# The resonance $X(4630)$

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

###### Abstract

We investigate the structure $X(4630)$ discovered by the LHCb collaboration in
the process $B^{+}\rightarrow J/\psi\phi K^{+}$ as a resonance in the
$J/\psi\phi$ mass distribution. We explore this resonance as a diquark-
antidiquark state $X=[cs][\overline{c}\overline{s}]$ with spin-parities
$J^{\mathrm{PC}}=1^{-+}$. Its mass and current coupling are calculated using
the QCD two-point sum rule method by taking into account vacuum condensates up
to dimension $10$. We also study decays of this tetraquark to mesons
$J/\psi\phi$, $\eta_{c}\eta^{(\prime)}$, and $\chi_{c1}\eta^{(\prime)}$, and
compute partial widths of these channels. To this end, we employ the light-
cone sum rule approach and technical methods of soft-meson approximation to
extract strong coupling at relevant tetraquark-meson-meson vertices. Our
predictions for the mass $m=(4632\pm 60)~{}\mathrm{MeV}$ and width
$\Gamma=(159\pm 31)~{}\mathrm{MeV}$ of $X$ are in a very nice agreement with
recent measurements of the LHCb collaboration. These results allow us to
interpret the resonance $X(4630)$ as the tetraquark $X$ with spin-parities
$J^{\mathrm{PC}}=1^{-+}$.

##  I Introduction

Recently the LHCb collaboration informed about new charmonium-like resonances
$Z_{cs}$ and $X$ observed in the process $B^{+}\rightarrow J/\psi\phi K^{+}$
in $J/\psi K^{+}$ and $J/\psi\phi$ invariant mass distributions LHCb:2021uow .
The new resonances $Z_{cs}(4000)^{+}$ and $Z_{cs}(4220)^{+}$ were discovered
in the $J/\psi K^{+}$ channel, and are presumably exotic mesons with a quark
content $c\overline{c}u\overline{s}$. States fixed in the $J/\psi\phi$ channel
should be composed of $c\overline{c}s\overline{s}$ quarks provided they are
four-quark structures. New resonances in this channel $X(4630)$ and $X(4685)$
enriched a list of vector, axial-vector and scalar states discovered by LHCb
during last years Aaij:2016iza ; LHCb:2016nsl . The collaboration also updated
parameters of states seen at early stages of investigations.

These experimental results generated a theoretical activity aimed to explain
obtained information in the context of various approaches of high energy
physics. Studies were concentrated mainly around the resonances $X(4630)$ and
$Z_{cs}$, in which authors calculated masses, magnetic moments of these
states, explored their decay channels Liu:2021xje ; Ozdem:2021yvo ;
Yang:2021sue ; Wang:2021ghk ; Turkan:2021ome . Some of new states were
explained as threshold effects as well Ge:2021sdq .

The structure $X(4630)$ is a wide resonance with the mass

$m_{\exp}=(4626\pm 16_{-110}^{+18})~{}\mathrm{MeV},$ (1)

and width

$\Gamma_{\exp}=(174\pm 27_{-73}^{+134})~{}\mathrm{MeV},$ (2)

respectively. The LHCb determined also the spin-parity of $X(4630)$ and fixed
them $J^{\mathrm{P}}=1^{-}$.

It should be noted that, a vector structure $Y(4626)$ with the mass
$4625.9_{-6.0}^{+6.2}(\mathrm{stat.})\pm 0.4(\mathrm{sys.})~{}\mathrm{MeV}$
and the width $49.8_{-11.5}^{+13.9}(\mathrm{stat.})\pm
4.0(\mathrm{sys.})~{}\mathrm{MeV}$ was seen by the Belle collaboration
recently in the process $e^{+}e^{-}\rightarrow D_{s}^{\ast}D_{s1}(2536)$
Belle:2019qoi . This resonance can be considered as a member of $Y$ family of
vector states discovered in electron-positron annihilations. Another members
of this group are resonances $Y(4630)$ and $Y(4660)$. First of them was
detected by Belle in the process
$e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\Lambda_{c}^{-}$ as a peak in the
$\Lambda_{c}^{+}\Lambda_{c}^{-}$ invariant mass distribution Pakhlova:2008vn .
Its parameters
$m=4634_{-7}^{+8}(\mathrm{stat.})_{-8}^{+5}(\mathrm{sys.})~{}\mathrm{MeV}$ and
$\Gamma=92_{-24}^{+40}(\mathrm{stat.})_{-21}^{+10}(\mathrm{sys.})~{}\mathrm{MeV}$
are close to ones of the resonance $Y(4626)$, and whether they are different
states or not is under studies. Besides different assumptions about its
structure, $Y(4630)$ is usually identified with the vector state $Y(4660)$.

The resonance $Y(4660)$, as a particle produced in $e^{+}e^{-}$ annihilation,
bears the quantum numbers $J^{\mathrm{PC}}=1^{--}$. It was modeled as excited
$5{}^{3}S_{1}$ and $6{}^{3}S_{1}$ charmonia, as a compound of the scalar
$f_{0}(980)$ and vector $\psi(2S)$ mesons, or as a baryonium state. In our
work Sundu:2018toi , we explored $Y(4660)$ by treating it as the diquark-
antidiquark vector state $[cs][\overline{c}\overline{s}]$. We calculated the
mass and current coupling of the tetraquark $[cs][\overline{c}\overline{s}]$,
and also evaluated its full width. Our results for the mass and full width of
the state $[cs][\overline{c}\overline{s}]$ allowed us to interpret it as the
observed resonance $Y(4660)$.

From analysis of the decay channel $X(4630)\rightarrow J/\psi\phi$, it is
clear, that $X(4630)$ is charmonium-like state probably with hidden strange
component $s\overline{s}$. Then, in the four-quark model its quark content
should be $c\overline{c}s\overline{s}$. It is also evident that $C$-parity
conservation implies that $X(4630)$ is $C$ parity positive particle, i.e., the
quantum numbers of this resonance should be $J^{\mathrm{PC}}=1^{-+}$. In other
words, it can be considered as $C=+1$ counterpart of the resonance $Y(4626)$.
Spin-parities $J^{\mathrm{PC}}=1^{-+}$ exclude interpretation of $X(4630)$ as
an ordinary meson, because these quantum numbers are not accessible in the
conventional quark-antiquark model. In other words, the resonance $X(4630)$
maybe is a double-exotic state: It is composed of four quarks and carry exotic
quantum numbers.

Four valence quarks can be grouped in different ways to form a single
structure. Indeed, they may form two conventional colorless mesons and
constitute a hadronic molecule. Alternatively, four quarks
$c\overline{c}s\overline{s}$ may build a diquark-antidiquark state
$[cs][\overline{c}\overline{s}]$. The resonance $X(4630)$ was examined in the
context of both these models. Thus, it was considered in Ref. Yang:2021sue  as
the molecule $D_{s}^{\ast}\overline{D}_{s1}(2536)$ with required spin-
parities. An analysis was performed there using the one-boson-exchange method.
The mass of the molecule $D_{s}^{\ast}\overline{D}_{s1}$ was found equal to
$4644~{}\mathrm{MeV}$ which is consistent with the LHCb data. The authors also
emphasized, that a decay to a meson pair $J/\psi\phi$ is the main decay
channel of the molecule $D_{s}^{\ast}\overline{D}_{s1}$.

The molecule model for $Y(4626)$ was used in Ref. He:2019csk , in which it was
examined as a system $J^{\mathrm{PC}}=1^{--}$ appearing from the interaction
$D_{s}^{\ast}\overline{D}_{s1}-D_{s}\overline{D}_{s1}$. In this article
structures with spin-parities $J^{\mathrm{PC}}=0^{--},\ 0^{-+},\ 1^{-+}$ and
others were explored as well. This treatment for the masses of the molecules
$D_{s}^{\ast}\overline{D}_{s1}$ with $J^{\mathrm{PC}}=1^{--}$ and
$J^{\mathrm{PC}}=1^{-+}$ leads to predictions $4646~{}\mathrm{MeV}$ and
$4648~{}\mathrm{MeV}$, respectively. In the context of the Bethe-Salpeter
method similar problems were addressed in Ref. Dong:2021juy .

