# New scalar resonance $X_{0}(2900)$ as a $\overline{D}^{*}K^{*}$ molecule:
Mass and width

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, Acibadem-Kadiköy, 34722 Istanbul, Turkey  School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box
19395-5531, Tehran, Iran  H. Sundu  Department of Physics, Kocaeli University,
41380 Izmit, Turkey

###### Abstract

We explore features of the scalar structure $X_{0}(2900)$, which is one of the
two resonances discovered recently by LHCb in the $D^{-}K^{+}$ invariant mass
distribution in the decay $B^{+}\to D^{+}D^{-}K^{+}$. We treat $X_{0}(2900)$
as a hadronic molecule composed of the conventional mesons $\overline{D}^{*0}$
and $K^{*0}$ and calculate its mass, coupling and width. The mass and coupling
of $X_{0}(2900)$ are determined using the QCD two-point sum rule method by
taking into account quark, gluon, and mixing vacuum condensates up to
dimension $15$. The decay of this structure to final state $D^{-}K^{+}$ is
investigated in the context of the light-cone sum rule approach supported by a
soft-meson technique. To this end, we evaluate strong coupling $G$
corresponding to vertex $X_{0}D^{-}K^{+}$, which allows us to find width of
the decay $X_{0}(2900)\to D^{-}K^{+}$. Obtained predictions for the mass of
the hadronic molecule $\overline{D}^{*0}K^{*0}$ $m=(2868\pm
198)~{}\mathrm{MeV}$ and for its width $\Gamma=(49.6\pm 9.3)~{}\mathrm{MeV}$
can be considered as arguments in favor of molecule interpretation of
$X_{0}(2900)$.

##  I Introduction

The LHCb collaboration recently announced about observation of two resonant
structures $X_{0}(2900)$ and $X_{1}(2900)$ (hereafter $X_{0}$ and $X_{1}$,
respectively) in the invariant $D^{-}K^{+}$ mass distribution of the decay
$B^{+}\rightarrow D^{+}D^{-}K^{+}$ LHCb:2020 . In accordance with LHCb,
obtained results constitute the clear evidence for exotic mesons with full
open flavors. At the same time, the collaboration did not exclude models which
explain experimental data using hadronic rescattering effects.

The LHCb extracted the masses and widths of these structures, as well as
determined their spin-parities. Thus, it was shown that the $X_{0}$ is the
scalar resonance $J^{\mathrm{P}}=0^{+}$ with parameters

$\displaystyle m_{0}$ $\displaystyle=$ $\displaystyle(2866\pm 7\pm
2)~{}\mathrm{MeV},\ $ $\displaystyle\Gamma_{0}$ $\displaystyle=$
$\displaystyle(57\pm 12\pm 4)~{}\mathrm{MeV},$ (1)

whereas $X_{1}$ is the vector state $J^{\mathrm{P}}=1^{-}$ and has the mass
and width

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle(2904\pm 5\pm
1)~{}\mathrm{MeV},\ $ $\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle(110\pm 11\pm 4)~{}\mathrm{MeV}.$ (2)

From decay channels of these resonances $X_{0(1)}\rightarrow$ $D^{-}K^{+}$, it
is evident that they are built of four different valence quarks
$ud\overline{s}\overline{c}$. These circumstances place $X_{0}$ and $X_{1}$ to
distinguishable position in the $XYZ$ family of exotic mesons. The LHCb’s
discovery is doubly remarkable, because existence of the resonance $X(5568)$,
seen by the D0 collaboration D0:2016mwd  and presumably composed of quarks
$sd\overline{b}\overline{u}$, was not later confirmed by other experiments.

The LHCb information generated interesting theoretical investigations to
explain structure of new resonances $X_{0}$ and $X_{1}$, calculate their
masses, and if possible, estimate widths Karliner:2020vsi ; Wang:2020xyc ;
Hu:2020mxp ; Chen:2020aos ; He:2020jna ; Liu:2020orv ; Liu:2020nil ;
Lu:2020qmp ; Zhang:2020oze ; Huang:2020ptc ; Xue:2020vtq ; Molina:2020hde . In
these papers the authors made different suggestions about internal
organization of these resonances, and used various methods and schemes to
compute their parameters. The diquark-antidiquark and hadronic molecule
pictures are dominant models to account for collected experimental data. For
example, in Refs. Karliner:2020vsi ; Wang:2020xyc  $X_{0}$ was considered as a
scalar tetraquark $X_{0}=[sc][\overline{u}\overline{d}]$ using a
phenomenological approach and the sum rule method, respectively. In Ref.
Chen:2020aos  $X_{0}$ was interpreted as $S$-wave hadronic
$D^{\ast-}K^{\ast+}$ molecule, whereas $X_{1}$ considered $P$-wave diquark-
antidiquark state $X_{1}=[ud][\overline{s}\overline{c}]$. In Ref. Liu:2020orv
, on the contrary, it was asserted that two resonance-like peaks generated in
the process $B^{+}\rightarrow D^{+}D^{-}K^{+}$ due to rescattering effects may
emerge in LHCb experiment as the states $X_{0}$ and $X_{1}$.

It is worth noting that the exotic scalar meson with open flavor structure
$X_{c}=[su][\overline{c}\overline{d}]$ was studied in our work Agaev:2016lkl ,
in which it was explored as a charmed partner of the resonance $X(5568)$. The
mass and width of this tetraquark were calculated using the sum rule method
and two interpolating currents. These currents correspond to structures
$C\gamma_{5}\otimes\gamma_{5}C$ and $C\gamma_{\mu}\otimes\gamma^{\mu}C$, and
are scalar-scalar (S) and axial-axial currents (A), respectively. The width of
$X_{c}$ was evaluated by analyzing decay channels $X_{c}\rightarrow
D_{s}^{-}\pi^{+}$ and $X_{c}\rightarrow D^{0}\overline{K}^{0}$. Performed
studies led to the following results

$m_{\mathrm{S}}=(2634\pm 62)~{}\mathrm{MeV},\ \Gamma_{\mathrm{S}}=(57.7\pm
11.6)~{}\mathrm{MeV},$ (3)

and

$m_{\mathrm{A}}=(2590\pm 60)~{}\mathrm{MeV},\ \Gamma_{\mathrm{A}}=(63.4\pm
14.2)~{}\mathrm{MeV}.$ (4)

The prediction $(2.55\pm 0.09)~{}\mathrm{GeV}$ for the mass of $X_{c}$ was
made in Ref. Chen:2016mqt , as well.

