# Flavor Exotic Triply-Heavy Tetraquark States in AdS/QCD Potential

Halil Mutuk<EMAIL_ADDRESS>Department of Physics, Faculty of Science,
Ondokuz Mayis University, Atakum, 55200 Samsun, Türkiye

###### Abstract

We study the $S$-wave mass spectra of flavor exotic triply-heavy tetraquark
states $cc\bar{c}\bar{q}$, $cc\bar{b}\bar{q}$, $bb\bar{c}\bar{q}$ and
$bb\bar{b}\bar{q}$. We adopt a diquark-antidiquark scheme to solve Schrödinger
equation. The calculations are carried out in a nonrelativistic quark model
with a color interaction described by a potential computed in AdS/QCD. The
AdS/QCD potential model consists of a central potential which reflects short
distance and large distance behaviour of QCD, spin dependent term for
hyperfine splitting and a constant term. We find stable state candidates in
the $cc\bar{c}\bar{q}$ sector whereas in the $cc\bar{b}\bar{q}$,
$bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ sectors all the states lie above
corresponding $S$-wave meson-meson thresholds.

##  I Introduction

Quark model describes ordinary mesons as ($q\bar{q}$) systems and baryons as
($qqq$) systems in terms of quark $q$ and antiquark $\bar{q}$. It explained
physical properties of observed hadrons up to the millennium. In the original
work of Murray Gell-Mann [1], multiquark states were conjectured. Definition
of an exotic state stems from the quark model. According to the quark model,
the parity for a meson is $P=(-1)^{L+1}$ and $C$ parity for a neutral meson is
$C=(-1)^{L+S}$, where $L$ and $S$ are the orbital and spin angular momentum,
respectively. In this picture the following combinations are allowed:
$J^{PC}=0^{-+},0^{++},1^{--},1^{+-},1^{++},\cdots$. A meson with quantum
numbers that cannot be included by any combination spin-ortibal state of a
quark-antiquark, such as $J^{PC}=0^{--},0^{+-},1^{-+},2^{+-}$, is exotic.
Besides that, state(s) with unusual properties but ordinary quantum numbers
are called “crypto-exotic” states [2]. These states can have unconventional
content of valence quarks or gluons. Some hadrons do not have exotic parity,
charge parity, or flavor quantum numbers but may have exotic color or flavor
or spatial configurations. In addition to $J^{PC}$ exotics, flavor-exotics
could also be in case.The investigation of exotic mesons which are particles
either have unusual quantum numbers that are not possible in the conventional
quark model or built up of four valence quarks (i.e. tetraquarks) became an
important topic in the high energy physics field, especially with the
observation of exotic XYZ states. The milestone was the observation of
$X(3872)$ (now referred to name as $\chi_{c1}(3872)$) with a mass of
($3872.0\pm 0.6$ MeV) in the charmonium sector [3]. Existence of multiquark
states does not contradict to first principles of Quantum chromodynamics
(QCD). The studies for exotic hadrons may open a new window for understanding
of the nuclear matter and most important may provide important hints to the
understanding of the nonperturbative aspects of QCD.

After the discovery of $X(3872)$, many more charmoniumlike and bottomoniumlike
states were observed. In 2020, the LHCb Collaboration observed two states,
called $X_{0}(2900)$ and $X_{1}(2900)$ in the invariant mass distribution of
$B^{+}\to D^{+}D^{-}K^{+}$ [4]. These are the first states that have four
different flavors of quarks. In the same year, the observation of a charged
state referred as $Z_{cs}^{-}(3985)$ is anonunced by the BESIII Collaboration
[5]. Very recently the first doubly-charmed tetraquark $T_{cc}^{+}$ is
observed by the LHCb Collaboration [6, 7]. These exotic states contain at
least one or two heavy quarks and triggered many phenomenological studies to
enlighten their internal structures and dynamics.

A recent observation of a narrow structure around 6.9 GeV, the so called
$X(6900)$, and a broad structure around 6.2 GeV to 6.8 GeV in the $J/\psi$
mass spectrum is a good candidate for fully-heavy tetraquarks [8]. This
observation led to plenty of studies in the literature. In Ref. [9], this
narrow structure can be explained by the $P$-wave $cc\bar{c}\bar{c}$ states in
nonrelativistic quark model. In dynamical diquark model, $X(6900)$ state is
assigned as $2S$ multiplet [10]. Using chiral quark model and quark
delocalization color screening model, $X(6900)$ is explained as a compact
resonance with the quantum number $J^{P}=0^{+}$ [11]. It is argued in Ref.
[12] that $X(6900)$ may be interpreted as radially excited $0^{++}(3S)$ or
$2^{++}(3S)$ or orbitally excited $2P$ state with quark content
$cc\bar{c}\bar{c}$. Using diquark-antidiquark interpolating currents in QCD
sum rule, the narrow structure is assigned to be a $P$-wave state with
$J^{PC}=0^{-+}$ or $J^{PC}=1^{-+}$ and broad structure to be an $S$-wave
$cc\bar{c}\bar{c}$ state with $J^{PC}=0^{++}$ or $J^{PC}=2^{++}$ [13]. Ref.
[14] found via QCD sum rules that broad structure around 6.2-6.8 GeV can be
interpreted as the $J^{PC}=0^{++}$ octet–octet tetraquark state and narrow
structure around 6.9 GeV can be interpreted as $J^{PC}=0^{++}$ the octet–octet
tetraquark state. The debate about the internal structure of these states is
an ongoing endeavour. For more details about these structures, see a recent
review [15].

If $X(6900)$ is a genuine fully-heavy tetraquark state, it is natural to think
that tetraquarks with three heavy quarks $(Q)$ and one light quark $(q)$,
$(QQ\bar{Q}\bar{q})$ may also form compactly. Triply-heavy tetraquarks can be
obtained from fully-heavy tetraquarks by replacing one heavy quark with one
light antiquark, and the strong interactions resume to be provided by short
range one-gluon exchange (OGE) and quark confinement. As mentioned in Ref.
[16], triply-heavy tetraquarks can be related to the doubly-heavy ones and may
present a possible discrimination between singly-heavy tetraquarks and
traditional heavy-light mesons. For example the inner structure of
$D_{s0}^{\ast}(2317)$ is still undetermined. The quark content can be
$c\bar{s}$, $cq\bar{s}\bar{q}$, or even $c\bar{s}-cq\bar{s}\bar{q}$ [17]. The
problem for the nature of $D_{s0}^{\ast}(2317)$ arises from the fact that the
light quark pair can create or annihilate dynamically in this energy region.
However, the genuine tetraquarks can be easily recognized if the light quark
pair is replaced with the charmonium $c\bar{c}$ or bottomonium $b\bar{b}$
pair. Traditional heavy-light mesons can become hidden charm and bottom
triply-heavy tetraquarks with the excitation of one heavy-quark pair. The
associated production of bottomonia and open charm hadrons $\Upsilon D/D_{s}$
in $pp$ collisions was reported [18].

There are few studies regarding triply-heavy tetraquarks in the literature
compared to doubly and fully-heavy tetraquarks. Triply-heavy tetraquarks were
investigated by an extended relativized quark model in Ref. [16] where no
stable sate was reported. Ref. [19] studied ($Q^{2}\bar{Q}^{2}$) systems with
chromagnetic interaction and reported that some configurations of triply-heavy
tetraquark are possible. Ref. [20] did not find any bound triply-heavy
tetraquarks using a nonrelativistic quark model. Ref. [21] obtained masses of
possible heavy tetraquarks including ($qQ\bar{Q}\bar{Q}$) by using the color-
magnetic interaction. Ref. [22] calculated mass splittings and masses of the
$(QQ\bar{Q}\bar{q})$ tetraquark states in the framework of the color-magnetic
interaction. Ref. [23] studied triply-heavy tetraquarks within the framework
of QCD sum rule and $bb\bar{b}\bar{q}$ state was found to be very narrow. It
was found in lattice QCD studies that the lowest spin-1 $uc\bar{b}\bar{b}$ and
$sc\bar{b}\bar{b}$ states are near to their corresponding meson-meson
thresholds [24, 25]. Nonstrange multiquark exotics as Efimov states were
studied in a holographic description [26]. They found that hidden-flavor
tetraquark exotics such as $Q\bar{Q}q\bar{q}$, $QQ\bar{Q}\bar{q}$, and
$QQ\bar{Q}\bar{Q}$ are unbound as compact topological molecules. Masses,
lifetimes and weak decays of the triply heavy $b\bar{c}b\bar{q}$ tetraquarks
were studied [27]. The mass spectrum of $S$-wave triply-heavy tetraquarks was
studied by an extended chromomagnetic model [28]. They found no stable state
lie below the thresholds of two pseudoscalar mesons. Triply-heavy tetraquark
states are investigated in the framework of the chiral quark model with the
resonance ground method [29]. Some stable state candidates were obtained. The
results of abovementioned works are controversial: a consistent conclusion on
triply-heavy tetraquarks is still absent.

QCD is known to be the fundamental quantum field theory of strong
interactions. Charmonium $(c\bar{c})$ and bottomonium $(b\bar{b})$ states are
‘old boys’ for the understanding of perturbative and nonperturbative natures
of QCD. There are rich and precise experimental data for the traditional heavy
quarkonium which help us to clarify the quark-antiquark forces [30, 31].
Potential models are useful tools to describe the forces acting between quarks
in a phenomenologically acceptable manner [32]. Short-distance behaviour of
the quark-antiquark potential can be represented by a Coulomblike term while
the long-distance behaviour can be represented by a linear term in the
separation $r$. The quark-antiquark potential have contributed to the
understanding of strong interactions and quark confinement which is beyond the
reach of perturbation theory. Apart from conventional mesons and baryons, QCD
foresees multiquark states which may provide important hints for discerning of
the nonperturbative aspects of QCD. There are many studies for tetraquark
states in the literature using potential models, see for the recents [33, 34,
35, 36].

