# Non-linear Regge trajectories with AdS/QCD

Miguel Angel Martin Contreras<EMAIL_ADDRESS>Instituto de Física y
Astronomía,
Universidad de Valparaíso,
A. Gran Bretaña 1111, Valparaíso, Chile  Alfredo Vega<EMAIL_ADDRESS>Instituto de Física y Astronomía,
Universidad de Valparaíso,
A. Gran Bretaña 1111, Valparaíso, Chile

(August 27, 2024)

###### Abstract

In this work, we consider a non-quadratic dilaton
$\Phi(z)=(\kappa\,z)^{2-\alpha}$ in the context of the static soft wall model
to describe the mass spectrum of a wide range of vector mesons from the light
up to the heavy sectors. The effect of this non-quadratic approach is
translated into non-linear Regge trajectories with the generic form
$M^{2}=a\,(n+b)^{\nu}$. We apply this sort of fits for the isovector states of
$\omega$, $\phi$, $J/\psi$, and $\Upsilon$ mesons and compare with the
corresponding holographic duals. We also extend these ideas to the heavy-light
sector by using the isovector set of parameters to extrapolate the proper
values of $\kappa$ and $\alpha$ through the average constituent mass $\bar{m}$
for each mesonic specie considered. In the same direction, we address the
description of possible non-$q\,\bar{q}$ candidates using $\bar{m}$ as a
holographic threshold, associated with the structure of the exotic state, to
define the values of $\kappa$ and $\alpha$. We study the $\pi_{1}$ mesons in
the light sector and the $Z_{c}$, $Y$, and $Z_{b}$ mesons in the heavy sector
as possible exotic vector states. Finally, the RMS error for describing these
twenty-seven states with fifteen parameters (four values for $\kappa$ and
$\alpha$ respectively and seven values for $\bar{m}$) is $12.61\%$.

††preprint: APS/123-QED

##  I Introduction

Nowadays, there is no doubt that hadrons are bound states of quarks and
gluons, whose interactions are described by _quantum chromodynamics_ (QCD).
This quantum field theory is endowed with a coupling constant that controls
the energy of the hadronic processes. At high energies, the smallness of the
coupling constant makes the theory perturbative. On the other hand, the low
energy behavior is non-perturbative. It is precisely in the latter regime
where several hadronic properties are found. Also, the developed perturbative
theoretical tools are insufficient to describe this particular hadronic
physics. This issue motivated the development of techniques and tools that
allows the direct use of QCD in the study of hadrons, such as Lattice QCD
(e.g.Aoki et al. (2014)) or the use of the Dyson Schwinger equations to study
hadrons (e.g.Roberts and Williams (1994)).

This picture has also prompted the development of phenomenological models
inspired by QCD, capturing important properties of the interaction between
quarks and gluons, offering us alternatives to perform calculations of
hadronic properties.

A successful example of phenomenological models for the study of hadrons is
the so-called _quark potential models_ Lucha et al. (1991); Bykov et al.
(1984); Dib and Neill (2012); Rai et al. (2008); Patel and Vinodkumar (2009),
which have been remained valid since the middle seventies when the first heavy
quark mesons, the $J/\psi$ meson was observed. In this approach, the
Schrodinger equation, with a potential describing the interaction between
constituent heavy quarks inside the meson, provides good results describing
the mesonic spectra and other properties related to the hadronic wave
function, such as the decay constants Van Royen and Weisskopf (1967).

From the QCD point of view, it is possible to infer the behavior of the
potential when the constituent quarks are close or far between them. In the
former case, the large $Q^{2}$ limit, the coupling constant is small enough,
allowing to use perturbative techniques to describe the quark interaction by
considering the one-gluon exchange only. As a result, the potential is found
to be Coulomb-like in this limit. On the other case, In the long-inter-quark
distance. or small $Q^{2}$ limit, the strong coupling constant becomes large,
preventing any perturbative machinery. In this case, quarks are considered as
_confined partons_. This part of the potential cannot be explored by
analytical QFT methods. But, extensive developments in _lattice QCD_ proved
that this term seems to be linear Kawanai and Sasaki (2015). Similar results
were found on the holographic side, where the dictionary establishes that a
closed string world-sheet is dual to the Wilson loop, which accounts for
confinement on the boundary theory Andreev and Zakharov (2006); White (2007);
Jugeau (2010).

Summarizing, today we know that the constituent quark interaction potential is
well-motivated from QCD: it must interpolate between a Coulomb-like potential
at short distances and a linear-like potential at long distances. In order to
fit this phenomenological suggestion, several alternatives have been proposed
(See Lucha et al. (1991); Bykov et al. (1984); Dib and Neill (2012)). The
simplest realization of these sorts of ideas, giving excellent results, is
just the sum of both contributions.

Another succesfull possibility of building phenomenological models is to study
hadron properties using _gauge/gravity correspondence_. Namely the so-called
bottom-up _AdS/QCD_ approach allows us to calculate hadronic properties by
capturing the main strong interaction features of hadrons in different mediums
in an 5-dimensional AdS-like metric tensor and other background fields, as the
dilaton.