It is clear that $X_{c}$ and $X_{0}$ are different particles and their decay
channels differ from each another. Nevertheless, it is convenient to compare
parameters of $X_{c}$ with LHCb data to make some preliminary assumptions on
structure of $X_{0}$. The mass of the ground-state tetraquark $X_{c}$ is not
large enough to account for LHCb data. We must also take into account that,
the tetraquark $X_{c}$ is composed of a relatively heavy diquark $[su]$ and
heavy antidiquark $[\overline{c}\overline{d}]$, whereas $X_{0}$ would has a
light diquark $[ud]$-heavy antidiquark $[\overline{c}\overline{s}]$ structure.
Heavy-light tetraquarks are more compact and lighter than ones with the same
quark content but other compositions Agaev:2018khe . Therefore, the mass of
the resonance $X_{0}$ should be within or even below limits (3)-(4) provided
we treat it as a ground-state tetraquark: In the diquark-antidiquark picture
$X_{0}$ may be considered as a radially excited
$[ud][\overline{c}\overline{s}]$ state He:2020jna .

Alternatively, one may analyze it as a hadronic molecule, i.e., as a bound
state of conventional $D$ and $K$ mesons. Mesons $D^{-}$ and $K^{+}$ may form
a bound state if the mass of a molecule $D^{-}K^{+}$ is less than
corresponding two-meson threshold $2365~{}\mathrm{MeV}$. But this estimate is
considerably below the mass of $X_{0}$, therefore, it is difficult to expect
that the molecule $D^{-}K^{+}$ can be considered as the $X_{0}$ state. For
compounds $\overline{D}^{\ast 0}K^{\ast 0}$ (hereafter
$\overline{D}^{\ast}K^{\ast}$) and $D^{\ast-}K^{\ast+}$ relevant two-particle
thresholds are equal to $\sim 2900~{}\mathrm{MeV}$, and hence they cannot
dissociate to vector mesons $D^{\ast}$ and $K^{\ast}$ provided masses of these
molecules are below this limit: An estimation for the mass
$2848~{}\mathrm{MeV}$ of the scalar molecule $D^{\ast}\overline{K}^{\ast}$
obtained in Ref. Molina:2010tx  supports this assumption. But such hadronic
molecules can decay to a pair of pseudoscalar $D$ and $K$ mesons. Then,
structures $\overline{D}^{\ast}K^{\ast}$ and $D^{\ast-}K^{\ast+}$ may be
interpreted as $X_{0}$ if their masses are compatible with $m_{0}$. The
scenario with $D^{\ast-}K^{\ast+}$ as $X_{0}$ was realized in Ref.
Chen:2020aos , in which the authors calculated the mass of the molecule
$D^{\ast-}K^{\ast+}$ in the framework of the QCD sum rule method. Result
obtained there $2.87_{-0.14}^{+0.19}~{}\mathrm{GeV}$ indicates that an
assumption about molecule structure of $X_{0}$ deserves detailed studies.

In the present work, we treat $X_{0}$ as a hadronic molecule
$\overline{D}^{\ast}K^{\ast}$ composed of two vector mesons
$\overline{D}^{\ast 0}=\overline{c}u$ and $K^{\ast 0}=d\overline{s}$, and
compute not only its mass, but also width. The mass of $X_{0}$ is evaluated in
the context of the sum rule method, where we take into account quark, gluon
and mixed condensates up to dimension $15$. The width of $X_{0}$ is found by
considering the decay channel $X_{0}\rightarrow D^{-}K^{+}$. To this end, we
calculate the coupling $G$ that describes strong vertex $X_{0}D^{-}K^{+}$ in
the context of the light-cone sum rule (LCSR) approach using technical tools
of the soft-meson approximation. Information on this coupling obtained from
analysis allows us to estimate the width of $X_{0}$.

This work is structured in the following manner: In Section II, we calculate
the mass and coupling of the hadronic molecule $\overline{D}^{\ast}K^{\ast}$.
In Section III, we compute the strong coupling $G$ by employing the LCSR
method and soft-meson technique. In this section we find also the width of the
decay $X_{0}\rightarrow D^{-}K^{+}$. Section IV is reserved for discussion and
our conclusions.

##  II Spectroscopic parameters of the $\overline{D}^{\ast}K^{\ast}$

The mass $m$, and current coupling $f$ of the hadronic molecule
$\overline{D}^{*}K^{*}$ are necessary to check the assumption about a molecule
nature of the resonance $X_{0}$. These spectroscopic parameters are required
also to study its strong decay. We compute $m$, and $f$ using the QCD two-
point sum rule method Shifman:1978bx ; Shifman:1978by , which is one of the
effective nonperturbative approaches to determine parameters of the ordinary
and exotic hadrons.

The required sum rules can be derived from analysis of the two-point
correlation function

$\Pi(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{J(x)J^{{\dagger}}(0)\\}|0\rangle,$ (5)

where $\mathcal{T}$ denotes the time-ordered product and $J(x)$ is the
interpolating current for the scalar particle $X_{0}$. For molecule state
$\overline{D}^{*}K^{*}$ this current is given by the expression

$J(x)=[\overline{c}_{a}(x)\gamma^{\mu}u_{a}(x)][\overline{s}_{b}(x)\gamma_{\mu}d_{b}(x)].$
(6)

In Eq. (6) $c(x)$, $s(x)$, $u(x)$ and $d(x)$ stand for the corresponding quark
fields, whereas $a$ and $b$ are color indices.

Within the sum rule method the masses of various tetraquarks were analyzed in
numerous articles (see, for example, the review papers Chen:2016qju ;
Chen:2016spr ; Albuquerque:2018jkn ; Agaev:2020zad ), therefore below we
present only crucial points of performed analysis. In the sum rule method the
correlation function $\Pi(p)$ should be expressed both in terms of physical
parameters of $X_{0}$ and quark-gluon degrees of freedom. In the first case,
one finds the phenomenological side of the sum rules $\Pi^{\mathrm{Phys}}(p)$
from Eq. (5) by inserting a complete set of intermediate states. As a result
we get

$\Pi^{\mathrm{Phys}}(p)=\frac{\langle 0|J|X_{0}\rangle\langle
X_{0}|J^{\dagger}|0\rangle}{m^{2}-p^{2}}+\cdots,$ (7)

where the contribution of only the ground-state particle $X_{0}$ is shown
explicitly: Dots denote effects of higher resonances and continuum states in
the $X_{0}$ channel.