The AdS/CFT correspondence relates nonperturbative aspects of QCD with string
theory in order to get an analytical first order approach to nonperturbative
effects. It also provides insights into the nonperturbative part of QCD such
as the orbital and radial spectra of hadrons [37]. In this respect AdS/QCD
models find place in the literature. In Ref. [38], charmonium vector states
were studied in hard-wall, soft-wall and brainless setup of AdS/QCD models at
zero and finite temperatures. Using the static potential inspired by
holographic principle with both the vanila AdS-Schwarzschild metric and the
one with an infrared cutoff, dissociation temperatures of heavy quarkonia were
calculated [39]. In Ref. [40], a holographic model of heavy-light mesons by
extending the AdS/QCD correspondence to incorporate the behavior of the heavy
quark limit was constructed. Baryon spectrum at finite temperature by using
AdS/QCD correspondence is calcualted in Ref. [41]. Ref. [42] studied meson,
nucleon and $\Delta$-baryon properties at zero and finite temperatures in the
hard-wall AdS/QCD model. Phenomenological analysis of the scalar glueball and
scalar meson spectra within an AdS/QCD framework is presented in Ref. [43].
$S$-wave charmonium and bottomonium states were described at zero and finite
temperatures by using bottom-up AdS/QCD model [44].

In this present paper, we use AdS/QCD inspired potential in diquark-
antidiquark approach to study spectroscopy of triply-heavy tetraquark states.
This potential is derived in the context of AdS/QCD correspondence and have
similar behavior as expected from QCD [45]. It was successfully used to study
pentaquark states [46], meson and tetraquark states [47], radiative decays of
$\eta_{c}$ and $\eta_{b}$ mesons [48], and doubly heavy baryons [49]. In a
recent study, the energy configuration in a quark-antiquark pair from Nambu-
Goto action in a deformed AdS space is calculated [50]. It was shown that this
configuration has the shape of a Cornell potential. To the best of our
knowledge, there is no reported work on triply-heavy tetraquark states using
AdS/QCD potential.

This paper is organized as follows: In Sections II and III, we describe
AdS/QCD potential model and diquark-antidiquark model of this work,
respectively. In Section IV, the masses of the triply-heavy tetraquark systems
are calculated and comparison of triply-heavy tetraquark masses with previous
studies within different approaches are given. Section V is reserved for
conclusion and summary of the obtained results.

##  II AdS/QCD Potential

Heavy quarkonium bound states are important systems for our understanding of
the QCD due to the their nonrelativistic nature. In this perspective, it is
possible to describe the interaction between quark $q$ and antiquark $\bar{q}$
in terms of a local potential $V(r)$, where $r$ is the interquark separation.
The mostly used potential in literature is the so called Cornell potential
[51]. There are also other potential models in the literature which gave
successful results [52, 53, 54, 55]. These potentials may include some spin
dependent and constant terms in addition to Coulombic and linear parts.

Using AdS/QCD correspondence, the $(Q\bar{Q})$ potential was derived in [45].
This potential was used to obtain meson spectra as mentioned before. The
AdS/QCD correspondence is a new approach for studying the nonperturbative
regime of QCD. We shall use this potential with the corresponding parameters
of Ref. [47]. The potential can be written as

$V(r)=V_{AdS/QCD}(\lambda)+V_{spin}(r)+V_{0},$ (1)

where $V_{AdS/QCD}$ is the AdS/QCD potential, $V_{spin}$ is the spin-spin
interaction term and $V_{0}$ is a constant. The $V_{AdS/QCD}$ is given as a
parametric equation

$\displaystyle V_{AdS/QCD}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{g}{\pi}\sqrt{\frac{c}{\lambda}}\left(-1+\int_{0}^{1}dvv^{-2}\left[e^{\lambda
v^{2}/2}\left(1-v^{4}e^{\lambda(1-v^{2})}\right)^{-1/2}-1\right]\right),$
$\displaystyle r(\lambda)$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{\lambda}{c}}\int_{0}^{1}dvv^{2}e^{\lambda
v^{2}/2}\left(1-v^{4}e^{\lambda(1-v^{2})}\right)^{-1/2},$ (2)

where $r$ is the interquark distance and $\lambda$ is in the range of
$0\leq\lambda<2$. This interquark potential derivation is conducted in the
framework of the gauge/string duality approach [56, 57, 58]. The $V_{spin}(r)$
spin term is responsible for the spin-spin interactions

$V_{spin}(r)=A\frac{\delta(r)}{m_{1}m_{2}}(\textbf{S}_{1}\cdot\textbf{S}_{2})~{}\text{with}~{}\delta(r)=\left(\frac{\sigma}{\sqrt{\pi}}\right)^{3}e^{-\sigma^{2}r^{2}},$
(3)

where $\sigma$ is a parameter of the smeared delta function and $A$ is the
parameter proportional to the QCD running coupling constant and gets two
different values: $A_{b}$ in the case of mesons comprising at least bottom
quark and $A_{c}$ for otherwise.

In the center of mass frame, it is possible to factorize the angular and
radial parts of the Schrödinger equation as

$\bigg{\\{}\frac{1}{2\mu}\left[-\frac{d^{2}}{dr^{2}}+\frac{L(L+1)}{r^{2}}\right]+V(r)\bigg{\\}}\psi(r)=E\psi(r).$
(4)

Here $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ is the reduced masses of quark 1 and quark
2. In Refs. [46, 47, 48, 49] a cutoff is introduced in the context of Bethe-
Salpeter equation to cure the singularity since as $\lambda\to 0$ the
potential diverges in Eq. II. This divergence is harmless in the context of
Schrödinger equation [47]. We solve Schrödinger equation numerically by
assuming Dirichlet boundary conditions.

##  III Diquark Model

The history of diquark is old as quarks. Diquark structure was first suggested
by Gell-Mann [1]. A diquark is a bound quark-quark $(qq)$ pair, whereas an
antidiquark is a bound antiquark-antiquark $(\bar{q}\bar{q})$ pair. This
binding is a result of OGE between the quarks.

The diquark structure is an important object for the underlying structure of
baryons. Diquark model for baryon was introduced in Refs. [59, 60, 61] such
that a diquark can form inside the baryon. This quark-diquark approach was
successfully applied for baryons, see for example [62, 63, 64, 65, 66].
Diquarks are also important for the understanding of tetraquarks and mesons
outside of scope of the quark model. Many models in the literature explain the
inner structure of tetraquarks to be compact and to consist of diquark-
antidiquark pairs [67]. In this perspective it can be said that tetraquarks
are composed of diquarks and antidiquarks. These diquarks and antidiquarks are
not by themselves colorless but are introduced as constituent objects of
tetraquark systems to form colorless combinations. Indeed, there are some
indications of diquark correlations in the hadrons (for a recent review see
[68]).

In the study of triply-heavy tetraquarks, we will treat the four-body problem
as three two-body problem. Considering the diquarks and antidiquarks as
constituents of the tetraquarks to predict the tetraquark masses were done in
many works, see for example Refs. [69, 70, 71, 72, 73, 74, 75, 76, 77, 10]. At
first step we compute the mass of the diquark, then we do the same for
antidiquark and finally we solve the Schrödinger equation once again for a
two-body system composed of a diquark and an antidiquark. For the single-
flavor heavy tetraquarks, the second step is simultaneously done when the
first step is over, i.e. the mass of the diquark equals to the mass of the
antidiquark. But in the triply-heavy tetraquarks, the mass of the diquark is
not equal to the antidiquark due to the quark content.

Diquark masses are needed as input to compute triply-tetraquark masses. When
we combine two quarks in the fundamental color representation, it reduces to
$|qq\rangle:\text{3}\bigotimes\text{3}=\bar{\text{3}}\bigoplus\text{6}$, a
color antitriplet $\bar{\text{3}}$ and a color sextet 6. In a similar way,
combining two antiquarks reduces to
$|\bar{q}\bar{q}\rangle:\bar{\text{3}}\bigotimes\bar{\text{3}}=\text{3}\bigoplus\bar{\text{6}}$,
a triplet 3 and antisextet $\bar{\text{6}}$. The antitriplet state is
attractive whereas the sextet state is repulsive. Combining an antitriplet
diquark and a triplet antidiquark form a color singlet tetraquark,
$3-\bar{3}$. In addition to this configuration, $6-\bar{6}$ configuration can
also yield a tetraquark state.

In OGE approximation, the attractive potential between two quarks in diquark
is half the one between a quark and antiquark. Therefore, it is natural to use
the following relation for the potential of diquark

$V_{QQ}=\frac{1}{2}V_{Q\bar{Q}}.$ (5)

Many authors also extended this factor 1/2 to the whole potential describing
the quark-quark interaction in different models [78, 73, 74]. In the sextet
configuration, the form of the diquark potential derived from the quark-
antiquark potential changes. This will be explained in the following section.

In this paper, we consider a diquark with internal orbital quantum number
$\ell=0$. We also assume that diquarks are pointlike particles. The mass
spectrum for triply-heavy tetraquarks is obtained using

$M=m_{d}+m_{\bar{d}}+E,$ (6)

where $m_{d}$ is the mass of diquark, $m_{\bar{d}}$ is the mass of
antidiquark, and $E$ is the energy eigenvalue.