Isovector meson masses (PDG)
---
$n$ |  $\omega$ (MeV) |  $\phi$ (MeV) |  $\psi$ (MeV) |  $\Upsilon$ (MeV)
1 | $782.65\pm 0.12$ | $1019.461\pm 0.016$ | $3096.916\pm 0.011$ | $9460.3\pm 0.26$
2 | $1.400-1450$ | $1698\pm 20$ | $3686.109\pm 0.012$ | $10023.26\pm 0.32$
3 | $1670\pm 30$ | $2135\pm 8\pm 9$ | $4039\pm 1$ | $10355\pm 0.5$
4 | $1960\pm 25$ | $---$ | $4421\pm 4$ | $10579.4\pm 1.2$
5 | $2290\pm 20$ | $---$ | $---$ | $10889.9^{+3.2}_{-2.6}$
6 | $---$ | $---$ | $---$ | $10992.9^{+10.0}_{-3.1}$
Table 1: This table summarises the experimental masses (Tanabashi et al.
(2018)) for isovector mesons families consisting on $\omega$, $\phi$, $\psi$
and $\Upsilon$ radial states.

In these bottom-up sorts of models, a dilaton field is used to induce
confinement on the dual boundary theory. If the dilaton considered is static
and quadratic Karch et al. (2006), confinement is manifest by the appearance
of linear Regge trajectories in the mesonic sector. Further works consider
other possible forms of the dilaton field that interpolates the quadratic
dilaton at high $z$, keeping linear trajectories (for higher quantum numbers)
and allowing the study of other phenomena as chiral symmetry breaking. Other
approaches in the AdS/QCD context includes deforming the AdS background as it
was done in Folco Capossoli et al. (2019) or the use of a D$p$/D$q$ background
with a static quadratic dilaton He et al. (2010) to induce linear confinement.

Linear Regge trajectories are a good description of the mesonic mass spectra
in the light sector, and this has been used traditionally as a guideline in
order to catch hadron properties in the AdS side of AdS/QCD models. But if
hadrons contain $s$ or heavy quarks, linearity in trajectory is lost Afonin
and Pusenkov (2014); Chen (2018a, b, c); Gershtein et al. (2006). Moreover,
starting from the quadratic form of Bethe Salpeter equation Chen (2018c);
Anisovich et al. (2004) , it is possible to infer that mesons consisting of
heavier quarks have Regge trajectories deviated from the linearity. In this
sort of analysis, heavy quarkonium is expected to have radial trajectories
scaling the excitation number as $n^{2/3}$. This feature suggests that the
linear behavior of the Regge trajectory should be dependent on the quark
constituent mass, implying linearity for the light flavored mesons and non-
linearity for the heavier ones. We will explore this hypothesis in this work.

For this reason, we explore other kind of dilatons in order to describe
hadrons where linear Regge trajectories disagree with experimental data. This
could be interesting at moment to study, for example, heavy mesons in
holographic models, because as it can be seen in literature Afonin and
Pusenkov (2013); Braga et al. (2016a, b, c); Braga and Ferreira (2017a); Braga
et al. (2017), AdS/QCD models applied to charmonium or bottomonium spectra are
no so good enough to describe them, despite the fact that other observables
(as the melting temperature) have the proper qualitative behaviour.

| Linear Regge Trajectory: $M^{2}=a(n+b)$ | Non Linear Regge Trajectory ($M^{2}=a(n+b)^{\nu}$)
---|---|---
Meson | a | b | $R^{2}$ | a | b | $\nu$ | $R^{2}$
$\omega$ | $1.1074$ | $-0.3781$ | $0.9978$ | $1.1078$ | $-0.3784$ | $0.9998$ | $0.9978$
$\phi$ | $1.7595$ | $-0.4048$ | $0.9999$ | $1.8545$ | $-0.4524$ | $0.9617$ | $1.000$
$\psi$ | $3.2607$ | $2.0259$ | $0.9997$ | $7.6516$ | $0.4460$ | $0.6249$ | $0.9999$
$\Upsilon$ | $6.2015$ | $13.9182$ | $0.9996$ | $85.3116$ | $0.2849$ | $0.1917$ | $0.9999$
Table 2: Summary of linear and non linear fits for isovector meson Regge
trajectories drawn in Fig. 1. We expose parameters for each parametrization
considered, altogether with the correlation coefficient $R^{2}$. Observe that
linear fits bring good description of the trajectories, but $R^{2}$ decrease
from unity when we increase the quark constituent mass. Also notice that the
non-linear fit is more precise since $R^{2}$ is bigger than the linear one in
each case.
---
Figure 1: This plot shows $M^{2}$ vs $n$ for different vector mesons
($\omega,\phi,\psi$ and $\Upsilon$s). Dots represent experimental data, and in
each panel there are two continuous lines, one represent the best linear fit
($M^{2}=a(n+b)$) and the other corresponding to a non linear fit
($M^{2}=a(n+b)^{\nu}$).

