# Doubly charged vector tetraquark
$Z_{\mathrm{V}}^{++}=[cu][\overline{s}\overline{d}]$

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  H. Sundu  Department of
Physics, Kocaeli University, 41380 Izmit, Turkey

###### Abstract

We explore properties of the doubly charged vector tetraquark
$Z_{\mathrm{V}}^{++}=[cu][\overline{s}\overline{d}]$ built of four quarks of
different flavors using the QCD sum rule methods. The mass and current
coupling of $Z_{\mathrm{V}}^{++}$ are computed in the framework of the QCD
two-point sum rule approach by taking into account quark, gluon and mixed
vacuum condensates up to dimension $10$. The full width of this tetraquark is
saturated by $S$-wave $Z_{\mathrm{V}}^{++}\to\pi^{+}D_{s1}(2460)^{+},\
\rho^{+}D_{s0}^{\ast}(2317)^{+}$, and $P$-wave
$Z_{\mathrm{V}}^{++}\to\pi^{+}D_{s}^{+},\ K^{+}D^{+}$ decays. Strong couplings
required to find partial widths of aforementioned decays are calculated in the
context of the QCD light-cone sum rule method and a soft-meson approximation.
Our predictions for the mass $m=(3515\pm 125)~{}\mathrm{MeV}$ and full width
$\Gamma_{\mathrm{full}}=156_{-30}^{+56}~{}\mathrm{MeV}$ of
$Z_{\mathrm{V}}^{++}$ are useful to search for this exotic meson in various
processes. Recently, the LHCb collaboration discovered neutral states
$X_{0(1)}(2900)$ as resonance-like peaks in $D^{-}K^{+}$ invariant mass
distribution in the decay $B^{+}\to D^{+}D^{-}K^{+}$. We argue that mass
distribution of $D^{+}K^{+}$ mesons in the same $B$ decay can be used to
observe the doubly charged scalar $Z_{\mathrm{S}}^{++}$ and vector
$Z_{\mathrm{V}}^{++}$ tetraquarks.

##  I Introduction

Hadrons with unusual quantum numbers and/or multi quark-gluon contents
attracted interest of researches starting from first days of the quark-parton
model and Quantum Chromodynamics (QCD). It is known that conventional hadrons
are composed of quark-antiquark pairs $q\overline{q}$, or made of three
valence quarks $qq^{\prime}q^{\prime\prime}$. Masses and quantum parameters of
ordinary mesons and baryons agree with predictions obtained in this scheme and
can be calculated using standard methods of high energy physics. But
fundamental principles of QCD do not forbid existence of particles built of
four, five, etc. quarks, containing valence gluons or composed of exclusively
valence gluons. Such hadrons may be produced in decays of $B$ meson, in the
electron-positron and proton-antiproton annihilations $e^{+}e^{-}$ and
$\overline{p}p$ , in the double charmonium production processes, in the two-
photon fusion and $pp$ collisions.

The concept of multiquark hadrons was applied by R. Jaffe to explain a mass
hierarchy inside the lowest scalar multiplet Jaffe:1976ig . In accordance with
this assumption the nonet of the light scalars are four-quark states
$q^{2}\overline{q}^{2}$, which explain exprementally observed features of
these particles. Another interesting result about multiquark hadrons was
obtained also by R. Jaffe Jaffe:1976yi . Thus, he considered six-quark states
composed of only light quarks, and using the MIT quark-bag model calculated
parameters of flavor-singlet and neutral bound state $uuddss$
($\mathrm{H}$-dibaryon) with isospin $I=0$ and spin-parity
$J^{\mathrm{P}}=0^{+}$. If exist, this double-strange structure would be
stable against strong decays, and have mean lifetime $\tau\approx
10^{-10}\mathrm{s}$, which is considerably longer than that of conventional
mesons.

Properties of exotic hadrons were investigated in the context of QCD-inspired
nonperturbative methods already at eighties of the last century. One of such
effective methods is the QCD sum rule approach Shifman:1978bx ; Shifman:1978by
, which was employed to perform quantitative analyses and calculate masses and
other parameters of the glueballs, hybrid $q\overline{q}g$ and four-quark
mesons (tetraquarks) Shifman:1978bx ; Balitsky:1982ps ; Govaerts:1984hc ;
Govaerts:1985fx ; Balitsky:1986hf ; Braun:1985ah ; Braun:1988kv .
Unfortunately, achievements obtained at early stages of theoretical studies
were not accompanied by reliable experimental measurements connected mainly
with difficulties in detecting heavy resonances. As a result, existence of
exotic hadrons was not then certainly established.

This situation changed starting from the Belle’s discovery of the
charmoniumlike resonance $X(3872)$ Choi:2003ue , which was confirmed later by
other collaborations Abazov:2004kp ; Acosta:2003zx ; Aubert:2004ns . During
followed years different experimental groups collected information about
masses, widths, and quantum numbers of numerous heavy resonances, which may be
considered as four-quark exotic mesons. There were attempts to describe new
charmoniumlike states as excitations of the ordinary charmonium
$c\overline{c}$: It is worth noting that some of them really allows such
interpretation. But the bulk of available experimental data cannot be included
into this scheme. These resonances may be considered in a tetraquark model,
which treats them either as two-meson molecules or diquark-antidiquark states.

Observation of the charged resonances $Z^{\pm}(4430)$ and $Z_{c}^{\pm}(3900)$
had important impact on the physics of multiquark hadrons, because they could
not be interpreted as neutral charmonia and became real candidates to
tetraquarks Choi:2007wga ; Ablikim:2013mio . Exotic mesons containing four
quarks of different flavors also differ from charmoniumlike states and are
promising candidates to genuine four-quark mesons. Analyses of such structures
were inspired by information of the $D0$ collaboration about evidence for the
state $X(5568)$ composed presumably of four different quarks D0:2016mwd .
Later, other collaborations could not confirm existence of this state
Aaij:2016iev ; CMS:2016fvl , but knowledge gained due to theoretical studies
of $X(5568)$, methods and schemes elaborated during this process played an
important role in our understanding of exotic mesons. The resonances which are
candidates to four-quark exotic mesons now form the wide family of $XYZ$
states. Detailed information on $XYZ$ particles including a history of the
problem, as well as experimental results and theoretical achievements of last
years are collected in numerous interesting review articles Jaffe:2004ph ;
Swanson:2006st ; Klempt:2007cp ; Chen:2016qju ; Chen:2016spr ;
Esposito:2016noz ; Ali:2017jda ; Olsen:2017bmm ; Albuquerque:2018jkn ;
Brambilla:2019esw ; Agaev:2020zad .

Recently the LHCb collaboration announced about new resonance-like structures
$X_{0(1)}(2900)$ observed in the process $B^{+}\rightarrow D^{+}D^{-}K^{+}$
Aaij:2020hon ; Aaij:2020ypa . They were seen in the $D^{-}K^{+}$ mass
distribution, and can be considered as first strong evidence for exotic mesons
composed of four quarks of different flavors. Indeed, from decay modes of
these states, it is clear that they contain valence quarks
$ud\overline{c}\overline{s}$. But one should bear in mind that
$X_{0(1)}(2900)$ may have alternative origin, and appear due to rescattering
diagrams $\chi_{c1}D^{\ast-}K^{\ast+}$ and $D_{sJ}\overline{D}_{1}^{0}K^{0}$
in the decay $B^{+}\rightarrow D^{+}D^{-}K^{+}$ Liu:2020orv .

The LHCb determined masses and spin-parities of these resonances, that were
used in various models to explain internal organizations of $X_{0(1)}(2900)$.
As usual, they were interpreted as hadronic molecules, diquark-antidiquark
states, and rescattering effects (see, Refs. Agaev:2020nrc ; Agaev:2021knl
and references therein). In our articles Agaev:2020nrc ; Agaev:2021knl , we
treated $X_{0}(2900)$ and $X_{1}(2900)$ as a scalar hadronic molecule
$\overline{D}^{\ast 0}K^{\ast 0}$ and diquark-antidiquark vector states
$[ud][\overline{c}\overline{s}]$, respectively. Predictions for their masses
and widths extracted from sum rule analyses seem confirm assumptions made on
their structures.