Constructing $\text{color}\otimes\text{spin}$ wave functions for the triply-
heavy tetraquark states can be done under the Pauli exclusion principle as

$\displaystyle\phi_{1}\chi_{1}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{0}^{\bar{3}}(\bar{Q}_{3}\bar{q}_{4})^{3}_{0}\rangle_{0}\delta_{12},$
$\displaystyle\phi_{1}\chi_{2}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{\bar{3}}(\bar{Q}_{3}\bar{q}_{4})^{3}_{1}\rangle_{0},$
$\displaystyle\phi_{1}\chi_{3}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{0}^{\bar{3}}(\bar{Q}_{3}\bar{q}_{4})^{3}_{1}\rangle_{1}\delta_{12},$
$\displaystyle\phi_{1}\chi_{4}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{\bar{3}}(\bar{Q}_{3}\bar{q}_{4})^{3}_{0}\rangle_{1},$
$\displaystyle\phi_{1}\chi_{5}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{\bar{3}}(\bar{Q}_{3}\bar{q}_{4})^{3}_{1}\rangle_{1},$
$\displaystyle\phi_{1}\chi_{6}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{\bar{3}}(\bar{Q}_{3}\bar{q}_{4})^{3}_{1}\rangle_{2},$
$\displaystyle\phi_{2}\chi_{1}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{0}^{6}(\bar{Q}_{3}\bar{q}_{4})_{0}^{\bar{6}}\rangle_{0},$
(7) $\displaystyle\phi_{2}\chi_{2}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{6}(\bar{Q}_{3}\bar{q}_{4})_{1}^{\bar{6}}\rangle_{0}\delta_{12},$
$\displaystyle\phi_{2}\chi_{3}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{0}^{6}(\bar{Q}_{3}\bar{q}_{4})_{1}^{\bar{6}}\rangle_{1},$
$\displaystyle\phi_{2}\chi_{4}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{6}(\bar{Q}_{3}\bar{q}_{4})_{0}^{\bar{6}}\rangle_{1}\delta_{12},$
$\displaystyle\phi_{2}\chi_{5}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{6}(\bar{Q}_{3}\bar{q}_{4})_{1}^{\bar{6}}\rangle_{1}\delta_{12},$
$\displaystyle\phi_{2}\chi_{6}$ $\displaystyle=$
$\displaystyle|(Q_{1}Q_{2})_{1}^{6}(\bar{Q}_{3}\bar{q}_{4})_{1}^{\bar{6}}\rangle_{2}\delta_{12},$

where $\delta_{12}=0$ if $Q_{1}$ and $Q_{2}$ are identical flavor quarks, and
$\delta_{12}=1$ for the other cases. Here superscript 3, $\bar{3}$, 6 or
$\bar{6}$ denotes the color, and the subscript 0, 1 or 2 denotes the spin. For
a good discussion of the color and spin wave functions of the heavy
tetraquarks, see Ref. [79].

To exhaust all possible configurations of the triply-heavy tetraquarks
$(QQ\bar{Q}\bar{q})$, it is legitimate to replace heavy quark $Q$ by either
charm $c$ or bottom $b$ quark and light quark $q$ by $u$, $d$ and $s$. The
possible cases we have are $bb\bar{b}\bar{q}$, $bb\bar{c}\bar{q}$,
$bc\bar{b}\bar{q}$, $bc\bar{c}\bar{q}$, $cc\bar{c}\bar{q}$, and
$cc\bar{b}\bar{q}$. In this paper we study flavor-exotic $cc\bar{c}\bar{q}$,
$cc\bar{b}\bar{q}$, $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ states. The
$cb\bar{c}\bar{q}$ and $cb\bar{b}\bar{q}$ systems are quite different systems.
$cb\bar{c}\bar{q}$ and $cb\bar{b}\bar{q}$ tetraquarks are hidden charm and
hidden bottom tetraquarks, respectively and no flavor exotic state exists in
these configurations.

##  IV Mass Spectrum

The parameters of the model are as follows: $c=0.3~{}\text{GeV}^{2}$,
$g=2.75$, $V_{0}=-0.49~{}\text{GeV}$, $A_{c}=7.92~{}\text{GeV}^{3}$,
$A_{b}=3.09~{}\text{GeV}^{3}$, $k=1.48$, $k^{\prime}=2.15$,
$\sigma=1.21~{}\text{GeV}^{2}$, $m_{q}=0.302~{}\text{GeV}$,
$m_{s}=0.454~{}\text{GeV}$, $m_{c}=1.733~{}\text{GeV}$,
$m_{b}=5.139~{}\text{GeV}$ [47]. The potential in Eq. 1 can be used in a quark
model to calculate tetraquark spectra under the hypothesis that a tetraquark
can be considered as a bound state of a diquark and an antidiquark.

###  IV.1 Diquark Masses

Before calculating masses of tetraquark states, we will first obtain diquark
masses. The results are listed in Table 1. As mentioned before, the parameter
$A$ is related to the strong coupling constant $\alpha_{s}$, in our case
$A\sim\kappa\alpha_{s}$, and string tension is related to $c$. For triplet
configurations of diquarks, the potential is half of the quark-antiquark
potential. For sextet configurations, color factor and string tension change
as in the triplet configuration. It is given in Ref. [74] that color factor
$\kappa$ of Coulomb term in the Cornell potential,
$V(r)=\frac{\kappa\alpha_{s}}{r}+br$, corresponding to the sextet
configurations is +1/3. For the string tension $b$ of linear term in the
Cornell potential, it is divided by four and sign is also changed. For more
discussion see the mentioned reference. In this work, we rely on these
observations and make the relevant changes in the potential for sextet
configurations. Having calculated the masses for the diquarks and
antidiquarks, one can use the initial stage of the model which describe quark-
antiquark systems, to study tetraquark systems.

Table 1: Results for the diquark masses. The braces $\\{\\}$ show symmetric wave functions of the subsystems and the parantheses () show without permutation symmetries. All results are in MeV. Configuration | Mass | Configuration | Mass
---|---|---|---
$(\bar{c}\bar{n})^{3}_{1}$ | 2021 | $(\bar{b}\bar{n})^{3}_{1}$ | 5342
$(\bar{c}\bar{n})^{3}_{0}$ | 1982 | $(\bar{b}\bar{n})^{3}_{0}$ | 5325
$(\bar{c}\bar{n})^{\bar{6}}_{1}$ | 1834 | $(\bar{b}\bar{n})^{\bar{6}}_{1}$ | 5095
$(\bar{c}\bar{n})^{\bar{6}}_{0}$ | 1836 | $(\bar{b}\bar{n})^{\bar{6}}_{0}$ | 5098
$(\bar{c}\bar{s})^{3}_{1}$ | 2103 | $(\bar{b}\bar{s})^{3}_{1}$ | 5414
$(\bar{c}\bar{s})^{3}_{0}$ | 2073 | $(\bar{b}\bar{s})^{3}_{0}$ | 5403
$(\bar{c}\bar{s})^{\bar{6}}_{1}$ | 1900 | $(\bar{b}\bar{s})^{\bar{6}}_{1}$ | 5300
$(\bar{c}\bar{s})^{\bar{6}}_{0}$ | 1901 | $(\bar{b}\bar{s})^{\bar{6}}_{0}$ | 5302
$\\{cc\\}^{\bar{3}}_{1}$ | 3130 | $\\{bb\\}^{\bar{3}}_{1}$ | 9646
$\\{cc\\}^{6}_{0}$ | 2917 | $\\{bb\\}^{6}_{0}$ | 9590

###  IV.2 The $cc\bar{c}\bar{q}$ and $cc\bar{b}\bar{q}$ Tetraquark States

The mass spectra and possible $S$-wave thresholds are listed in Table 2. Here
we assume $SU(2)$ flavor symmetry exist for up and down quarks as $n={u,d}$
and $q=s$. We also show values of the difference of the triply-heavy
tetraquark $(M)$ and threshold masses $(T)$, $\Delta=M-T$. If this quantity is
negative, then the triply-heavy tetraquark state lies below the threshold of
the fall-apart decay into two-meson and thus should be a narrow state. The
states with small positive values of $\Delta$ could be also observed as
resonances, since their decay rates will be suppressed by the phase space. All
other states are expected to be broad and thus difficult to observe in the
experiments.