In the framework of the QCD sum rule method a diquark-antidiquark option was
considered in Ref. Wang:2013exa . The result of this article for the mass of
the tetraquark $[cs][\overline{c}\overline{s}]$ with $J^{\mathrm{PC}}=1^{-+}$
equals to $4.63_{-0.08}^{+0.11}~{}\mathrm{GeV}$ and agrees with the new LHCb
data. As is seen, almost all models for $X(4630)$ and predictions for its mass
extracted using various methods within errors are consistent with experimental
data. Stated differently, masses of exotic states do not provide information
sufficient to verify different models by confronting them with each another
or/and experimental data. Therefore, besides computations of the mass, there
is a necessity to evaluate the full width of $X(4630)$ as precise as possible.

In the present work, we are going to fulfill this program and calculate the
mass and width of the resonance $X(4630)$. We treat $X(4630)$ as diquark-
antidiquark vector state $X=[cs][\overline{c}\overline{s}]$ with spin-parities
$J^{\mathrm{PC}}=1^{-+}$. Investigations are performed in the context of the
QCD sum rule method Shifman:1978bx ; Shifman:1978by , which is one of powerful
nonperturbative tools of high energy physics. It allows one to compute
parameters not only of conventional mesons and baryons, but also of multiquark
hadrons Agaev:2020zad ; Albuquerque:2018jkn .

The mass and current coupling of the tetraquark $X$ are calculated in the
framework of the QCD two-point sum rule approach. In these calculations, we
take into account various quark, gluon, and mixed vacuum condensates up to
dimension $10$. To investigate numerous decay channels of $X$, we use the
light-cone sum rule (LCSR) method Balitsky:1989ry . Most of tetraquarks are
strong-interaction unstable particles and decay to two conventional mesons.
The resonance $X(4630)$ decays primarily to a pair of mesons $J/\psi\phi$
which is experimentally confirmed fact. In the present work, we study decays
of the tetraquark $X$ not only to $J/\psi\phi$, but also to
$\eta_{c}\eta^{(\prime)}$, and $\chi_{c1}\eta^{(\prime)}$ mesons saturating by
these five channels its full width. The process $X\rightarrow J/\psi\phi$ is
the dominant decay channel of the tetraquark $X$, whereas remaining modes are
subdominant ones, but their contributions are important to evaluate the full
width of $X$.

Partial widths of aforementioned decays are determined by strong couplings at
relevant vertices. For instance, in the case of the dominant decay this is a
strong coupling $G$ at the vertex $XJ/\psi\phi$. Calculation of the strong
coupling at the tetraquark and two mesons vertex $XJ/\psi\phi$ in the LCSR
method necessitates usage of complementary technical tools. A reason is that
$X$ is built of four valence quarks, and the light-cone expansion of the
relevant nonlocal correlator leads to expressions which instead of
distribution amplitudes of the $\phi$ meson depend on its local matrix
elements. To preserve the four-momentum at the vertex $XJ/\psi\phi$, in this
situation one needs to impose additional kinematical restriction on the
momentum of the $\phi$ meson. Troubles encountered afterward can be handled by
including into analysis technical methods known as a soft-meson approximation
Ioffe:1983ju ; Braun:1995 . The soft-meson approximation was adapted for
investigation of tetraquarks in Ref. Agaev:2016dev , and applied to explore
decays some of such particles (see, for example, Ref. Agaev:2020zad ). In
present article, strong couplings at relevant vertices are computed by
including into analysis nonperturbative terms up to dimension $8$. The
coupling $G$ receives a contribution also from twist $4$ matrix element of the
$\phi$ meson.

This article is organized in the following way: The mass and current coupling
of the tetraquark $X$ are computed in Section II. We calculate the strong
coupling $G$ of particles at the vertex $XJ/\psi\phi$ in Section III. Here, we
find also the partial width of the decay $X\rightarrow J/\psi\phi$. Section IV
is devoted to analysis of the processes $X\rightarrow\eta_{c}\eta^{(\prime)}$
and $X\rightarrow\chi_{c1}\eta^{(\prime)}$, and to computation of their
partial widths. To this end, we calculate couplings $g_{1}$ and $g_{2}$
corresponding to vertices $X\eta_{c}\eta$ and $X\eta_{c}\eta^{\prime}$,
respectively. Strong couplings $g_{3}$ and $g_{4}$ required to study decays
$X\rightarrow\chi_{c1}\eta^{(\prime)}$ are found also in this section. In Sec.
V, we confront our results with LHCb data for the resonance $X(4630)$. This
section contains also our concluding remarks.

##  II Mass and current coupling of the tetraquark $X$

Sum rules to calculate the mass $m$ and current coupling $f$ of the tetraquark
$X$ can be derived from analysis of the correlation function

$\Pi_{\mu\nu}(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{{\dagger}}(0)\\}|0\rangle,$ (3)

where $J_{\mu}(x)$ is the interpolating current for the $X$ state, and
$\mathcal{T}$ is time-ordered product of two currents.

The current with required properties has the following form

$\displaystyle J_{\mu}(x)$ $\displaystyle=$
$\displaystyle\epsilon\widetilde{\epsilon}\left[s_{b}^{T}(x)C\gamma_{5}c_{c}(x)\overline{s}_{d}(x)\gamma_{5}\gamma_{\mu}C\overline{c}_{e}^{T}(x)\right.$
(4)
$\displaystyle\left.-s_{b}^{T}(x)C\gamma_{\mu}\gamma_{5}c_{c}(x)\overline{s}_{d}(x)\gamma_{5}C\overline{c}_{e}^{T}(x)\right],$

where $\epsilon\widetilde{\epsilon}=\epsilon_{abc}\epsilon_{ade}$, and $a$,
$b$, $c$, $d$, and $e$ are color indices. In expression above $C$ is the
charge conjugation matrix.

The current $J_{\mu}(x)$ describes the tetraquark composed of the color
antitriplet scalar diquark $\epsilon s^{T}C\gamma_{5}c$ and color triplet
vector antidiquark
$\widetilde{\epsilon}\overline{s}\gamma_{5}\gamma_{\mu}C\overline{c}^{T}$ and
belongs to
$\mathbf{[}\overline{\mathbf{3}_{\mathbf{c}}}\mathbf{]}_{cs}\mathbf{[3}_{\mathbf{c}}\mathbf{]}_{\overline{c}\overline{s}}$
representation of the color group $SU_{c}(3)$. Because the scalar diquark
configuration is most attractive and stable two-quark system Jaffe:2004ph ,
the current $J_{\mu}(x)$ corresponds to a ground-state vector particle with
lowest mass and required spin-parities.

To derive desired sum rules, we write down the correlation function
$\Pi_{\mu\nu}(q)$ using the mass and current coupling of the state $X$. For
these purposes, we insert into the correlation function a complete set of
states with quantum numbers of $X$ and carry out in Eq. (3) integration over
$x$. As a result, we get

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

with $m$ being the mass of $X$. In Eq. (5) dots stand for contributions of
higher resonances and continuum states. We introduce the current coupling $f$
by means of the matrix element

$\langle 0|J_{\mu}|X(p,\varepsilon)\rangle=fm\varepsilon_{\mu},$ (6)

where $\varepsilon_{\mu}$ is the polarization vector of the tetraquark $X$. In
terms of $m$ and $f$, the correlation function can be rewritten in the
following form

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

The Borel transformation applied to Eq. (7) yields

$\mathcal{B}\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=m^{2}f^{2}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m^{2}}\right)e^{-m^{2}/M^{2}}+\cdots,$
(8)

where $M^{2}$ is the Borel parameter.