We have approximated the phenomenological side of the sum rule
$\Pi^{\mathrm{Phys}}(p)$ in Eq. (7) using a simple-pole term. But, in the case
of the multiquark systems, this approach must be applied with some caution,
because the physical side may receive a contribution also from two-hadron
reducible terms. In fact, the relevant interpolating current couples not only
to the multiquark hadron, but interacts also with two conventional hadron
states lying below the mass of the multiquark system Kondo:2004cr ; Lee:2004xk
. These contributions can be either subtracted from the sum rules or included
into parameters of the pole term. In the case of the tetraquarks the second
approach is preferable and was applied in articles Wang:2015nwa ;
Agaev:2018vag ; Sundu:2018nxt . It appears that, the two-meson states generate
the finite width $\Gamma(p)$ of the tetraquark and lead to modification

$\frac{1}{m^{2}-p^{2}}\rightarrow\frac{1}{m^{2}-p^{2}-i\sqrt{p^{2}}\Gamma(p)}.$
(8)

These effects, properly taken into account in the sum rules, rescale the
coupling $f$ leaving stable the mass $m$ of the tetraquark. Detailed analyses
proved that two-hadron contributions as a whole, and two-meson ones in
particular are small, and can be neglected Lee:2004xk ; Wang:2015nwa ;
Agaev:2018vag ; Sundu:2018nxt . Therefore, to derive the phenomenological side
of the sum rules, we use in Eq. (7) the zero-width single-pole approximation.

Introducing the spectroscopic parameters of $X_{0}$ through the matrix element

$\langle 0|J|X_{0}\rangle=fm,$ (9)

we rewrite $\Pi^{\mathrm{Phys}}(p)$ in the final form

$\Pi^{\mathrm{Phys}}(p)=\frac{f^{2}m^{2}}{m^{2}-p^{2}}+\cdots.$ (10)

The function $\Pi^{\mathrm{Phys}}(p)$ has a simple Lorentz structure
proportional to $\sim I$, and the term in Eq. (10) is the invariant amplitude
$\Pi^{\mathrm{Phys}}(p^{2})$ corresponding to this structure.

The second component of the sum rules $\Pi^{\mathrm{OPE}}(p)$, is calculated
in the operator product expansion ($\mathrm{OPE}$) with some accuracy. To find
$\Pi^{\mathrm{OPE}}(p)$, we employ the explicit expression of the
interpolating current $J(x)$ in Eq. (5), and contract relevant heavy and light
quark fields. After these manipulations, for $\Pi^{\mathrm{OPE}}(p)$ we get

$\displaystyle\Pi^{\mathrm{OPE}}(p)=i\int
d^{4}xe^{ipx}\mathrm{Tr}\left[\gamma^{\mu}S_{u}^{aa^{\prime}}(x)\gamma^{\nu}S_{c}^{a^{\prime}a}(-x)\right]$
$\displaystyle\times$
$\displaystyle\mathrm{Tr}\left[\gamma_{\mu}S_{d}^{bb^{\prime}}(x)\gamma_{\nu}S_{s}^{b^{\prime}b}(-x)\right],$
(11)

where $S_{c}(x)$ and $S_{u(s,d)}(x)$ are the heavy $c$\- and light
$u(s,d)$-quark propagators, respectively. Their explicit expressions can be
found, for instance, in Ref. Agaev:2020zad . The correlation function has also
a trivial structure: We denote the relevant invariant amplitude by
$\Pi^{\mathrm{OPE}}(p^{2})$.

The sum rules for $m$ and $f$ can be found by equating
$\Pi^{\mathrm{Phys}}(p^{2})$ and $\Pi^{\mathrm{OPE}}(p^{2})$ and performing
standard manipulations of the method. First of all, one should apply the Borel
transformation to both sides of this equality to suppress contributions of
higher resonances and continuum states. At the next stage, by using the
hypothesis about quark-hadron duality, one subtracts higher resonances and
continuum terms from the physical side of the equality. As a result, the sum
rule equality starts to depend on the Borel $M^{2}$ and continuum threshold
$s_{0}$ parameters.

The second equality required to find sum rules is obtained by applying the
operator $d/d(-1/M^{2})$ to the first expression. Then the sum rules for $m$
and $f$ are

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

and

$f^{2}=\frac{e^{m^{2}/M^{2}}}{m^{2}}\Pi(M^{2},s_{0}).$ (13)

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

The function $\Pi(M^{2},s_{0})$ has the following form

$\Pi(M^{2},s_{0})=\int_{\mathcal{M}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}).$
(14)

Throughout this article, we neglect the mass of the quarks $u$ and $d$, and
set $\mathcal{M}=m_{c}+m_{s}$ in Eq. (14). The two-point spectral density
$\rho^{\mathrm{OPE}}(s)$ is computed as an imaginary part some of terms in the
correlation function $\Pi^{\mathrm{OPE}}(p)$. The component $\Pi(M^{2})$ is
the Borel transformation of remaining terms in $\Pi^{\mathrm{OPE}}(p)$, and
are obtained directly from their expressions. Calculations are carried out by
taking into account vacuum condensates up to dimension $15$. The
dimension-$15$ contribution to the correlation function is proportional to
product of light quark condensates $\langle\overline{s}g_{s}\sigma
Gs\rangle\langle\overline{q}g_{s}\sigma Gq\rangle^{2}$. This and other higher
dimensional terms in $\Pi(M^{2},s_{0})$ are obtained as products of basic
vacuum condensates using factorization procedure by assuming that it does not
lead to essential ambiguities. Analytical expressions of
$\rho^{\mathrm{OPE}}(s)$ and $\Pi(M^{2})$ are rather lengthy to be presented
here explicitly.

