+}$, and calculated its mass and width. The mass
$(2924\pm 107)~{}\mathrm{MeV}$ of $D^{\ast+}K^{\ast+}$ is also consistent with
the LHCb data. Comparing these two models with each other, one sees that they
lead to very close results and can be applied to describe the resonance
$T_{cs0}^{a++}$. Therefore, at this level of our knowledge, we can interpret
the molecules $D_{s}^{\ast+}\rho^{+}$ and $D^{\ast+}K^{\ast+}$, or their
superposition as candidates to the resonance $T_{cs0}^{a++}$.

The neutral molecules $\mathcal{M}^{0}=D_{s}^{\ast+}\rho^{-}$ and $D^{\ast
0}K^{\ast 0}$ are possible models for the resonance $T_{cs0}^{a0}$. A linear
superposition of these molecules also may be used to model $T_{cs0}^{a0}$. The
mass of $T_{cs0}^{a0}$ as an isoscalar partner of $T_{cs0}^{a++}$ should be
close to the extracted value $m$. In fact, experimentally measured mass and
width differences between $T_{cs0}^{a++}$ and $T_{cs0}^{a0}$ are equal to
$\Delta m\approx 28~{}\mathrm{MeV}$ and $\Delta\Gamma\approx
15~{}\mathrm{MeV}$, respectively. But, to be accepted as a reliable model for
$T_{cs0}^{a0}$ the molecule picture should be successfully confronted with the
LHCb data.

As we have noted in Sec. II, the $J(x)$ couples to various diquark-antidiquark
states, and their specially chosen superposition gives the molecule current.
The fixed ground-level diquark-antidiquark state, as it has been discussed in
a detailed form in Sec. I, does not describe parameters of the resonances
$T_{cs0}^{a0/++}$. In these circumstances, assumptions about molecule nature
of $T_{cs0}^{a0/++}$ seem are more realistic than other four-quark models.
Number of alternative molecule models for $T_{cs0}^{a0/++}$ is restricted by
the masses and widths of these states, as well as by the fact that the
resonance $T_{cs0}^{a++}$ carries two units of electric charge. This question
requires additional detailed analysis, which is beyond the scope of the
present work.

The structures $T_{cs0}^{a0/++}$ were studied in the context of other models
as well Chen:2022svh ; Ge:2022dsp ; Wei:2022wtr ; Liu:2022hbk . At present
there are no definite conclusion on the quark structure not only of resonances
$T_{cs0}^{a0/++}$ but also other exotic hadrons. This paper and Ref.
Agaev:2022eyk  are attemps to clarify a situation around of very interesting
structure $T_{cs0}^{a++}$.

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