The resonances $X_{0(1)}(2900)$ emerge in intermediate phase of the decay
chain $B^{+}\rightarrow D^{+}X\rightarrow D^{+}D^{-}K^{+}$, and are neutral
states. These processes occur due to color-favored and color-suppressed
transformations of the $B^{+}$ meson Burns:2020xne . But weak decays of
$B^{+}$ with the same topologies may trigger also processes $B^{+}\rightarrow
D^{-}Z^{++}\rightarrow D^{-}D^{+}K^{+}$, where $Z^{++}$ is a doubly charged
exotic meson with quark content $cu\overline{s}\overline{d}$. In our view,
experimental data collected by LHCb about weak decays of $B$ meson may be used
to uncover doubly charged tetraquarks $Z^{++}=[cu][\overline{s}\overline{d}]$
with different spin-parities. In fact, the scalar and vector particles
$Z_{\mathrm{S}}^{++}$ and $Z_{\mathrm{V}}^{++}$ may appear as resonances in
the $D^{+}K^{+}$ mass distribution, and provide valuable information on new
exotic mesons.

Let us note that doubly charged diquark-antidiquarks already attracted
interests of physicists, and some of them was studied in a rather detailed
form. Thus, spectroscopic parameters and strong decays of pseudoscalar
tetraquarks $cc\overline{s}\overline{s}$ and $cc\overline{d}\overline{s}$ were
calculated in Ref. Agaev:2018vag . Doubly charged scalar, pseudoscalar and
axial-vector states $[sd][\overline{u}\overline{c}]$ were considered in our
article Agaev:2017oay . The tetraquarks $Z^{++}$ are positively charged
counterparts of states $[sd][\overline{u}\overline{c}]$ and have the same
masses and decay widths. Therefore, one can safely use their parameters to
study exotic mesons $Z^{++}$.

For completeness and following analysis, we provide masses and widths of
various tetraquarks $Z^{++}$ using, for these purposes, our results from Ref.
Agaev:2017oay

$\displaystyle m_{\mathrm{Z}_{\mathrm{S}}}$ $\displaystyle=$ $\displaystyle
2628_{-153}^{+166}~{}\mathrm{MeV},\ \Gamma_{\mathrm{Z}_{\mathrm{S}}}=(66.89\pm
15.11)~{}\mathrm{MeV,}$ $\displaystyle m_{\mathrm{Z}_{\mathrm{PS}}}$
$\displaystyle=$ $\displaystyle 2719_{-156}^{+144}~{}\mathrm{MeV},\
\Gamma_{\mathrm{Z}_{\mathrm{PS}}}=(38.1\pm 7.1)~{}\mathrm{MeV,}$
$\displaystyle m_{\mathrm{Z}_{\mathrm{AV}}}$ $\displaystyle=$ $\displaystyle
2826_{-157}^{+134}~{}\mathrm{MeV},\ \Gamma_{\mathrm{Z}_{\mathrm{AV}}}=(47.3\pm
11.1)~{}\mathrm{MeV.}$

Here, subscripts $Z_{\mathrm{S}}$, $Z_{\mathrm{PS}}$ and $Z_{\mathrm{AV}}$
refer to the scalar, pseudoscalar and axial-vector $Z^{++}$, respectively.

As is seen, there are not predictions for parameters of the vector tetraquark
$Z_{\mathrm{V}}^{++}$, but its mass should be around or larger than
$m_{\mathrm{Z}_{\mathrm{AV}}}$. Using this preliminary estimate for the mass
of $Z_{\mathrm{V}}^{++}$ and $m_{\mathrm{Z}_{\mathrm{S}}}$ from Eq.
(LABEL:eq:DCh) , we see that decays of $Z_{\mathrm{S}}^{++}$ and
$Z_{\mathrm{V}}^{++}$ to ordinary mesons $D_{s}^{+}\pi^{+}$ and $D^{+}K^{+}$
are kinematically allowed processes, i.e., the tetraquarks
$Z_{\mathrm{S}}^{++}$ and $Z_{\mathrm{V}}^{++}$ may be seen in $D^{+}K^{+}$
mass distribution in the decay $B^{+}\rightarrow D^{-}D^{+}K^{+}$.

In the present article, we compute spectroscopic parameters and total width of
the vector tetraquark $Z_{\mathrm{V}}^{++}$ using various versions of the QCD
sum rule method. Our interest to this particle is twofold: First, it is a
fully open flavor tetraquark, and additionally bears two units of electric
charge. By theoretical exploration of $Z_{\mathrm{V}}^{++}$, we will complete
list of such states (LABEL:eq:DCh). Second reason is that, there are
opportunities to fix the tetraquarks $Z_{\mathrm{S}}^{++}$ and
$Z_{\mathrm{V}}^{++}$ using existing or future LHCb data.

We evaluate the mass and current coupling of $Z_{\mathrm{V}}^{++}$ by
employing the QCD two-point sum rule approach Shifman:1978bx ; Shifman:1978by
. In our analysis, we take into account contributions of various vacuum
condensates up to dimension $10$. Prediction for the mass of
$Z_{\mathrm{V}}^{++}$, as well as its quantum numbers $J^{\mathrm{P}}=1^{-}$
permit us to classify kinematically allowed decay modes of this tetraquark.
The mass and coupling of $Z_{\mathrm{V}}^{++}$ are also input parameters
necessary to calculate partial widths of the decays
$Z_{\mathrm{V}}^{++}\rightarrow D_{s1}(2460)^{+}\pi^{+},\
D_{s0}^{\ast}(2317)^{+}\rho^{+}$, $\ D^{+}K^{+}$, and $D_{s}^{+}\pi^{+}$. To
this end, we explore vertices of $Z_{\mathrm{V}}^{++}$ with ordinary mesons
(for example, $Z_{\mathrm{V}}^{++}D^{+}K^{+}$), and find corresponding strong
couplings. Relevant investigations are carried out using the QCD light-cone
sum rule (LCSR) method, which is one of effective nonperturbative tools to
study conventional hadrons Balitsky:1989ry . In the case of tetraquark-
ordinary meson vertices the standard methods of LCSR have to be applied in
conjunction with a soft-meson approximation Belyaev:1994zk ; Ioffe:1983ju .
For analysis of the exotic mesons the light-cone sum rule method and soft
approximation was suggested in Ref. Agaev:2016dev , and used to investigate
decays of various tetraquarks Agaev:2020zad .

This paper is structured in the following manner: Section II is devoted to
calculations of the mass and coupling of the vector tetraquark
$Z_{\mathrm{V}}^{++}=[cu][\overline{s}\overline{d}]$. In Section III, we
compute strong couplings in relevant tetraquark-meson vertices, and evaluate
partial widths of $Z_{\mathrm{V}}^{++}$ decays. The full width of
$Z_{\mathrm{V}}^{++}$ is found in this section as well. Section IV is reserved
for our conclusions and final notes.

##  II The spectroscopic parameters of $Z_{\mathrm{V}}$

The mass $m$ and current coupling $f$ are important parameters of the vector
tetraquark $Z_{\mathrm{V}}^{++}=[cu][\overline{s}\overline{d}]$ (in what
follows, we omit superscripts denoting charges of the tetraquark and various
mesons). We use the QCD two-point sum rule method to evaluate values of these
parameters.

We begin calculations from analysis of the two-point correlation function

$\Pi_{\mu\nu}(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{{\dagger}}(0)\\}|0\rangle,$ (2)

where $J_{\mu}(x)$ is the interpolating current for $Z_{\mathrm{V}}$, and
$\mathcal{T}$ denotes the time-ordered product of two currents. The vector
tetraquark $Z_{\mathrm{V}}$ can be modeled using a scalar diquark
$uC\gamma_{5}c$ and vector antidiquark
$\overline{s}\gamma_{\mu}\gamma_{5}C\overline{d}$. It is known that color-
antitriplet diquarks and color-triplet antidiquarks are most stable two-quark
structures Jaffe:2004ph . Therefore, we chose as building blokes of
$Z_{\mathrm{V}}$ structures $\varepsilon^{abc}u_{b}C\gamma_{5}c_{c}$ and
$\varepsilon^{amn}\overline{s}_{m}\gamma_{\mu}\gamma_{5}C\overline{d}_{n}^{T}$,
which belongs to $[\overline{\mathbf{3}}_{c}]$ and $[\mathbf{3}_{c}]$
representations of the color group $SU_{c}(3)$, respectively. Then, the
colorless interpolating current $J_{\mu}(x)$ takes the form

$J_{\mu}(x)=\varepsilon\widetilde{\varepsilon}[u_{b}^{T}(x)C\gamma_{5}c_{c}(x)][\overline{s}_{m}(x)\gamma_{\mu}\gamma_{5}C\overline{d}_{n}^{T}(x)],$
(3)

where $\varepsilon\widetilde{\varepsilon}=\varepsilon^{abc}\varepsilon^{amn}$,
and $a$, $b$, $c$, $m$ and $n$ denote quark colors. In Eq. (3) $u(x)$, $c(x)$,
$s(x)$ and $d(x)$ are the quark fields, and $C$ stands for the charge-
conjugation operator.