This work has been structured as follows: in section II we consider four
families of isovector mesons with different constituent quarks and we show
that these mass spectra agree with a non-linear Regge trajectory, parametrized
by $M^{2}=a(n+b)^{\nu}$ inspired by the parametrization suggested in Chen
(2018b), where a primer interpretation for the exponent $\nu$ is to account
for the linearity deviation in the radial trajectory. Notice that this
suggested parametrization is far different from the ones proposed in the
context of quantum corrections to the string formulation Kruczenski et al.
(2005), and is also different from the ones suggested in Sergeenko (1994);
Gershtein et al. (2006). We associate the index $\nu$ with the average
constituent quark mass in each case, and then we propose an expression for
this index.

In section III we review holographic recipe to describe mesonic masses. In the
standard AdS/QCD scheme, the energy scale associated with the dilaton field,
which could be static or dynamical, defines the slope in the Regge trajectory.
Moreover, the _hadronic identity_ of the given hadronic state is defined by
the bulk mass of the corresponding dual bulk field. This bulk mass carries
information about the scaling information of the operators that define hadrons
at the conformal boundary. Beyond these two parameters, there is no other bulk
quantity suitable to define the mesonic state at hand. This fact sets a
drawback in the formulation. For instance, in the holographic Regge trajectory
associated to vector mesons written in the soft-wall model context, i.e.,
$M_{n}^{2}=4\,\kappa^{2}(n+1)$, it is not possible to differentiate among the
elements in this nonet by holographic means. In other words, the trajectory
written above could be for $\rho$ or $\omega$ mesons if we fix $\kappa$ in the
light unflavored sector. This issue can be translated into the fact that
AdS/QCD models do not deal with the mesonic inner structure directly. A
possible form to circumvent this issue is to consider the effect of quark
constituent masses in the Regge trajectory. According to the Bethe-Salpeter
analysis, constituent quark masses induces non-linearities to the trajectory.
Therefore, we conjecture that the linearity deviation in Regge trajectories is
associated, in the AdS side, with the deviation of the dilaton field from the
quadratic profile proposed in Karch et al. (2006) We propose a dilaton
deformation of the form $z^{2-\alpha}$, where $\alpha$ encodes the effect of
the average constituent quark mass on the Regge trajectory. In section IV, we
apply these ideas to the description of radial isovector states ($\omega$,
$\phi$, J$/\psi$ and $\Upsilon$), with quantum numbers defined as
$I^{G}\,J^{PC}=0^{-}(1^{--})$. We will use this fit to establish how the
parameters $\kappa$ and $\alpha$ run with the constituent mass. This will
allow us test this approximation with other mesonic species by using their
constituent configuration as an entry.

In section V we used the isovector non-linear trajectories fitted to
extrapolate the values of $\kappa$ and $\alpha$ for $K^{*}$ and the heavy-
light vector mesons. We also make a description of non-$q\,\bar{q}$ states by
testing at the holographic level some of the proposals to describe exotic
mesons as multiquark states or gluonic excitations. This exotic states can be
described by considering the conformal dimension $\Delta$ associated with the
operator that creates these states and how $\Delta$ affects the bulk mass term
in the associated holographic potential. In this case, we use $\bar{m}$ as a
holographic threshold, defined in terms of the structure of each exotic state,
to define the values of $\kappa$ and $\alpha$. We consider exotic candidates
in the light sector ($\pi_{1}$ meson) as well as in the heavy one ($Z_{c}$,
$Z_{b}$ and $X$ mesons).

Finally, in section VI we expose the conclusions and final comments about the
present work.

##  II Non-linear Trajectories

The relation between hadronic squared mass and radial (and orbital) quantum
number is considered usually as linear. This affirmation in general, accepted
due to experimental evidence, is especially true in the light sector, but when
quark masses are increased, a non-linear Regge trajectory seems better to
describe hadron spectra Afonin and Pusenkov (2014); Chen (2018a, b, c);
Gershtein et al. (2006).

For instance, in the Bethe-Salpeter analysis, by including the the quark mass
directly in the radial trajectory

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

where $a$ is a universal slope and $b$ accounts for the effect of the mesonic
quantum numbers, it is expected that non-linearities associated with the
constituent mass emerge Afonin and Pusenkov (2014); Chen (2018c). In the
holographic AdS/QCD context, we can parametrize this 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 non-linearity deviation in the trajectory should be captured in a non-
quadratic static dilaton, in the same form as the original soft-wall model
dilaton encloses the linearity of Regge trajectoriesKarch et al. (2006).

In this work we consider four families of isovector mesons labeled as
$I^{G}\,J^{PC}=0^{-}(1^{--})$, and investigate linear and non-linear
expressions for $M^{2}$. In table 1 we summarize the experimental masses of
all four isovector meson families considered in our analysis. In table 2 we
show our fits for a linear and non-linear Regge trajectory. In Fig. 1, we
summarize experimental data fitted using linear and non-linear fits.