The QCD side of the sum rules $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ should be
calculated in the operator product expansion ($\mathrm{OPE}$) with some
certain accuracy. We calculate the correlation function
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ using explicit expression of the current
$J_{\mu}(x)$ and express it in terms of heavy and light quark propagators.
Then for $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$, we get the following formula

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p)=i\int
d^{4}xe^{ipx}\epsilon\widetilde{\epsilon}\epsilon^{\prime}\widetilde{\epsilon}^{\prime}\left\\{\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{s}^{bb^{\prime}}(x)\gamma_{5}\right.\right.$
$\displaystyle\left.\times
S_{c}^{cc^{\prime}}(x)\right]\mathrm{Tr}\left[\gamma_{5}\gamma_{\mu}\widetilde{S}_{c}^{ee^{\prime}}(-x)\gamma_{\nu}\gamma_{5}S_{s}^{dd^{\prime}}(-x)\right]$
$\displaystyle-\mathrm{Tr}\left[\gamma_{5}\gamma_{\mu}\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{s}^{d^{\prime}d}(-x)\right]\mathrm{Tr}\left[\gamma_{5}\gamma_{\nu}\widetilde{S}_{s}^{bb^{\prime}}(x)\right.$
$\displaystyle\times\left.\gamma_{5}S_{c}^{cc^{\prime}}(x)\right]-\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{\nu}\gamma_{5}S_{s}^{d^{\prime}d}(-x)\right]$
$\displaystyle\times\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{s}^{bb^{\prime}}(x)\gamma_{\mu}\gamma_{5}S_{c}^{cc^{\prime}}(x)\right]+\mathrm{Tr}\left[\gamma_{5}\gamma_{\nu}\widetilde{S}_{s}^{bb^{\prime}}(x)\right.$
$\displaystyle\left.\left.\times\gamma_{\mu}\gamma_{5}S_{c}^{cc^{\prime}}(x)\right]\mathrm{Tr}\left[\gamma_{5}\widetilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{s}^{dd^{\prime}}(-x)\right]\right\\},$
(9)

where
$\epsilon^{\prime}\widetilde{\epsilon}^{\prime}=\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\epsilon_{a^{\prime}d^{\prime}e^{\prime}}$.
In Eq. (9) $S_{s}^{ab}(x)$ and $S_{c}^{ab}(x)$ are the $s$ and $c$-quark
propagators, respectively. Their explicit expressions are collected in Ref.
Agaev:2020zad . Here, we also use the notation

$\widetilde{S}_{s(c)}(x)=CS_{s(c)}^{T}(x)C.$ (10)

The QCD sum rules can be obtained by isolating the same Lorentz structures in
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ and $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ and
performing required manipulations. We work with the terms proportional to
$g_{\mu\nu}$, which receive contributions only from spin$-1$ particles and are
free of contaminations. To derive sum rules, it is necessary to equate
invariant amplitudes $\Pi^{\mathrm{Phys}}(p^{2})$ and
$\Pi^{\mathrm{OPE}}(p^{2})$ that correspond to these structures. At the next
step, one has to apply the Borel transformation to both sides of obtained
expression: This is necessary to suppress contributions of higher resonances
and continuum states. At the final stage of manipulations, we use the
assumption about the quark-hadron duality, and subtract from the physical side
contributions of higher resonances and continuum terms. But this equality is
not enough to determine both the mass and coupling of $X$. To complete a
system of equations, we act by the operator $d/d(-1/M^{2})$ to both sides of
the first equality, and by this way find missed second expression. This system
can be solved to find sum rules for the mass $m$ and coupling $f$, which read

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

and

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

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

These sum rules depend on the function $\Pi(M^{2},s_{0})$ which is Borel
transformed and continuum subtracted invariant amplitude
$\Pi^{\mathrm{OPE}}(p^{2})$. It is a function of the Borel and continuum
threshold parameters $M^{2}$ and $s_{0}$, and has the following form

$\Pi(M^{2},s_{0})=\int_{4\mathcal{M}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}),$
(13)

where $\mathcal{M=}m_{c}+m_{s}$, and $\rho^{\mathrm{OPE}}(s)$ is the two-point
spectral density. The second component of the invariant amplitude $\Pi(M^{2})$
contains nonperturbative contributions calculated directly from
$\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. In the present article, $\Pi(M^{2},s_{0})$
is calculated by taking into account quark, gluon and mixed vacuum condensates
up to dimension $10$. Details of computations required to determine the
spectral density $\rho^{\mathrm{OPE}}(s)$ can be found, for instance, in Ref.
Agaev:2016dev . Therefore, we do not consider here these usual operations, and
do not provide lengthy explicit expression of the function $\Pi(M^{2},s_{0})$.

The sum rules for the mass and coupling given by Eqs. (11) and (12) contain
quark, gluon and mixed condensates which are universal parameters of
computations. They depend also masses of $c$ and $s$ quarks. Numerical values
all of these parameters are listed below

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\ \langle\overline{s}s\rangle=(0.8\pm
0.1)\langle\overline{q}q\rangle,$ $\displaystyle\langle\overline{s}g_{s}\sigma
Gs\rangle=m_{0}^{2}\langle\overline{s}s\rangle,\ m_{0}^{2}=(0.8\pm
0.1)~{}\mathrm{GeV}^{2},\ $
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle\langle
g_{s}^{3}G^{3}\rangle=(0.57\pm 0.29)~{}\mathrm{GeV}^{6},$ $\displaystyle
m_{c}=(1.275\pm 0.025)~{}\mathrm{GeV}.\ \ \
m_{s}=93_{-5}^{+11}~{}\mathrm{MeV}.\text{ }$ (14)

The sum rules are functions also of auxiliary parameters $M^{2}$ and $s_{0}$,
which have to obey standard constraints imposed on them by the sum rule
method. This means, that in the working regions of the parameters $M^{2}$ and
$s_{0}$ a pole contribution ($\mathrm{PC}$) should dominate in the sum rules
and the operator product expansion should converge rapidly. To quantify these
constraints and use them to fix working windows for $M^{2}$ and $s_{0}$, we
introduce expressions

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

and

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

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

The formula Eq. (15) determines a contribution of the pole term to the
function $\Pi(M^{2},s_{0})$. In our present study, we adopt the limit
$\mathrm{PC\geqslant 0.2}$, which is typical for multiquark particles. The
convergence of the operator product expansion is examined by means of the
expression Eq. (16): The convergence of $\mathrm{OPE}$ is fulfilled if at the
minimum of the Borel parameter the ratio $R(M^{2})$ does not exceed $0.01$.
The mass and current coupling of $X$ obtained by means of the sum rules, in
general, have not to depend on the Borel parameter, but in actual
computations, one can only limit its influence on obtained predictions. Thus,
a stability of extracted results is among employed constraints to get the
parameters $M^{2}$ and $s_{0}$.

Computations show that the working regions that meet all these constraints are

$M^{2}\in[5.5,6.5]~{}\mathrm{GeV}^{2},\ s_{0}\in[24,25]~{}\mathrm{GeV}^{2}.$
(17)

In fact, in these regions the pole contribution varies within a range
$0.66\leq\mathrm{PC}\leq 0.26$. The convergence of $\mathrm{OPE}$ is also
satisfied, because at $M^{2}=5.5~{}\mathrm{GeV}^{2}$, we fix $R(M^{2})\leq
0.01$.

Figure 1:  The mass of the tetraquark $X(4630)$ as a function of the Borel
parameter $M^{2}$ at fixed $s_{0}$ (left panel), and as a function of the
continuum threshold $s_{0}$ at fixed $M^{2}$ (right panel).

To extract numerical values of the mass $m$ and coupling $f$, we calculate
them at different choices of the parameters $M^{2}$ and $s_{0}$, and find
their mean values averaged over the working regions Eq. (17). For $m$ and $f$
these calculations yield

$\displaystyle m$ $\displaystyle=$ $\displaystyle(4632\pm 60)~{}\mathrm{MeV},$
$\displaystyle f$ $\displaystyle=$ $\displaystyle(9.2\pm 0.8)\times
10^{-3}~{}\mathrm{GeV}^{4}.$ (18)

The values from Eq. (18) correspond to sum rules’ results computed at middle
point of the working regions, i.e., to results at the points
$M^{2}=6~{}\mathrm{GeV}^{2}$ and $s_{0}=24.5~{}\mathrm{GeV}^{2}$. At this
point the pole contribution is $\mathrm{PC}\approx 0.51$, which guarantees
reliability of obtained predictions, and a ground-state nature of $X$.

In Fig. 1, we plot the mass of the tetraquark $X$ as functions of the
parameters $M^{2}$ and $s_{0}$. As is seen, the mass $m$ is sensitive to a
choice of $M^{2}$ and $s_{0}$. It is also evident that within limits
$M^{2}\in[5.5,6.5]~{}\mathrm{GeV}^{2}$ this dependence is weak and theoretical
errors do not exceed $1.5\%$, whereas similar estimate for the coupling gives
$9\%$. This effect has simple explanation: The mass of the tetraquark is
determined by the ratio of the correlation functions Eq. (11). As a result,
this ratio smooths dependence of $m$ on the parameter $M^{2}$, which is not a
case for the coupling Eq. (12).