The sum rules for $m$ and $f$ can be derived by calculating $\Pi_{\mu\nu}(p)$
in terms of the physical parameters of $Z_{\mathrm{V}}$, as well as in the
operator product expansion ($\mathrm{OPE}$) with certain accuracy. Having
equated these two expressions, applied the Borel transformation to suppress
contributions of higher resonances and continuum states, and subtracted these
contributions using the quark-hadron duality assumption Shifman:1978bx ;
Shifman:1978by , it is possible to get required sum rules for the mass and
coupling of the tetraquark $Z_{\mathrm{V}}$.

Expression of $\Pi_{\mu\nu}(p)$ in terms of parameters of $Z_{\mathrm{V}}$ is
obtained by saturating the correlation function $\Pi_{\mu\nu}(p)$ with a
complete set of $J^{\mathrm{P}}=1^{-}$ states and performing in Eq. (2)
integration over $x$

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{\langle
0|J_{\mu}|Z_{\mathrm{V}}\rangle\langle
Z_{\mathrm{V}}|J_{\nu}^{\dagger}|0\rangle}{m^{2}-p^{2}}+\cdots,$ (4)

where the dots stand for contributions of higher resonances and continuum
states.

To proceed, it is convenient to introduce the matrix element

$\langle 0|J_{\mu}|Z_{\mathrm{V}}\rangle=fm\epsilon_{\mu},$ (5)

where $\epsilon_{\mu}$ is the polarization vector of the tetraquark
$Z_{\mathrm{V}}$. Then $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ can be rewritten in
the following form

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{m^{2}f^{2}}{m^{2}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m^{2}}\right)+\cdots.$
(6)

The two-component Lorentz structure in the parentheses in Eq. (6) corresponds
to a vector particle. The first component of this structure is proportional to
$g_{\mu\nu}$ and does not contain effects of scalar particles: It is formed
due to only vector states. Therefore, in our studies we utilize the structure
$\sim g_{\mu\nu}$, and label corresponding invariant amplitude by
$\Pi^{\mathrm{Phys}}(p^{2})$.

Expression in Eq. (4) is obtained in the zero-width single-pole approximation.
In general, the correlation function $\Pi_{\mu\nu}(p)$ receives contributions
also from two-meson reducible terms, because the current $J_{\mu}$ couples not
only to four-quark structures, but also to two mesons with relevant quantum
numbers Kondo:2004cr ; Lee:2004xk . Interactions with mesons lying below the
mass of the $Z_{\mathrm{V}}$ generate a finite width of the tetraquark
$Z_{\mathrm{V}}$, and modify the quark propagator in Eq. (4)

$\frac{1}{m^{2}-p^{2}}\rightarrow\frac{1}{m^{2}-p^{2}-i\sqrt{p^{2}}\Gamma(p)}.$
(7)

The two-meson 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 . 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-meson contributions are small
and can be neglected.

The QCD side of the sum rules $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ is found by
inserting the interpolating current $J_{\mu}(x)$ into Eq. (2), and contracting
relevant quark fields

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p)=i\int
d^{4}xe^{ipx}\varepsilon\widetilde{\varepsilon}\varepsilon^{\prime}\widetilde{\varepsilon}^{\prime}\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{u}^{bb^{\prime}}(x)\gamma_{5}S_{c}^{cc^{\prime}}(x)\right]$
$\displaystyle\times\mathrm{Tr}\left[\gamma_{\mu}\gamma_{5}\widetilde{S}_{d}^{n^{\prime}n}(-x)\gamma_{5}\gamma_{\nu}S_{s}^{m^{\prime}m}(-x)\right],$
(8)

where

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

with $S_{c}(x)$ and $S_{u(s,d)}(x)$ being the heavy $c$\- and light
$u(s,d)$-quark propagators, respectively (for explicit expressions, see Ref.
Agaev:2020zad ). The invariant amplitude in $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$
corresponding to the structure $g_{\mu\nu}$ in our following analysis will be
denoted by $\Pi^{\mathrm{OPE}}(p^{2})$.

Calculations performed in accordance with a scheme briefly explained above
yield

$m^{2}=\frac{\Pi^{\prime}(M^{2},s_{0})}{\Pi(M^{2},s_{0})},$ (10)

and

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

which are sum rules for $m$ and $f$, respectively. Here, $\Pi(M^{2},s_{0})$ is
the Borel transformed and subtracted amplitude $\Pi^{\mathrm{OPE}}(p^{2})$,
which depends on the Borel $M^{2}$ and continuum threshold $s_{0}$ parameters.
In Eq. (10) $\Pi^{\prime}(M^{2},s_{0})$ is the derivative of the amplitude
over $d/d(-1/M^{2})$.

It is clear that the amplitude $\Pi(M^{2},s_{0})$ is a key ingredient of the
obtained sum rules. Calculations of this function give the following result

$\Pi(M^{2},s_{0})=\int_{\mathcal{M}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}),$
(12)

where $\mathcal{M}=m_{c}+m_{s}$. Computations are carried out by taking into
account vacuum condensates up to dimension $10$. The amplitude
$\Pi(M^{2},s_{0})$ has two components: First of them is expressed using the
spectral density $\rho^{\mathrm{OPE}}(s)$, which is computed as an imaginary
part of $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. The Borel transformation of another
terms are found directly from $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ and included
into $\Pi(M^{2})$. The first component in Eq. (12) contains a main part of the
amplitude, whereas $\Pi(M^{2})$ is formed from higher dimensional terms.
Analytical expressions of $\rho^{\mathrm{OPE}}(s)$ and $\Pi(M^{2})$, are
rather lengthy, therefore we do not write down them here.

The sum rules depend on vacuum condensates up to dimension $10$. The basic
condensates

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

were estimated from analysis of numerous hadronic processes Shifman:1978bx ;
Shifman:1978by ; Ioffe:2005ym ; Narison:2015nxh , and are known and universal
parameters. Higher dimensional condensates are factorized and expressed in
terms of basic ones: we assume that factorization does not lead to large
uncertainties. The masses of $s$ and $c\ $quarks are also among input
parameters, for which we use $m_{s}=93_{-5}^{+11}~{}\mathrm{MeV}$, and
$m_{c}=1.27\pm 0.2~{}\mathrm{GeV}$, respectively.

The Borel and continuum threshold parameters $M^{2}$ and $s_{0}$ are auxiliary
quantities of computations and their choice is controlled by constraints
imposed on the pole contribution ($\mathrm{PC}$) and convergence of
$\mathrm{OPE}$, as well as by minimum sensitivity of the extracted quantities
to the Borel parameter $M^{2}$. Thus, the maximum allowed value of $M^{2}$
should be fixed to meet the restriction on $\mathrm{PC}$

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

The lower limit of the working window for the Borel parameter is found from
convergence of the operator product expansion, which is quantified by the
ratio

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

Here $\Pi^{\mathrm{DimN}}(M^{2},s_{0})$ denotes contribution of the last term
(or a sum of last few terms) in the $\mathrm{OPE}$. In the present work, at
the maximum of $M^{2}$ we apply the constraint $\mathrm{PC>0.2}$, which is
typical for the multiquark hadrons. To guarantee the convergence of the
operator product expansion at minimum of $M^{2}$, we use a requirement
$R(M^{2})\leq 0.01$.

Computations demonstrate that working regions for the parameters $M^{2}$ and
$s_{0}$

$M^{2}\in[4.5,6.5]\ \mathrm{GeV}^{2},\ s_{0}\in[14,16]\ \mathrm{GeV}^{2},$
(16)

satisfy aforementioned constraints. In fact, in these regions $\mathrm{PC}$
changes on average within limits

$0.78\leq\mathrm{PC}\leq 0.28.$ (17)

At the minimum $M^{2}=4.5~{}\mathrm{GeV}^{2}$, a contribution to
$\Pi(M^{2},s_{0})$ arising from a sum of last three terms in $\mathrm{OPE}$
does not exceed $1\%$ of the full result. In fact, at
$M^{2}=4.5~{}\mathrm{GeV}^{2}$ the ratio $R(M^{2})$ for
$\mathrm{DimN=Dim(8+9+10)}$ is equal to $0.007$, which proves convergence of
the sum rules.