$\omega$ with $\alpha=0.04$ and $\kappa=498$ MeV |  $\phi$ with $\alpha=0.07$ and $\kappa=585$ MeV
---|---
n |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | R. E. (%) | n |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | R. E. (%)
$1$ | $782.65\pm 0.12$ | $981.43$ | $25.4$ | $1$ | $1019.461\pm 0.016$ | $1139.43$ | $11.8$
$2$ | $1400-1450$ | $1374$ | $3.6$ | $2$ | $1698\pm 20$ | $1583$ | $5.8$
$3$ | $1670\pm 30$ | $1674$ | $0.25$ | $3$ | $2135\pm 8\pm 9$ | $1921$ | $10$
$4$ | $1960\pm 25$ | $1967$ | $1.7$ | $4$ | Not Seen | $-$ | $-$
$5$ | $2290\pm 20$ | $2149$ | $6.2$ | $5$ | Not Seen | $-$ | $-$
$M^{2}=0.9514(0.012+n)^{0.9798}$ with $R^{2}=0.999$ |  $M^{2}=1.268(0.0244+n)^{0.9650}$ with $R^{2}=0.999$
$\psi$ with $\alpha=0.54$ and $\kappa=2150$ MeV |  $\Upsilon$ with $\alpha=0.863$ and $\kappa=11209$ MeV
n |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | R. E. (%) | n |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | R. E. (%)
$1$ | $3096.916\pm 0.011$ | $3077.09$ | $0.61$ | $1$ | $9460.3\pm 0.26$ | $9438.5$ | $0.23$
$2$ | $3686.109\pm 0.012$ | $3689.62$ | $0.1$ | $2$ | $10023.26\pm 0.32$ | $9923.32$ | $0.78$
$3$ | $4039\pm 1$ | $4137.5$ | $2.44$ | $3$ | $10355\pm 0.5$ | $10277.2$ | $0.75$
$4$ | $4421\pm 4$ | $4499.4$ | $1.77$ | $4$ | $10579.4\pm 1.2$ | $10558.6$ | $0.19$
$5$ | Not Seen | $-$ | $-$ | $5$ | $10889.9^{+3.2}_{-2.6}$ | $10793.5$ | $0.88$
$6$ | Not Seen | $-$ | $-$ | $6$ | $10992.9^{+10.0}_{-3.1}$ | $10995.7$ | $0.03$
$M^{2}=8.07(0.287+n)^{0.6315}$ with $R^{2}=0.999$ |  $M^{2}=76.511(0.901+n)^{0.2369}$ with $R^{2}=0.999$
Table 3: Summary of results for different families of isovector radial mesonic
states considered in this work. All of the mass spectra displayed in this
table are calculated with the parameters mentioned on each sub-table header,
using (15). The Regge trajectories are also presented in units of GeV2. The
last column on each set of data is the relative error per state. Experimental
results are read from PDG Tanabashi et al. (2018). Figure 2: This plot shows
the behavior of the dilaton exponent $\alpha$ as a function of the quark
constituent mass. Notice that for low masses, dilaton should be quadratic,
implying the appearance of linear Regge trajectories for such states.

If we see closely the fits done in table 2, the non-linear proposal could
provide better results for the low lying states in each isovector trajectory.
We have to keep in mind that linearity is enforced in the high excitation
levels Gershtein et al. (2006); Ebert et al. (2009). Besides, this is expected
by analyzing the Cornell potential: high excited states are dominated by the
confinement term.

Following the Bethe-Saltpeter approximation, deviations in the linearity
should be produced by the constituent mass. In the study of charmonium, we can
get $M_{n}^{2}\propto n^{2/3}$, thus intermediate constituent masses would
imply values for $\nu$ ranging between one and $2/3$. Bottomonium states
should have a exponent $\nu$ lower than $2/3$. This particular behaviour
suggest that $\nu$ should be running with the constituent quark mass. At
holographic level, this idea will allow us to infer information about the
internal structure of the hadron at hand.

We suggest that $\nu$ is a function of the average constituent quark masses
and, also propose an expression to fit the four exponents $\nu$ appearing in
table 2 plus an additional point suggested by the chiral limit, i.e.,
additionally, we consider $\nu=1$ when constituent quarks are massless.

The average constituent quark masses mentioned above can be defined as

$\bar{m}(q_{1},q_{2})=\frac{1}{2}\left(m_{q_{1}}+m_{q_{2}}\right).$

For the constituent quark masses, we use the following set of values

$m_{u}=0.336\leavevmode\nobreak\ \text{GeV}\leavevmode\nobreak\
,\leavevmode\nobreak\ m_{d}=0.340\leavevmode\nobreak\
\text{GeV}\leavevmode\nobreak\ ,\leavevmode\nobreak\
m_{s}=0.486\leavevmode\nobreak\ \text{GeV}$ $m_{c}=1.550\leavevmode\nobreak\
\text{GeV}\leavevmode\nobreak\ ,\leavevmode\nobreak\
m_{b}=4.730\leavevmode\nobreak\ \text{GeV}$

For the exponent $\nu$ introduced in the non-linear fit (2), we propose the
following parameterization in terms of the average constituent quark mass
given

$\nu=a_{\nu}+b_{\nu}e^{\left(-c_{\nu}\bar{m}^{2}\right)},$ (3)

with the following model parameters

$a_{\nu}=0.1893\leavevmode\nobreak\ ,\leavevmode\nobreak\
b_{\nu}=0.8221\leavevmode\nobreak\ ,\leavevmode\nobreak\ c_{\nu}=-0.2634.$

This fit model mimics the effect of considering the constituent masses in the
non-linearity of the Regge trajectory. Notice that in the massless constituent
quark case, i.e. $\bar{m}=0$, we recover linearity.