Table 2:  Predicted mass spectra for the $cc\bar{c}\bar{q}$ and $cc\bar{b}\bar{q}$ systems, corresponding thresholds and $\Delta$ values. The meson masses are taken from [80] and the predicted mass for $M_{B_{c}^{\ast}}=6338$ MeV is taken from [81]. All results are in units of MeV. $J^{P}$ |  Configuration |  Mass |  $S$-wave threshold |  $\Delta$
---|---|---|---|---
$0^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}5005\\\ 5154\end{bmatrix}$ |  $\begin{bmatrix}\eta_{c}D,J/\psi D^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(156,-99)\\\ (305,50)\end{bmatrix}$
|  $|\\{cc\\}^{6}_{0}(\bar{c}\bar{n})^{\bar{6}}_{0}\rangle_{0}$ |  |
$1^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}5053\\\ 5081\\\ 5106\end{bmatrix}$ |  $\begin{bmatrix}\eta_{c}D^{\ast},J/\psi D^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(62,-51)\\\ (90,-23)\\\ (115,2)\end{bmatrix}$
|  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{1}\rangle_{1}$ |
|  $|\\{cc\\}^{6}_{0}(\bar{c}\bar{n})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}5116\end{bmatrix}$ |  $\begin{bmatrix}J/\psi D^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}12\end{bmatrix}$
$0^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}5112\\\ 5249\end{bmatrix}$ |  $\begin{bmatrix}\eta_{c}D_{s},J/\psi D^{\ast}_{s}\end{bmatrix}$ |  $\begin{bmatrix}(160,-97)\\\ (297,40)\end{bmatrix}$
|  $|\\{cc\\}^{6}_{0}(\bar{c}\bar{s})^{\bar{6}}_{0}\rangle_{0}$ |
$1^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}5118\\\ 5173\\\ 5190\end{bmatrix}$ |  $\begin{bmatrix}\eta_{c}D_{s}^{\ast},J/\psi D^{\ast}_{s}\end{bmatrix}$ |  $\begin{bmatrix}(22,-91)\\\ (77,-36)\\\ (94,-19)\end{bmatrix}$
|  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{1}\rangle_{1}$ |
|  $|\\{cc\\}^{6}_{0}(\bar{c}\bar{s})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}5202\end{bmatrix}$ |  $\begin{bmatrix}J/\psi D^{\ast}_{s}\end{bmatrix}$ |  $\begin{bmatrix}-7\end{bmatrix}$
$0^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}8526\\\ 8638\end{bmatrix}$ |  $\begin{bmatrix}B_{c}D,B_{c}^{\ast}D^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(382,181)\\\ (494,293)\end{bmatrix}$
|  $|\\{cc\\}^{6}_{0}(\bar{b}\bar{n})^{\bar{6}}_{0}\rangle_{0}$ |
$1^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}8541\\\ 8598\\\ 8613\end{bmatrix}$ |  $\begin{bmatrix}B_{c}D^{\ast},B_{c}^{\ast}D^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(310,196)\\\ (367,253)\\\ (382,268)\end{bmatrix}$
|  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{1}\rangle_{1}$ |
|  $|\\{cc\\}^{6}_{0}(\bar{b}\bar{n})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}8630\end{bmatrix}$ |  $\begin{bmatrix}B_{c}^{\ast}D^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}285\end{bmatrix}$
$0^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}8576\\\ 8670\end{bmatrix}$ |  $\begin{bmatrix}B_{c}D_{s},B_{c}^{\ast}D_{s}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(333,126)\\\ (427,220)\end{bmatrix}$
|  $|\\{cc\\}^{6}_{0}(\bar{b}\bar{s})^{\bar{6}}_{0}\rangle_{0}$ |
$1^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}8610\\\ 8644\\\ 8651\end{bmatrix}$ |  $\begin{bmatrix}B_{c}D_{s}^{\ast},B_{c}^{\ast}D_{s}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(223,160)\\\ (257,194)\\\ (264,201)\end{bmatrix}$
|  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{1}\rangle_{1}$ |
|  $|\\{cc\\}^{6}_{0}(\bar{b}\bar{s})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{cc\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}8674\end{bmatrix}$ |  $\begin{bmatrix}B_{c}^{\ast}D_{s}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}224\end{bmatrix}$

As can be seen from Table 2, the masses of the hidden charm $cc\bar{c}\bar{n}$
states lie in the range of 5005-5154 MeV and $cc\bar{c}\bar{s}$ states lie in
the range of 5112-5249 MeV. The mass splitting between the $cc\bar{c}\bar{n}$
tetraquarks is 149 MeV whereas it is 137 MeV in the $cc\bar{c}\bar{s}$
tetraquarks. The mass splittings for these triply-heavy tetraquarks arises
from spin-spin interactions, as in the case of conventional mesons. For the
$cc\bar{b}\bar{n}$ system, the masses lie in the range of 8526-8638 MeV
whereas for $cc\bar{b}\bar{s}$ system the masses lie in the range of 8576-8674
MeV. The mass splittings between $cc\bar{b}\bar{n}$ and $cc\bar{b}\bar{s}$
tetraquarks are 112 and 98 MeV, respectively. As mentioned in Ref. [22], the
quantum numbers of $cc\bar{c}\bar{n}$ and $cc\bar{c}\bar{s}$ tetraquark states
are the same of $D$ and $D_{s}$ mesons. However the masses of $D$ and $D_{s}$
mesons, which are around 2.0-2.7 GeV, are much lower than the predicted mass
values of $cc\bar{c}\bar{n}$ and $cc\bar{c}\bar{s}$ states. No orbital or
radial excitation can induce such a mass difference. Therefore once these
states could be observed, it is easy to identify them as $D-$ or $D_{s}-$like
meson with an excited $c\bar{c}$ pair. It will be good to touch on the color
configurations of triply-heavy tetraquark states. In Refs. [22, 28] no
explicit color configurations are presented however it was mentioned that
ground states of $cc\bar{c}\bar{q}$ and $cc\bar{b}\bar{q}$ are both dominated
by color-sextet $6\otimes\bar{6}$ color configurations. For example in Ref.
[28], it was pointed out that the $nc\bar{c}\bar{c}$ state with $J^{P}=0^{+}$
has a mass of $4936.7$ MeV with $66.7\%$ of the $6\otimes\bar{6}$ component,
and the state $sc\bar{c}\bar{c}$ with $J^{P}=0^{+}$ has a mass of $5040.1$ MeV
with $66.6\%$ $6\otimes\bar{6}$ component. In Ref. [16], $6\otimes\bar{6}$
color configurations have higher mass values of both $J^{P}=0^{+}$ and
$J^{P}=1^{+}$ for $cc\bar{c}\bar{q}$ and $cc\bar{b}\bar{q}$ tetraquarks. Our
results also support this conclusion except $cc\bar{b}\bar{s}$ case where
$\bar{3}\otimes 3$ with $J^{P}=2^{+}$ state is just 4 MeV above than the
$6\otimes\bar{6}$ state with $J^{P}=0^{+}$ pattern.

Comparing to the thresholds of $cc\bar{c}\bar{q}$ and $cc\bar{b}\bar{q}$
systems we find that there are some bound state candidates in the
$cc\bar{c}\bar{q}$ sector. There are some configurations in the
$cc\bar{c}\bar{q}$ sector which are just few MeVs above than the corresponding
two-meson thresholds. There may be a chance for observing them as resonances.
The predicted masses of $cc\bar{b}\bar{q}$ states are much higher than their
corresponding two-meson thresholds which refer that they may easily decay into
two mesons via the fall-apart mechanism.

We compare our mass results with the predictions of other studies in Table 3.
Before going into discussion, it will be good to mention that we present mass
results starting from lower values for each spin state. It can be observed
that for the $cc\bar{c}\bar{q}$ tetraquark states our results agree well with
the results of Ref. [28] in which an extended chromomagnetic model is used and
with the results of Ref. [23] which is a QCD sum rule study. In Ref. [23] for
the $J^{P}=0^{+}$ and $J^{P}=1^{+}$, both $6\otimes\bar{6}$ and
$\bar{3}\otimes 3$ type interpolating currents gave mass value as $5.1\pm 0.2$
GeV for the $cc\bar{c}\bar{q}$ tetraquark states. Our results are quite lower
than the predicted masses of Refs. [22, 16] for $cc\bar{c}\bar{q}$
tetraquarks. Ref. [16] used an extended relativistic quark model whereas in
Ref. [22] color-magnetic interaction model is used. In the $cc\bar{b}\bar{q}$
tetraquark states, our results are compatible with the results of Ref. [16].
Results of Ref. [22] are approximately 200 MeV below than our results whereas
results of Ref. [28] are approximately 300 MeV above than our results.
Extracted central mass values of Ref. [23] are lower than our mass values. For
$J^{P}=0^{+}$, the extracted masses are $8.0\pm 0.3$ GeV for $6\otimes\bar{6}$
current and $8.2\pm 0.3$ GeV for $\bar{3}\otimes 3$ current whereas for
$J^{P}=1^{+}$, $6\otimes\bar{6}$ type current gave mass as $8.1\pm 0.3$ GeV
and $\bar{3}\otimes 3$ type current gave mass as $8.2\pm 0.3$ GeV.

Table 3:  Comparison of $cc\bar{c}\bar{q}$ and $cc\bar{b}\bar{q}$ tetraquark masses with other studies. All results are in units of MeV except Ref. [23] where they are in GeV.  Configuration | $J^{P}$ | This work | [22] | [23] | [16] | [28]
---|---|---|---|---|---|---
$cc\bar{c}\bar{n}$ |  |  |  |  |  |
| $0^{+}$ | 5005-5154 | 5411.6-5658.8 |  $5.1\pm 0.2$-$5.1\pm 0.2$ | 5403-5533 | 4936.7-5185.3
| $1^{+}$ | 5053-5081-5106 | 5463.4-5530.5-5605.6 |  $5.1\pm 0.2$-$5.1\pm 0.2$ | 5400-5473-5482 | 4968.1-5135.5-5154.4
| $2^{+}$ | 5116 | 5593 | $\cdots$ | 5506 | 5198.4
$cc\bar{c}\bar{s}$ |  |  |  |  |  |
| $0^{+}$ | 5112-5249 | 5592.1-5839.5 | $\cdots$ | 5476-5599 | 5040.1-5290.6
| $1^{+}$ | 5118-5173-5190 | 5638.8-5710.5-5781.9 | $\cdots$ | 5481-5552-5554 | 5069.2-5239.9-5254.3
| $2^{+}$ | 5202 | 5773.2 | $\cdots$ | 5584 | 5303.3
$cc\bar{b}\bar{n}$ |  |  |  |  |  |
| $0^{+}$ | 8526-8638 | 8756.8-8962.4 | $8.0\pm 0.3-8.2\pm 0.3$ | 8658-8761 | 8199.1-8444.8
| $1^{+}$ | 8541-8598-8613 | 8803.7-8868.3-8935.0 | $8.1\pm 0.3-8.2\pm 0.3$ | 8667-8734-8740 | 8217.5-8430.0-8454.4
| $2^{+}$ | 8630 | 8908.6 | $\cdots$ | 8755 | 8477.9
$cc\bar{b}\bar{s}$ |  |  |  |  |  |
| $0^{+}$ | 8576-8670 | 8935.1-9140.7 | $\cdots$ | 8751-8845 | 8300.5-8538.7
| $1^{+}$ | 8610-8644-8651 | 8982.6-9046.9-9114.0 | $\cdots$ | 8761-8820-8825 | 8324.8-8521.9-8553.3
| $2^{+}$ | 8674 | 9086.9 | $\cdots$ | 8840 | 8569.1