The sum rules (12) and (13) depend on universal quark
$\langle\overline{q}q\rangle=\langle 0|\overline{q}q|0\rangle$, gluon
$\langle\alpha_{s}G^{2}/\pi\rangle=\langle 0|\alpha_{s}G^{2}/\pi|0\rangle$ and
mixed quark-gluon $\langle\overline{q}g_{s}\sigma Gq\rangle=\langle
0|\overline{q}g_{s}\sigma Gq|0\rangle$ vacuum condensates ($q\equiv u,~{}d$,
and similar expressions for the strange quark $s$) Shifman:1978bx ;
Shifman:1978by ; Ioffe:1981kw ; Ioffe:2005ym  and masses of $c$ and $s$ quarks

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\ \langle\overline{s}s\rangle=(0.8\pm
0.1)\langle\overline{q}q\rangle,$ $\displaystyle\langle\overline{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle,\
\langle\overline{s}g_{s}\sigma
Gs\rangle=m_{0}^{2}\langle\overline{s}s\rangle,$ $\displaystyle
m_{0}^{2}=(0.8\pm 0.2)~{}\mathrm{GeV}^{2}$
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle
m_{s}=93_{-5}^{+11}~{}\mathrm{MeV},\ m_{c}=1.27\pm 0.2~{}\mathrm{GeV}.$ (15)

As is seen, the vacuum condensate of strange quarks is different from $\langle
0|\overline{q}q|0\rangle$ Ioffe:1981kw . The mixed condensates
$\langle\overline{q}g_{s}\sigma Gq\rangle$ and $\langle\overline{s}g_{s}\sigma
Gs\rangle$ can be expressed in terms of the corresponding quark condensates
and parameter $m_{0}^{2}$, numerical value of which was extracted from
analysis of baryonic resonances Ioffe:2005ym .

Figure 1: The mass $m$ of the $X_{0}$ as a function of the Borel parameter
$M^{2}$ at fixed $s_{0}$ (left panel), and as a function of the continuum
threshold parameter $s_{0}$ at fixed $M^{2}$ (right panel).

The $m$ and $f$ are functions of the parameters $M^{2}$ and $s_{0}$, as well.
The correct choice for $M^{2}$ and $s_{0}$ is an important problem of sum rule
computations. The working regions for $M^{2}$ and $s_{0}$ should satisfy usual
constraints imposed on the pole contribution ($\mathrm{PC}$) and convergence
of the operator product expansion. To analyze these questions, we introduce
the quantities

$\mathrm{PC}=\frac{\Pi(M^{2},s_{0})}{\Pi(M^{2},\infty)},$ (16)

and

$R(M^{2})=\frac{\Pi^{\mathrm{DimN}}(M^{2},s_{0})}{\Pi(M^{2},s_{0})}.$ (17)

In Eq. (17) $\Pi^{\mathrm{DimN}}(M^{2},s_{0})$ is a last term (or a sum of
last few terms) in the correlation function. In the present analysis, we use
the sum of last three terms in $\mathrm{OPE}$, and hence $\mathrm{DimN\equiv
Dim(13+14+15)}$.

The $\mathrm{PC}$ is used to fix upper bound for $M^{2}$, whereas $R(M^{2})$
is necessary to find low limit for the Borel parameter. These two values of
$M^{2}$ fix the boundaries of a region where the Borel parameter can be
varied. Our analysis shows that the working regions for the parameters $M^{2}$
and $s_{0}$ are

$M^{2}\in[2,3]\ \mathrm{GeV}^{2},\ s_{0}\in[11.3,13.3]\ \mathrm{GeV}^{2},$
(18)

and they obey restrictions on $\mathrm{PC}$ and convergence of $\mathrm{OPE}$.
Thus, at $M^{2}=3~{}\mathrm{GeV}^{2}$ the pole contribution is $0.5$, whereas
at $M^{2}=2~{}\mathrm{GeV}^{2}$ it becomes equal to $0.8$. At the minimum of
$M^{2}=2~{}\mathrm{GeV}^{2}$, we find $R\approx 0.01$, which guarantees the
convergence of the sum rules. We extract the parameters $m$ and $f$
approximately at a middle point of the window (18),
$M^{2}=2.5~{}\mathrm{GeV}^{2}$ and $s_{0}=12~{}\mathrm{GeV}^{2}$, where the
pole contribution is $\mathrm{PC}\approx 0.65$. This fact ensures the ground
state nature of $X_{0}$.

Our predictions for $m$ and $f$ are

$\displaystyle m$ $\displaystyle=$ $\displaystyle(2868~{}\pm
198)~{}\mathrm{MeV},$ $\displaystyle f$ $\displaystyle=$ $\displaystyle(3.0\pm
0.7)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (19)

The sum rule results, in general, should not depend on the parameter $M^{2}$.
But in real calculations $m$ and $f$ are sensitive to the choice of $M^{2}$.
From inspection of Eq. (19) it is seen, that theoretical uncertainties in the
case of $m$ equal to $\pm 6.9\%$, whereas for the coupling $f$ they amount to
$\pm 23.3\%$. Theoretical ambiguities of $f$ are larger, because $f$ is
determined directly in terms of $\Pi(M^{2},s_{0})$, whereas the sum rule for
$m$ depends on the ratio of such functions and is exposed to smaller
variations. Nevertheless, uncertainties even for the coupling $f$ remain
within limits accepted in sum rule computations. In Fig. 1 we display the sum
rule’s predictions for $m$ as a function of $M^{2}$, where one can see its
residual dependence on the Borel parameter.

The continuum threshold parameter $s_{0}$ separates a ground-state
contribution from effects due to higher resonances and continuum states. In
other words, $\sqrt{s_{0}}$ has to be smaller than the mass $m^{\ast}$ of the
first excitation of the $X_{0}$. The self-consistent sum rule analysis implies
that the difference $\sqrt{s_{0}}-m$ is around or less than $m^{\ast}-m$.
Excited states of conventional hadrons and their parameters are known either
from experimental measurements or from alternative theoretical studies. In the
case of the multiquark hadrons there is a deficiency of relevant information.
The mass gap $\sqrt{s_{0}}-m\approx(500-600)~{}\mathrm{MeV}$ found in the
present work can be considered as a reasonable estimate
$m^{\ast}\approx(m+500)~{}\mathrm{MeV}$ for the hadronic molecule
$\overline{D}^{\ast}K^{\ast}$ containing one heavy quark. Dependence of
extracted value of $m$ on the scale $s_{0}$ is also shown in Fig. 1.

Obtained prediction for the mass of the state $\overline{D}^{\ast}K^{\ast}$ is
in a nice agreement with new LHCb measurements. This is necessary, but not
enough to interpret $X_{0}$ as the hadronic molecule. For reliable
conclusions, we need to calculate the width of the molecule
$\overline{D}^{\ast}K^{\ast}$ and confront it with data: only together these
parameters can support or not assumptions about the structure of $X_{0}$.

