# Holographic stability for non-$q\bar{q}$ candidates

Miguel Angel Martin Contreras<EMAIL_ADDRESS>School of Nuclear
Science and Technology
University of South China
Hengyang, China
No 28, West Changsheng Road, Hengyang City, Hunan Province, China.  Alfredo
Vega<EMAIL_ADDRESS>Instituto de Física y Astronomía,
Universidad de Valparaíso,
A. Gran Bretaña 1111, Valparaíso, Chile

###### Abstract

In the context of bottom-up AdS/QCD models, we discuss how the configurational
entropy can describe heavy non-$q\,\bar{q}$ states. Using the non-quadratic
softwall model, introduced to describe non-linear Regge trajectories, we
parametrize different multiquark and exotic meson structures to describe
$Z_{c}$, $\psi$, and $Z_{b}$ states as non-$q\,\bar{q}$ hadrons in terms of
stability. We found that $Z_{c}$ is better described as a hybrid meson with
one gluon tube, $\psi$ as hadrocharmonium, and $Z_{b}$ as hadronic molecule.

††preprint: APS/123-QED

##  I Introduction

Since the seminal work [1], the softwall model has become one of the most
famous bottom-up models in AdS/QCD to describe a plethora of hadronic
properties such as form factors [2, 3], deconfinement phase transition [4, 5,
6], chiral symmetry breaking [7], scattering processes [8, 9, 10] and hadron
spectroscopy [11, 12, 13, 14, 15]. This model has also been extended to
dynamical versions [16, 17, 18], finite density [19, 20], finite temperature
[21, 22, 23], and magnetic field [24, 25, 26].

The main idea behind this model is the emergence of bounded states in the AdS
bulk due to a dilaton field that breaks conformal invariance. These bounded
states are dual to hadrons at the boundary. The matching is achieved by
comparing the 2-point function calculated in large $N$ QCD at high $q^{2}$
with the holographic dual object. The most particular feature of the softwall
model is the linear confinement, i.e., the bounded states form a linear mass
spectrum, dual to Regge trajectories. These trajectories are suitable for
describing light-unflavored mesons. However, when constituent masses start to
be considered, linearity in the trajectories disappears [27].

An alternative to describe these non-linearities in hadrons spectra is a
deformation of the quadratic structure in the dilaton field, inherited from
the softwall model. This idea was applied to describe isovector mesons in
[28]. With the inclusion of the constituent mass, it is possible also to test
other hadronic species as the non-$q\,\bar{q}$ states. In particular, this
work wants to explore the configurational entropy as a test tool to describe
these states, as it was exposed initially in [29, 30]. In the first work [11],
the authors explore how configurational entropy (CE) is related to multiquark
hybrid meson states with a high CE value. On the other hand, [30] discusses
how CE is a tool to describe the abundance of these non-$q\,\bar{q}$ states in
nature. This work follows a different approach. We will connect the
configurational entropy with the hadron stability by discussing how bulk
locality is equivalent to confinement at the conformal boundary. This will
derivate the possibility of using CE to distinguish what multiquark or hybrid
meson models are suitable for describing heavy quark exotica.

This manuscript is organized as follows: In Section II, we give a summary of
the ideas behind the non-quadratic softwall model introduced in [28], and how
it works describing non-$q\,\bar{q}$ states. Section III describes the
differential configurational entropy (DCE) algorithm. Section IV discusses the
connection between DCE and hadron stability. The application of DCE to what
non-$q\,\bar{q}$ structures are suitable to describe heavy vector exotica is
written in Section V. Finally, we summarized our work in section VI.

##  II Non-quadratic softwall model

###  II.1 Non-quadratic dilaton in a nutshell

The non-quadratic dilaton is a holographic proposal developed to address heavy
quarkonium masses with better accuracy [28]. The motivation comes from the
Bethe-Salpeter formalism, where Regge trajectories, when including constituent
quark masses, can be written as [31]

$(M_{n}-m_{q_{1}}-m_{q_{2}})^{2}=a(n+b),$ (1)

where $a$ is a universal slope and $b$ accounts for the effect of the mesonic
quantum numbers, it is expected that nonlinearities associated with the
constituent mass emerge [32, 33]. In the holographic AdS/QCD context, it is
possible to include these constituent quark mass effects by adding an extra
$\nu$ exponent to the radial trajectory as follows

$M_{n}^{2}=a(n+b)^{\nu}.$ (2)

This nonlinearity in Regge trajectories is translated into deformations on the
static quadratic dilaton

$\Phi(z)=\left(\kappa\,z\right)^{2-\alpha},$ (3)

where the SWM dilaton gains an extra parameter $\alpha$ that captures the
constituent mass effects. In this sense, $\kappa$ and $\alpha$ sets a running
with the constituent mass of hadrons in a particular family. In the original
work [28], the isovector mesons, composed of $\rho$, $\omega$, $J/\psi$, and
$\Upsilon$ mesons, labeled $I^{G}\,J^{PC}=0^{-}(1^{--})$. The $\rho$ meson
corresponds to the light unflavored case, where mesons, under chiral symmetry,
have zero constituent mass. This mass scheme sets the following
parametrizations for $\kappa$ and $\alpha$

$\displaystyle\alpha(\bar{m})$ $\displaystyle=$ $\displaystyle
a_{\alpha}-b_{\alpha}e^{-c_{\alpha}\bar{m}^{2}}$ (4)
$\displaystyle\kappa(\bar{m})$ $\displaystyle=$ $\displaystyle
a_{\kappa}-b_{\kappa}e^{-c_{\kappa}\bar{m}^{2}},$ (5)

where the fit coefficients are given by

$\begin{gathered}a_{\alpha}=0.8454,~{}b_{\alpha}=0.8485,~{}c_{\alpha}=0.4233\,\,\text{GeV}^{2},\\\
a_{\kappa}=15.2085\,\text{GeV},~{}b_{\kappa}=14.8082\,\text{GeV},~{}c_{\kappa}=0.0524\,\text{GeV}^{2}\end{gathered}$

From the running of $\kappa$ and $\alpha$ with the hadron constituent mass,
i.e. $\kappa(\bar{m})$ and $\alpha(\bar{m})$, we can infer specific values for
$\kappa$ and $\alpha$ for a given vector hadron, as a non-$q\,\bar{q}$
candidate. The input in this procedure is the parametrization of the
constituent mass $\bar{m}$, which contains information on the inner structure
of the hadron. Thus, we will fix the constituent mass parametrization instead
of setting a Regge scale and a quadratic deformation exponent.