Central values of the mass $m$ and coupling $f$ are evaluated at the middle
point of regions (16), in other words, at $M^{2}=5.5~{}\mathrm{GeV}^{2}$ and
$s_{0}=15~{}\mathrm{GeV}^{2}$. At this point the pole contribution is
$\mathrm{PC}\approx 0.56$, which ensures the ground-state nature of
$Z_{\mathrm{V}}$. Results for $m$ and $f$ read

$\displaystyle m$ $\displaystyle=$ $\displaystyle(3515~{}\pm
125)~{}\mathrm{MeV},$ $\displaystyle f$ $\displaystyle=$
$\displaystyle(5.24\pm 1.10)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (18)

As it has been emphasized above, predictions extracted from sum rules should
not depend on a choice of $M^{2}$. In real analysis, however, there is a
residual dependence on this parameter. In Fig. 1, we plot the mass of the
tetraquark $Z_{\mathrm{V}}$ for wide range of $M^{2}$ and $s_{0}$. It is seen,
that only the region between two vertical lines in the left panel can be
considered as a relatively stable plateau, where parameters of
$Z_{\mathrm{V}}$ are evaluated. The current coupling $f$ of the tetraquark
$Z_{\mathrm{V}}$ is depicted in Fig. 2 as functions of $M^{2}$ and $s_{0}$. In
the case under discussion, one observes that $f$ is almost stable in the
explored range of the Borel parameter (16).

Figure 1: Dependence of the $Z_{\mathrm{V}}$ tetraquark’s mass $m$ on the
Borel parameter $M^{2}$ (left panel), and on the continuum threshold parameter
$s_{0}$ (right panel). Vertical lines fix boundaries of working regions for
$M^{2}$ and $s_{0}$ used in numerical computations.

Figure 2: The coupling $f$ of the tetraquark $Z_{\mathrm{V}}$ as a function of
$M^{2}$ (left panel), and $s_{0}$ (right panel). Variations of $f$ are shown
within working limits of the parameters $M^{2}$ and $s_{0}$.

The second source of theoretical errors is connected with a choice of the
continuum threshold parameter $s_{0}$. Its working window should also satisfy
limits stemming from dominance of $\mathrm{PC}$ and convergence of
$\mathrm{OPE}$. Despite $M^{2}$, $s_{0}$ bears physical information about
first excitation of the tetraquark $Z_{\mathrm{V}}$. The self-consistent
analysis implies that $\sqrt{s_{0}}$ has to be smaller than mass of such
state. There are only a few samples when two observed resonances were assumed
to be ground and radially excited states of the same tetraquark. The
resonances $Z_{c}(3900)$ and $Z_{c}(4430)$ may form one of such pairs
Agaev:2017tzv . The difference between masses of $Z_{c}(3900)$ and
$Z_{c}(4430)$ is equal to $\approx 530~{}\mathrm{MeV}$, therefore a mass gap
$\sqrt{s_{0}}-m\approx(400-600)~{}\mathrm{MeV}$ can be considered as a
reasonable estimate for tetraquarks. Here, we get on average
$\sqrt{s_{0}}-m\approx 400~{}\mathrm{MeV}$ which is in accord with this
general analysis.

Effects connected with a choice of parameters $M^{2}$ and $s_{0}$ are two main
sources of theoretical uncertainties in sum rule computations. In the case of
the mass $m$ these ambiguities equal to $\pm 3.6\%$, whereas for the coupling
$f$ they are $\pm 21\%$ of the full result. Theoretical uncertainties for $f$
are larger than for the mass, nevertheless, they do not overshoot accepted
limits.

It is interesting to analyze a gap between masses of axial-vector and vector
tetraquarks with the same content. Our present calculations demonstrate that
for tetraquarks $[cu][\overline{s}\overline{d}]$ the difference between masses
of the axial-vector and vector particles $Z_{\mathrm{AV}}$ and
$Z_{\mathrm{V}}$ is $m-m_{\mathrm{Z}_{\mathrm{AV}}}\approx
690~{}\mathrm{MeV}$. This result can be compared with similar predictions for
other tetraquarks. Thus, resonances $Y(4660)$ and $X(4140)$ with spin-parities
$J^{\mathrm{PC}}=1^{--}$ and $1^{++}$, quark content
$[cs][\overline{c}\overline{s}]$ and color structure
$[\overline{\mathbf{3}}_{c}]\otimes[\mathbf{3}_{c}]$ were investigated in
Refs. Sundu:2018toi ; Agaev:2017foq , respectively. An estimate for
$m_{Y}-m_{X_{1}}$ extracted from these studies is approximately equal to
$500~{}\mathrm{MeV}$. In other words, radially excited axial-vector and
ground-state vector tetraquarks, in this special case, are close in mass. This
fact maybe useful to explain numerous charged and neutral resonances from
$XYZ$ family in the mass range of $4-5~{}\mathrm{GeV}$.

Another issue to be addressed here, is a mass gap between vector tetraquarks
$Z_{\mathrm{V}}$ and $X_{1}(2900)$, which amounts to $\approx
600~{}\mathrm{MeV}$ and is quite large. It has been noted in section I, that
$X_{1}$ can be modeled as a vector tetraquark
$[ud][\overline{c}\overline{s}]$, and hence both $Z_{\mathrm{V}}$ and $X_{1}$
are composed of same quarks. But there are two reasons, which may lead to
aforementioned mass effect. First of them is internal organizations of these
states: The tetraquark $Z_{\mathrm{V}}$ is composed of a heavy diquark $[cu]$
and relatively heavy antidiquark $[\overline{s}\overline{d}]$, whereas $X_{1}$
is strongly heavy-light compound. The latter is more tightly connected
structure and should be lighter than $Z_{\mathrm{V}}$. Besides, diquarks in
$Z_{\mathrm{V}}$ have fractional positive electric charges which generate
repulsive forces between them. In the case of $X_{1}$, between $[ud]$ and
$[\overline{c}\overline{s}]$ exists attractive electromagnetic interaction.
Whether these features of $Z_{\mathrm{V}}$ and $X_{1}$ are enough to explain a
mass gap between them or there are other sources of this effect, requires
additional studies.

##  III Strong decays of the tetraquark $Z_{\mathrm{V}}$

The sum of partial widths of $Z_{\mathrm{V}}$ tetraquark’s different decay
channels constitutes its full width. The result for $m$ obtained in the
previous section is necessary to fix kinematically allowed strong decay modes
of $Z_{\mathrm{V}}$. Decays to final states $D_{s1}(2460)\pi$ and
$D_{s0}^{\ast}(2317)\rho$ (below, simply $D_{s1}\pi$ and $D_{s0}^{\ast}\rho$)
are among allowed $S$-wave process for the tetraquark $Z_{\mathrm{V}}$. The
$P$-wave processes, which will be explored, are decays
$Z_{\mathrm{V}}\rightarrow DK$ and $Z_{\mathrm{V}}\rightarrow D_{s}\pi$.

###  III.1 Processes $Z_{\mathrm{V}}\rightarrow D_{s1}\pi$ and
$Z_{\mathrm{V}}\rightarrow D_{s0}^{\ast}\rho$

Here, we study the decays $Z_{\mathrm{V}}\rightarrow D_{s1}\pi$ and
$Z_{\mathrm{V}}\rightarrow D_{s0}^{\ast}\rho$, and compute their partial
widths. We provide details of calculations for the first process and write
down only essential formulas and final results for the second decay. It should
be noted that the mass $m=(3515~{}\pm 125)~{}\mathrm{MeV}$ makes possible
$S$-wave decays of $Z_{\mathrm{V}}$ to final states $DK_{1}(1270),$
$D_{s}b_{1}(1235)$ and $D_{s}a_{1}(1260)$ as well. But widths of these
processes, as it will be explained below, are suppressed relative to
aforementioned two decays due to kinematical factors. Therefore, we restrict
ourselves by investigation of two dominant $S$-wave decays.

The width of the process $Z_{\mathrm{V}}\rightarrow D_{s1}\pi$ can be found
using the coupling $g_{1}$ that describes strong interactions of particles
$Z_{\mathrm{V}}$,$\ D_{s1}$, and $\pi$ at the vertex
$Z_{\mathrm{V}}D_{s1}\pi$. In order to evaluate $g_{1}$, we use the QCD light-
cone sum rule method Balitsky:1989ry  and a soft-meson approximation
Belyaev:1994zk ; Ioffe:1983ju .

Starting point in LCSR method is the correlation function

$\Pi_{\mu\nu}(p,q)=i\int
d^{4}xe^{ipx}\langle\pi(q)|\mathcal{T}\\{J_{\mu}^{D_{1}}(x)J_{\nu}^{{\dagger}}(0)\\}|0\rangle,$
(19)

where by $D_{1}$ in the current $J_{\mu}^{D_{1}}$ we denote the meson
$D_{s1}$. The current $J_{\nu}(0)$ in the correlation function
$\Pi_{\mu\nu}(p,q)$ is introduced in Eq. (3). As interpolating current
$J_{\mu}^{D_{1}}(x)$ for the axial-vector meson $D_{s1}$, we use the
expression

$J_{\mu}^{D_{1}}(x)=\overline{s}_{l}(x)i\gamma_{5}\gamma_{\mu}c_{l}(x),$ (20)

with $l$ being the color index.