##  III The decay $X_{0}\rightarrow D^{-}K^{+}$

In this section we explore the strong decay $X_{0}\rightarrow D^{-}K^{+}$ in
order to find width of the resonance $X_{0}$. Strictly speaking, there are
other decay channel of the scalar state $X_{0}$, namely the $S$-wave decay to
a pair of mesons $\overline{D}^{0}K^{0}$. Because $X_{0}$ was observed as
enhancement in the $D^{-}K^{+}$ mass distribution, we concentrate on the first
process and saturate full width of the resonance $X_{0}$ by this channel.

The width of the decay $X_{0}\rightarrow D^{-}K^{+}$ is determined by the
strong coupling $G$ corresponding to the vertex $X_{0}D^{-}K^{+}$. We are
going to calculate $G$ using method of the QCD sum rule on the light-cone
Balitsky:1989ry ; Belyaev:1994zk  and methods of the soft-meson approximation
Ioffe:1983ju . To this end, we start from analysis of the correlation function
Belyaev:1994zk ; Cragie:1982ng

$\Pi(p,q)=i\int d^{4}xe^{ipx}\langle
K(q)|\mathcal{T}\\{J^{D}(x)J^{{\dagger}}(0)\\}|0\rangle,$ (20)

where by $K$ and $D$ we denote, in short forms, the mesons $K^{+}$ and
$D^{-}$, respectively. In the correlation function $\Pi(p,q)$, the
interpolating current $J(x)$ is given by Eq. (6), whereas for $J^{D}(x)$ we
use

$J^{D}(x)=\overline{c}_{n}(x)i\gamma_{5}d_{n}(x),$ (21)

with $n$ being the color index. It is not difficult to determine $\Pi(p,q)$ in
terms of the physical parameters of the particles involved into the decay
Belyaev:1994zk . By taking into account the ground states in the $D$ and
$X_{0}$ channels, we get

$\displaystyle\Pi^{\mathrm{Phys}}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|J^{D}|D\left(p\right)\rangle}{p^{2}-m_{D}^{2}}\langle
D\left(p\right)K(q)|X_{0}(p^{\prime})\rangle$ (22)
$\displaystyle\times\frac{\langle
X_{0}(p^{\prime})|J^{\dagger}|0\rangle}{p^{\prime 2}-m^{2}}+\cdots,$

where $p$, $q$ and $p^{\prime}=p+q$ are the momenta of the particles $D$, $K$,
and $X_{0}$, and $m_{D}$ is the mass of $D$ meson. The ellipses in Eq. (22)
refer to contributions of higher resonances and continuum states in the $D$
and $X_{0}$ channels.

In order to finish computation of the correlation function, we introduce the
matrix elements

$\displaystyle\langle
0|J^{D}|D\left(p\right)\rangle=\frac{f_{D}m_{D}^{2}}{m_{c}},\ \langle
X_{0}(p^{\prime})|J^{\dagger}|0\rangle=fm,$ $\displaystyle\langle
D\left(p\right)K(q)|X_{0}(p^{\prime})\rangle=Gp\cdot p^{\prime}.$ (23)

In expressions above $f_{D}\,$ is decay constant of the meson $D^{-}$. Then
for $\Pi^{\mathrm{Phys}}(p,q)$ we find

$\Pi^{\mathrm{Phys}}(p,q)=\frac{Gfmf_{D}m_{D}^{2}}{m_{c}(p^{2}-m_{D}^{2})(p^{\prime
2}-m^{2})}p\cdot p^{\prime}+\cdots.$ (24)

To continue, we have to calculate $\Pi^{\mathrm{QCD}}(p,q)$ in terms of the
quark-gluon degrees of freedom and find the QCD side of the sum rule.
Contractions in Eq. (20) of $c$ and $d$ quark and antiquark fields yield

$\displaystyle\Pi^{\mathrm{OPE}}(p,q)=\int
d^{4}xe^{ipx}\left[\gamma^{\mu}S_{c}^{bn}(-x){}\gamma_{5}\right.$
$\displaystyle\left.\times
S_{d}^{na}(x){}\gamma_{\mu}\right]_{\alpha\beta}\langle
K(q)|\overline{u}_{\alpha}^{b}(0)s_{\beta}^{a}(0)|0\rangle,$ (25)

where $\alpha$ and $\beta$ are the spinor indexes.

Apart from quark propagators the function $\Pi^{\mathrm{OPE}}(p,q)$ depends
also on local matrix elements of the quark operator $\overline{u}s$ sandwiched
between the vacuum and $K$ meson. To express $\langle
K(q)|\overline{u}_{\alpha}^{b}(0)s_{\beta}^{a}(0)|0\rangle$ using the $K$
meson’s local matrix elements, we expand $\overline{u}(0)s(0)$ over the full
set of Dirac matrices $\Gamma^{J}$ and project them onto the color-singlet
states

$\overline{u}_{\alpha}^{b}(0)s_{\beta}^{a}(0)\rightarrow\frac{1}{12}\delta^{ba}\Gamma_{\beta\alpha}^{J}\left[\overline{u}(0)\Gamma^{J}s(0)\right],$
(26)

where

$\Gamma^{J}=\mathbf{1},\ \gamma_{5},\ \gamma_{\mu},\ i\gamma_{5}\gamma_{\mu},\
\sigma_{\mu\nu}/\sqrt{2}.$ (27)

The expression (26) reveals a main difference between vertices composed of
conventional mesons and vertices containing a tetraquark and two ordinary
mesons. Indeed, in the vertices of ordinary mesons the correlation function
depends on distribution amplitudes (DAs) of one of the final-state mesons, for
example, on DAs of $K$ meson. The DAs of the mesons are determined as non-
local matrix elements of relevant quark fields placed between the meson and
vacuum states. In the case under discussion, it is evident that instead of
non-local matrix elements, we have $\Pi^{\mathrm{QCD}}(p,q)$ that contains
local matrix elements of $K$ meson. The reason is that $X_{0}$ and
interpolating current Eq. (6) are built of four quark fields at the same
space-time location. Substitution of this current into the correlation
function and contractions of $c$ and $d$ quark fields yield expressions, where
the remaining light quarks are sandwiched between the $K$ meson and vacuum
states, forming local matrix elements. In other words, we encounter the
situation when dependence of the correlation function on the meson DAs
disappears and integrals over the meson DAs reduce to overall normalization
factors. In the framework of the LCSR method such situation is possible in the
kinematical limit $q\rightarrow 0$, when the light-cone expansion is replaced
by the short-distant one. As a result, instead of the expansion in terms of
DAs, one gets expansion over the local matrix elements Belyaev:1994zk . The
limit $q\rightarrow 0$ is known as the soft-meson approximation. In this
approximation $p=p^{\prime}$ and invariant amplitudes
$\Pi^{\mathrm{Phys}}(p^{2})$ and $\Pi^{\mathrm{OPE}}(p^{2})$ depend only on
the variable $p^{2}$. For our purposes a decisive fact is the observation made
in Ref. Belyaev:1994zk : the soft-meson approximation and full LCSR treatment
of the conventional mesons’ vertices lead to predictions which are numerically
very close to each other.