The mass of the tetraquark $X$ obtained in the present work is in excellent
agreement with the LHCb data for the mass of the resonance $X(4630)$. At this
phase of our studies, we can conclude that $X(4630)$ is the diquark-
antidiquark state $X=[cs][\overline{c}\overline{s}]$ with spin-parities
$J^{\mathrm{PC}}=1^{-+}$.

##  III  Decay $X\rightarrow J/\psi\phi$

The resonance $X(4630)$ was observed in the invariant mass distribution of the
$J/\psi\phi$ mesons. Hence, the process $X(4630)\rightarrow J/\psi\phi$ can be
considered as its dominant decay channel. In this section, we consider this
decay and calculate partial width of the process $X\rightarrow J/\psi\phi$,
which is determined by the strong coupling $G$ at the vertex $XJ/\psi\phi$.
Hence, our aim is to find a sum rule for this parameter.

In the context of the LCSR method the vertex $XJ/\psi\phi$ can be analyzed by
employing the non-local correlator

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

with $J_{\nu}$ and $J_{\mu}^{J/\psi}$ being the interpolating currents of the
tetraquark $X$ and vector meson $J/\psi$, respectively. The $J_{\nu}$ is given
by Eq. (4), and the current $J_{\mu}^{J/\psi}$ has the form

$J_{\mu}^{J/\psi}(x)=\overline{c}_{l}(x)\gamma_{\mu}c_{l}(x),$ (20)

where $l=1,2,3$ is color index.

We calculate $\Pi_{\mu\nu}(p,q)$ using the QCD sum rule on the light-cone and
soft-meson approximation. For these purposes, one should rewrite this function
using the masses, decay constants (or couplings) of involved particles and
find physical side of the sum rule $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)$. It is
not difficult to see that

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|J_{\mu}^{J/\psi}|J/\psi\left(p\right)\rangle}{p^{2}-m_{1}^{2}}\langle
J/\psi\left(p\right)\phi(q)|X(p^{\prime})\rangle$ (21)
$\displaystyle\times\frac{\langle
X(p^{\prime})|J_{\nu}^{\dagger}|0\rangle}{p^{\prime 2}-m^{2}}+\cdots,$

where, we introduce short-hand notation $m_{1}$ for the mass of the meson
$J/\psi$. In Eq. (21) $p^{\prime}$, and $p$ are the momenta of the tetraquark
$X$ and meson $J/\psi$, respectively: Then the 4-momentum of the meson $\phi$
is equal to $q=p^{\prime}-p$.

The correlation function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)$ can be further
simplified using matrix elements of particles involved into this decay. The
matrix element of the tetraquark $X$ has been given above in Eq. (6). We
introduce the matrix element of the meson $J/\psi$ in the following form

$\langle
0|J_{\mu}^{J/\psi}|J/\psi\left(p\right)\rangle=f_{1}m_{1}\varepsilon_{\mu}(p),$
(22)

where $f_{1}$ and $\varepsilon_{\mu}(p)$ are the decay constant and
polarization vector of $J/\psi$, respectively. We define the matrix element of
the vertex $XJ/\psi\phi$ by the expression

$\displaystyle\langle
J/\psi\left(p\right)\phi(q)|X(p^{\prime})\rangle=G\left[(q-p)_{\gamma}g_{\alpha\beta}-(p^{\prime}+q)_{\alpha}g_{\gamma\beta}\right.$
$\displaystyle\left.+(p^{\prime}+p)_{\beta}g_{\gamma\alpha}\right]\varepsilon^{\gamma}(p^{\prime})\varepsilon^{\ast\alpha}(p)\varepsilon^{\ast\beta}(q).$
(23)

Here $\varepsilon^{\gamma}(p^{\prime})$, $\varepsilon^{\alpha}(p)$, and
$\varepsilon^{\beta}(q)$ are polarization vectors of the tetraquark $X$, and
mesons $J/\psi$ and $\phi$, respectively. Then the contribution arising from
the ground state particles is equal to

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)=G\frac{mff_{1}m_{1}}{\left(p^{\prime
2}-m^{2}\right)\left(p^{2}-m_{1}^{2}\right)}$
$\displaystyle\times\left(\frac{m_{1}^{2}-m^{2}-m_{\phi}^{2}}{m^{2}}\varepsilon_{\mu}^{\ast}(q)p_{\nu}+\frac{m^{2}-m_{1}^{2}-m_{\phi}^{2}}{m_{1}^{2}}\varepsilon_{\nu}^{\ast}(q)p_{\mu}\right.$
$\displaystyle{}\left.-\frac{m^{2}+m_{1}^{2}-m_{\phi}^{2}}{m^{2}m_{1}^{2}}\varepsilon^{\ast}(q)\cdot
pp_{\mu}p_{\nu}+2\varepsilon^{\ast}(q)\cdot pg_{\mu\nu}+\cdots\right),$ (24)

where $m_{\phi}$ is the mass of the $\phi$ meson. Here by dots we indicate
terms that vanish in the soft-meson limit $p=p^{\prime}$ Agaev:2016dev  (see
also an explanation below).

The QCD expression for the correlator $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$ is
found using the quark propagators. After some calculations, we get

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)=-i\int
d^{4}xe^{ipx}\epsilon\widetilde{\epsilon}\left\\{\left[\gamma_{5}\widetilde{S}_{c}^{lc}(x)\gamma_{\mu}\right.\right.$
$\displaystyle\left.\left.\times\widetilde{S}_{c}^{el}(-x)\gamma_{\nu}\gamma_{5}\right]+\left[\gamma_{\nu}\gamma_{5}\widetilde{S}_{c}^{lc}(x){}\gamma_{\mu}\widetilde{S}_{c}^{el}(-x){}\gamma_{5}\right]\right\\}_{\alpha\beta}$
$\displaystyle\times\langle\phi(q)|\overline{s}_{\alpha}^{b}(0)s_{\beta}^{d}(0)|0\rangle,$
(25)

where $\alpha$ and $\beta$ are spinor indices.

As is seen, the function $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,q)$ depends on the
local matrix elements of the $\phi$ meson. In the full LCSR method there would
be non-local matrix elements, which after some transformations can be
expressed using $\phi$ meson’s distribution amplitudes of various twists and
quark-gluon content. A reason behind of emerged situation is rather simple.
Indeed, the meson $J/\psi$ is composed of $c$-quark and antiquark at $x$ which
can be contracted only with $c$-antiquark and quark from $X$. As a result,
remaining $s$-quark fields located at the same space-time position $x=0$
establish local matrix elements of the $\phi$ meson.

To understand consequences of this situation, it is convenient to perform
following transformations

$\overline{s}_{\alpha}^{b}s_{\beta}^{d}\rightarrow\frac{1}{12}\delta^{bd}\Gamma_{\beta\alpha}^{j}\left(\overline{s}\Gamma^{j}s\right),$
(26)

where $\Gamma^{j}$ is the full set of Dirac matrices,

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

Let us note, that in Eq. (26), we use also the projector onto a color-singlet
state $\delta^{bd}/3$.

After these manipulations, it is easy to carry out the color summation.
Afterwards, we substitute quark propagators into obtained expression and
perform $4-$dimensional integration over $x$. This integration creates in the
integrand the delta function $\delta^{4}(p^{\prime}-p)$, which as an argument
contains only four-momenta of the tetraquark $X$ and meson $J/\psi$.
Therefore, subsequent integration over $p$ or $p^{\prime}$ sets
$p=p^{\prime}$, which is the consequence of the four-momentum preservation at
the vertex $XJ/\psi\phi$. Stated differently, to preserve at the tetraquark-
meson-meson vertex the four-momentum one has to choose $q=0$. In the full LCSR
this is known as the soft-meson approximation Braun:1995 . At vertices of
ordinary mesons $q\neq 0$, and only in the soft-meson limit, one equates $q$
to zero, whereas the tetraquark-meson-meson vertex can be explored in the
framework of the LCSR method only for $q=0$. It is worth emphasizing that
tetraquark-tetraquark-meson vertices can be explored using the full LCSR
method: Correlation function of a such vertex depends on distribution
amplitudes of a final meson Agaev:2016srl ; Sundu:2017xct ; Agaev:2019coa .
For our purposes, it is important, that both the soft-meson approximation and
full LCSR treatment of the ordinary mesons’ vertices lead for the strong
couplings to very close numerical predictions Braun:1995 , hence our treatment
of the coupling $G$ should give a reliable result.