In the next section we will develop the holographic machinery to deal with the
non-linear fits for isovector mesons and, also we will extend these ideas to
other mesonic species.

Figure 3: $\nu$ exponent as a function of the average constituent quark mass.
For the massless case, it should be expected to recover $\nu=1$.

##  III Geometric Background

Let us consider a five-dimensional AdS Poincarè patch defined by the following
metric

$dS^{2}=e^{2A(z)}\left[dz^{2}+\eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\right],$ (4)

$K^{*}$ with $\bar{m}=413$ MeV, $\alpha=0.055$, and $\kappa=531.24$ MeV
---
n | State |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | R. E. (%)
1 | $K^{*}(892)$ | $895.55\pm 0.8$ | $1038.4$ | $16.2$
2 | $K^{*}(1410)$ | $1414\pm 15$ | $1451.0$ | $2.6$
3 | $K^{*}(1680)$ | $1718\pm 18$ | $1754.5$ | $2.1$
Experimental Linear R. T.: | $M^{2}=1.075(-0.2157+n)$ with $R^{2}=0.9992$.
Experimental Non-Linear R. T.: | $M^{2}=1.157(-0.6102+n)^{0.718}$ with $R^{2}=1$.
Theoretical Non-Linear R. T.: | $M^{2}=1.175(-0.0911+n)^{0.902}$ with $R^{2}=1$.

Table 4: Summary of results for the vector kaon $K^{*}$ radial mesonic states,
with $I(J^{P})=1/2(1^{-})$. The last column is the relative error per state.
Notice that we also show the linear and non-linear radial experimental Regge
trajectories (R. T) altogether with the theoretical fit for the sake of
clarity. The values of $\kappa$ and $\alpha$ are extrapolated using the fits
(12) and (13) with $\bar{m}$ as an input. Experimental results are read from
PDG

Tanabashi et al. (2018).

Also, we consider a bulk vector field $A_{m}(z,x)$ dual to isovector mesonic
states interacting with a static dilaton $\Phi(z)$, as in the original soft-
wall model proposal Karch et al. (2006). This dilaton is motivated by the
particle phenomenology to model confinement through the appearance of Regge
trajectories. The action for such fields is

$I_{V}=-\frac{1}{4\,g_{5}^{2}}\int{d^{5}x\,\sqrt{-g}\,e^{-\Phi(z)}\,F_{mn}\,F^{mn}}.$
(5)

We have assumed the bulk vector field as massless since for mesons this
quantity is fixed to be zero.

From this action, and imposing the gauge fixing $A_{z}=0$, we arrive at the
following equation of motion for the bulk field

$\partial_{z}\left[e^{-B(z)}\,\partial_{z}\,A_{\mu}(z,q)\right]+(-q^{2})e^{-B(z)}\,A_{\mu}(z,q)=0,$
(6)

where we have introduced $B(z)=\Phi(z)-A(z)$. Let us span the bulk vector
field as $A_{\mu}(z,q)=A_{\mu}(q)\,\psi(z,q)$ in order to transform the
equation of motion into a Schrodinger-like one. Performing the Bogoliubov
transformation $\psi(z)=e^{\Phi(z)/2}\phi(z,q)$ we arrive to the the following
expression

$-\phi^{\prime\prime}(z,q)+V(z)\,\phi(z,q)=(-q^{2})\,\phi(z,q)$ (7)

where the holographic potential $V(z)$ is defined as

$\displaystyle V(z)$ $\displaystyle=$
$\displaystyle\frac{1}{4}B^{\prime}(z)^{2}-\frac{1}{2}B^{\prime\prime}(z)$ (8)
$\displaystyle=$
$\displaystyle\frac{3}{4\,z^{2}}+\frac{\Phi^{\prime}(z)}{2\,z}+\frac{\Phi^{\prime}(z)^{2}}{4}-\frac{\Phi^{\prime\prime}(z)}{2},$
(9)

where we have used the warp factor $A(z)=\log(R/z)$. Recall that in the
original spirit of the soft wall model, the soft breaking done by the
inclusion of the dilaton in the geometric background is translated into the
appearance of bound states organized in a Regge trajectory with the slope
defined by the static dilaton. Furthermore, the emergent bulk eigenmodes are
dual to the hadronic states at the conformal boundary. A similar situation can
be seen in top/down models Erdmenger et al. (2008). where confinement is
achieved by the intersection of geometrical defects, as in the Dp/Dq system
Kruczenski et al. (2003). A matter of example, in the latter model, the
associated Sturm-Liouville spectrum arising from geometric fluctuations
behaves quadratically with the excitation number, i.e., $M_{n}^{2}\propto
n^{2}$, which is clearly non-linear by construction.