The soft-meson approximation simplifies the QCD component of the sum rule, but
leads to additional complications in its phenomenological side. In this limit
for invariant amplitude $\Pi^{\mathrm{Phys}}(p^{2})$ we get

$\Pi^{\mathrm{Phys}}(p^{2})=G\frac{fmf_{D}m_{D}^{2}\widetilde{m}^{2}}{m_{c}(p^{2}-\widetilde{m}^{2})^{2}}+\cdots,$
(28)

where $\widetilde{m}^{2}=(m^{2}+m_{D}^{2})/2$. This amplitude contains the
double pole at $p^{2}=\widetilde{m}^{2}$, therefore its Borel transformation
is given by the formula

$\displaystyle\Pi^{\mathrm{Phys}}(M^{2})=G\frac{fmf_{D}m_{D}^{2}}{m_{c}}\frac{\widetilde{m}^{2}e^{-\widetilde{m}^{2}/M^{2}}}{M^{2}}+\cdots.$
(29)

In the standard approach the invariant amplitude
$\Pi^{\mathrm{Phys}}(p^{2},p^{\prime 2})$ depends on two variables $p^{2}$ and
$p^{\prime 2}$, and the Borel transformations over $p^{2}$ and $p^{\prime 2}$
suppress contributions of higher resonances and continuum states. The
suppressed terms afterwards can be subtracted using assumption on quark-hadron
duality. But in the soft limit even after Borel transformation besides ground-
state term $\Pi^{\mathrm{Phys}}(M^{2})$ contains additional unsuppressed
contributions. This is a price to be paid in the soft approximation for simple
$\Pi^{\mathrm{OPE}}(p^{2})$ expression. To remove contaminating contributions
from the phenomenological side of the sum rule, one has to act on
$\Pi^{\mathrm{Phys}}(M^{2})$ by the operator Belyaev:1994zk ; Ioffe:1983ju

$\mathcal{P}(M^{2},m^{2})=\left(1-M^{2}\frac{d}{dM^{2}}\right)M^{2}e^{m^{2}/M^{2}},$
(30)

that singles out the ground-state term. Contributions remaining in
$\Pi^{\mathrm{Phys}}(M^{2})$ after this prescription can be subtracted by the
usual way. Then the sum rule for the strong coupling $G$ reads

$G=\frac{m_{c}\widetilde{m}^{2}}{fmf_{D}m_{D}^{2}}\mathcal{P}(M^{2},\widetilde{m}^{2})\Pi^{\mathrm{OPE}}(M^{2},s_{0}).$
(31)

Returning to the calculation of $\Pi^{\mathrm{OPE}}(p,q)$, it is worth noting
that by substituting the expansion (26) into Eq. (20), one has to perform
summations over color indices and fix local matrix elements of $K$ meson that
contribute to $\Pi^{\mathrm{OPE}}(p^{2})$ in the soft limit. There are only a
few such elements: Two-particle matrix elements of twist-2 and twist-3

$\displaystyle\langle
0|\overline{u}\gamma_{\mu}\gamma_{5}s|K(q)\rangle=if_{K}q_{\mu},$
$\displaystyle\langle
0|\overline{u}i\gamma_{5}s|K\rangle=\frac{f_{K}m_{K}^{2}}{m_{s}}.$ (32)

There are also three-particle local matrix elements of $K$ meson, for an
example,

$\langle
0|\overline{u}\gamma^{\nu}\gamma_{5}igG_{\mu\nu}s|K(q)\rangle=iq_{\mu}f_{K}m_{K}^{2}\kappa_{4K},$
(33)

which may contribute to $\Pi^{\mathrm{OPE}}(p^{2})$. The elements given by Eq.
(32) appear due to the propagator $S_{d}$, perturbative term of $S_{c}$, and
the expansion (26), whereas three-particle matrix elements may contribute
after gluon insertions to $\overline{u}\Gamma^{J}s$ stemming from
nonperturbative components of the propagator $S_{c}$. In matrix elements (32)
and (33) quark and gluon fields are fixed at the same position $x=0$, and
$\kappa_{4K}$ is the twist-4 matrix element of the $K$ meson.

Procedures to calculate the correlation function in the soft approximation
were presented in Refs. Agaev:2016ijz ; Agaev:2016dev , therefore we skip
further technical details and provide final expression for the correlation
function, which is calculated with dimension-$9$ accuracy and given as a sum
of the perturbative and nonperturbative components

$\displaystyle\Pi^{\mathrm{OPE}}(M^{2},s_{0})$ $\displaystyle=$
$\displaystyle\frac{\mu_{K}}{8\pi^{2}}\int_{\mathcal{M}^{2}}^{s_{0}}\frac{ds(m_{c}^{2}-s)^{2}}{s}e^{-s/M^{2}}+\Pi_{\mathrm{NP}}(M^{2}).$

The nonperturbative component of the correlation function
$\Pi_{\mathrm{NP}}(M^{2})$ has the following form

$\displaystyle\Pi_{\mathrm{NP}}(M^{2})=-\frac{\langle\overline{d}d\rangle\mu_{K}m_{c}}{3}e^{-m_{c}^{2}/M^{2}}$
$\displaystyle+\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\frac{\mu_{K}m_{c}^{4}}{72M^{4}}\int_{0}^{1}\frac{dz}{z^{3}(z-1)^{3}}e^{-m_{c}^{2}/[M^{2}z(1-z)]}$
$\displaystyle+\frac{\langle\overline{d}g\sigma
Gd\rangle\mu_{K}m_{c}^{3}}{6M^{4}}e^{-m_{c}^{2}/M^{2}}-\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\langle\overline{d}d\rangle$
$\displaystyle\times\frac{\mu_{K}m_{c}(m_{c}^{2}+3M^{2})\pi^{2}}{54M^{6}}e^{-m_{c}^{2}/M^{2}}+\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\langle\overline{d}g\sigma
Gd\rangle$
$\displaystyle\times\frac{\mu_{K}m_{c}(m_{c}^{4}+6M^{2}m_{c}^{2}+6M^{4})\pi^{2}}{216M^{10}}e^{-m_{c}^{2}/M^{2}}.$
(35)