Equation (26) applied to $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,0)$ generates
different local matrix elements of the $\phi$ meson, which are known and can
be used to find analytical expression and carry out numerical computations.
Analysis confirms, that only two matrix elements of the $\phi$ meson
contribute to the correlation function. First of them is twist-$2$ matrix
element

$\langle\phi(q)|\overline{s}(0)\gamma_{\mu}s(0)|0\rangle=f_{\phi}m_{\phi}\varepsilon_{\mu}^{\ast}(q),$
(28)

where $f_{\phi}$ is the decay constant of $\phi$ meson. Second matrix element,
which survives in the soft-meson limit, has twist $4$ and is given by the
expression

$\langle\phi(q)|\overline{s}(0)g\widetilde{G}_{\mu\nu}\gamma^{\nu}\gamma_{5}s(0)|0\rangle=f_{\phi}m_{\phi}^{3}\zeta_{4\phi}\varepsilon_{\mu}^{\ast}(q).$
(29)

Here,
$\widetilde{G}_{\mu\nu}=1/2\varepsilon_{\mu\nu\alpha\beta}G^{\alpha\beta}$ is
the gluon dual field-strength tensor. The parameter $\zeta_{4\phi}=\pm 0.02$
was determined from the sum rule analysis in Ref. Ball:2007zt , and is small.

But before deriving the sum rule for the strong coupling $G$, the soft limit
should be implemented also in the physical expression of the correlation
function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,q)$. The limit $q\rightarrow 0$
modifies $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,0)$ in accordance with the recipe

$\frac{1}{\left(p^{\prime
2}-m^{2}\right)\left(p^{2}-m_{1}^{2}\right)}\rightarrow\frac{1}{\left(p^{2}-\widetilde{m}^{2}\right)^{2}},$
(30)

where $\widetilde{m}^{2}$ is equal to $(m^{2}+m_{1}^{2})/2$. After this
transformation instead of two single poles at $p^{\prime 2}=m^{2}$ and
$p^{2}=m_{1}^{2}$ the function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,0)$ acquires a
double-pole at $p^{2}=\widetilde{m}^{2}$.

With this information at hands, it is possible to derive a desired sum rule
for the strong coupling $G$. To this end, we equate the invariant amplitudes
$\Pi^{\mathrm{Phys}}(p^{2})$ and $\Pi^{\mathrm{OPE}}(p^{2})$ corresponding to
structures $\varepsilon\sim\varepsilon_{\mu}^{\ast}(q)p_{\nu}$ in the physical
and $\mathrm{OPE}$ expressions of the correlation function. In accordance with
recipes of the sum rule analysis, we have to perform the Borel transformation
to suppress contributions of higher resonances and continuum states, as well
as subtract these terms from the QCD side of the sum rule using assumption
about quark-hadron duality.

In the soft-meson approximation the physical side of the sum rule has a more
complicated structure than in the case of the full LCSR method. A reason is
that, in the soft limit contributions of higher resonances and continuum
states demonstrate different behavior. Thus, some of them even after the Borel
transformation remain unsuppressed and appear as contamination in the physical
side Braun:1995 . Therefore, before carrying out the continuum subtraction
they should be excluded from $\mathcal{B}\Pi^{\mathrm{Phys}}(p^{2})$ by means
of some prescription. This problem is solved by the operator Ioffe:1983ju ;
Braun:1995

$\mathcal{P}(M^{2},\widetilde{m}^{2})=\left(1-M^{2}\frac{d}{dM^{2}}\right)M^{2}e^{\widetilde{m}^{2}/M^{2}},$
(31)

that acts to both sides of the sum rule. It eliminates unsuppressed terms in
the physical side, but modifies also QCD side of the sum rule. Then
contributions of higher resonances with regular behavior can be subtracted
from the QCD side using the quark-hadron duality assumption.

The sum rule for the strong coupling $G$ reads

$G=\frac{m}{fm_{1}f_{1}}\frac{\mathcal{P}(M^{2},\widetilde{m}^{2})\mathcal{B}\Pi^{\mathrm{OPE}}(p^{2})}{m_{1}^{2}-m^{2}-m_{\phi}^{2}}.$
(32)

The Borel transformed and subtracted correlation function
$\mathcal{B}\Pi^{\mathrm{OPE}}(p^{2})$ can be presented in the following form

$\mathcal{B}\Pi^{\mathrm{OPE}}(p^{2})=\int_{4\mathcal{M}^{2}}^{s_{0}}ds\rho^{\mathrm{pert.}}(s)e^{-s/M^{2}}+\Pi(M^{2}).$
(33)

The integral in Eq. (33) is a perturbative term, where the spectral density
$\rho^{\mathrm{pert.}}(s)$ is determined by the expression

$\rho^{\mathrm{pert.}}(s)=\frac{f_{\phi}m_{\phi}m_{c}}{4\pi^{2}}\frac{\sqrt{s(s-4m_{c}^{2})}}{s}.$
(34)

The second component of $\mathcal{B}\Pi^{\mathrm{OPE}}(p^{2})$, i.e., the
function $\Pi(M^{2})$ contains the twist-4 and nonperturbative contributions,

$\displaystyle\Pi(M^{2})$ $\displaystyle=$
$\displaystyle\frac{f_{\phi}m_{\phi}^{3}m_{c}\zeta_{4\phi}}{16\pi^{2}}\int_{0}^{1}\frac{dx}{x(x-1)}e^{-m_{c}^{2}/M^{2}x(1-x)}$
(35)
$\displaystyle+\frac{f_{\phi}m_{\phi}m_{c}}{4}\mathcal{F}^{\mathrm{n.-pert.}}(M^{2}),$

where $\mathcal{F}^{\mathrm{n.-pert.}}(M^{2})$ is given by the formula

$\displaystyle\mathcal{F}^{\mathrm{n.-pert.}}(M^{2})=\Big{\langle}\frac{\alpha_{s}G^{2}}{\pi}\Big{\rangle}\int_{0}^{1}f_{1}(x,M^{2})dx-\Big{\langle}g_{s}^{3}G^{3}\Big{\rangle}$
$\displaystyle\times\int_{0}^{1}f_{2}(x,M^{2})dx-\Big{\langle}\frac{\alpha_{s}G^{2}}{\pi}\Big{\rangle}^{2}\int_{0}^{1}f_{3}(x,M^{2})dx.$
(36)

The nonperturbative contributions of four, six and eight dimensions are
proportional to $\langle\alpha_{s}G^{2}/\pi\rangle$, $\langle
g_{s}^{3}G^{3}\rangle$ and $\langle\alpha_{s}G^{2}/\pi\rangle^{2}$,
respectively. The functions $f_{i}(x,M^{2})$, $i=1,2,3$ are explicitly given
below:

$\displaystyle
f_{1}(x,M^{2})=\frac{1}{18M^{4}x^{2}(1-x)^{2}}\left[8m_{c}^{2}(1-x)^{2}\right.$
$\displaystyle\left.+M^{2}(2-7x+9x^{2}-4x^{3}+2x^{4})\right]e^{-m_{c}^{2}/M^{2}x(1-x)},$
(37) $\displaystyle
f_{2}(x,M^{2})=\frac{1}{240M^{8}\pi^{2}x^{5}(x-1)^{5}}\left[2M^{4}x^{2}(1-x)^{2}\right.$
$\displaystyle\times\left(3-11x+15x^{2}-8x^{3}+4x^{4}\right)+24m_{c}^{4}(1-2x)^{2}$
$\displaystyle\times\left(-2-7x+17x^{2}-20x^{3}+10x^{4}\right)-3m_{c}^{2}M^{2}x$
$\displaystyle\times\left(4-49x+176x^{2}-293x^{3}+218x^{4}-26x^{5}\right.$
$\displaystyle\left.\left.-40x^{6}+10x^{7}\right)\right]e^{-m_{c}^{2}/M^{2}x(1-x)},$
(38)

and

$\displaystyle f_{3}(x,M^{2})$ $\displaystyle=$
$\displaystyle\frac{16\pi^{2}m_{c}^{2}}{9M^{10}x^{3}(x-1)^{3}}\left(25m_{c}^{2}+6M^{2}x\right.$
(39) $\displaystyle\left.-6M^{2}x^{2}\right)e^{-m_{c}^{2}/M^{2}x(1-x)}.$