The hadronic mass spectra, and the Regge trajectories, are constructed from
the eigenvalues of this potential, which is fixed by the structure of the
$B(z)$ function. In the context of the original soft wall model Karch et al.
(2006), the potential is fixed by $B(z)=\kappa^{2}\,z^{2}-\log(R/z)$ obtaining
the linear spectrum

$M_{n}^{2}=4\kappa^{2}(n+1),$ (10)

State | $I(J^{P})$ | $q_{1}\,q_{2}$ |  $\bar{m}$ (MeV) |  $\kappa$ (MeV) | $\alpha$ |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | R. E. (%)
---|---|---|---|---|---|---|---|---
$K^{*}(782)$ | $1/2(1^{-})$ | $d\,\bar{s}$ | $413$ | $531.24$ | $0.055$ | $895.55\pm 0.8$ | $1038.4$ | $16.2$
$D^{*0}(2007)$ | $1/2(1^{-})$ | $c\,\bar{u}$ | $943$ | $1070.8$ | $0.261$ | $2006.85\pm 0.05$ | $1902.5$ | $5.20$
$D^{+0}(2010)$ | $1/2(1^{-})$ | $c\,\bar{d}$ | $945$ | $1073.6$ | $0.262$ | $2010.26\pm 0.05$ | $1906.4$ | $5.16$
$D_{s}^{*+}$ | $0(?^{?})$ | $c\,\bar{s}$ | $1018$ | $1179.1$ | $0.296$ | $2112.2\pm 0.4$ | $2051.7$ | $2.86$
$B^{*+}$ | $1/2(1^{-})$ | $u\,\bar{b}$ | $2533$ | $4681.2$ | $0.800$ | $5324.70\pm 0.22$ | $4561.2$ | $14.3$
$B^{*0}$ | $1/2(1^{-})$ | $d\,\bar{b}$ | $2535$ | $4687.3$ | $0.801$ | $5324.70\pm 0.22$ | $4564.4$ | $14.27$
$B_{s}^{*0}$ | $0(1^{-})$ | $s\,\bar{b}$ | $2608$ | $4901.2$ | $0.809$ | $5415^{+1.8}_{-1.5}$ | $4683.0$ | $13.52$

Table 5: red

Summary of results for vector heavy-light mesonic states contrasting our
theoretical results with the available experimental data. The last column is
the relative error per state. Experimental results are read from PDG Tanabashi
et al. (2018). Although $D_{s}^{*+}$ has not been fully identified, their
decay modes are consistent with $J^{P}=1^{-}$. See Tanabashi et al. (2018) for
further details.

associated with vector mesons with massless constituent quarks. The Regge
slope is identified to the $\kappa$, in units of GeV, that fixes the scale of
the trajectory. The linearity observed in (10) is achieved by the specific
quadratic form of the dilaton, which induces a $z^{2}$ behavior at high-$z$ in
the confining potential, that is translated into the linear dependence with
the radial excitation number $n$.

Figure 4: This plot shows how the dilaton scale $\kappa$ runs with the average
constituent mass. For the massless case we have used $\kappa=0.388$ GeV Karch
et al. (2006).

If we want to consider the case when the constituent quarks are massive, we
must extend to non-linear Regge trajectories Afonin and Pusenkov (2014); Chen
(2018a, b, c); Gershtein et al. (2006). As we mentioned early, we consider
non-linearity connected with the quark constituent mass. Therefore, massless
quarks are tied to linear trajectories. Beyond the chiral symmetry breaking,
once the quarks get mass, the trajectory ceases to be linear but remains as a
good approximation in the light sector. A deviation in the linearity for the
spectrum must be associated with a change from the usual quadratic static
dilaton. This deviation, related to the constituent quark mass, can be
parametrized into a shift in the quadratic exponent of the dilaton

$\Phi(z)=(\kappa\,z)^{2-\alpha}.$ (11)

Fixing $\alpha$ to be zero, we have the massless constituent quarks and
recover the soft-wall model result. In the following sections, we will discuss
the massive constituent quarks case. The exponent for the dilaton (considering
an additional point suggested by chiral limit) can be fitted as

$\alpha(\bar{m})=a_{\alpha}-b_{\alpha}e^{\left(-c_{\alpha}\bar{m}^{2}\right)},$
(12)

with the following set of parameters:

$a_{\alpha}=0.8454\leavevmode\nobreak\ ,\leavevmode\nobreak\
b_{\alpha}=0.8485\leavevmode\nobreak\ ,\leavevmode\nobreak\
c_{\alpha}=0.4233,$

and for the energy scale $\kappa$ we have the following fit

$\kappa(\bar{m})=a_{\kappa}-b_{\kappa}e^{-c_{\kappa}\bar{m}^{2}},$ (13)

with the following fit coefficients:

$a_{\kappa}=15.2085\leavevmode\nobreak\ ,\leavevmode\nobreak\
b_{\kappa}=14.8082\leavevmode\nobreak\ ,\leavevmode\nobreak\
c_{\kappa}=0.0524$

Notice that the fit for $\alpha$ runs consistently with the constituent mass,
as we can expect from the analysis done for the linearity deviation parameter
$\nu$ defined above.