###  IV.3 The $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ Tetraquark States

The masses of $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ tetraquarks with
corresponding thresholds and $\Delta$ values are presented in Table 4. The
predicted masses of $bb\bar{c}\bar{q}$ tetraquarks lie around 12 GeV,
specifically in the range of 11949-12088 MeV for $bb\bar{c}\bar{n}$ tetraquark
states and in the range of 12022-12143 MeV for $bb\bar{c}\bar{s}$ tetraquark
states. The mass splittings are 139 and 121 MeV for the $bb\bar{c}\bar{n}$ and
$bb\bar{c}\bar{s}$ tetraquarks, respectively. In the $bb\bar{b}\bar{q}$
tetraquarks, the mass values lie around 15 GeV, specifically from 15564 to
15618 MeV for $bb\bar{b}\bar{n}$ tetraquarks and from 15669 to 15715 MeV for
$bb\bar{b}\bar{s}$ tetraquarks. The mass splittings are 54 and 46 MeV for
$bb\bar{b}\bar{n}$ and $bb\bar{b}\bar{s}$ tetraquarks, respectively. All the
predicted masses for the $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ tetraquarks
are above than their corresponding $S$-wave meson-meson thresholds. They can
easily decay into two mesons via the fall-apart mechanism.

Table 4: Predicted mass spectra for the $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ systems, corresponding thresholds and $\Delta$ values. The meson masses are taken from [80] and the predicted mass for $M_{B_{c}^{\ast}}=6338$ MeV is taken from [81]. All results are in units of MeV. $J^{P}$ |  Configuration |  Mass |  $S$-wave threshold |  $\Delta$
---|---|---|---|---
$0^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}11949\\\ 12088\end{bmatrix}$ |  $\begin{bmatrix}\bar{B}_{c}\bar{B},\bar{B}_{c}^{\ast}\bar{B}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(395,286)\\\ (534,425)\end{bmatrix}$
|  $|\\{bb\\}^{6}_{0}(\bar{c}\bar{n})^{\bar{6}}_{0}\rangle_{0}$ |
$1^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}11954\\\ 11970\\\ 11981\end{bmatrix}$ |  $\begin{bmatrix}\bar{B}_{c}\bar{B}^{\ast},\bar{B}_{c}^{\ast}\bar{B}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(355,291)\\\ (371,307)\\\ (382,318)\end{bmatrix}$
|  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{1}\rangle_{1}$ |
|  $|\\{bb\\}^{6}_{0}(\bar{c}\bar{n})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{n})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}11987\end{bmatrix}$ |  $\begin{bmatrix}\bar{B}_{c}^{\ast}\bar{B}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}324\end{bmatrix}$
$0^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}12022\\\ 12143\end{bmatrix}$ |  $\begin{bmatrix}\bar{B}_{c}\bar{B}_{s},\bar{B}_{c}^{\ast}\bar{B}^{\ast}_{s}\end{bmatrix}$ |  $\begin{bmatrix}(381,269)\\\ (502,390)\end{bmatrix}$
|  $|\\{bb\\}^{6}_{0}(\bar{c}\bar{s})^{\bar{6}}_{0}\rangle_{0}$ |
$1^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}12031\\\ 12049\\\ 12058\end{bmatrix}$ |  $\begin{bmatrix}\bar{B}_{c}\bar{B}_{s}^{\ast},\bar{B}_{c}^{\ast}\bar{B}_{s}\end{bmatrix}$ |  $\begin{bmatrix}(341,278)\\\ (369,296)\\\ (368,305)\end{bmatrix}$
|  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{1}\rangle_{1}$ |
|  $|\\{bb\\}^{6}_{0}(\bar{c}\bar{s})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{c}\bar{s})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}12063\end{bmatrix}$ |  $\begin{bmatrix}\bar{B}_{c}^{\ast}\bar{B}_{s}\end{bmatrix}$ |  $\begin{bmatrix}310\end{bmatrix}$
$0^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}15564\\\ 15618\end{bmatrix}$ |  $\begin{bmatrix}\eta_{b}\bar{B},\Upsilon\bar{B}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(886,779)\\\ (940,833)\end{bmatrix}$
|  $|\\{bb\\}^{6}_{0}(\bar{b}\bar{n})^{\bar{6}}_{0}\rangle_{0}$ |
$1^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}15573\\\ 15588\\\ 15595\end{bmatrix}$ |  $\begin{bmatrix}\eta_{b}\bar{B}^{\ast},\Upsilon\bar{B}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(850,788)\\\ (865,803)\\\ (872,810)\end{bmatrix}$
|  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{1}\rangle_{1}$ |
|  $|\\{bb\\}^{6}_{0}(\bar{b}\bar{n})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{n})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}15607\end{bmatrix}$ |  $\begin{bmatrix}\Upsilon\bar{B}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}822\end{bmatrix}$
$0^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{1}\rangle_{0}$ |  $\begin{bmatrix}15669\\\ 15715\end{bmatrix}$ |  $\begin{bmatrix}\eta_{b}\bar{B}_{s},\Upsilon\bar{B}_{s}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(903,793)\\\ (949,839)\end{bmatrix}$
|  $|\\{bb\\}^{6}_{0}(\bar{b}\bar{s})^{\bar{6}}_{0}\rangle_{0}$ |
$1^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{0}\rangle_{1}$ |  $\begin{bmatrix}15672\\\ 15683\\\ 15685\end{bmatrix}$ |  $\begin{bmatrix}\eta_{b}\bar{B}_{s}^{\ast},\Upsilon\bar{B}_{s}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}(858,796)\\\ (869,807)\\\ (871,809)\end{bmatrix}$
|  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{1}\rangle_{1}$ |
|  $|\\{bb\\}^{6}_{0}(\bar{b}\bar{s})^{\bar{6}}_{1}\rangle_{1}$ |
$2^{+}$ |  $|\\{bb\\}^{\bar{3}}_{1}(\bar{b}\bar{s})^{3}_{1}\rangle_{2}$ |  $\begin{bmatrix}15691\end{bmatrix}$ |  $\begin{bmatrix}\Upsilon B_{s}^{\ast}\end{bmatrix}$ |  $\begin{bmatrix}815\end{bmatrix}$

We compare our mass results with the predictions of other studies in Table 5.
For the $bb\bar{c}\bar{n}$ tetraquarks, we see that our results agree with
those of Refs. [16, 22]. Our predicted masses are quite higher than masses of
Ref. [28]. For the $bb\bar{c}\bar{s}$ tetraquarks, our results agree well with
mass values of Ref. [16]. Mass predictions of Ref. [22] are quite higher
whereas predictions of Ref. [28] are quite lower than our results. In the case
of $bb\bar{b}\bar{n}$ states, our results agree with results of Ref. [22].
Mass predictions of Ref. [23, 16, 28] are rather different than our predicted
mass values. This argument can be transferred to the $bb\bar{b}\bar{s}$
tetraquark states.

Table 5:  Comparison of $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ tetraquark masses with other studies. All results are in units of MeV except Ref. [23] where they are in GeV.  Configuration | $J^{P}$ | This work | [22] | [23] | [16] | [28]
---|---|---|---|---|---|---
$bb\bar{c}\bar{n}$ |  |  |  |  |  |
| $0^{+}$ | 11949-12083 | 12144.8-12257.2 | $\cdots$ | 11906-12013 | 11581.7-11694.9
| $1^{+}$ | 11954-11970-11981 | 12147.6-12196.5-12219.1 | $\cdots$ | 11873-11933-11984 | 11582.2-11624.7-11658.6
| $2^{+}$ | 11987 | 12225.2 | $\cdots$ | 11948 | 11651.7
$bb\bar{c}\bar{s}$ |  |  |  |  |  |
| $0^{+}$ | 12022-12143 | 12322.7-12439.4 | $\cdots$ | 11985-12067 | 11675.1-11787.2
| $1^{+}$ | 12031-12049-12058 | 12322.1-12375.8-12396.5 | $\cdots$ | 11950-12003-12040 | 11673.3-11722.2-11746.3
| $2^{+}$ | 12063 | 12405.9 | $\cdots$ | 12018 | 11750.2
$bb\bar{b}\bar{n}$ |  |  |  |  |  |
| $0^{+}$ | 15564-15618 | 15468.6-15572.7 | $13.5\pm 0.4-13.7\pm 0.3$ | 15109-15158 | 14706.1-14850.9
| $1^{+}$ | 15573-15588-15595 | 15493.5-15519.7-1553.4 | $13.5\pm 0.4-13.5\pm 0.4$ | 15107-15137-15142 | 14712.1-14485.1-14851.6
| $2^{+}$ | 15607 | 15545.0 | $\cdots$ | 15154 | 14871.8
$bb\bar{b}\bar{s}$ |  |  |  |  |  |
| $0^{+}$ | 15669-15715 | 15752.5-15645.3 | $\cdots$ | 15185-15232 | 14793.1-14936.2
| $1^{+}$ | 15672-15683-15685 | 15672.2-15697.0-15733.3 | $\cdots$ | 15184-15213-15219 | 14804.5-14929.2-14943.7
| $2^{+}$ | 15691 | 15723.9 | $\cdots$ | 15231 | 14956.6

Considering color configurations, QCD sum rule analysis of Ref. [23] gave mass
values with $J^{P}=0^{+}$ as $13.5\pm 0.4$ GeV for $6\otimes\bar{6}$ type
interpolating current and $13.7\pm 0.3$ GeV for $\bar{3}\otimes 3$ type
interpolating current of $bb\bar{b}\bar{q}$ tetraquarks. For the
$J^{P}=1^{+}$, both $6\otimes\bar{6}$ and $\bar{3}\otimes 3$ type
interpolating currents gave mass value as $13.5\pm 0.4$ GeV for the
$bb\bar{b}\bar{q}$ tetraquarks. In Ref. [28], the ground states of
$nb\bar{b}\bar{b}$ and $sb\bar{b}\bar{b}$ have $J^{P}=0^{+}$ quantum number
and are dominated by $6\otimes\bar{6}$ color configurations with a rate of
$(>85\%)$. In the $nc\bar{b}\bar{b}$ and $sc\bar{b}\bar{b}$ states,
$\bar{3}\otimes 3$ color configurations dominate $J^{P}=0^{+}$ ground states
with a rate of $(>70\%)$. In Ref. [16], ground states of $bb\bar{c}\bar{q}$
and $bb\bar{b}\bar{q}$ tetraquarks are in the $\bar{3}\otimes 3$
configurations with $J^{P}=1^{+}$ whereas in our results ground states of
$bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ tetraquarks are in $6\otimes\bar{6}$
color configurations but with $J^{P}=0^{+}$.