The width of the process $X\rightarrow J/\psi\phi$ is determined by the
formula

$\Gamma(X\rightarrow J/\psi\phi)=G^{2}\frac{\lambda(m,m_{1},m_{\phi})}{24\pi
m^{2}}|M|^{2},$ (40)

where

$\displaystyle|M|^{2}=\frac{1}{4m^{2}m_{1}^{2}m_{\phi}^{2}}\left[m_{1}^{8}+8m_{1}^{6}(m^{2}+m_{\phi}^{2})\right.$
$\displaystyle+(m_{\phi}^{2}-m^{2})^{2}\left(m_{\phi}^{4}+10m_{\phi}^{2}m^{2}+m^{4}\right)$
$\displaystyle-2m_{1}^{4}\left(9m_{\phi}^{4}+16m_{\phi}^{2}m^{2}+9m^{4}\right)+8m_{1}^{2}$
$\displaystyle\left.\times\left(m_{\phi}^{6}-4m_{\phi}^{4}m^{2}-4m_{\phi}^{2}m^{4}+m^{6}\right)\right],$
(41)

and $\lambda(a,b,c)$ is the function

$\lambda(a,b,c)=\frac{\sqrt{a^{4}+b^{4}+c^{4}-2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})}}{2a}.$
(42)

Parameters | Values (in $\mathrm{MeV}$ units)
---|---
$m_{1}[m_{J/\psi}]$ | $3096.900\pm 0.006$
$f_{1}[f_{J/\psi}]$ | $409\pm 15$
$m_{2}[m_{\eta_{c}}]$ | $2983.9\pm 0.5$
$f_{2}[f_{\eta_{c}}]$ | $320\pm 40$
$m_{3}[m_{\chi_{1c}}]$ | $3510.67\pm 0.05$
$f_{3}[f_{\chi_{1c}}]$ | $344\pm 27$
$m_{\phi}$ | $1019.461\pm 0.019$
$f_{\phi}$ | $228.5\pm 3.6$
$m_{\eta}$ | $547.862\pm 0.017$
$m_{\eta^{\prime}}$ | $957.78\pm 0.06$
Table 1: Masses and decay constants of mesons, which have been used in
numerical computations.

The sum rule Eq. (32) depends on the mass and decay constant of the $J/\psi$
and $\phi$ mesons: Their values are collected in Table 1. This table contains
also spectroscopic parameters of other mesons which will be used in the next
section. The masses of all mesons are borrowed from Ref. PDG:2020 . As the
decay constants $f_{\phi}$ and $f_{1}$ of the vector mesons $\phi$ and
$J/\psi$, we use their experimental values reported in Refs.
Chakraborty:2017hry ; Kiselev:2001xa , respectively. For the decay constants
$f_{2}$ and $f_{3}$ of the $\eta_{c}$ and $\chi_{1c}$ mesons, we utilize
relevant sum rules’ predictions from Refs. Colangelo:1992cx  and
VeliVeliev:2012cc , respectively.

In numerical analysis, the parameters $M^{2}$ and $s_{0}$ are chosen as in Eq.
(17). Computations allow us to find numerical value of the strong coupling $G$

$G=0.85\pm 0.12.$ (43)

In Fig. 2, we depict $G$ as a function of the Borel parameter $M^{2}$ at fixed
$s_{0}$. One sees that, the coupling $G$ is sensitive to $M^{2}$ and $s_{0}$,
which are main sources of theoretical ambiguities of the analysis: Ambiguities
arising due to variations of the parameters $M^{2}$ and $s_{0}$ are equal to
$\Delta^{\mathrm{(M}^{\mathrm{2}}\mathrm{,s}_{\mathrm{0}}\mathrm{)}}G=\pm
0.11$. Uncertainties in the decay constants $f_{1}$ and $f_{\phi}$ generates
$\Delta^{\mathrm{(f}_{\mathrm{1}}\mathrm{)}}G=\pm 0.03$ and
$\Delta^{\mathrm{(f}_{\mathrm{\phi}}\mathrm{)}}G=\pm 0.02$, respectively.
Errors connected with various vacuum condensates are very small and can be
neglected.

For the partial width of the process $X\rightarrow J/\psi\phi$, we get

$\Gamma(X\rightarrow J/\psi\phi)=(113\pm 30)~{}\mathrm{MeV}.$ (44)

This information will be used below to evaluate full width of the tetraquark
$X$.

Figure 2: The strong coupling $G$ as a function of the Borel parameter $M^{2}$
at fixed $s_{0}$.

##  IV Processes $X\to\eta_{c}\eta^{(\prime)}$ and
$X\to\chi_{1c}\eta^{(\prime)}$

In this section, we consider processes $X\rightarrow\eta_{c}\eta^{(\prime)}$
and $X\rightarrow\chi_{1c}\eta^{(\prime)}$ and calculate their partial widths.
It is not difficult to see, that decays to pseudoscalar mesons $\eta_{c}(1S)$
and $\eta^{(\prime)}$ with spin-parities $J^{\mathrm{PC}}=0^{-+}$ are $P$-wave
modes of the tetraquark $X$. The second pair of decays to axial-vector meson
$\chi_{1c}(1P)$ with $J^{\mathrm{PC}}=1^{++}$ and $\eta^{(\prime)}$ are its
$S$-wave modes. In all of these processes conservation of $C$-parity is the
case.

###  IV.1 Decays $X\rightarrow\eta_{c}\eta$ and
$X\rightarrow\eta_{c}\eta^{\prime}$

We start from analysis of the processes $X\rightarrow\eta_{c}\eta$ and
$X\rightarrow\eta_{c}\eta^{\prime}$, and extract couplings $g_{1}$ and $g_{2}$
which describe strong interaction at the vertices $X\eta_{c}\eta$ and
$X\eta_{c}\eta^{\prime}$, respectively.

To find the coupling $g_{1}$, one should analyze the following correlation
function on the light-cone

$\widetilde{\Pi}_{\mu}(p,q)=i\int
d^{4}xe^{ipx}\langle\eta(q)|\mathcal{T}\\{J^{\eta_{c}}(x)J_{\mu}^{{\dagger}}(0)\\}|0\rangle,$
(45)

where $J^{\eta_{c}}(x)$ is the interpolating current for the $\eta_{c}$ meson

$J^{\eta_{c}}(x)=\overline{c}_{l}(x)i\gamma_{5}c_{l}(x).$ (46)

A sum rule for the coupling $g_{1}$ is determined by calculating the
correlator $\widetilde{\Pi}_{\mu}(p,q)$ using both physical parameters of the
particles and quark-gluon degrees of freedom. The physical side of the sum
rule is given by the formula

$\displaystyle\widetilde{\Pi}_{\mu}^{\mathrm{Phys}}(p,q)$ $\displaystyle=$
$\displaystyle\frac{\langle
0|J^{\eta_{c}}|\eta_{c}\left(p\right)\rangle}{p^{2}-m_{2}^{2}}\langle\eta_{c}\left(p\right)\eta(q)|X(p^{\prime})\rangle$
(47) $\displaystyle\times\frac{\langle
X(p^{\prime})|J_{\mu}^{\dagger}|0\rangle}{p^{\prime 2}-m^{2}}+\cdots,$

where $m_{2}$ is the mass of the $\eta_{c}$ meson. Using the matrix elements
of the particles $X$ and $\eta_{c}$, and modeling the matrix element for the
vertex $X\eta_{c}\eta$, one can considerably simplify
$\widetilde{\Pi}_{\mu}^{\mathrm{Phys}}(p,q)$. We use the following new matrix
elements

$\langle
0|J^{\eta_{c}}|\eta_{c}\left(p\right)\rangle=\frac{f_{2}m_{2}^{2}}{2m_{c}},$
(48)

and

$\langle\eta_{c}\left(p\right)\eta(q)|X(p^{\prime})\rangle=g_{1}p\cdot\varepsilon(p^{\prime}),$
(49)

as well as, the matrix element of the tetraquark $X$ from Eq. (6). In
expression Eq. (48), $f_{2}$ is the decay constant of the $\eta_{c}$ meson.
Then for the correlation function
$\widetilde{\Pi}_{\mu}^{\mathrm{Phys}}(p,q)$, we find

$\displaystyle\widetilde{\Pi}_{\mu}^{\mathrm{Phys}}(p,q)=g_{1}\frac{fm_{2}^{2}f_{2}}{4m_{c}m\left(p^{\prime
2}-m^{2}\right)\left(p^{2}-m_{2}^{2}\right)}$
$\displaystyle\times\left[(m_{2}^{2}-m_{\eta}^{2}-m^{2})p_{\mu}+(m_{2}^{2}+m_{\eta}^{2}+m^{2})q_{\mu}\right],$
(50)

with $m_{\eta}$ being the mass of the $\eta$ meson.