##  IV Meson Masses

In AdS/QCD models, mesonic states are identified with the bulk field mass dual
to the hadronic states in consideration. This connection is made via the
conformal dimension $\Delta$ that fixes how the bulk field scales at the
boundary. On the field theory side, the matching is realized by considering
$\Delta$ as the scaling dimension of the operator that creates hadrons.The
Regge slope for a given mesonic trajectory is defined by the dilaton energy
scale $\kappa$. In other bottom-up approaches, when the higher excitations in
the angular momentum $L$ are considered, the scaling dimension is written in
terms of the _twist operator_ as $\Delta=\tau+L+S$. See for example Gutsche et
al. (2012). Beyond these two parameters, there is no other bulk information
that allows to identify a particular mesonic family in these sorts of bottom-
up models. Thus, the constituent information, that in the spectroscopy of
mesons allows to classify mesonic, is absent in the AdS/QCD formulation. A
similar situation can be observed in the top-down approach. Despite the fact
that $q\,\bar{q}$ states are introduced as the Chan-Paton form factors for the
open strings attached to the intersecting D-branes, the light vector meson
trajectories do not differentiate between unflavored states with different
isospin. See Erdmenger et al. (2008) for further details.

In general, from the standard AdS/CFT dictionary, for meson states with spin
$S$ and $L=0$ we have the following relation

$M_{5}^{2}\,R^{2}=(\Delta-S)(\Delta+S-4).$ (14)

This relation for the bulk field mass is counting the number of constituents
in the scaling dimension $\Delta$, but it says nothing about the type of
constituent or the mass fraction inside the meson. In the particular case of
isovector mesons, we have $\Delta=3$ and $S=1$ implying that such bulk vector
fields are massless.

Now let us turn our attention to the holographic potential (8) constructed
with the non-quadratic dilaton suggested in expression (11). This potential
has the generic form

$V_{q\,\bar{q}}(z,\kappa,\alpha)=\frac{3}{4z^{2}}-\frac{1}{2}\alpha^{2}\,\kappa^{2}\,(\kappa\,z)^{-\alpha}+\frac{1}{4}\alpha^{2}\,\kappa^{2}(\kappa\,z)^{2-2\alpha}\\\
+\frac{3}{2}\alpha\,\kappa^{2}\,(\kappa\,z)^{-\alpha}-\kappa^{2}(\kappa\,z)^{-\alpha}-\alpha\,\kappa^{2}\,(\kappa\,z)^{2-2\alpha}\\\
+\kappa^{2}\,(\kappa\,z)^{2-2\alpha}+\frac{\kappa}{z}(\kappa\,z)^{1-\alpha}-\frac{\alpha\,\kappa}{2\,z}(\kappa\,z)^{1-\alpha}.$
(15)

Notice that the massless constituent quark case, when $\alpha=0$, we recover
the potential for vector mesons Karch et al. (2006). In the non-quadratic
case, the meson constituent information regarding the mass fraction and the
quark flavor is consigned in the parameter $\alpha$.

At this point, $\alpha$ becomes an extra parameter to be considered in the
model. But, as we discussed in section II, it is possible to parametrize a
specific form depending on $\bar{m}$. This also will define a _running_ for
the slope $\kappa$. To properly construct these radial states, we will solve
the Schrodinger equation associated with the potential (15), with $\alpha$ and
$\kappa$ as entries, for the chosen isovector family. Numerical results for
each family are summarized in table 3.

##  V Extrapolation to other mesonic species

Let us apply the ideas developed above to other vector mesonic systems at
hand: Regge trajectory for vector kaons and masses for vector heavy-light
mesons. Since we have fitted the isovector mesons from $\omega$ up to
$\Upsilon$, we have now a wider picture to study mesons with masses ranging in
this interval.

The key idea is to use the running of $\alpha$ and $\kappa$ with the quark
constituent mass as _calibration curve_ to extrapolate the proper values for
other mesonic samples. This methodology will allows to include the constituent
mass fraction as part of the model in order to find the proper pair of
parameters $\kappa$ and $\alpha$.

###  V.1 Kaons

Vector kaons are mesonic states labeled by $I(J^{P})=1/2(1^{-})$, with $S=\pm
1$ and $C=B=0$. In order to set the values for $\alpha$ and $\kappa$, we will
use the following definition for the quark constituent mass as the average of
the masses of s and d quarks:

$\bar{m}_{K^{*}}=\frac{m_{s}+m_{d}}{2}.$ (16)

With this mass, we found for the $K^{*}$ system the following values for
$\kappa$ and $\alpha$ from the calibration curves:

$\displaystyle\kappa_{K^{*}}$ $\displaystyle=$ $\displaystyle
531.24\,\text{GeV},$ $\displaystyle\alpha_{K^{*}}$ $\displaystyle=$
$\displaystyle 0.0555.$

In table 4, we summarized the experimental non-linear and linear fits,
altogether with the theoretical one. Also, we have shown the associated mass
spectrum.