In expressions above, we introduce $\mu_{K}=f_{K}m_{K}^{2}/m_{s}$. It turns
out that only the twist-3 matrix element from Eq. (32) contributes to the
function $\Pi_{\mathrm{NP}}(M^{2})$. The last term in
$\Pi_{\mathrm{NP}}(M^{2})$ is proportional to
$\langle\alpha_{s}G^{2}/\pi\rangle\langle\overline{d}g\sigma Gd\rangle$ with
dimension $9$, and is suppressed additionally by the factor $1/M^{6}$.
Therefore, dimension-$9$ accuracy for computation of
$\Pi^{\mathrm{OPE}}(M^{2},s_{0})$ adopted in the present article is high and
enough to get reliable result.

Quantities | Values (in $\mathrm{MeV}$ units)
---|---
$m_{D}$ | $1869.61\pm 0.10$
$m_{K}$ | $493.68\pm 0.02$
$f_{D}$ | $211.9\pm 1.1$
$f_{K}$ | $155.6\pm 0.4$
Table 1: Parameters of the mesons $D^{-}$ and $K^{+}$ used in numerical
computations.

The sum rule (31) depends on the various vacuum condensates written down above
(15). It contains the masses and decay constants of the final-state mesons
$D^{-}$ and $K^{+}$: relevant spectroscopic parameters are collected in Table
1. All of them are borrowed from Ref. Tanabashi:2018oca . For decay constants
$f_{D}$ and $f_{K}$ Particle Data Group’s averages are used.

To carry out numerical analysis one also needs to fix $M^{2}$ and $s_{0}$. The
restrictions imposed on these auxiliary parameters are standard for sum rule
computations and have been discussed above. Our analysis demonstrates that
working regions (18) meet all constraints necessary for computations of
$\Pi^{\mathrm{OPE}}(M^{2},s_{0})$. Numerical calculations lead to the
following result

$|G|=(0.66\pm 0.06)\ \mathrm{GeV}^{-1}.$ (36)

This prediction for the strong coupling $G$ is typical for tetraquark-meson
vertices. Its numerical value and dimension are determined by definition of
matrix element $\langle D\left(p\right)K(q)|X_{0}(p^{\prime})\rangle$:
Modification of the vertex $DKX_{0}$ in Eq. (23) changes the value and
dimension of $G$. In general, it is possible to rewrite $\langle
D\left(p\right)K(q)|X_{0}(p^{\prime})\rangle$ in such a way that to make $G$
dimensionless. In our analysis $G$ is an intermediate parameter, whereas the
physical quantity to be found is the width $\Gamma$ of the decay
$X_{0}\rightarrow D^{-}K^{+\text{ }}$. The $\Gamma$ is calculated by taking
into account Eq. (23), and its expression depends on these matrix elements.
But regardless used convention for the vertex $DKX_{0}$ and analytical form of
the width, numerical computations lead to the same final result with correct
dimension, as it should be for a physical quantity.

Having used the matrix elements given by Eq. (23), we derive for the width of
the decay $X_{0}\rightarrow D^{-}K^{+\text{ }}$

$\Gamma\left[X_{0}\rightarrow D^{-}K^{+\text{
}}\right]=\frac{G^{2}m_{D}^{2}\lambda}{8\pi}\left(1+\frac{\lambda^{2}}{m_{D}^{2}}\right),$
(37)

where $\lambda=\lambda\left(m,m_{D},m_{K}\right)$ and

$\displaystyle\lambda\left(a,b,c\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2a}\left(a^{4}+b^{4}+c^{4}\right.$ (38)
$\displaystyle\left.-2\left(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2}\right)\right)^{1/2}.$

Then it is not difficult to find that

$\Gamma\left[X_{0}\rightarrow D^{-}K^{+}\right]=(49.6\pm 9.3)~{}\mathrm{MeV}.$
(39)

This prediction for the width of the resonance $X_{0}$ is in a reasonable
agreement with new LHCb data (1).

The molecule $\overline{D}^{\ast 0}K^{\ast 0}$ is composed of two neutral
vector mesons, which are strong-interaction unstable particles. The meson
$\overline{D}^{\ast}(2007)^{0}$ is relatively narrow state
$\Gamma_{\overline{D}^{\ast}}<2.1~{}\mathrm{MeV}$, whereas the width
$\Gamma_{K^{\ast}}=(47.4\pm 0.5)~{}\mathrm{MeV}$ of $K^{\ast}(892)^{0}$ within
experimental errors is comparable with LHCb data $\Gamma_{0}$. In Ref.
Chen:2020aos  $X_{0}$ was modeled as hadronic $D^{\ast-}K^{\ast+}$ molecule,
and width of the meson $K^{\ast+}$ was used to estimate roughly the $X_{0}$
resonance’s width. The hadronic molecules $D^{\ast-}K^{\ast+}$ and
$\overline{D}^{\ast 0}K^{\ast 0}$ are bound states, and partial widths of
their decay channels are determined by quark-gluon interactions inside of
these particles. Due to multiquark nature of molecules their internal dynamics
obviously differs from those of free mesons $D^{\ast}$ and $K^{\ast}$.
Therefore, estimation of the $D^{\ast-}K^{\ast+}$ and $\overline{D}^{\ast
0}K^{\ast 0}$ molecules’ widths using directly widths of constituent mesons
seems us to be problematic. One can suggest, that after dissociation of
$\overline{D}^{\ast 0}K^{\ast 0}$ to two vector mesons, decays of $K^{\ast 0}$
may be used for such analysis. But the mass of the $X_{0}$ is below relevant
two-meson thresholds in both pictures, i.e., $X_{0}$ does not decay to mesons
$\overline{D}^{\ast 0}+K^{\ast 0}$ or $D^{\ast-}+K^{\ast+}$. The $P$-wave
decay $\overline{D}^{\ast 0}K^{\ast
0}\rightarrow\overline{D}_{0}^{\ast}(2400)^{0}+K^{\ast 0}$ with the vector
meson $K^{\ast 0}$ in the final state is forbidden kinematically. Another two-
body $P$-wave decays of $\overline{D}^{\ast 0}K^{\ast 0}$, for example, to
mesons $\overline{D}^{\ast}(2007)^{0}+K_{0}^{\ast}(1430)$ and
$\overline{D}_{1}(2420)^{0}+K^{0}$ are not allowed because of the same
reasons. Nevertheless, there are multibody decays of $X_{0}$ which contribute
to its full width. For example, processes $X_{0}\rightarrow
D^{-}K^{+}\pi^{0}\pi^{0}$ and $X_{0}\rightarrow D^{-}K^{+}\pi^{+}\pi^{-}$ can
improve our prediction for $\Gamma$. But these processes imply production of
$4$ new valence quarks through different mechanisms, which suppress relevant
amplitudes by the factor $\alpha_{s}$ or additional strong couplings. As a
result, widths of such subdominant decays would be small.