###  II.2 Holographic set-up

As customary in these models, the starting point is a density action for the
bulk field dual to vector hadrons:

$I_{\text{H}}=-\frac{1}{2}\int{d^{5}x\,\sqrt{-g}\,e^{-\phi(z)}\,\left(\frac{1}{2\,g^{2}_{5}}\,F_{mn}F^{mn}-M_{5}^{2}\,A_{m}\,A^{m}\right)}.$
(6)

This action lives in an AdS5 space described by the Poincarè patch.

$dS^{2}=\frac{R^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\,\nu}\,dx^{\mu}\,dx^{\nu}\right),$
(7)

where $R$ is the AdS curvature radius and
$\eta_{\mu\,\nu}=\text{diag}(-1,\vec{1})$.

At this point, mentioning the gauge invariance is essential since we now have
massive vector bulk fields. Recall that the gauge invariance should manifest
at the conformal boundary, where all dual fields are massless [34]. The non-
zero bulk mass does not affect the gauge $A_{z}=0$, which is imposed to
decouple the bulk from the boundary information. If we pay attention to the
massive e.o.m for the vector bulk fields, i.e., for the $z$ component

$\Box\,A_{z}-\partial_{z}\left(\partial_{\mu}\,A^{\mu}\right)+M_{5}^{2}\,e^{2\,A}\,A_{z}=0,$
(8)

imposing the $A_{z}=0$ gauge still implies the transverse gauge
$\partial_{\mu}\,A^{\mu}=0$ at the conformal boundary, so Ward-Takahashi
identities hold.

From this configuration, after writing the equations of motion in the
$A_{z}=0$ gauge, imposing the on-shell mass condition $M_{n}^{2}=-q^{2}$,
Fourier transformed the massive bulk field
$A_{\mu}(z,q)=\tilde{A}_{\mu}(q)\,\psi_{z,q}$ and performed the standard
Boguliubov transformation $\psi(z)=e^{B(z)/2}\phi(z)$[1], we can reduce the
hadron spectroscopy problem to solve a Schrödinger-like equation

$-\phi^{\prime\prime}(z)+V(z)\phi(z)=M_{n}^{2}\,\phi(z),$ (9)

where the holographic potential $V(z)$ is defined in terms of the Bogoliubov
transformation $B(z)$ function as

$V(z)=\frac{B^{\prime}(z)^{2}}{4}-\frac{B^{\prime\prime}(z)}{2}+\frac{M_{5}(\Delta)^{2}\,R^{2}}{z^{2}}$
(10)

with $B(z)=(\kappa\,z)^{2-\alpha}-\log(R/z)$. This $B$ function is fundamental
in the bottom-up AdS/QCD scenario since it encloses how the background models
hadronic physics at the boundary. The other ingredient is the bulk mass
$M_{5}$.

From the field/operator duality, the operator creating hadrons at the boundary
is dual to the bulk field. The most general operator that creates hadrons is a
function of the constituent fields (quarks and gluons) and the covariant
derivatives, that is,$\mathcal{O}=f(\bar{q},q,G_{\mu\,\nu},D_{\mu})$. This
operator definition allows us to consider more general structures, such as
_hadronic molecules_ or _hadroquakonium_ in the non-$q\,\bar{q}$
phenomenology.

The dimension of $\mathcal{O}$, $\text{dim}\,\mathcal{O}=\Delta$ controls how
the bulk normalizable mode behaves near the conformal boundary, i.e.,
$\psi(z\to 0)\to z^{\Delta-1}$ for vector fields. Imposing that the bulk field
spin is equivalent to the hadronic spin $S$, we can write the most general
scaling dimension $\Delta$ in terms of hadronic information [13]

$\Delta=\Delta_{0}+L+\text{anomalous dimensions},$ (11)

where $\Delta_{0}$ is associated with pure constituent information and $L$,
counting covariant derivatives, sets the hadronic angular momentum number. In
the most general situation, including data from the renormalization group is
possible. This information can be encoded in these anomalous dimensions terms
[35, 36, 37]. For simplicity, we will not include anomalous dimensions.

This relation for the scaling dimension, via the equations of motion for the
vector field, allows us to write an expression for the hadronic mass identity,
encoded into the bulk mass as follows

$M_{5}^{2}\,R^{2}=\left(\Delta_{0}+L-1\right)\left(\Delta_{0}+L-3\right).$
(12)

However, this expression for vector hadrons does not define univocally vector
hadrons. Consider non-$q\,\bar{q}$ structures with spin one as diquark-
antidiquark pairs, hadronic molecules, and hadroquarkonium. For a given $L$,
all of these tetraquark structures have $\Delta_{0}=6$. Thus, they have the
same bulk mass, i.e., $M_{5}^{3}\,R^{2}=15$. This _degeneracy_ can be lifted
in the nonquadratic dilaton scenario since $\kappa$ and $\alpha$ are sensible
to the hadron structure.

Once the hadronic identity is defined, the holographic potential for the
non-$q\,\bar{q}$ vector states in this non-quadratic model has the following
form

$V_{\text{non-}q\,\bar{q}}(z,\kappa,\,\Delta)=V_{q\,\bar{q}}(z,\kappa,\alpha)+\frac{M_{5}^{2}(\Delta)\,R^{2}}{z^{2}},$
(13)

where $V_{q\,\bar{q}}(z)$ potential is the potential used to compute the
masses of the isovector family in the s-wave. It is defined as

$V_{q\,\bar{q}}(z,\kappa,\alpha)=\frac{3}{4z^{2}}-\frac{1}{2}\alpha^{2}\,\kappa^{2}\,(\kappa\,z)^{-\alpha}+\frac{1}{4}\alpha^{2}\,\kappa^{2}(\kappa\,z)^{2-2\alpha}\\\
+\frac{3}{2}\alpha\,\kappa^{2}\,(\kappa\,z)^{-\alpha}-\kappa^{2}(\kappa\,z)^{-\alpha}-\alpha\,\kappa^{2}\,(\kappa\,z)^{2-2\alpha}\\\
+\kappa^{2}\,(\kappa\,z)^{2-2\alpha}+\frac{\kappa}{z}(\kappa\,z)^{1-\alpha}-\frac{\alpha\,\kappa}{2\,z}(\kappa\,z)^{1-\alpha}.$
(14)

Setting $\alpha=0$ in $V_{q\,\bar{q}}(z)$ reduces the potential to the well-
known softwall model potential [1].