The function $\Pi_{\mu\nu}(p,q)$ has to be rewritten in terms of the physical
parameters of the initial and final particles involved into the decay. By
taking into account the ground states in the $D_{s1}$ and $Z_{\mathrm{V}}$
channels, we get

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)=\frac{\langle
0|J_{\mu}^{D_{1}}|D_{s1}\left(p\right)\rangle}{p^{2}-m_{1}^{2}}\langle
D_{s1}\left(p\right)\pi(q)|Z_{\mathrm{V}}(p^{\prime})\rangle$
$\displaystyle\times\frac{\langle
Z_{\mathrm{V}}(p^{\prime})|J_{\nu}^{\dagger}|0\rangle}{p^{\prime
2}-m^{2}}+\cdots,$ (21)

where $p$, $q$ and $p^{\prime}=p+q$ are the momenta of $D_{s1}$, $\pi$ , and
$Z_{\mathrm{V}}$, respectively. In Eq. (21) $m_{1}$ is the mass of the meson
$D_{s1}$, and the ellipses stand for contributions of higher resonances and
continuum states in the $D_{s1}$ and $Z_{\mathrm{V}}$ channels.

To get more detailed expression for $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)$, we
utilize the matrix elements

$\langle 0|J_{\mu}^{D_{1}}|D_{s1}\rangle=f_{1}m_{1}\epsilon_{\mu},\ \langle
Z_{\mathrm{V}}(p^{\prime})|J_{\nu}^{\dagger}|0\rangle=fm\epsilon_{\nu}^{\prime\ast},$
(22)

and model the vertex $\langle
D_{s1}\left(p\right)K(q)|Z_{\mathrm{V}}(p^{\prime})\rangle$ in the following
form

$\displaystyle\langle
D_{s1}\left(p\right)\pi(q)|Z_{\mathrm{V}}(p^{\prime})\rangle=g_{1}\left[\left(p\cdot
p^{\prime}\right)\left(\epsilon^{\ast}\cdot\epsilon^{\prime}\right)\right.$
$\displaystyle\left.-\left(p\cdot\epsilon^{\prime}\right)\left(p^{\prime}\cdot\epsilon^{\ast}\right)\right].$
(23)

In Eqs. (22) and (23) $f_{1}$ and $\epsilon_{\mu}$ are the decay constant and
polarization vector of the meson $D_{s1}$, respectively: Polarization vector
of the tetraquark $Z_{\mathrm{V}}$ in this section is denoted by
$\epsilon_{\nu}^{\prime}$. Using these matrix elements in Eq. (21), it is not
difficult to find

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)=g_{1}\frac{m_{1}f_{1}mf}{\left(p^{2}-m_{1}^{2}\right)\left(p^{\prime
2}-m^{2}\right)}$
$\displaystyle\times\left(\frac{m^{2}+m_{1}^{2}}{2}g_{\mu\nu}-p_{\mu}p_{\nu}^{\prime}\right).$
(24)

The function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)$ contains two structures
proportional to $g_{\mu\nu}$ and $p_{\mu}p_{\nu}^{\prime}$, respectively.
These structures and relevant invariant amplitudes can be employed to extract
the sum rule for the strong coupling $g_{1}$. We choose to work with the term
$\sim g_{\mu\nu}$ and corresponding invariant amplitude.

The QCD side of the sum rule can be obtained using explicit expressions of the
correlation function $\Pi_{\mu\nu}(p,q)$ and interpolating currents. Having
contracted quark and antiquark fields in the correlation function, we get

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)=\int
d^{4}xe^{ipx}\varepsilon\widetilde{\varepsilon}\left[\gamma_{5}\widetilde{S}_{c}^{lc}(x){}\gamma_{\mu}\gamma_{5}\right.$
$\displaystyle\left.\times\widetilde{S}_{s}^{ml}(-x){}{}\gamma_{\nu}\gamma_{5}\right]_{\alpha\beta}\langle\pi(q)|\overline{u}_{\alpha}^{b}(0)d_{\beta}^{n}(0)|0\rangle,$
(25)

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

The function $\Pi^{\mathrm{OPE}}(p,q)$ contains local matrix elements of the
quark operator $\overline{u}d$ sandwiched between the vacuum and pion. Simple
transformations allow one to express
$\langle\pi(q)|\overline{u}_{\alpha}^{b}(0)d_{\beta}^{n}(0)|0\rangle$ in terms
of the pion’s known local matrix elements. To this end, we expand
$\overline{u}(0)d(0)$ by employing full set of Dirac matrices
$\Gamma^{J}=\mathbf{1},\ \gamma_{5},\ \gamma_{\mu},\ i\gamma_{5}\gamma_{\mu},\
\sigma_{\mu\nu}/\sqrt{2}$, and project obtained operators onto the color-
singlet states. These manipulations lead to replacement

$\overline{u}_{\alpha}^{b}(0)d_{\beta}^{n}(0)\rightarrow\frac{1}{12}\delta^{bn}\Gamma_{\beta\alpha}^{J}\left[\overline{u}(0)\Gamma^{J}d(0)\right],$
(26)

which should be fulfilled in $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$. The operators
$\overline{u}(0)\Gamma^{J}d(0)$ placed between the vacuum and pion are matrix
elements of the $\pi$ meson.

The QCD expression of the correlation function (25) contains hard-scattering
and soft parts. The hard-scattering part of $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$
is expressed in terms of quark propagators, whereas matrix elements of the
pion form its soft component. For vertices built of conventional mesons
correlation functions depend on non-local matrix elements of a final meson
which are expressible in terms of its distribution amplitudes (DAs). In the
case under analysis, due to four-quark nature of $Z_{\mathrm{V}}$,
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$ contains only local matrix elements of the
pion. As a result, integrals over DAs which are typical for LCSR method,
reduce to overall normalization factors. In this case, the correlation
function has to be found by means of the soft-meson approximation which
implies computation of the hard-scattering part of
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$ in the limit $q\rightarrow 0$ Agaev:2016dev
. It is worth to emphasize that soft-meson approximation is necessary to
analyze tetraquark-meson-meson vertices: Strong couplings at vertices composed
of two tetraquarks and a meson can be calculated by employing full version of
the LCSR method Agaev:2016srl .

It is evident, that soft approximation should by applied also to the
phenomenological side of the sum rule. In the limit $q\rightarrow 0$ for the
amplitude $\Pi^{\mathrm{Phys}}(p^{2})$, we get

$\Pi^{\mathrm{Phys}}(p^{2})=g_{1}\frac{f_{1}m_{1}fm}{(p^{2}-\widetilde{m}^{2})^{2}}\widetilde{m}^{2}+\cdots,$
(27)

where $\widetilde{m}^{2}=(m^{2}+m_{1}^{2})/2$. The function
$\Pi^{\mathrm{Phys}}(p^{2})$ depends on one variable $p^{2}=p^{\prime 2}$ and
has the double pole at $p^{2}=\widetilde{m}^{2}$. The Borel transformation of
$\Pi^{\mathrm{Phys}}(p^{2})$ is given by the formula

$\Pi^{\mathrm{Phys}}(M^{2})=g_{1}f_{1}m_{1}fm\widetilde{m}^{2}\frac{e^{-\widetilde{m}^{2}/M^{2}}}{M^{2}}+\cdots.$
(28)

Besides ground-state contribution, the amplitude $\Pi^{\mathrm{Phys}}(M^{2})$
in the soft limit contains unsuppressed terms which survive even after Borel
transformation. These contaminations should be removed from
$\Pi^{\mathrm{Phys}}(M^{2})$ by applying the operator Belyaev:1994zk ;
Ioffe:1983ju

$\mathcal{P}(M^{2},\widetilde{m}^{2})=\left(1-M^{2}\frac{d}{dM^{2}}\right)M^{2}e^{\widetilde{m}^{2}/M^{2}}.$
(29)

After this operation, remaining undesired terms in
$\Pi^{\mathrm{Phys}}(M^{2})$ can be subtracted by usual manner in the context
of quark-hadron duality assumption. It is clear, that we have to apply the
operator $\mathcal{P}(M^{2},\widetilde{m}^{2})$ to QCD side of the sum rule as
well. Then, the sum rule for the strong coupling $g_{1}$ reads

$g_{1}=\frac{1}{f_{1}m_{1}fm\widetilde{m}^{2}}\mathcal{P}(M^{2},\widetilde{m}^{2})\Pi^{\mathrm{OPE}}(M^{2},s_{0}),$
(30)

where $\Pi^{\mathrm{OPE}}(M^{2},s_{0})$ is Borel transformed and subtracted
invariant amplitude $\Pi^{\mathrm{OPE}}(p^{2})$ corresponding to the structure
$g_{\mu\nu}$ in $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$.