Except some states in the $cc\bar{c}\bar{q}$ sector, all the states of
$cc\bar{b}\bar{q}$, $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ sectors are
above than their corresponding two-meson thresholds. They may decay into two
mesons through the fall-apart mechanism. Although these states are not stable,
some of them may be observed in the future experiments as resonances with
finite decay widths. In addition to fall-apart decay mechanism, triply-heavy
tetraquark states may decay through other two-body strong decay modes. In the
hidden charm and hidden bottom states, $c\bar{c}$ and $b\bar{b}$ quark pairs
may decay into the meson-meson or even baryon-antibaryon final states.

Before ending this section, mass spectra of $S$-wave triply-heavy tetraquark
states are given in Figs. 1 and 2. We can see from both figures that they have
similar mass patterns. This is an indication of light flavor $SU(3)$ symmetry
and heavy quark symmetry which have been successfully used in traditional
hadrons as well as in singly-heavy tetraquarks and doubly-heavy tetraquarks.
This heavy quark symmetry can be an ideal tool for investigating triply-heavy
tetraquarks. In the heavy quark symmetry if one increase the mass of heavy
quark, technically in the $m_{Q}\to\infty$ limit, QCD can be handled
independent of the heavy quark flavor and spin. The recently observed
$T_{cc}^{+}$ state has a quark content of $cc\bar{u}\bar{d}$ and is the
longest living exotic hadron discovered till now [6, 7]. For the doubly-heavy
tetraquarks $Q_{1}Q_{2}\bar{q}_{3}\bar{q}_{4}$, it was shown in Refs. [82, 83,
84] that the mass ratio between the heavy $Q_{1}Q_{2}$ diquark and
$\bar{q}_{3}\bar{q}_{4}$ light antidiquark is one of the determining factors
for being a stable tetraquark. For example, if we replace charm quarks with
bottom quarks in $T_{cc}^{+}$ state, the forming $bb\bar{u}\bar{d}$ state has
the largest mass ratio between heavy quarks and light antiquarks and is a
compact stable tetraquark candidate suggested by Ref. [85] after the discovery
of $\Xi_{cc}^{++}$ baryon [86]. If we replace $\bar{u}$ or $\bar{d}$ quark
with $\bar{c}$ or $\bar{b}$ in the $bb\bar{u}\bar{d}$ state, the ratio between
heavy quarks and antiquarks decrease. This may be another indication that why
$bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ triply-heavy tetraquark states
cannot be stable. Contrary to this, some bound state candidates in the
$bb\bar{c}\bar{q}$ sector was found in Ref. [29]. In the same work, a bound
state candidate in the $cc\bar{c}\bar{q}$ was found which supports our result
in the $cc\bar{c}\bar{q}$ sector.

[$cc\bar{c}\bar{n}$ ] [$cc\bar{c}\bar{s}$ ] [$cc\bar{b}\bar{n}$ ]
[$cc\bar{b}\bar{s}$ ]

Figure 1: Predicted mass spectra of $cc\bar{c}\bar{q}$ and $cc\bar{b}\bar{q}$
states

[$bb\bar{c}\bar{n}$ ] [$bb\bar{c}\bar{s}$ ] [$bb\bar{b}\bar{n}$ ]
[$bb\bar{b}\bar{s}$ ]

Figure 2: Predicted mass spectra of $bb\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$
states

##  V Summary and Concluding Remarks

In this work, we consider flavor exotic triply-heavy tetraquarks as composed
of diquarks and antidiquarks and systematically study $S$-wave triply-heavy
tetraquarks by solving Schrödinger equation numerically. The calculations are
carried out in a nonrelativistic quark model with a color interaction
described by a potential computed in AdS/QCD. The model used in this paper
consists of a central potential $V_{AdS}$ which contains a linear part at the
large distances for confining and the short-distance behaviour of perturbative
QCD, spin dependent term $V_{spin}$, and a constant term $V_{0}$.

From the predicted mass spectra, we can see that in the $cc\bar{c}\bar{q}$
systems we have found narrow state candidates. For $cc\bar{b}\bar{q}$,
$bb\bar{c}\bar{q}$, and $bb\bar{b}\bar{q}$ systems all the triply-heavy
tetraquarks lie above the corresponding meson-meson thresholds, therefore no
stable state exists in this system. The mass spectra for the flavor exotic
triply-heavy tetraquark states have quite similar patterns preserving the
light flavor $SU(3)$ and heavy quark symmetries. Our results are quite
different than Refs. [22, 23, 16, 28] since we have some bound state
candidates in the $cc\bar{c}\bar{q}$ tetraquark configurations. These studies
are conducted with different models explained before. The differences among
these studies stem from the different choices of interactions, parameters or
assumptions in the models. It should be also mentioned that up to now
$B_{c}^{\ast}$ meson has not been observed in the experiments. Therefore a
precise measurement of its mass is important for the corresponding threshold
values.

For the $XYZ$ states, it is not clear that whether a tetraquark or molecule
configuration is plausible for explaining even the same particles, especially
one or more light quarks do have place in these states. In this manner triply-
heavy tetraquark states may open a way to identify genuine tetraquark states.
Furthermore, triply-heavy tetraquarks are intermediate states between the
doubly- and fully-heavy tetraquarks due to the mass relations of quarks,
$m_{q}{\ll}m_{c}{\ll}m_{b}$. The binding force of the triply-heavy tetraquark
states is dominantly provided by the gluon exchange mechanism. Besides the
phenomenological models relying on gluon exchange interactions, other
nonperturbative methods such as lattice QCD, effective field theories, and QCD
sum rule can be used for further studies. For example, lattice QCD calculation
is relatively easy for the $bb\bar{c}\bar{q}$ abd $cc\bar{b}\bar{q}$ triply-
heavy tetraquarks since annihilation effects of the same flavor quark does not
contaminate the simulation.

The studies regarding with these triply-heavy tetraquark states may help to
understand underlying structures of the flavor exotic and multiquark states.
We hope that our results can be of help for the future experimental studies.

###### Acknowledgements.

The author of this work would like to thank Soner Albayrak, Rodrigo C. L.
Bruni and Henrique Boschi-Filho for their valuable discussions about AdS/QCD
potential. The author also thanks to the anonymous referee for his/her useful
comments.