The holographic algorithm will now focus on defining a threshold mass
$\bar{m}$, capturing the hadron structure information in terms of their
constituents. This threshold mass will define a pair $\kappa(\bar{m})$ and
$\alpha(\bar{m})$ for each structure. With these inputs, we can compute the
hadronic squared mass from the holographic potentials. With the eigenmodes, it
is straightforward to compute the associated configurational entropy.

###  II.3 Holographic non-$q\,\bar{q}$ candidates

_Exotic hadrons_ are all the mesonic states with quantum numbers not allowed
by the usual $q\,\bar{q}$ model. A good review of the physics of such states
can be found in [38, 39, 40] and references therein. In top-down AdS/QCD, the
work [41] addresses the exotic meson spectra for $Z_{c}$ and $Z_{b}$ in the
context of Sakai-Sugimoto models.

This section will focus on three tetraquark candidates: $Z_{c}$, $\psi$, and
$Z_{b}$ states. We will test diquark-antidiquark, hadroquarkonium, hadronic
molecule, and hybrid meson configurations. The first three configurations,
known as _multi-quark states_ , have $\Delta=6$. The last configuration, known
as _gluonic excitations_ , has $\Delta=5$ for one single gluonic tube and
$\Delta=7$ for two gluon tubes. All of these states are considered in
$s$-wave.

To give a parametric description of the hadronic constituent mass threshold
$\bar{m}_{eh}$, we will consider a generic exotic hadron with $N$ constituents
that can be quarks, gluons, or mesons. Each of these constituents will
contribute in a _weighted_ form to $\bar{m}_{eh}$ as [28]

$\bar{m}_{eh}=\sum_{i=1}^{N}({P^{\text{quark}}_{i}\,\bar{m}_{q_{i}}+P^{\text{Gluon}}_{i}\,m_{G_{i}}+P^{\text{meson}}_{i}\,m_{\text{meson}_{i}}}),$
(15)

along with the constrain condition

$\sum_{i=1}^{N}\,(P^{\text{quark}}_{i}+P^{\text{Gluon}}_{i}+P^{\text{meson}}_{i})=1.$
(16)

Notice that each weight $P_{i}$ measures the contribution of a given
constituent (quark, gluon, or meson) having a defined constituent mass. In our
case, we choose the following constituent masses

$\begin{gathered}m_{u}=0.336~{}\text{GeV},~{}m_{d}=0.340~{}\text{GeV}\\\
m_{c}=1.550~{}\text{GeV},~{}m_{b}=4.730~{}\text{GeV}\\\
m_{G}=0.7~{}\text{GeV},~{}m_{\rho(770)}=0.775~{}\text{GeV}\\\
m_{J/\psi}=3.097~{}\text{GeV},~{}m_{\Upsilon(1S)}=9.460~{}\text{GeV}.\end{gathered}$

Mesons masses are extracted from [42]. Gluon constituent mass is read from
[43].

####  II.3.1 Diquark configuration

_Diquarks_ are non-colored singlet objects used as essential building blocks
forming tetraquark mesons and pentaquark baryons. These fundamental blocks are
either a color anti-triplet or a color sextet in the SU(3) color
representation [44]. Spin-spin interactions bound these diquarks. The
constituent diquark approach helps describe the spectroscopy and decay of
multi-quark states. These diquark-composed candidates are expected to appear
as poles in the $S$-matrix, described by narrow widths.

Experimentally, charmonium and bottomonium tetraquark states can be identified
thanks to their decay into open-flavor states instead of a quarkonium with a
light meson due to the _spin-spin interaction dominance_ (See [38]).

In the case of charmonium, $Z_{c}$ states, with quantum numbers
$I^{G}(J^{CP})=1^{+}(1^{+-})$, are candidates to be vector tetraquarks.
Following Bambrilla, we consider the $Z_{c}$ states as a single trajectory.
Other studies, as [45], suggest $\psi(4230)$ with $0^{+}(1^{--})$ as a vector
tetraquark instead of $Z_{c}(4200)$. However, in this holographic picture,
considering both diquark-antidiquark structures implies the same threshold
mass, $\bar{m}_{4q}$, leading to degeneration in the mass.

Holographically, a hadron is conceived as a bag with $N$ constituents
characterized by $M_{5}$ despite its inner configuration. The holographic
approach to the tetraquark is equivalent to having four quarks with
$\Delta=3/2$, implying for the tetraquark $M_{5}^{2}\,R^{2}=15$.

For the threshold mass $\bar{m}_{4q}$, without any loss of generality, we can
assume that each $c$ quark contributes equally, i.e., $P_{c}=0.25$. Thus we
choose $\bar{m}_{4q}=m_{c}=1.55$ GeV as the threshold mass, implying that
$\kappa_{4q}(\bar{m})=2.151$ GeV and $\alpha_{4q}(\bar{m})=0.5387$. Table 1.

####  II.3.2 Hadroquarkonium

_Hadroquarkonium states_ are structures considering a vector meson core with a
_cloud_ of two quarks [46]. The experiments showed that most of the candidates
to be heavy exotic states appear as final states composed of heavy quarkonium
and light quarks, motivating the idea that these states are a compact heavy
quarkonium core surrounded by a light quark cloud [47]. This quarkonium core
interacts with the light quark cloud through a _colored Van der Waals force_ ,
allowing the decay of these states into the observed quarkonium core and the
light quarks [48].

At the holographic level, hadroquarkonium structures have the same behavior as
tetraquarks, i.e., $\Delta=6$ and $M_{5}^{2}\,R^{2}=15$, since essentially,
both models are a bag with four constituent quarks. The threshold mass settles
the difference. Thus, for the hadrocharmonium threshold mass, we have

$\displaystyle\bar{m}_{HQc}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{J/\psi}+\frac{1}{4}\,(\bar{m}_{u}+\bar{m}_{d})$
$\displaystyle=$ $\displaystyle 1.717~{}\text{GeV}.$

We will consider that states $\psi(4260)$, $\psi(4360)$ and $\psi(4660)$ with
$0^{+}(1^{--})$ are hadrocharmonium candidates [48], forming a vector
trajectory.

The summary of the holographic spectrum for $\psi$ states considered as
hadrocharmonium is given in Table 2.

####  II.3.3 Hadronic Molecule

_Hadronic molecules_ are states conformed by a pair of internal mesons bounded
by strong QCD forces, interacting between them via a residual weak QCD
colorless force [40]. These structures usually have two heavy quarkonia
interacting or one heavy quarkonium plus a light meson.