Prescriptions to calculate the correlation function
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$ in the soft approximation were presented in
Ref. Agaev:2016dev , and in many other publications, for this reason we do not
concentrate here on details. Computations of $\Pi^{\mathrm{OPE}}(M^{2},s_{0})$
performed in accordance with this scheme lead to the expression

$\displaystyle\Pi^{\mathrm{OPE}}(M^{2},s_{0})=\frac{f_{\pi}\mu_{\pi}}{48\pi^{2}}\int_{\mathcal{M}^{2}}^{s_{0}}\frac{ds(m_{c}^{2}-s)}{s^{2}}$
$\displaystyle\times\left(m_{c}^{4}+m_{c}^{2}s+6m_{c}m_{s}s-2s^{2}\right)e^{-s/M^{2}}$
$\displaystyle+\Pi_{\mathrm{NP}}(M^{2}),$ (31)

where the first term is the perturbative contribution to
$\Pi^{\mathrm{OPE}}(M^{2},s_{0})$. The nonperturbative component
$\Pi_{\mathrm{NP}}(M^{2})$ of the correlation function is calculated with
dimension-$9$ accuracy, and has the following form

$\displaystyle\Pi_{\mathrm{NP}}(M^{2})=\frac{\langle\overline{s}s\rangle
f_{\pi}\mu_{\pi}m_{c}\left(2M^{2}+m_{c}m_{s}\right)}{12M^{2}}e^{-m_{c}^{2}/M^{2}}$
$\displaystyle+\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\frac{f_{\pi}\mu_{\pi}m_{c}}{144M^{4}}\int_{0}^{1}\frac{dx}{X^{3}}\left[m_{c}^{2}m_{s}X-2m_{s}XM^{2}\right.$
$\displaystyle\left.+m_{c}^{3}x+m_{c}xXM^{2}\right]e^{m_{c}^{2}/[M^{2}X]}+\frac{\langle\overline{s}g\sigma
Gs\rangle f_{\pi}\mu_{\pi}}{72M^{6}}$
$\displaystyle\times\left(m_{s}M^{4}+m_{s}m_{c}^{2}M^{2}-3m_{c}^{3}M^{2}-m_{s}m_{c}^{4}\right)e^{-m_{c}^{2}/M^{2}}$
$\displaystyle+\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\langle\overline{s}s\rangle\frac{f_{\pi}\mu_{\pi}\pi^{2}m_{c}(3M^{2}-m_{c}^{2})}{216M^{8}}$
$\displaystyle\times\left(2M^{2}+m_{s}m_{c}\right)e^{-m_{c}^{2}/M^{2}}+\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\langle\overline{s}g\sigma
Gs\rangle$
$\displaystyle\times\frac{f_{\pi}\mu_{\pi}\pi^{2}m_{c}\left(m_{c}^{5}m_{s}+3M^{2}m_{c}^{4}-7m_{c}^{3}m_{s}M^{2}\right.}{1296M^{12}}$
$\displaystyle\left.-18m_{c}^{2}M^{4}+8m_{c}m_{s}M^{4}+18M^{6}\right)e^{-m_{c}^{2}/M^{2}},$
(32)

where $X=x-1$.

Contributions to $\Pi_{\mathrm{NP}}(M^{2})$ arise from the matrix element

$\langle 0|\overline{d}i\gamma_{5}u|\pi\rangle=f_{\pi}\mu_{\pi},$ (33)

where

$\mu_{\pi}=\frac{m_{\pi}^{2}}{m_{u}+m_{d}}=-\frac{2\langle\overline{q}q\rangle}{f_{\pi}^{2}}.$
(34)

The last equality in Eq. (34) is the relation between the mass $m_{\pi}$ and
decay constant $f_{\pi}$ of the pion, masses of quarks, and quark condensate
$\langle\overline{q}q\rangle$, which stems from the partial conservation of
the axial-vector current.

Parameters | Values (in $\mathrm{MeV}$ units)
---|---
$m_{1}=m[D_{s1}(2460)]$ | $2459.5\pm 0.6$
$f_{1}=f[D_{s1}(2460)]$ | $225\pm 25$
$m_{2}=m[D_{s0}^{*}(2317)]$ | $2317.8\pm 0.5$
$f_{2}=f[D_{s0}^{*}(2317)]$ | $202\pm 15$
$m_{D}$ | $1869.65\pm 0.05$
$f_{D}$ | $212.6\pm 0.7$
$m_{D_{s}}$ | $1968.34\pm 0.07$
$f_{D_{s}}$ | $249.9\pm 0.5$
$m_{K}$ | $493.677\pm 0.0016$
$f_{K}$ | $155.7\pm 0.3$
$m_{\pi}$ | $139.57039\pm 0.00018$
$f_{\pi}$ | $130.2\pm 1.2$
$m_{\rho}$ | $775.26\pm 0.25$
$f_{\rho}$ | $216\pm 3$
Table 1: Masses and decay constants of mesons used in numerical analyses.

In numerical computations of $\Pi^{\mathrm{OPE}}(M^{2},s_{0})$, we choose
$M^{2}$ and $s_{0}$ within limits given by Eq. (16). Besides $M^{2}$ and
$s_{0}$, Eq. (30) contains various vacuum condensates and spectroscopic
parameters of the final-state mesons $D_{s1}$ and $\pi$. The masses and decay
constants of the mesons $D_{s1}$ and $\pi$, as well as other parameters used
in numerical analyses are borrowed from Ref. PDG:2020  and presented in Table
1: For decay constants of the mesons $D_{s0}^{\ast}$ and $D_{s1}$, we utilize
the sum rule predictions from Refs. Narison:2015nxh  and Colangelo:2005hv ,
respectively.

For the coupling $g_{1}$, we get

$g_{1}=0.36_{-0.06}^{+0.09}~{}\mathrm{GeV}^{-1}$ (35)

The partial width of the decay $Z_{\mathrm{V}}\rightarrow D_{s1}\pi$ can be
found by means of the formula

$\Gamma_{1}\left[Z_{\mathrm{V}}\rightarrow
D_{s1}\pi\right]=\frac{g_{1}^{2}m_{1}^{2}\lambda}{24\pi}\left(3+\frac{2\lambda^{2}}{m_{1}^{2}}\right),$
(36)

where $\lambda\equiv\lambda\left(m,m_{1},m_{\pi}\right)$ and

$\displaystyle\lambda\left(a,b,c\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2a}\left[a^{4}+b^{4}+c^{4}\right.$ (37)
$\displaystyle\left.-2\left(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2}\right)\right]^{1/2}.$

By employing this expression, it is not difficult to obtain

$\Gamma_{1}\left[Z_{\mathrm{V}}\rightarrow
D_{s1}\pi\right]=29.8_{-9.4}^{+17.6}~{}\mathrm{MeV}.$ (38)

The strong coupling and partial width of the second process
$Z_{\mathrm{V}}\rightarrow D_{s0}^{\ast}\rho$ can be found by the same manner.
Here, one starts from the correlation function

$\widetilde{\Pi}_{\nu}(p,q)=i\int
d^{4}xe^{ipx}\langle\rho(q)|\mathcal{T}\\{J^{D_{0}}(x)J_{\nu}^{{\dagger}}(0)\\}|0\rangle,$
(39)

where the interpolating current of the scalar meson $D_{s0}^{\ast}$ is denoted
by $J^{D_{0}}(x)$ and determined by the expression

$J^{D_{0}}(x)=\overline{s}_{l}(x)c_{l}(x).$ (40)

In our studies, we use the matrix element $\langle
0|J^{D_{0}}|D_{s0}^{\ast}\rangle=f_{2}m_{2}$, with $m_{2}$ and $f_{2}$ being
the mass and decay constant of the $D_{s0}^{\ast}$. The vertex
$Z_{\mathrm{V}}D_{s0}^{\ast}\rho$ is modeled in the form