## References

  * [1] M. Gell-Mann, A Schematic Model of Baryons and Mesons, Phys. Lett. 8 (1964) 214–215.  doi:10.1016/S0031-9163(64)92001-3.
  * [2] J.-M. Richard, Exotic hadrons: review and perspectives, Few Body Syst. 57 (12) (2016) 1185–1212.  arXiv:1606.08593, doi:10.1007/s00601-016-1159-0.
  * [3] S. K. Choi, et al., Observation of a narrow charmonium-like state in exclusive $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays, Phys. Rev. Lett. 91 (2003) 262001\.  arXiv:hep-ex/0309032, doi:10.1103/PhysRevLett.91.262001.
  * [4] R. Aaij, et al., Amplitude analysis of the $B^{+}\to D^{+}D^{-}K^{+}$ decay, Phys. Rev. D 102 (2020) 112003.  arXiv:2009.00026, doi:10.1103/PhysRevD.102.112003.
  * [5] M. Ablikim, et al., Observation of a Near-Threshold Structure in the $K^{+}$ Recoil-Mass Spectra in $e^{+}e^{-}\rightarrow K^{+}(D_{s}^{-}D^{*0}+D_{s}^{*-}D^{0}$), Phys. Rev. Lett. 126 (10) (2021) 102001.  arXiv:2011.07855, doi:10.1103/PhysRevLett.126.102001.
  * [6] R. Aaij, et al., Observation of an exotic narrow doubly charmed tetraquark, Nature Phys. 18 (7) (2022) 751–754.  arXiv:2109.01038, doi:10.1038/s41567-022-01614-y.
  * [7] R. Aaij, et al., Study of the doubly charmed tetraquark $T_{cc}^{+}$, Nature Commun. 13 (1) (2022) 3351.  arXiv:2109.01056, doi:10.1038/s41467-022-30206-w.
  * [8] R. Aaij, et al., Observation of structure in the $J/\psi$ -pair mass spectrum, Sci. Bull. 65 (23) (2020) 1983–1993.  arXiv:2006.16957, doi:10.1016/j.scib.2020.08.032.
  * [9] M.-S. liu, F.-X. Liu, X.-H. Zhong, Q. Zhao, Full-heavy tetraquark states and their evidences in the LHCb di-$J/\psi$ spectrum (6 2020).  arXiv:2006.11952.
  * [10] J. F. Giron, R. F. Lebed, Simple spectrum of $c\bar{c}c\bar{c}$ states in the dynamical diquark model, Phys. Rev. D 102 (7) (2020) 074003.  arXiv:2008.01631, doi:10.1103/PhysRevD.102.074003.
  * [11] X. Jin, Y. Xue, H. Huang, J. Ping, Full-heavy tetraquarks in constituent quark models, Eur. Phys. J. C 80 (11) (2020) 1083.  arXiv:2006.13745, doi:10.1140/epjc/s10052-020-08650-z.
  * [12] R. Zhu, Fully-heavy tetraquark spectra and production at hadron colliders, Nucl. Phys. B 966 (2021) 115393.  arXiv:2010.09082, doi:10.1016/j.nuclphysb.2021.115393.
  * [13] H.-X. Chen, W. Chen, X. Liu, S.-L. Zhu, Strong decays of fully-charm tetraquarks into di-charmonia, Sci. Bull. 65 (2020) 1994–2000.  arXiv:2006.16027, doi:10.1016/j.scib.2020.08.038.
  * [14] B.-C. Yang, L. Tang, C.-F. Qiao, Scalar fully-heavy tetraquark states $QQ^{\prime}{\bar{Q}}\bar{Q^{\prime}}$ in QCD sum rules, Eur. Phys. J. C 81 (4) (2021) 324.  arXiv:2012.04463, doi:10.1140/epjc/s10052-021-09096-7.
  * [15] H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, S.-L. Zhu, An updated review of the new hadron states, Rept. Prog. Phys. 86 (2) (2023) 026201.  arXiv:2204.02649, doi:10.1088/1361-6633/aca3b6.
  * [16] Q.-F. Lü, D.-Y. Chen, Y.-B. Dong, E. Santopinto, Triply-heavy tetraquarks in an extended relativized quark model, Phys. Rev. D 104 (5) (2021) 054026.  arXiv:2107.13930, doi:10.1103/PhysRevD.104.054026.
  * [17] H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, S.-L. Zhu, A review of the open charm and open bottom systems, Rept. Prog. Phys. 80 (7) (2017) 076201.  arXiv:1609.08928, doi:10.1088/1361-6633/aa6420.
  * [18] R. Aaij, et al., Production of associated Y and open charm hadrons in pp collisions at $\sqrt{s}=7$ and 8 TeV via double parton scattering, JHEP 07 (2016) 052.  arXiv:1510.05949, doi:10.1007/JHEP07(2016)052.
  * [19] B. Silvestre-Brac, Systematics of Q**2 (anti-Q**2) systems with a chromomagnetic interaction, Phys. Rev. D 46 (1992) 2179–2189.  doi:10.1103/PhysRevD.46.2179.
  * [20] B. Silvestre-Brac, C. Semay, Systematics of L = 0 q-2 anti-q-2 systems, Z. Phys. C 57 (1993) 273–282.  doi:10.1007/BF01565058.
  * [21] Y. Cui, X.-L. Chen, W.-Z. Deng, S.-L. Zhu, The Possible Heavy Tetraquarks qQ anti-q anti-Q, qq anti-Q anti-Q and qQ anti-Q anti-Q, HEPNP 31 (2007) 7–13.  arXiv:hep-ph/0607226.
  * [22] K. Chen, X. Liu, J. Wu, Y.-R. Liu, S.-L. Zhu, Triply heavy tetraquark states with the $QQ\bar{Q}\bar{q}$ configuration, Eur. Phys. J. A 53 (1) (2017) 5.  arXiv:1609.06117, doi:10.1140/epja/i2017-12199-3.
  * [23] J.-F. Jiang, W. Chen, S.-L. Zhu, Triply heavy $QQ\bar{Q}\bar{q}$ tetraquark states, Phys. Rev. D 96 (9) (2017) 094022.  arXiv:1708.00142, doi:10.1103/PhysRevD.96.094022.
  * [24] P. Junnarkar, N. Mathur, M. Padmanath, Study of doubly heavy tetraquarks in Lattice QCD, Phys. Rev. D 99 (3) (2019) 034507.  arXiv:1810.12285, doi:10.1103/PhysRevD.99.034507.
  * [25] R. J. Hudspith, B. Colquhoun, A. Francis, R. Lewis, K. Maltman, A lattice investigation of exotic tetraquark channels, Phys. Rev. D 102 (2020) 114506.  arXiv:2006.14294, doi:10.1103/PhysRevD.102.114506.
  * [26] Y. Liu, M. A. Nowak, I. Zahed, Heavy Holographic Exotics: Tetraquarks as Efimov States, Phys. Rev. D 100 (12) (2019) 126023.  arXiv:1904.05189, doi:10.1103/PhysRevD.100.126023.
  * [27] Y. Xing, Weak decays of triply heavy tetraquarks ${b{\bar{c}}}{b{\bar{q}}}$, Eur. Phys. J. C 80 (1) (2020) 57.  arXiv:1910.11593, doi:10.1140/epjc/s10052-020-7625-3.
  * [28] X.-Z. Weng, W.-Z. Deng, S.-L. Zhu, Triply heavy tetraquark states, Phys. Rev. D 105 (3) (2022) 034026.  arXiv:2109.05243, doi:10.1103/PhysRevD.105.034026.
  * [29] X. Liu, Y. Tan, D. Chen, H. Huang, J. Ping, Possible triply heavy tetraquark states in a chiral quark model, Phys. Rev. D 107 (5) (2023) 054019.  arXiv:2205.08281, doi:10.1103/PhysRevD.107.054019.
  * [30] G. S. Bali, QCD forces and heavy quark bound states, Phys. Rept. 343 (2001) 1–136.  arXiv:hep-ph/0001312, doi:10.1016/S0370-1573(00)00079-X.
  * [31] N. Brambilla, et al., Heavy Quarkonium: Progress, Puzzles, and Opportunities, Eur. Phys. J. C 71 (2011) 1534.  arXiv:1010.5827, doi:10.1140/epjc/s10052-010-1534-9.
  * [32] W. Lucha, F. F. Schoberl, D. Gromes, Bound states of quarks, Phys. Rept. 200 (1991) 127–240.  doi:10.1016/0370-1573(91)90001-3.
  * [33] G. Yang, J. Ping, L. He, Q. Wang, Potential model prediction of fully-heavy tetraquarks $QQ\bar{Q}\bar{Q}$ ($Q=c,b$) (6 2020).  arXiv:2006.13756.
  * [34] Z. Zhao, K. Xu, A. Kaewsnod, X. Liu, A. Limphirat, Y. Yan, Determination of Parameters of a Potential Model for Tetraquark Study by Studying All S-Wave Mesons, Few Body Syst. 62 (4) (2021) 80, [Erratum: Few Body Syst. 63, 24 (2022)].  arXiv:2109.00164, doi:10.1007/s00601-021-01667-4.
  * [35] Z. Asadi, G. R. Boroun, Masses of fully heavy tetraquark states from a four-quark static potential model, Phys. Rev. D 105 (1) (2022) 014006.  arXiv:2112.11028, doi:10.1103/PhysRevD.105.014006.
  * [36] G. Yang, J. Ping, J. Segovia, Hidden-charm tetraquarks with strangeness in the chiral quark model, Phys. Rev. D 104 (9) (2021) 094035.  arXiv:2109.04311, doi:10.1103/PhysRevD.104.094035.
  * [37] S. J. Brodsky, G. F. de Teramond, AdS/CFT and Light-Front QCD, Subnucl. Ser. 45 (2009) 139–183.  arXiv:0802.0514, doi:10.1142/9789814293242_0008.
  * [38] Y. Kim, J.-P. Lee, S. H. Lee, Heavy quarkonium in a holographic QCD model, Phys. Rev. D 75 (2007) 114008.  arXiv:hep-ph/0703172, doi:10.1103/PhysRevD.75.114008.
  * [39] D. Hou, H.-c. Ren, Heavy Quarkonium States with the Holographic Potential, JHEP 01 (2008) 029.  arXiv:0710.2639, doi:10.1088/1126-6708/2008/01/029.
  * [40] Y. Bai, H.-C. Cheng, A Holographic Model of Heavy-light Mesons, JHEP 08 (2013) 074.  arXiv:1306.2944, doi:10.1007/JHEP08(2013)074.
  * [41] Z. Li, B.-Q. Ma, Baryon spectrum in a finite-temperature AdS/QCD model, Phys. Rev. D 89 (1) (2014) 015014.  arXiv:1312.3451, doi:10.1103/PhysRevD.89.015014.
  * [42] Z. Wang, B.-Q. Ma, A unified approach to hadron phenomenology at zero and finite temperatures in a hard-wall AdS/QCD model, Eur. Phys. J. A 52 (5) (2016) 122.  arXiv:1512.01957, doi:10.1140/epja/i2016-16122-2.