Recent works [39, 38] suggest that $Z_{c}$ or $\psi$ states are candidates to
be charmonium hadronic molecules, having at least one $c\,\bar{c}$ pair in the
molecule inner core. These states usually decay to $J/\psi\,\pi$.

On the holographic side, hadronic molecules have the same $\Delta$ and $M_{5}$
as tetraquarks and hadroquarkonium since the constituent content is
essentially the same. The degeneracy is broken by the threshold mass
$\bar{m}_{HM}$ that captures information about the inner structure. For these
charmonium hadronic molecules, we have

$\displaystyle\bar{m}_{HM,\,\psi}$ $\displaystyle=$
$\displaystyle\frac{1}{3}m_{J/\psi}+\frac{2}{3}m_{\rho}=1.549\,\text{GeV},$
$\displaystyle\bar{m}_{HM,\,Z_{c}}$ $\displaystyle=$ $\displaystyle
0.283\,m_{J/\psi}+0.717\,m_{\rho}=1.4323\,\text{GeV}.$

A similar situation occurs in the bottomonium sector [39], where $Z_{B}$
states, with $I^{G}(J^{CP})=1^{+}(1^{+-})$, are considered candidates for
hadronic molecules, with a $\Upsilon(1\,S)$ meson core. In this situation, the
threshold mass is

$\bar{m}_{HM,\,Z_{B}}=0.458\,m_{\Upsilon(1S)}+0.542\,m_{\rho}=4.753\,\text{GeV}.$
(17)

The summary of the holographic spectra for $Z_{c}$, $\psi$, and $Z_{c}$
states, considered as hadronic molecules, is given in Tables 1, 2, and 3
respectively.

####  II.3.4 Gluonic excitations

_Gluonic excitations_ are hadrons with constituent gluonic fields. Since QCD-
confined states are naturally non-perturbative, having constituent gluons
inside hadrons is unsurprising. In this category, we have glueballs $G\,G$,
hybrid mesons $q\,\bar{q}\,G$, and hybrid baryons $q\,q\,q\,G$, where $G$ is a
constituent degree of flavor. In other words, a color-octet $q\,\bar{q}$ pair
is color-neutralized by an excited gluon $G$. These hybrid configurations
allow the introduction of other quantum numbers that are impossible in the
quark constituent model, such as $J^{CP}=1^{+-}$ for hybrid vector mesons.

Holographically, in the case of vector hybrid mesons, we will consider mesons
with one gluon for $Z_{c}$ and two gluons for $Z_{b}$ candidates. In the case
of one constituent gluon, $\Delta=5$ implies $M_{5}^{2}\,R^{2}=8$.For two
gluons, $\Delta=7$ yields $M_{5}^{2}\,R^{2}=24$. The threshold masses for both
cases are the following

$\displaystyle\bar{m}_{Z_{c}}$ $\displaystyle=$ $\displaystyle
0.98\,m_{c}+0.2\,m_{G}=1.533\,\text{GeV}$ $\displaystyle\bar{m}_{Z_{b}}$
$\displaystyle=$ $\displaystyle 0.99\,m_{b}+0.01\,m_{G}=4.6897\,\text{GeV}$

where we have assumed $m_{q}=m_{\bar{q}}$ implicitly.

A summary of the mass spectrum for these hybrid meson candidates can be found
in Tables 1 for $Z_{c}$ and 3 for $Z_{b}$ states.

$Z_{c}$ states
---
Non-$q\bar{q}$ state | $Z_{c}(3900)$ |  $Z_{c}(4200)$) | $Z_{c}(4430)$
Exp. Masses (MeV) | $3887.1\pm 3.6$ | $4296^{+35}_{-32}$ | $4478^{+13}_{-18}$
Diquark-Antidiquark (MeV) $\bar{m}_{4Q}=1550$ MeV $\kappa=2151$ MeV and $\alpha=0.5387$ | $4004.8\,(3.0\%)$ | $4384.9\,(2.1\%)$ | $4706.6\,(5.1\%)$
Hadronic Molecule (MeV) $\bar{m}_{HM}=1432.3$ MeV $\kappa=1907$ MeV and $\alpha=0.4887$ | $3817.1\,(1.8\%)$ | $4214.7\,(1.9\%)$ | $4552.1\,(1.6\%)$
Hybrid Meson (MeV) $\bar{m}_{Z_{c}}=1533$ MeV $\kappa=2114.8$ MeV and $\alpha=0.5317$ | $3721.9\,(4.2\%)$ | $4156.4\,(3.2\%)$ | $4513.2\,(0.8\%)$
Table 1: Holographic mass spectrum for the $Z_{c}$ non-$q\bar{q}$ states
candidates according to their threshold mass $\bar{m}$. Parameters $\kappa$
and $\alpha$ are read from the expressions (4) and (5). In the case of $Z_{c}$
as a hybrid meson, we consider one constituent gluon, i.e., $\Delta=5$. The
quantity inside the parenthesis is the relative error. Experimental data come
from [42]. $\psi$ states
---
Non-$q\bar{q}$ state | $\psi(4230)$ |  $\psi(4360)$) | $\psi(4660)$
Exp. Masses (MeV) | $4222.5\pm 2.4$ | $4374\pm 7$ | $4630\pm 6$
Hadrocharmonium (MeV) $\bar{m}_{HQc}=1717$ MeV $\kappa=2522.6$ MeV and $\alpha=0.6036$ | $4222.5\,(0.2\%)$ | $4577.4\,(4.6\%)$ | $4871.8\,(5.2\%)$
Hadronic Molecule (MeV) $\bar{m}_{HM}=1549.1$ MeV $\kappa=2149.2$ MeV and $\alpha=0.5384$ | $4003.5\,(5.2\%)$ | $4383.7\,(0.2\%)$ | $4705.6\,(1.6\%)$
Table 2: Holographic mass spectrum for the $\psi$ (or $Y$) non-$q\bar{q}$
states candidates according to their threshold mass $\bar{m}$. Parameters
$\kappa$ and $\alpha$ are read from the expressions (4) and (5). The quantity
inside the parenthesis is the relative error. Experimental data come from
[42]. $Z_{b}$ states
---
Non-$q\bar{q}$ state | $Z_{b}(10610)$ |  $Z_{b}(10650)$)
Exp. Masses (MeV) | $10609\pm 6$ | $10652.2\pm 1.5$
Diquark-Antidiquark (MeV) $\bar{m}_{HM}=4753$ MeV $\kappa=11269.5$ MeV and $\alpha=0.8633$ | $10224.5\,(3.6\%)$ | $10517.6\,(1.3\%)$
Hybrid Meson (MeV) $\bar{m}_{Zb}=4689.7$ MeV $\kappa=11102.3$ MeV and $\alpha=0.8633$ | $10257.9\,(3.3\%)$ | $10512.5\,(1.3\%)$
Table 3: Holographic mass spectrum for the $Z_{b}$ non-$q\bar{q}$ states
candidates according to their threshold mass $\bar{m}$. Parameters $\kappa$
and $\alpha$ are read from the expressions (4) and (5). For $Z_{b}$ as a
hybrid meson, we consider two constituent gluons, i.e., $\Delta=7$. The
quantity inside the parenthesis is the relative error. Experimental data come
from [42].