$\displaystyle\langle
D_{s0}^{\ast}\left(p\right)\rho(q)|Z_{\mathrm{V}}(p^{\prime})\rangle=g_{2}\left[\left(q\cdot
p^{\prime}\right)\left(\epsilon^{\ast}\cdot\epsilon^{\prime}\right)\right.$
$\displaystyle\left.-\left(q\cdot\epsilon^{\prime}\right)\left(p^{\prime}\cdot\epsilon^{\ast}\right)\right],$
(41)

where $\epsilon_{\mu}$ is the polarization vector of the $\rho$ meson. Then,
the phenomenological and QCD sides of the sum rule are given by expressions

$\displaystyle\widetilde{\Pi}_{\nu}^{\mathrm{Phys}}(p,q)=g_{2}\frac{m_{2}f_{2}mf}{\left(p^{2}-m_{2}^{2}\right)\left(p^{\prime
2}-m^{2}\right)}$
$\displaystyle\times\left(\frac{m_{2}^{2}-m^{2}}{2}\epsilon_{\nu}^{\ast}+p^{\prime}\cdot\epsilon^{\ast}q_{\nu}\right),$
(42)

and

$\displaystyle\widetilde{\Pi}_{\nu}^{\mathrm{OPE}}(p,q)=-i\int
d^{4}xe^{ipx}\varepsilon\widetilde{\varepsilon}\left[\gamma_{5}\widetilde{S}_{c}^{lc}(x){}\widetilde{S}_{s}^{ml}(-x)\right.$
$\displaystyle\left.\times{}{}\gamma_{\nu}\gamma_{5}\right]_{\alpha\beta}\langle\rho(q)|\overline{u}_{\alpha}^{b}(0)d_{\beta}^{n}(0)|0\rangle,$
(43)

respectively. We perform calculations using the structures
$\sim\epsilon_{\nu}^{\ast}$ in $\widetilde{\Pi}_{\nu}^{\mathrm{Phys}}$ and
$\widetilde{\Pi}_{\nu}^{\mathrm{OPE}}$. The relevant amplitude
$\widetilde{\Pi}^{\mathrm{OPE}}(M^{2},s_{0})$ receives contributions from the
matrix elements of the $\rho$ meson

$\displaystyle\langle 0|\overline{d}\gamma_{\mu}u|\rho\rangle$
$\displaystyle=$ $\displaystyle f_{\rho}m_{\rho}\epsilon_{\mu},$
$\displaystyle\langle
0|\overline{d}g\widetilde{G}_{\mu\nu}\gamma_{\nu}\gamma_{5}u|\rho\rangle$
$\displaystyle=$ $\displaystyle f_{\rho}m_{\rho}^{3}\zeta_{4}\epsilon_{\mu},$
(44)

where $m_{\rho}$ and $f_{\rho}$ are the mass and decay constant of the $\rho$
meson, and $\widetilde{G}_{\mu\nu}$ is the dual gluon field tensor
$\widetilde{G}_{\mu\nu}=\varepsilon_{\mu\nu\alpha\beta}G^{\alpha\beta}/2$. The
second equality in Eq. (44) is the matrix element of the twist-4 operator
Ball:1998ff . The parameter $\zeta_{4\rho}$ was evaluated in the context of
QCD sum rule approach at the renormalization scale $\mu=1\,\,{\mathrm{GeV}}$
in Ref. Ball:2007zt  and is equal to $\zeta_{4\rho}=0.07\pm 0.03$.

The correlation function $\widetilde{\Pi}^{\mathrm{OPE}}(M^{2},s_{0})$ has the
form

$\displaystyle\widetilde{\Pi}^{\mathrm{OPE}}(M^{2},s_{0})=\frac{f_{\rho}m_{\rho}}{16\pi^{2}}\int_{\mathcal{M}^{2}}^{s_{0}}\frac{ds(s-m_{c}^{2})}{s}$
$\displaystyle\times\left(m_{c}^{2}+2m_{c}m_{s}-s\right)e^{-s/M^{2}}+\frac{f_{\rho}m_{\rho}^{3}\zeta_{4\rho}}{32\pi^{2}}$
$\displaystyle\times\int_{\mathcal{M}^{2}}^{s_{0}}\frac{ds(m_{c}^{4}-s^{2})}{s^{2}}e^{-s/M^{2}}+\widetilde{\Pi}_{\mathrm{NP}}(M^{2}),$
(45)

where $\widetilde{\Pi}_{\mathrm{NP}}(M^{2})$ is nonperturbative component of
$\widetilde{\Pi}^{\mathrm{OPE}}$: We refrain from providing its explicit
expression here.

Omitting further details, we write down results for the strong coupling
$g_{2}$ and partial width of the process $Z_{\mathrm{V}}\rightarrow
D_{s0}^{\ast}\rho$

$\displaystyle g_{2}=0.61_{-0.12}^{+0.18}~{}\mathrm{GeV}^{-1},$
$\displaystyle\Gamma_{2}\left[Z_{\mathrm{V}}\rightarrow
D_{s0}^{\ast}\rho\right]=9.3_{-3.1}^{+6.5}~{}\mathrm{MeV}.$ (46)

Returning to other $S$-wave decays listed above, we assume that couplings
corresponding to vertices $Z_{\mathrm{V}}DK_{1}(1270)$ etc., are the same
order of $g_{1}$ and $g_{2}$. Then widths of these decays are suppressed,
because the factor
$m_{\ast}^{2}$$\lambda\left(3+2\lambda^{2}/m_{\ast}^{2}\right)$ ($m_{\ast}$ is
a mass of a final meson) is smaller for two final-state mesons of
approximately equal mass than in the case of light and heavy mesons. We also
take into account that the full width $\Gamma_{\mathrm{full}}$ of the
tetraquark $Z_{\mathrm{V}}$ is formed mainly due to $P$-wave processes
$Z_{\mathrm{V}}\rightarrow DK$ and $Z_{\mathrm{V}}\rightarrow D_{s}\pi$, and
for this reason consider only two dominant $S$-wave decays.

###  III.2 Decays $Z_{\mathrm{V}}\rightarrow DK$ and
$Z_{\mathrm{V}}\rightarrow D_{s}\pi$

In this subsection, we consider the $P$-wave decays $Z_{\mathrm{V}}\rightarrow
DK$ and $Z_{\mathrm{V}}\rightarrow D_{s}\pi$ of the tetraquark
$Z_{\mathrm{V}}$. Treatments of these processes do not differ from analysis
carried out above, differences being mainly in meson-tetraquark vertices and
matrix elements of final-state mesons employed in calculations.

Let us concentrate on the decay $Z_{\mathrm{V}}\rightarrow DK$. The
correlation function to find a sum rule for the strong coupling $G_{1}$ of
vertex $Z_{\mathrm{V}}DK$ is given by the formula

$\Pi_{\nu}(p,q)=i\int d^{4}xe^{ipx}\langle
K(q)|\mathcal{T}\\{J^{D}(x)J_{\nu}^{{\dagger}}(0)\\}|0\rangle,$ (47)

where $J^{D}(x)$ is the interpolating current

$J^{D}(x)=\overline{d}_{l}(x)i\gamma_{5}c_{l}(x),$ (48)

for the pseudoscalar meson $D$.

Then, the physical side of the sum rule has the form

$\displaystyle\Pi_{\nu}^{\mathrm{Phys}}(p,q)=\frac{\langle
0|J^{D}|D\left(p\right)\rangle}{p^{2}-m_{D}^{2}}\langle
D\left(p\right)K(q)|Z_{\mathrm{V}}(p^{\prime})\rangle$
$\displaystyle\times\frac{\langle
Z_{\mathrm{V}}(p^{\prime})|J_{\nu}^{\dagger}|0\rangle}{p^{\prime
2}-m^{2}}+\cdots.$ (49)

Using the matrix elements

$\langle 0|J^{D}|D\rangle=\frac{f_{D}m_{D}^{2}}{m_{c}},\ \langle
D\left(p\right)K(q)|Z_{\mathrm{V}}(p^{\prime})\rangle=G_{1}p\cdot\epsilon^{\prime},$
(50)

for $\Pi_{\nu}^{\mathrm{Phys}}$, we find

$\displaystyle\Pi_{\nu}^{\mathrm{Phys}}(p,q)=G_{1}\frac{f_{D}m_{D}^{2}fm}{2m_{c}(p^{2}-m_{D}^{2})(p^{\prime
2}-m^{2})}$
$\displaystyle\times\left[\left(-1+\frac{m_{D}^{2}-m_{K}^{2}}{m^{2}}\right)p_{\nu}+\left(1+\frac{m_{D}^{2}-m_{K}^{2}}{m^{2}}\right)q_{\nu}\right]$
$\displaystyle+\cdots.$ (51)

In expressions above, $m_{D}$ and $f_{D}$ are the mass and decay constant of
the $D$ meson, respectively.