The QCD side of the sum rule reads

$\displaystyle\widetilde{\Pi}_{\mu}^{\mathrm{OPE}}(p,q)=i\int
d^{4}xe^{ipx}\epsilon\widetilde{\epsilon}\left\\{\left[\gamma_{5}\widetilde{S}_{c}^{lc}(x)\gamma_{5}\right.\right.$
$\displaystyle\left.\left.\times\widetilde{S}_{c}^{el}(-x)\gamma_{\mu}\gamma_{5}\right]+\left[\gamma_{\mu}\gamma_{5}\widetilde{S}_{c}^{lc}(x){}\gamma_{5}\widetilde{S}_{c}^{el}(-x){}\gamma_{5}\right]\right\\}_{\alpha\beta}$
$\displaystyle\times\langle\eta(q)|\overline{s}_{\alpha}^{b}(0)s_{\beta}^{d}(0)|0\rangle.$
(51)

It is clear that $\widetilde{\Pi}_{\mu}^{\mathrm{OPE}}(p,q)$ contains only
local matrix elements of $\eta$, therefore remaining calculations have to be
carried out in the context of the soft-meson approximation. Technical methods
of such treatment have been explained in the previous section. Therefore, we
do not concentrate on further details, and note that in the soft limit
$\widetilde{\Pi}_{\mu}^{\mathrm{OPE}}(p,0)$ receives contributions only from
the matrix element

$2m_{s}\langle\eta|\overline{s}i\gamma_{5}s|0\rangle=h_{\eta}^{s}.$ (52)

The parameter $h_{\eta}^{s}$ in Eq. (52) can be defined theoretically
Agaev:2014wna , but for our purposes it is enough to use its phenomenological
value extracted from analysis of relevant exclusive processes. Thus, we have

$h_{\eta}^{s}=-h_{s}\sin\varphi,\ h_{s}=(0.087\pm 0.006)\ \mathrm{GeV}^{3},$
(53)

where $\varphi=39^{\circ}.3\pm 1^{\circ}.0$ is the $\eta-\eta^{\prime}$ mixing
angle in the quark-flavor basis (for details, see Ref. Agaev:2014wna ).

The sum rule for $g_{1}$ is derived by making use of invariant amplitudes
corresponding to structures $p_{\mu}$ in
$\widetilde{\Pi}_{\mu}^{\mathrm{Phys}}(p)$ and
$\widetilde{\Pi}_{\mu}^{\mathrm{OPE}}(p)$. It reads

$g_{1}=\frac{4mm_{c}}{fm_{2}^{2}f_{2}}\frac{\mathcal{P}(M^{2},m^{\prime
2})\mathcal{B}\widetilde{\Pi}^{\mathrm{OPE}}(p^{2})}{m_{2}^{2}-m^{2}-m_{\eta}^{2}},$
(54)

where $m^{\prime 2}=(m^{2}+m_{2}^{2})/2$. The Borel transformed and subtracted
invariant amplitude $\mathcal{B}\widetilde{\Pi}^{\mathrm{OPE}}(p^{2})$ is
given by the following expression

$\displaystyle\mathcal{B}\widetilde{\Pi}^{\mathrm{OPE}}(p^{2})$
$\displaystyle=$
$\displaystyle-\frac{h_{\eta}^{s}m_{c}}{4\pi^{2}m_{s}}\int_{4\mathcal{M}^{2}}^{s_{0}}ds\frac{\sqrt{s(s-4m_{c}^{2})}}{s}e^{-s/M^{2}}$
(55) $\displaystyle+\widetilde{\Pi}(M^{2}).$

The nonperturbative component $\widetilde{\Pi}(M^{2})$ is calculated with
dimension-$8$ accuracy and determined by formulas similar to ones from Eqs.
(36)-(39). Therefore, there is no need to write down their explicit
expressions.

The width of the mode $X\rightarrow\eta_{c}\eta$ can be calculated using the
formula

$\Gamma(X\rightarrow\eta_{c}\eta)=g_{1}^{2}\frac{\lambda^{3}(m,m_{2},m_{\eta})}{24\pi
m^{2}}.$ (56)

Numerical computations yield

$|g_{1}|=3.75\pm 0.78,$ (57)

and

$\Gamma(X\rightarrow\eta_{c}\eta)=(18.0\pm 5.4)~{}\mathrm{MeV.}$ (58)

The second process $X\rightarrow\eta_{c}\eta^{\prime}$ can be considered in a
similar manner, difference being in the matrix element of the $\eta^{\prime}$
meson

$2m_{s}\langle\eta^{\prime}|\overline{s}i\gamma_{5}s|0\rangle=h_{s}\cos\varphi,$
(59)

that contributes to the corresponding correlation function. For this decay, we
find

$g_{2}=4.38\pm 0.90,$ (60)

and

$\Gamma(X\rightarrow\eta_{c}\eta)=(15.5\pm 4.5)~{}\mathrm{MeV}.$ (61)

Effects of these processes on the full width of $X$ are not small, and will be
taken into account.

###  IV.2 Decays $X\rightarrow\chi_{1c}\eta$ and
$X\rightarrow\chi_{1c}\eta^{\prime}$

Processes $X\rightarrow\chi_{1c}\eta$ and $X\rightarrow\chi_{1c}\eta^{\prime}$
can be explored in accordance with a scheme described above. Here, we have to
evaluate the strong couplings $g_{3}$ and $g_{4}$ which correspond to vertices
$X\chi_{1c}\eta$ and $X\chi_{1c}\eta^{\prime}$.

Let us consider the decay $X\rightarrow\chi_{1c}\eta$ and write down some
principal expressions. Thus, the correlation function which is necessary to
extract the coupling $g_{3}$ is

$\widehat{\Pi}_{\mu\nu}(p,q)=i\int
d^{4}xe^{ipx}\langle\eta(q)|\mathcal{T}\\{J_{\nu}^{\chi_{1c}}(x)J_{\mu}^{{\dagger}}(0)\\}|0\rangle,$
(62)

where $J_{\mu}^{\chi_{1c}}(x)$ is the interpolating current for the axial-
vector meson $\chi_{1c}$

$J_{\nu}^{\chi_{1c}}(x)=\overline{c}_{l}(x)i\gamma_{5}\gamma_{\nu}c_{l}(x).$
(63)

The function $\widehat{\Pi}_{\mu\nu}(p,q)$ in terms of particles’ parameters
has the form

$\displaystyle\widehat{\Pi}_{\mu\nu}^{\mathrm{Phys}}(p,q)=\frac{\langle
0|J_{\nu}^{\chi_{1c}}|\chi_{1c}\left(p\right)\rangle}{p^{2}-m_{3}^{2}}\langle\chi_{1c}\left(p\right)\eta(q)|X(p^{\prime})\rangle$
$\displaystyle\times\frac{\langle
X(p^{\prime})|J_{\mu}^{\dagger}|0\rangle}{p^{\prime 2}-m^{2}}+\cdots,$ (64)

where $m_{3}$ is the $\chi_{1c}$ meson’s mass.