##  III Diferential Configurational Entropy

Configurational Entropy (CE) constitutes a fundamental concept with far-
reaching implications across various scientific disciplines, including
statistical mechanics, information theory, and holography. This concept
encapsulates the diverse arrangements or microstates that a given macrostate
may embody. A higher CE value signifies a correspondingly greater multiplicity
of potential microstate configurations. From a thermodynamic standpoint, CE
pertains to the work performed by a system with a non-exchange of temperature.
Thus, CE could be helpful to infer when a given physical system, at zero
temperature, is stable: we will say that the configuration with less CE is
more stable than the others.

In information theory, CE bridges the informational content of physical
solutions to the equations of motion governing them. Notably, CE assumes the
role of a _logarithmic measure_ quantifying the spatial complexity inherent in
solutions localized within pre-defined energy boundaries. Conceptually, it
quantifies the informational content inherent in solutions corresponding to a
specified set of e.o.m. Consequently, CE may be construed as assessing the
information required to comprehensively characterize functions localized
within a given parameter space. In a broader context, dynamic solutions emerge
from the extremization of actions, while CE functions as a metric capturing
the underlying information content within these dynamically evolving states.

In the domain of discrete variables, CE is rigorously articulated as an
application of Shannon entropy, succinctly formulated as [49, 50, 51]:

$S_{C}=-\sum_{n}\,p_{n}\,\text{log}\,p_{n}.$ (18)

In the transition to continuous variables, the concept of Differential
Configurational Entropy (DCE) emerges, characterized by:

$S_{C}\left[f\right]=-\int{d^{d}\,k\,\tilde{f}\left(k\right)\,\text{log}\,\tilde{f}\left(k\right)},$
(19)

where $\tilde{f}\left(k\right)=f\left(k\right)/f\left(k\right)_{\text{Max}}$.
Here, $f(k)_{\text{Max}}$ represents the maximum value assumed by $f(k)$, and
$f(k)$ itself belongs to the square-integrable space
$L^{2}\left(\mathbb{R}^{2}\right)$, ensuring its Fourier transformability.
Mathematically, $f(k)$ corresponds to the energy density within momentum
space, denoted as $\rho(k)$.

In the context of AdS/CFT duality, the application of CE finds fertile ground
within both bottom-up and top-down AdS/QCD models, as initially delineated by
[52]. The versatility of CE becomes apparent in its applicability to a
spectrum of physical scenarios, spanning from hadronic states to heavy
quarkonium stability under varying thermal, magnetic, and density conditions.
Furthermore, CE’s utility extends to investigating the holographic
deconfinement phase transition, as elucidated in [53, 54, 55, 56], and
recently in the exploration of holographic stability in light nuclides [57].

Within this discourse, the holographic dictionary serves as a central
construct, enabling the translation of spatial configurations of boundary
particles into the holographic configuration of the dual bulk field. This
process encapsulates the information encoded in the arrangement of
constituents within a hadron, briefly represented by the energy density
inherent in the temporal component of the energy-momentum tensor, denoted as
$\rho(z)\equiv T_{00}(z)$.

The procedure to compute CE, as delineated by [52, 57], emanates from the on-
shell bulk action, yielding the energy-momentum tensor as

$T_{mn}=\frac{2}{\sqrt{-g}}\,\frac{\partial\left[\sqrt{-g}\,\mathcal{L}_{\text{Hadron}}\right]}{\partial\,g^{mn}}.$
(20)

From the holographic potential $V(z)$ (14) and inverting the transformation
$\psi(z)=e^{B(z)/2}\,\phi(z)$, it is possible to obtain the Sturm-Liouville
modes to feed up the on-shell energy-momentum tensor

$\rho(z)\equiv
T_{00}=\frac{e^{-B\left(z\right)}}{2}\left(\frac{z}{R}\right)^{3}\times\\\
\left\\{\left[\frac{1}{\mathcal{K}^{2}}\left(M_{n}^{2}\,\psi_{n}^{2}+\psi_{n}^{\prime
2}\right)-\frac{M_{5}^{2}\,R^{2}}{z^{2}}\psi_{n}^{2}\right]\right\\}\,\Omega,$
(21)

The next step in the recipe for the CE is to define the _modal fraction_. To
do so is necessary to compute a Fourier-transformed representation of $\rho$,
i.e., $\bar{\rho}(k)$. Thus, the modal fraction emerges as

$f(k)=\frac{|\bar{\rho}(k)|^{2}}{\int dk|\bar{\rho}(k)|^{2}}.$ (22)

The differential configurational entropy (DCE) for the non-$q\,\bar{q}$ state
is written from the modal fraction as

$S_{DCE}=-\int dk\,\tilde{f}(k)\log\,\tilde{f}(k)$ (23)

where $\tilde{f}\left(k\right)=f\left(k\right)/f\left(k\right)_{\text{Max}}$.
Notice that we normalize the modal fraction with
$f\left(k\right)_{\text{Max}}$.

In the following sections, we will use this tool to explore the stability of
tetraquark candidates from their holographic duals constructed in the non-
quadratic SWM.

##  IV Configurational entropy and hadron stability

Since the first applications of DCE to AdS/QCD, the connection between hadron
stability and DCE has been open since there is no apparent direct connection
from fundamental grounds.

Stability in hadrons is an issue following confinement. Once constituents are
bounded in a hadronic structure, it is natural to wonder if such a structure
is stable enough (in time) to be considered a hadronic state.

The connection with configurational entropy arises with the locality as a
synonym for confinement: a QCD-bounded state of constituent quarks and gluons
has highly well-spatially localized wave functions for these constituents.
Regarding configurational entropy, these systems are more stable (less CE)
than delocalized ones. In other words, confinement can be understood as a
transition from localization to delocalization in space.

Let us turn these ideas to the AdS/QCD realm. Mesons in bottom-up models
appear from the field/operator duality: operators creating hadrons at the
boundary are dual to bulk fields representing these hadrons.