In terms of quark propagators the same correlation function
$\Pi_{\upsilon}(p,q)$ is determined by the expression

$\displaystyle\Pi_{\nu}^{\mathrm{OPE}}(p,q)=\int
d^{4}xe^{ipx}\varepsilon\widetilde{\varepsilon}\left[\gamma_{5}\widetilde{S}_{c}^{lc}(x){}\gamma_{5}\widetilde{S}_{d}^{nl}(-x)\right.$
$\displaystyle\left.\times\gamma_{5}{}{}\gamma_{\nu}\right]_{\alpha\beta}\langle
K(q)|\overline{u}_{\alpha}^{b}(0)s_{\beta}^{m}(0)|0\rangle.$ (52)

Operations necessary to derive the sum rule for the coupling $G_{1}$ have just
been explained above, that is why we do not consider these questions. Let us
note that the sum rule for $G_{1}$ has been obtained using the structures
proportional $p_{\nu}$. The local matrix element of $K$ meson which
contributes to $\Pi_{\nu}^{\mathrm{OPE}}(M^{2},s_{0})$ is

$\langle 0|\overline{d}i\gamma_{5}u|\pi\rangle=\frac{f_{K}m_{K}^{2}}{m_{s}},$
(53)

where $m_{K}$ and $f_{K}$ are the mass and decay constant of the $K$ meson.

The width of the process $Z_{\mathrm{V}}\rightarrow DK$ can be found by means
of the formula

$\Gamma_{3}\left[Z_{\mathrm{V}}\rightarrow
DK\right]=\frac{G_{1}^{2}\lambda^{3}(m,m_{D,}m_{K})}{24\pi m^{2}}.$ (54)

In sum rule computations of the coupling $G_{1}$ the Borel and continuum
threshold parameters are chosen as in Eq. (16). The spectroscopic parameters
of the mesons $D$ and $K$ are collected in Table 1. Our predictions read

$\displaystyle G_{1}=4.3_{-0.7}^{+1.2},$
$\displaystyle\Gamma_{3}\left[Z_{\mathrm{V}}\rightarrow
DK\right]=34.6_{-10.9}^{+20.6}~{}\mathrm{MeV}.$ (55)

For the second $P$-wave decay $Z_{\mathrm{V}}\rightarrow D_{s}\pi$, we find

$\displaystyle G_{2}=6.6_{-1.1}^{+1.8},$
$\displaystyle\Gamma_{4}\left[Z_{\mathrm{V}}\rightarrow
D_{s}\pi\right]=81.8_{-25.8}^{+48.9}~{}\mathrm{MeV}.$ (56)

Decay channels of the tetraquark $Z_{\mathrm{V}}$ considered in this section
allow us to evaluate its full width

$\Gamma_{\mathrm{full}}=156_{-30}^{+56}~{}\mathrm{MeV}.$ (57)

It is clear that $Z_{\mathrm{V}}$ can be classified as an exotic meson of wide
width. Its main decay modes are processes $Z_{\mathrm{V}}\rightarrow D_{s}\pi$
and $Z_{\mathrm{V}}\rightarrow DK$ with branching ratios
$\mathcal{BR}(Z_{\mathrm{V}}\rightarrow D_{s}\pi)\approx 0.52$ and
$\mathcal{BR}(Z_{\mathrm{V}}\rightarrow DK)\approx 0.22$, respectively.
Contribution of the decay $Z_{\mathrm{V}}\rightarrow D_{s1}\pi$ is also
considerable with estimate $\mathcal{BR}(Z_{\mathrm{V}}\rightarrow
D_{s}\pi)\approx 0.19$.

##  IV Conclusions and final notes

In this article, we have studied the doubly charged vector tetraquark
$Z_{\mathrm{V}}^{++}=[cu][\overline{s}\overline{d}]$ and calculated its mass
$m$ and width $\Gamma_{\mathrm{full}}$. The parameters of
$Z_{\mathrm{V}}^{++}$ have been evaluated using the QCD two-point and light-
cone sum rules. The doubly charged tetraquarks
$[sd][\overline{u}\overline{c}]$ with spin-parities $J^{\mathrm{P}}=0^{+}$,
$0^{-}$ and $1^{+}$ were investigated in our paper Agaev:2017oay . The exotic
mesons $Z^{++}$ are built of four quarks of different flavors, and moreover
bear two units of electric charge. These particles were not found till now,
but due to unique features are interesting objects for both theoretical and
experimental studies.

The LHCb collaboration recently observed structures $X_{0(1)}(2900)$ which may
be interpreted as exotic mesons containing four different quarks Aaij:2020hon
; Aaij:2020ypa . The structures $X_{0(1)}$ were seen as resonance-like peaks
in the mass distribution of $D^{-}K^{+}$ mesons. The master process to
discover $X_{0(1)}$ was the weak decay of the $B$ meson $B^{+}\rightarrow
D^{+}D^{-}K^{+}$. It is remarkable that the process $B^{+}\rightarrow
D^{+}D^{-}K^{+}$ and data collected during its exploration can be employed to
observe another tetraquarks, namely states with quark content
$cu\overline{s}\overline{d}$. In fact, analysis of the invariant mass
distribution of mesons $D^{+}K^{+}$ may lead to information about the scalar
and vector tetraquarks $Z_{\mathrm{S}}^{++}$ and $Z_{\mathrm{V}}^{++}$. Decays
to final mesons $D^{+}K^{+}$ are among favored channels of these particles. In
fact, relevant branching ratios are equal to
$\mathcal{BR}(Z_{\mathrm{S}}\rightarrow DK)\approx 0.86$ and
$\mathcal{BR}(Z_{\mathrm{V}}\rightarrow DK)\approx 0.22$, respectively.

The decay of the $B$ meson $B^{+}\rightarrow D^{-}D_{s}^{+}\pi^{+}$ is also
useful to see states $Z_{\mathrm{S}}^{++}$ and $Z_{\mathrm{V}}^{++}$, because
processes $Z_{\mathrm{S}}\rightarrow D_{s}\pi$ and $Z_{\mathrm{V}}\rightarrow
D_{s}\pi$ have considerable branching ratios
$\mathcal{BR}(Z_{\mathrm{S}}\rightarrow D_{s}\pi)\approx 0.12$ and
$\mathcal{BR}(Z_{\mathrm{V}}\rightarrow D_{s}\pi)\approx 0.52$. The decay
$B^{+}\rightarrow D^{-}D_{s}^{+}\pi^{+}$ can be used to discover also
tetraquarks $[ud][\overline{c}\overline{d}]$: These neutral states may be
fixed in the invariant mass distribution of $D^{-}\pi^{+}$ mesons.

We have modeled $Z_{\mathrm{V}}^{++}$ as a four-quark exotic meson with
diquark-antidiquark structure. Our analyses have proved that the interpolating
current (3) used in the framework of the QCD sum rule method correctly
describes the particle $Z_{\mathrm{V}}^{++}$ and leads to reliable results for
its parameters. In fact, existence of the working windows for parameters
$M^{2}$ and $s_{0}$ that satisfy standard requirements of the sum rules, and
self-consistency of used $\sqrt{s_{0}}$ and extracted $m$ argue in favor of
this conclusion. It is known that a diquark-antidiquarks are tightly bound
states, and may be favorite forms for doubly charged tetraquarks. But such
four-quark systems may exist also as hadronic molecules. Thus, doubly charged
molecular compounds with the quark content $\overline{Q}\overline{Q}qq$, where
$Q$ is $c$ or $b$-quark were considered in Ref. Ohkoda:2012hv . In this
article the authors used the heavy quark effective theory to derive
interactions between heavy mesons, and coupled channel Schrodinger equations
to find the bound and/or resonant states with various quantum numbers. It was
demonstrated that, for example, $D$ and $D^{\ast}$ mesons can form doubly
charged resonant state with $I(J^{\mathrm{P}})=1(0^{-})$. Similar analysis of
a molecule counterpart of $Z_{\mathrm{V}}^{++}$ in the context of the QCD sum
rule method implies usage of molecular-type interpolating current. Whether
such current would lead to strong sum rule predictions or not requires
detailed analysis, which however is beyond the scope of the present paper.

It is seen, that three-meson weak decays of $B$ meson are sources of valuable
information on four-quark exotic states. Data collected by various
collaborations in running experiments can be utilized for these purposes. New
decays of $B$ meson can open wide prospects to study numerous four-quark
states. Indeed, final-state mesons in such processes can be combined to form
different pairs and their invariant mass distributions can be explored to
detect resonance-type enhancements. In any case, additional experimental and
theoretical studies are necessary to take advantage of emerging opportunities.

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