We have explored the dominant decay channel $X_{0}\rightarrow D^{-}K^{+}$ of
the resonance $X_{0}$. The result for its width in Eq. (39) has been obtained
in the context of the LCSR method by applying first principles of the QCD, and
is reliable prediction for this parameter. It can be improved further by
including into analysis other decay channels of the $X_{0}$, which are beyond
the scope of the present article.

##  IV Discussion and conclusions

In the present work we have explored one of two new resonances $X_{0}$ and
$X_{1}$ reported by the LHCb collaboration. Namely, we have considered $X_{0}$
as a scalar hadronic molecule $\overline{D}^{\ast}K^{\ast}$, and calculated
its mass and width. For these purposes, we have used the QCD sum rule method.
The spectroscopic parameters of the state $\overline{D}^{\ast}K^{\ast}$ have
been extracted from two-point sum rules, whereas for analysis of its strong
decay channel, we used LCSR method and soft-meson approximation. Obtained
predictions for $m$ and $\Gamma$ are in nice agreement with reported LHCb
data, which can be interpreted in favor of molecule nature of the resonance
$X_{0}$.

The suggestion about molecular structure of $X_{0}$ was made in Ref.
Chen:2020aos , in which the authors computed the mass of the molecule
$D^{\ast-}K^{\ast+}$ using the sum rule method. Calculations were done by
taking into account nonperturbative terms up to dimension $8$. Prediction
obtained there for $\widetilde{m}$

$\widetilde{m}=2870_{-140}^{+190}~{}\mathrm{MeV}$ (40)

is very close to our result. The molecule composition for the $X_{0}$ in
different frameworks was proposed in Refs. Hu:2020mxp ; Liu:2020nil ;
Huang:2020ptc , as well.

Alternatively, the resonance $X_{0}$ was analyzed in Refs. Karliner:2020vsi ;
Wang:2020xyc ; He:2020jna ; Zhang:2020oze  by assuming that it is a diquark-
antidiquark state. The sum rule prediction for the mass of the scalar
tetraquark $T_{1}=[sc][\overline{u}\overline{d}]$ with an axial-axial
$C\gamma_{\mu}\otimes\gamma^{\mu}C$ type structure reads Wang:2020xyc

$m_{T_{1}}=(2910\pm 120)~{}\mathrm{MeV}.$ (41)

The similar sum rule investigations were performed in Ref. Zhang:2020oze .
Results for masses of the scalar tetraquark
$T_{2}=[ud][\overline{s}\overline{c}]$ with scalar-scalar and axial-axial
structures are equal to

$\displaystyle m_{T_{2S}}$ $\displaystyle=$ $\displaystyle
2750_{-190}^{+180}~{}\mathrm{MeV},$ $\displaystyle m_{T_{2A}}$
$\displaystyle=$ $\displaystyle 2770_{-200}^{+190}~{}\mathrm{MeV},$ (42)

respectively. By taking into account uncertainties of calculations, the author
concluded that $X_{0}$ could be interpreted as a tetraquark
$J^{\mathrm{P}}=0^{+}$ with either scalar-scalar or axial-axial
configurations. It is seen that states $T_{1}$ and $T_{2}$ are connected by
the relations $T_{1}=\overline{T_{2}}$ or $\overline{T_{1}}=T_{2}$ as
conventional mesons, for example, $D^{0}=c\overline{u}$ and
$\overline{D^{0}}=\overline{c}u$. Masses of such particles should be equal to
each other, which is not the case for $m_{T_{1}}$ and $m_{T_{2A}}$. In our
view, additional studies are necessary to solve problems existing in the QCD
sum rule analyses of the resonance $X_{0}$ in the diquark-antidiquark picture.

Interesting analysis of the ground-state and radially excited tetraquark
$T_{2}$ was performed in Ref. He:2020jna . In this paper the mass of the
particles $T_{2}(1S)$ and $T_{2}(2S)$ were found equal to
$2360~{}\mathrm{MeV}$ and $2860~{}\mathrm{MeV}$, respectively. As a result,
the resonance $X_{0}$ was interpreted there as the excited tetraquark
$T_{2}(2S)$.

The enhancements in the $D^{-}K^{+}$ mass distribution labeled by $X_{0}$ and
$X_{1}$ may have alternative origin Liu:2020orv . Thus, the authors of Ref.
Liu:2020orv  investigated the process $B^{+}\rightarrow D^{+}D^{-}K^{+}$ via
$\chi_{c1}D^{\ast-}K^{\ast+}$ and $D_{sJ}\overline{D}_{1}^{0}K^{0}$
rescattering diagrams. It was argued that, two resonance-like peaks obtained
around thresholds $D^{\ast-}K^{\ast+}$ and $\overline{D}_{1}^{0}K^{0}$ may
simulate the states $X_{0}$ and $X_{1}$ without a necessity to introduce
genuine four-quark mesons. The observed experimental peaks were explained
there by the triangle singularities in the scattering amplitudes located in
the vicinity of the physical boundary.

Experimental results obtained by the LHCb collaboration do not raise doubts
about existence of the resonance-like enhancements $X_{0}$ and $X_{1}$ in the
$D^{-}K^{+}$ mass distribution. These structures were already considered as
meson molecules, diquark-antidiquark systems, rescattering effects. In other
words, there are different and controversial interpretations of the structures
$X_{0}$ and $X_{1}$ in the literature. Additional theoretical efforts seem are
required to clarify nature of these states.

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