The connection is done with the operator dimension set as the bulk field
scaling dimension (minus the spin), $\Delta$ [58, 59]. This information
defines the bulk field mass $M_{5}$. Thus, for a given hadron, defined by an
operator $\mathcal{O}$ with dimension $\Delta$ at the boundary, it will be
dual to a bulk field with mass $M_{5}^{2}\,R^{2}=(\Delta-S)(\Delta+S-4)$. In
other terms, _a hadron in bottom-up models is a bag with $N$ constituents
characterized by $M_{5}^{2}$_. Any information about the hadronic inner
structure is not present _ab initio_.

This map between boundary and bulk physics also implies that normalizable bulk
modes are dual to mesons at the boundary. From the AdS/CFT correspondence
ground, this statement conflicts with field/operator duality since spatially
localized wave functions at the boundary should be delocalized at the bulk due
to the IR/UV behavior. However, this holds for $\mathcal{N}=4$ SYM theory dual
to a Type IIB supergravity, where bulk states are unbounded [60]. Following
[61, 62], one of the consequences of lifting the bulk conformal invariance (by
placing a hard cutoff or using a dilaton field) is precisely this
reinterpretation of the operator/field duality where extended and localized
objects at the boundary are dual to localized objects at the bulk. This
locality in the bulk objects seems natural in non-conformal theories [63].
This idea supports computing hadronic properties (mass spectrum, form factors,
decay constants, thermal densities, etc.) in bottom-up models.

Therefore, for bottom-up models (or models where non-conformality exists),
local operators at the boundary can have local dual objects at the bulk. This
affirmation connects with the configurational entropy directly. Thus, it is
correct to infer that _locality at the bulk (involving lower configurational
entropy) implies stability at the boundary_.

This hypothesis is strongly supported by the thermal or chemical analysis for
these bottom-up models [51, 64, 65], where the melting criterion is dictated
by when the spectral peak disappears. This peak broadening can be understood
as a thermal (chemical) delocalization of the bulk quasinormal modes.

Figure 1: Differential configurational entropy, in natural entropy units, as a
function of excitation number for non-$q\,\bar{q}$ candidates. The left-upper
panel plots the $Z_{c}$ holographic comparison between diquark, hadronic
molecule, and exotic gluon structure. We present an analysis of $\psi$ or
($Y$) with hadronic molecule and hadrocharmonium structures in the right
panel. The bottom panel presents the DCE for gluonic excitation (with two
gluons) and hadronic molecule structures for $Z_{b}$ states. From DCE grounds,
those structures with less entropy are the most stable and, therefore,
preferred.

##  V DCE for non-$q\,\bar{q}$ states

We can discuss stability further once we have established that normalizable
modes mimic hadrons at the boundary in locality terms. One of the main
features of the non-quadratic dilaton is the inclusion of constituent mass
dependence. Inspired by the well-known [33] fact that constituent mass affects
Regge trajectory concavity, i.e., linearity, this dilaton encloses the mass
behavior in the quadratic deformation parameter $\alpha$. As in the original
softwall proposal [1], the slope $\kappa$ is still flavor dependent. However,
this slope also runs with the constituent mass threshold $\bar{m}$. In
practical terms, despite having two parameters at hand, we only have to deal
with the mass threshold. Thus, at this point, we have increased the hadronic
information encoded into the model since we have structural parameters
included.

In the original non-quadratic softwall work [28], the holographic prediction
of which non-$q\bar{q}$ structure was focused on minimizing the RMS error in
the mass spectrum. Following this prescription, for $Z_{c}$, the lowest RMS
error (1.5%) is achieved in the hadronic molecule case. For $\psi$, the lowest
error (2.7%) is for the hadronic molecule case. And, for $Z_{b}$, the lowest
RMS error (3.5%) appears for hybrid meson (with two gluon flux tubes). Let us
compare this criterion with the DCE analysis.

In the case of DCE, even though multiquark states for vector hadrons have the
same bulk mass, the non-quadratic dilaton provides a mechanism to distinguish
between structures through the threshold mass. This constituent mass
parametrization allows extrapolating a pair $(\kappa,\alpha)$ for each
structure. Tables 1, 2, and 3 summarize the extrapolated values for each
structure, according to the non-$q\,\bar{q}$ candidate.

The DCE analysis calculates the configurational entropy for each structure
using the procedure exposed in section III. Following the idea that bulk modes
capture locality information about hadron modes at the boundary, the
trajectory with less DCE will be the most stable. This analysis differs
entirely from the one done in [30]. In this work, authors predicted the heavy
non-$q\,\bar{q}$ next generation by constructing Regge trajectories inspired
by the DCE behavior of the associated bulk modes. They also discussed the
possible existence of excited non-$q\,\bar{q}$ states from DCE holographic
grounds, arguing that the lower production of these highly excited states
agrees with the DCE holographic result, interpreted as hadronic information
compressed, according to Shannon’s information theory. Thus, only lower
excitation modes are allowed and expected to dominate experiments.

The analysis proposed in this work deals with the structural stability encoded
in the threshold mass $\bar{m}$. Thus, we will compare the entropic effect of
one parametrization over another for each family of states. The results of
this analysis are summarized in figure 1.

For $Z_{c}$ states, we compare three possible structures: diquark-antidiquark,
hadronic molecule (multiquark states), and hybrid meson with one gluonic flux
tube. For the ground state, the lowest DCE is achieved by the diquark
structure. However, hybrid mesons (one exotic gluon flux tube) become more
stable for excited states. Performing a quadrature analysis, i.e., looking for
the smallest total DCE per structure, we found that the most stable
configuration is the hybrid meson.

We compare two multiquark structures in the $\psi$ states: hadronic molecule
and hadrocharmonium. In this scenario, the smallest DCE is observed for
hadrocharmonium.

Finally, for the $Z_{b}$ states, following hadron phenomenology[38], we tested
the hybrid meson (with two gluon tubes) and the hadronic molecule structure.
We found the lowest DCE and, therefore, the most stable structure for the
hadronic molecule.

The DCE holographic analysis brings different results from the pure
spectroscopic analysis, which does not account for the configurational
information encoded into the threshold mass.

##  VI Conclusions

As the vast and extensive literature demonstrates, in recent years,
configurational entropy has proven to be an interesting tool to address in
holographic models [29, 30, 55, 56, 54, 66, 67, 68]. This work explored the
DCE to analyze stability when information regarding the hadronic inner
structure is provided. In parallel, we also discussed the connection between
hadronic stability and configurational entropy. Even though thermodynamically
speaking, CE does not account for processes where energy transfers change the
configuration of a system. It can provide insights into configurational
stability. Later, we can complement this discussion with thermal or chemical
approaches to the concept.