  * [43] M. Rinaldi, V. Vento, Scalar spectrum in a graviton soft wall model, J. Phys. G 47 (12) (2020) 125003.  arXiv:2002.11720, doi:10.1088/1361-6471/abba71.
  * [44] M. A. Martin Contreras, S. Diles, A. Vega, Heavy quarkonia spectroscopy at zero and finite temperature in bottom-up AdS/QCD, Phys. Rev. D 103 (8) (2021) 086008.  arXiv:2101.06212, doi:10.1103/PhysRevD.103.086008.
  * [45] O. Andreev, V. I. Zakharov, Heavy-quark potentials and AdS/QCD, Phys. Rev. D 74 (2006) 025023.  arXiv:hep-ph/0604204, doi:10.1103/PhysRevD.74.025023.
  * [46] F. Giannuzzi, Heavy pentaquark spectroscopy in the diquark model, Phys. Rev. D 99 (9) (2019) 094006.  arXiv:1903.04430, doi:10.1103/PhysRevD.99.094006.
  * [47] M. V. Carlucci, F. Giannuzzi, G. Nardulli, M. Pellicoro, S. Stramaglia, AdS-QCD quark-antiquark potential, meson spectrum and tetraquarks, Eur. Phys. J. C 57 (2008) 569–578.  arXiv:0711.2014, doi:10.1140/epjc/s10052-008-0687-2.
  * [48] F. Giannuzzi, eta(b) and eta(c) radiative decays in the Salpeter model with the AdS/QCD inspired potential, Phys. Rev. D 78 (2008) 117501.  arXiv:0810.2736, doi:10.1103/PhysRevD.78.117501.
  * [49] F. Giannuzzi, Doubly heavy baryons in a Salpeter model with AdS/QCD inspired potential, Phys. Rev. D 79 (2009) 094002.  arXiv:0902.4624, doi:10.1103/PhysRevD.79.094002.
  * [50] R. C. L. Bruni, E. Folco Capossoli, H. Boschi-Filho, Quark-antiquark potential from a deformed AdS/QCD, Adv. High Energy Phys. 2019 (2019) 1901659.  arXiv:1806.05720, doi:10.1155/2019/1901659.
  * [51] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, T.-M. Yan, Charmonium: The Model, Phys. Rev. D 17 (1978) 3090, [Erratum: Phys.Rev.D 21, 313 (1980)].  doi:10.1103/PhysRevD.17.3090.
  * [52] J. L. Richardson, The Heavy Quark Potential and the Upsilon, J/psi Systems, Phys. Lett. B 82 (1979) 272–274.  doi:10.1016/0370-2693(79)90753-6.
  * [53] W. Buchmuller, S. H. H. Tye, Quarkonia and Quantum Chromodynamics, Phys. Rev. D 24 (1981) 132.  doi:10.1103/PhysRevD.24.132.
  * [54] R. K. Bhaduri, L. E. Cohler, Y. Nogami, A Unified Potential for Mesons and Baryons, Nuovo Cim. A 65 (1981) 376–390.  doi:10.1007/BF02827441.
  * [55] C. Semay, B. Silvestre-Brac, Diquonia and potential models, Z. Phys. C 61 (1994) 271–275.  doi:10.1007/BF01413104.
  * [56] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252.  arXiv:hep-th/9711200, doi:10.4310/ATMP.1998.v2.n2.a1.
  * [57] J. M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859–4862.  arXiv:hep-th/9803002, doi:10.1103/PhysRevLett.80.4859.
  * [58] S.-J. Rey, J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379–394.  arXiv:hep-th/9803001, doi:10.1007/s100520100799.
  * [59] M. Ida, R. Kobayashi, Baryon resonances in a quark model, Prog. Theor. Phys. 36 (1966) 846.  doi:10.1143/PTP.36.846.
  * [60] D. B. Lichtenberg, L. J. Tassie, Baryon Mass Splitting in a Boson-Fermion Model, Phys. Rev. 155 (1967) 1601–1606.  doi:10.1103/PhysRev.155.1601.
  * [61] M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson, D. B. Lichtenberg, Diquarks, Rev. Mod. Phys. 65 (1993) 1199–1234.  doi:10.1103/RevModPhys.65.1199.
  * [62] E. Santopinto, An Interacting quark-diquark model of baryons, Phys. Rev. C 72 (2005) 022201.  arXiv:hep-ph/0412319, doi:10.1103/PhysRevC.72.022201.
  * [63] D. Ebert, R. N. Faustov, V. O. Galkin, Masses of excited heavy baryons in the relativistic quark model, Phys. Lett. B 659 (2008) 612–620.  arXiv:0705.2957, doi:10.1016/j.physletb.2007.11.037.
  * [64] R. N. Faustov, V. O. Galkin, Strange baryon spectroscopy in the relativistic quark model, Phys. Rev. D 92 (5) (2015) 054005.  arXiv:1507.04530, doi:10.1103/PhysRevD.92.054005.
  * [65] S. M. Moosavi Nejad, A. Armat, Study of bottom and charmed baryons in quark-diquark model, Eur. Phys. J. A 56 (11) (2020) 287.  doi:10.1140/epja/s10050-020-00291-w.
  * [66] R. N. Faustov, V. O. Galkin, Triply heavy baryon spectroscopy in the relativistic quark model, Phys. Rev. D 105 (1) (2022) 014013.  arXiv:2111.07702, doi:10.1103/PhysRevD.105.014013.
  * [67] A. Ali, L. Maiani, A. D. Polosa, Multiquark Hadrons, Cambridge University Press, 2019.  doi:10.1017/9781316761465.
  * [68] M. Y. Barabanov, et al., Diquark correlations in hadron physics: Origin, impact and evidence, Prog. Part. Nucl. Phys. 116 (2021) 103835.  arXiv:2008.07630, doi:10.1016/j.ppnp.2020.103835.
  * [69] L. Maiani, F. Piccinini, A. D. Polosa, V. Riquer, Diquark-antidiquarks with hidden or open charm and the nature of X(3872), Phys. Rev. D 71 (2005) 014028\.  arXiv:hep-ph/0412098, doi:10.1103/PhysRevD.71.014028.
  * [70] D. Ebert, R. N. Faustov, V. O. Galkin, Masses of heavy tetraquarks in the relativistic quark model, Phys. Lett. B 634 (2006) 214–219.  arXiv:hep-ph/0512230, doi:10.1016/j.physletb.2006.01.026.
  * [71] P. Lundhammar, T. Ohlsson, Nonrelativistic model of tetraquarks and predictions for their masses from fits to charmed and bottom meson data, Phys. Rev. D 102 (5) (2020) 054018.  arXiv:2006.09393, doi:10.1103/PhysRevD.102.054018.
  * [72] M. Monemzadeh, N. Tazimi, P. Sadeghi, Tetraquarks as Diquark Antidiquark Bound Systems, Phys. Lett. B 741 (2015) 124–127.  arXiv:1507.07916, doi:10.1016/j.physletb.2014.12.001.
  * [73] Q.-F. Lü, Y.-B. Dong, Masses of open charm and bottom tetraquark states in a relativized quark model, Phys. Rev. D 94 (9) (2016) 094041.  arXiv:1603.06417, doi:10.1103/PhysRevD.94.094041.
  * [74] V. R. Debastiani, F. S. Navarra, A non-relativistic model for the $[cc][\bar{c}\bar{c}]$ tetraquark, Chin. Phys. C 43 (1) (2019) 013105.  arXiv:1706.07553, doi:10.1088/1674-1137/43/1/013105.
  * [75] M. A. Bedolla, J. Ferretti, C. D. Roberts, E. Santopinto, Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective, Eur. Phys. J. C 80 (11) (2020) 1004.  arXiv:1911.00960, doi:10.1140/epjc/s10052-020-08579-3.
  * [76] Z. Ghalenovi, M. M. Sorkhi, Spectroscopy of hidden-charm tetraquarks in diquark model, Eur. Phys. J. Plus 135 (5) (2020) 399.  doi:10.1140/epjp/s13360-020-00343-6.
  * [77] R. N. Faustov, V. O. Galkin, E. M. Savchenko, Masses of the $QQ\bar{Q}\bar{Q}$ tetraquarks in the relativistic diquark–antidiquark picture, Phys. Rev. D 102 (11) (2020) 114030.  arXiv:2009.13237, doi:10.1103/PhysRevD.102.114030.
  * [78] D. Ebert, R. N. Faustov, V. O. Galkin, W. Lucha, Masses of tetraquarks with two heavy quarks in the relativistic quark model, Phys. Rev. D 76 (2007) 114015\.  arXiv:0706.3853, doi:10.1103/PhysRevD.76.114015.
  * [79] W. Park, S. H. Lee, Color spin wave functions of heavy tetraquark states, Nucl. Phys. A 925 (2014) 161–184.  arXiv:1311.5330, doi:10.1016/j.nuclphysa.2014.02.008.
  * [80] P. A. Zyla, et al., Review of Particle Physics, PTEP 2020 (8) (2020) 083C01.  doi:10.1093/ptep/ptaa104.
  * [81] S. Godfrey, N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32 (1985) 189–231.  doi:10.1103/PhysRevD.32.189.
  * [82] J. Vijande, A. Valcarce, N. Barnea, Exotic meson-meson molecules and compact four–quark states, Phys. Rev. D 79 (2009) 074010.  arXiv:0903.2949, doi:10.1103/PhysRevD.79.074010.
  * [83] E. Hernández, J. Vijande, A. Valcarce, J.-M. Richard, Spectroscopy, lifetime and decay modes of the $T^{-}_{bb}$ tetraquark, Phys. Lett. B 800 (2020) 135073\.  arXiv:1910.13394, doi:10.1016/j.physletb.2019.135073.
  * [84] Q.-F. Lü, D.-Y. Chen, Y.-B. Dong, Masses of doubly heavy tetraquarks $T_{QQ^{\prime}}$ in a relativized quark model, Phys. Rev. D 102 (3) (2020) 034012\.  arXiv:2006.08087, doi:10.1103/PhysRevD.102.034012.
  * [85] M. Karliner, J. L. Rosner, Discovery of doubly-charmed $\Xi_{cc}$ baryon implies a stable ($bb\bar{u}\bar{d}$) tetraquark, Phys. Rev. Lett. 119 (20) (2017) 202001.  arXiv:1707.07666, doi:10.1103/PhysRevLett.119.202001.
  * [86] R. Aaij, et al., Observation of the doubly charmed baryon $\Xi_{cc}^{++}$, Phys. Rev. Lett. 119 (11) (2017) 112001.  arXiv:1707.01621, doi:10.1103/PhysRevLett.119.112001.