The function $\widehat{\Pi}_{\mu\nu}^{\mathrm{Phys}}(p,q)$ can be simplified
using the following matrix element

$\langle
0|J_{\mu}^{\chi_{1c}}|\chi_{1c}\left(p\right)\rangle=f_{3}m_{3}\varepsilon_{\mu}(p),$
(65)

with $f_{3}$ and $\varepsilon_{\mu}(p)$ being the decay constant and
polarization vector of the meson $J_{\mu}^{\chi_{1c}}$, respectively. The
matrix element for the vertex $X\chi_{1c}\eta$ is modeled in the form

$\displaystyle\langle\chi_{1c}\left(p\right)\eta(q)|X(p^{\prime})\rangle=g_{3}\left\\{\left[p\cdot
p^{\prime}\right]\left[\varepsilon^{\ast}(p)\cdot\varepsilon(p^{\prime})\right]\right.$
$\displaystyle\left.-[p\cdot\varepsilon(p^{\prime})][p^{\prime}\cdot\varepsilon^{\ast}(p)]\right\\}.$
(66)

Standard calculations of $\widehat{\Pi}_{\mu\nu}^{\mathrm{Phys}}(p,q)$ using
Eqs. (6), (65) and (66) yield

$\displaystyle\widehat{\Pi}_{\mu\nu}^{\mathrm{Phys}}(p,q)=g_{3}\frac{fmf_{3}m_{3}}{\left(p^{\prime
2}-m^{2}\right)\left(p^{2}-m_{3}^{2}\right)}$
$\displaystyle\times\left[\frac{1}{2}\left(m^{2}+m_{3}^{2}-m_{\eta}^{2}\right)g_{\mu\nu}-p_{\mu}p_{\nu}^{\prime}\right].$
(67)

The correlation function $\widehat{\Pi}_{\mu\nu}^{\mathrm{OPE}}(p,q)$ is given
by the expression

$\displaystyle\widehat{\Pi}_{\mu\nu}^{\mathrm{OPE}}(p,q)=i\int
d^{4}xe^{ipx}\epsilon\widetilde{\epsilon}\left\\{\left[\gamma_{5}\widetilde{S}_{c}^{lc}(x)\gamma_{\nu}\gamma_{5}\right.\right.$
$\displaystyle\left.\left.\times\widetilde{S}_{c}^{el}(-x)\gamma_{\mu}\gamma_{5}\right]+\left[\gamma_{\mu}\gamma_{5}\widetilde{S}_{c}^{lc}(x){}\gamma_{\nu}\gamma_{5}\widetilde{S}_{c}^{el}(-x){}\gamma_{5}\right]\right\\}_{\alpha\beta}$
$\displaystyle\times\langle\eta(q)|\overline{s}_{\alpha}^{b}(0)s_{\beta}^{d}(0)|0\rangle.$
(68)

The required sum rule for the coupling $g_{3}$ is derived by equating
invariant amplitudes of structures $\sim g_{\mu\nu}$ from the functions
$\widehat{\Pi}_{\mu\nu}^{\mathrm{Phys}}(p,q)$ and
$\widehat{\Pi}_{\mu\nu}^{\mathrm{OPE}}(p,q)$, and has the form

$g_{3}=\frac{2}{fmf_{3}m_{3}}\frac{\mathcal{P}(M^{2},\widehat{m}^{2})\mathcal{B}\widehat{\Pi}^{\mathrm{OPE}}(p^{2})}{m^{2}+m_{3}^{2}-m_{\eta}^{2}},$
(69)

where $\widehat{m}^{2}=(m^{2}+m_{3}^{2})/2$. Numerical analysis for $g_{3}$
gives

$g_{3}=(1.34\pm 0.23)\times 10^{-1}~{}\mathrm{GeV}^{-1}.$ (70)

Width of the process $X\rightarrow\chi_{1c}\eta$ is determined by the
expression

$\Gamma(X\rightarrow\chi_{1c}\eta)=g_{3}^{2}\frac{\lambda
m_{3}^{2}}{24\pi}\left(3+\frac{2\lambda^{2}}{m_{3}^{2}}\right),$ (71)

with $\lambda$ being equal to $\lambda(m,m_{3},m_{\eta})$. Then it is not
difficult to find that

$\Gamma(X\rightarrow\chi_{1c}\eta)=(7.9\pm 1.9)~{}\mathrm{MeV}.$ (72)

For the decay $X\rightarrow\chi_{1c}\eta^{\prime}$, we get

$|g_{4}|=(1.39\pm 0.23)\times 10^{-1}~{}\mathrm{GeV}^{-1},$ (73)

and

$\Gamma(X\rightarrow\chi_{1c}\eta^{\prime})=(4.9\pm 1.2)\ \ \mathrm{MeV},$
(74)

respectively.

##  V Summing up

The full width of the tetraquark $X$ can be evaluated using results for the
partial width of its five decay modes obtained in Secs. III and IV. One of
these modes $X\rightarrow J/\psi\phi$ is dominant decay channels of the
tetraquark $X$, whereas remaining processes are sub-dominant ones. After
simple computations, we get

$\Gamma=(159\pm 31)~{}\mathrm{MeV}.$ (75)

Our result for the full width $\Gamma$ of the tetraquark $X$ is in a very nice
agreement with $\Gamma_{\exp}=(174\pm 27_{-73}^{+134})~{}\mathrm{MeV}$ found
by the LHCb collaboration.

But by drawing such conclusions, we take into account that both theoretical
and experimental information on the full with of $X(4630)$ suffers from
errors. The uncertainties are large in the case of $\Gamma_{\exp}$ which limit
credibility of conclusions that are based on these data. Experimental errors
make also difficult detailed comparisons and choices between existing
theoretical models for $X(4630)$. In this sense, more precise measurements of
$\Gamma_{\exp}$ are required.

From another side, our present result can be further refined by including into
analysis other decay modes of $X$. There are a few processes which contribute
to the full width of the tetraquark $X$. Thus, decays to meson pairs
$D_{s}^{\ast\pm}D_{s0}(2317)^{\mp}$ and $D_{s}^{\pm}D_{s1}(2460)^{\mp}$ are
among kinematically allowed channels of $X$ . These processes belong to
$\mathrm{V\rightarrow V+S}$ and $\mathrm{V\rightarrow PS+AV}$ type $S$-wave
decay modes of $X$, respectively. Their partial widths are determined by the
expression Eq. (71) with relevant strong coupling. To make crude estimates for
partial widths of these decays, we may assume that strong couplings at
corresponding tetraquark-meson-meson vertices are a same order of $g_{3}$
($|g_{4}|$). Then widths of these modes are suppressed relative to decays
$X\rightarrow\chi_{1c}\eta^{(\prime)}$, because the factor
$\Lambda=m_{\ast}^{2}$ $\lambda\left(3+2\lambda^{2}/m_{\ast}^{2}\right)/24\pi$
($m_{\ast}$ is a mass of a heaviest final meson) is smaller for two final-
state mesons of approximately equal mass than in the case of light and heavy
mesons. For instance, in the decay $X\rightarrow\chi_{1c}\eta$ it is equal to
$\Lambda\approx 0.44$, while we find $\Lambda\approx 0.17$ for the process
$X\rightarrow D_{s}^{-}D_{s1}(2460)^{+}$. But these decay channels, in total,
may compensate a $15~{}\mathrm{MeV}$ gap between $\Gamma_{\exp}$ and $\Gamma$.

Another interesting field of future studies is an exploration of $X(4630)$ in
the molecule picture using the QCD sum rule method. This is necessary to
compare predictions for the molecule and diquark-antidiquark models with each
another, as well as with the LHCb data. In the context of the QCD sum rule
approach diquark-antidiquark and molecule models for the same resonance lead
to different results Agaev:2021vur ; Agaev:2022ast . As a rule, a molecule of
conventional mesons is heavier than a diquark-antidiquark structure with
identical content and spin-parities. A width of such molecule is also larger
than that of its diquark counterpart, i.e., a diquark structure is more stable
than a meson molecule. Therefore, despite existing investigations of $X(4630)$
in different approaches, it is necessary to examine the molecule model for
$X(4630)$ in the context of the QCD sum rule method as well.

Analysis performed in the present article and gained knowledge about the mass
and full width of the tetraquark $X$ , as well as a very nice agreement
between these parameters and LHCb measurements allows us to interpret
$X(4630)$ as the vector diquark-antidiquark state $X$ with the spin-parities
$J^{\mathrm{PC}}=1^{-+}$.

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