Speaking about hadronic stability is equivalent to talking about wave-function
locality. This discussion stretches the hypothesis that normalizable modes
living in the AdS bulk are dual to meson modes. This assumption is supported
by slightly breaking the bulk conformal invariance, inducing bounded KK-towers
dual to hadronic Regge trajectories. In bottom-up models, this is achieved by
introducing a dilaton field. Thus, well-localized (bounded) bulk modes imply
localized and confined objects at the boundary.

In this context, we found that, as a holographic prediction for heavy
non-$q\,\bar{q}$ hadrons, the $Z_{c}$ states should be cataloged as _hybrid
meson_ , $\psi$ states as _hadrocharmonium_ , and $Z_{b}$ as _hadronic
molecule_.

## References

  * Karch  _et al._ [2006] A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. D 74, 015005 (2006), arXiv:hep-ph/0602229 .
  * Grigoryan and Radyushkin [2007] H. R. Grigoryan and A. V. Radyushkin, Phys. Rev. D 76, 095007 (2007), arXiv:0706.1543 [hep-ph] .
  * Abidin and Carlson [2009] Z. Abidin and C. E. Carlson, Phys. Rev. D 79, 115003 (2009), arXiv:0903.4818 [hep-ph] .
  * Herzog [2007] C. P. Herzog, Phys. Rev. Lett. 98, 091601 (2007), arXiv:hep-th/0608151 .
  * Cai and Shock [2007] R.-G. Cai and J. P. Shock, JHEP 08, 095 (2007), arXiv:0705.3388 [hep-th] .
  * Afonin and Katanaeva [2018] S. S. Afonin and A. D. Katanaeva, Phys. Rev. D 98, 114027 (2018), arXiv:1809.07730 [hep-ph] .
  * Ballon-Bayona  _et al._ [2021] A. Ballon-Bayona, L. A. H. Mamani, and D. M. Rodrigues, Phys. Rev. D 104, 126029 (2021), arXiv:2107.10983 [hep-ph] .
  * Ballon Bayona  _et al._ [2008] C. A. Ballon Bayona, H. Boschi-Filho, and N. R. F. Braga, JHEP 03, 064 (2008), arXiv:0711.0221 [hep-th] .
  * Braga and Vega [2012] N. R. F. Braga and A. Vega, Eur. Phys. J. C 72, 2236 (2012), arXiv:1110.2548 [hep-ph] .
  * Vega  _et al._ [2011] A. Vega, I. Schmidt, T. Gutsche, and V. E. Lyubovitskij, Phys. Rev. D 83, 036001 (2011), arXiv:1010.2815 [hep-ph] .
  * Colangelo  _et al._ [2008] P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau, and S. Nicotri, Phys. Rev. D 78, 055009 (2008), arXiv:0807.1054 [hep-ph] .
  * Vega and Schmidt [2008] A. Vega and I. Schmidt, Phys. Rev. D 78, 017703 (2008), arXiv:0806.2267 [hep-ph] .
  * Vega and Schmidt [2009] A. Vega and I. Schmidt, Phys. Rev. D 79, 055003 (2009), arXiv:0811.4638 [hep-ph] .
  * Branz  _et al._ [2010] T. Branz, T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, Phys. Rev. D 82, 074022 (2010), arXiv:1008.0268 [hep-ph] .
  * Gutsche  _et al._ [2013] T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, Phys. Rev. D 87, 016017 (2013), arXiv:1212.6252 [hep-ph] .
  * Batell and Gherghetta [2008] B. Batell and T. Gherghetta, Phys. Rev. D 78, 026002 (2008), arXiv:0801.4383 [hep-ph] .
  * Li  _et al._ [2013] D. Li, M. Huang, and Q.-S. Yan, Eur. Phys. J. C 73, 2615 (2013), arXiv:1206.2824 [hep-th] .
  * Li and Huang [2013] D. Li and M. Huang, JHEP 11, 088 (2013), arXiv:1303.6929 [hep-ph] .
  * Colangelo  _et al._ [2011] P. Colangelo, F. Giannuzzi, and S. Nicotri, Phys. Rev. D 83, 035015 (2011), arXiv:1008.3116 [hep-ph] .
  * Sachan and Siwach [2012] S. Sachan and S. Siwach, Mod. Phys. Lett. A 27, 1250163 (2012), arXiv:1109.5523 [hep-th] .
  * Kim  _et al._ [2007] Y. Kim, J.-P. Lee, and S. H. Lee, Phys. Rev. D 75, 114008 (2007), arXiv:hep-ph/0703172 .
  * Fujita  _et al._ [2010] M. Fujita, T. Kikuchi, K. Fukushima, T. Misumi, and M. Murata, Phys. Rev. D 81, 065024 (2010), arXiv:0911.2298 [hep-ph] .
  * Fujita  _et al._ [2009] M. Fujita, K. Fukushima, T. Misumi, and M. Murata, Phys. Rev. D 80, 035001 (2009), arXiv:0903.2316 [hep-ph] .
  * Dudal  _et al._ [2016] D. Dudal, D. R. Granado, and T. G. Mertens, Phys. Rev. D 93, 125004 (2016), arXiv:1511.04042 [hep-th] .
  * Li  _et al._ [2017] D. Li, M. Huang, Y. Yang, and P.-H. Yuan, JHEP 02, 030 (2017), arXiv:1610.04618 [hep-th] .
  * Fang  _et al._ [2021] Z. Fang, Y.-Y. Li, and Y.-L. Wu, Eur. Phys. J. C 81, 545 (2021).
  * Chen [2018a] J.-K. Chen, Eur. Phys. J. C 78, 648 (2018a).
  * Martin Contreras and Vega [2020] M. A. Martin Contreras and A. Vega, Phys. Rev. D 102, 046007 (2020), arXiv:2004.10286 [hep-ph] .
  * Colangelo and Loparco [2019] P. Colangelo and F. Loparco, Phys. Lett. B 788, 500 (2019), arXiv:1811.05272 [hep-ph] .
  * Karapetyan and da Rocha [2021] G. Karapetyan and R. da Rocha, Eur. Phys. J. Plus 136, 993 (2021), arXiv:2103.10863 [hep-ph] .
  * Chen [2018b] J.-K. Chen, Phys. Lett. B 786, 477 (2018b), arXiv:1807.11003 [hep-ph] .
  * Afonin and Pusenkov [2014] S. S. Afonin and I. V. Pusenkov, Phys. Rev. D 90, 094020 (2014), arXiv:1411.2390 [hep-ph] .
  * Chen [2018c] J.-K. Chen, Eur. Phys. J. C 78, 235 (2018c).
  * Braga  _et al._ [2016] N. R. F. Braga, M. A. Martin Contreras, and S. Diles, EPL 115, 31002 (2016), arXiv:1511.06373 [hep-th] .
  * Gursoy and Kiritsis [2008] U. Gursoy and E. Kiritsis, JHEP 02, 032 (2008), arXiv:0707.1324 [hep-th] .
  * Cherman  _et al._ [2009] A. Cherman, T. D. Cohen, and E. S. Werbos, Phys. Rev. C 79, 045203 (2009), arXiv:0804.1096 [hep-ph] .
  * Boschi-Filho  _et al._ [2013] H. Boschi-Filho, N. R. F. Braga, F. Jugeau, and M. A. C. Torres, Eur. Phys. J. C 73, 2540 (2013), arXiv:1208.2291 [hep-th] .
  * Brambilla  _et al._ [2019] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, (2019), arXiv:1907.07583 [hep-ex] .
  * Guo  _et al._ [2018] F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. 90, 015004 (2018), arXiv:1705.00141 [hep-ph] .
  * Lebed  _et al._ [2017] R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017), arXiv:1610.04528 [hep-ph] .
  * Liu  _et al._ [2019a] Y. Liu, M. A. Nowak, and I. Zahed, Phys. Rev. D 100, 126023 (2019a).
  * Workman  _et al._ [2022] R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Hou  _et al._ [2001] W.-S. Hou, C.-S. Luo, and G.-G. Wong, Phys. Rev. D64, 014028 (2001), arXiv:hep-ph/0101146 [hep-ph] .
  * Jaffe [2005] R. L. Jaffe,  _Proceedings, 6th International Conference on Hyperons, charm and beauty hadrons (BEACH 2004): Chicago, USA, June 27-July 3, 2004_ , Phys. Rept. 409, 1 (2005), [,191(2004)], arXiv:hep-ph/0409065 [hep-ph] .
  * Wang [2018] Z.-G. Wang, Eur. Phys. J. C78, 933 (2018), arXiv:1809.10299 [hep-ph] .
  * Liu  _et al._ [2019b] Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Prog. Part. Nucl. Phys. 107, 237 (2019b), arXiv:1903.11976 [hep-ph] .
  * Voloshin [2008] M. B. Voloshin, Prog. Part. Nucl. Phys. 61, 455 (2008), arXiv:0711.4556 [hep-ph] .
  * Brambilla  _et al._ [2017] N. Brambilla, V. Shtabovenko, J. Tarrús Castellà, and A. Vairo, Phys. Rev. D95, 116004 (2017), arXiv:1704.03476 [hep-ph] .
  * Gleiser and Stamatopoulos [2012] M. Gleiser and N. Stamatopoulos, Phys. Rev. D 86, 045004 (2012), arXiv:1205.3061 [hep-th] .
  * Bernardini  _et al._ [2017] A. E. Bernardini, N. R. F. Braga, and R. da Rocha, Phys. Lett. B 765, 81 (2017), arXiv:1609.01258 [hep-th] .
  * Braga and da Rocha [2017] N. R. F. Braga and R. da Rocha, Phys. Lett. B 767, 386 (2017), arXiv:1612.03289 [hep-th] .
  * Bernardini and da Rocha [2016] A. E. Bernardini and R. da Rocha, Phys. Lett. B 762, 107 (2016), arXiv:1605.00294 [hep-th] .
  * Braga  _et al._ [2018] N. R. F. Braga, L. F. Ferreira, and R. a. Da Rocha, Phys. Lett. B 787, 16 (2018), arXiv:1808.10499 [hep-ph] .
  * Braga and da Mata [2020a] N. R. F. Braga and R. da Mata, Phys. Lett. B 811, 135918 (2020a), arXiv:2008.10457 [hep-th] .
  * Braga and da Mata [2020b] N. R. F. Braga and R. da Mata, Phys. Rev. D 101, 105016 (2020b), arXiv:2002.09413 [hep-th] .
  * Braga and Junqueira [2021] N. R. F. Braga and O. C. Junqueira, Phys. Lett. B 814, 136082 (2021), arXiv:2010.00714 [hep-th] .
  * Martin Contreras  _et al._ [2022] M. A. Martin Contreras, A. Vega, and S. Diles, Phys. Lett. B 835, 137551 (2022), arXiv:2206.01834 [hep-ph] .
  * Polchinski and Strassler [2002] J. Polchinski and M. J. Strassler, Phys. Rev. Lett. 88, 031601 (2002), arXiv:hep-th/0109174 .
  * Balasubramanian  _et al._ [1999] V. Balasubramanian, P. Kraus, A. E. Lawrence, and S. P. Trivedi, Phys. Rev. D 59, 104021 (1999), arXiv:hep-th/9808017 .
  * Hamilton  _et al._ [2006] A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, Phys. Rev. D 74, 066009 (2006), arXiv:hep-th/0606141 .
  * Rehren [2000] K.-H. Rehren, Phys. Lett. B 493, 383 (2000), arXiv:hep-th/0003120 .
  * Banks  _et al._ [1998] T. Banks, M. R. Douglas, G. T. Horowitz, and E. J. Martinec, (1998), arXiv:hep-th/9808016 .
  * Bena [2000] I. Bena, Phys. Rev. D 62, 066007 (2000), arXiv:hep-th/9905186 .
  * Braga  _et al._ [2017] N. R. F. Braga, L. F. Ferreira, and A. Vega, Phys. Lett. B 774, 476 (2017), arXiv:1709.05326 [hep-ph] .
  * Martin Contreras  _et al._ [2021] M. A. Martin Contreras, S. Diles, and A. Vega, Phys. Rev. D 103, 086008 (2021), arXiv:2101.06212 [hep-ph] .
  * Karapetyan [2022] G. Karapetyan, Eur. Phys. J. Plus 137, 590 (2022), arXiv:2112.11359 [nucl-th] .
  * Barreto and da Rocha [2022] W. Barreto and R. da Rocha, Phys. Rev. D 105, 064049 (2022), arXiv:2201.08324 [hep-th] .
  * Martin Contreras  _et al._ [2023] M. A. Martin Contreras, A. Vega, and S. Diles, (2023), arXiv:2308.16007 [hep-ph] .