###  V.2 Heavy-light mesons

Heavy-light mesons are defined as hadronic systems where one of the
constituent quarks is heavy (i.e., charm or bottom) whilst the other is light
flavored (up, down or strange). The physics of these heavy-light hadrons has
become one of the vastest fields of research in particle physics. Calculations
of the heavy-light mass spectra are done in the context of effective QFT
Alhakami (2016), potential methods Ebert et al. (2010), QCD sum rules
Gelhausen et al. (2013), Bethe-Salpeter equation Gutiérrez-Guerrero et al.
(2019) and lattice QCD Brambilla et al. (2018).

Following the same procedure done in the case of vector kaons, we can fix the
quark constituent mass $\bar{m}$ as an average of the pair of constituent
quarks inside the heavy-light meson. The mass spectrum and the corresponding
values of $\kappa$ and $\alpha$ are summarized in table 5.

###  V.3 Non-$q\bar{q}$ vector states

Holographic spectrum | Non-$q\,\bar{q}$ states
---|---
$\Delta=6$ and $\bar{m}_{\text{diquark-antidiquark}}$ | Multiquark state
$\alpha=0.539$ and $\kappa=2151$ MeV |  $I^{G}(J^{CP})=1^{+}(1^{+-})$ $Z_{c}$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $4004.8$ | 1 | $Z_{c}(3900)$ | $3887.2\pm 2.3$ | $3.0$
2 | $4384.9$ | 2 | $Z_{c}(4200)$ | $4196^{+35}_{-32}$ | $4.5$
3 | $4706.6$ | 3 | $Z_{c}(4430)$ | $4478^{+15}_{-18}$ | $5.1$
$\Delta=6$ and $\bar{m}_{\text{hadronic molecule}}$ | Multiquark state
$\alpha=0.539$ and $\kappa=2151$ MeV |  $I^{G}(J^{CP})=1^{+}(1^{+-})$ $Z_{c}$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $3816.3$ | 1 | $Z_{c}(3900)$ | $3887.2\pm 2.3$ | $1.82$
2 | $4213.9$ | 2 | $Z_{c}(4200)$ | $4196^{+35}_{-32}$ | $0.43$
3 | $4551.4$ | 3 | $Z_{c}(4430)$ | $4478^{+15}_{-18}$ | $1.64$
$\Delta=6$ and $\bar{m}_{\text{Hadrocharmonium}}$ | Multiquark state
$\alpha=0.604$ and $\kappa=2523$ MeV |  $I^{G}(J^{CP})=0^{+}(1^{--})$ $Y$ or $\psi$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $4228.3$ | 1 | $\psi(4260)$ | $4230\pm 8$ | $0.25$
2 | $4577.3$ | 2 | $\psi(4360)$ | $4368\pm 13$ | $4.8$
3 | $4871.8$ | 3 | $\psi(4660)$ | $4643\pm 9$ | $4.9$
$\Delta=6$ and $\bar{m}_{\text{Hadronic Molecule}}$ | Multiquark state
$\alpha=0.538$ and $\kappa=1548.7$ MeV |  $I^{G}(J^{CP})=0^{+}(1^{--})$ $Y$ or $\psi$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $40027.8$ | 1 | $\psi(4260)$ | $4230\pm 8$ | $5.37$
2 | $4383.1$ | 2 | $\psi(4360)$ | $4368\pm 13$ | $0.35$
3 | $4705.1$ | 2 | $\psi(4360)$ | $4643\pm 9$ | $1.34$
$\Delta=6$ and $\bar{m}_{\text{hadronic molecule}}$ | Multiquark state
$\alpha=0.863$ and $\kappa=11649$ MeV |  $I^{G}(J^{CP})=1^{+}(1^{+-})$ $Z_{B}$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $10410.9$ | 1 | $Z_{B}(10610)$ | $10607.2\pm 2$ | $1.85$
2 | $10669.3$ | 2 | $Z_{B}(10650)$ | $10652.2\pm 1.5$ | $0.16$
$\Delta=5$ and $\bar{m}_{\text{Hybrid Meson}}$ | Gluonic excitation state
$\alpha=0.0367$ and $\kappa=488$ MeV |  $I^{G}(J^{CP})=0^{-}(1^{+-})$ $\pi_{1}$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $1351.7$ | 1 | $\pi_{1}(1400)$ | $1354\pm 25$ | $0.16$
2 | $1646.6$ | 2 | $\pi_{1}(1600)$ | $1660^{+15}_{-11}$ | $0.8$
3 | $1901.7$ | 3 | $\pi_{1}(2015)$ | $2014\pm 20\pm 16$ | $5.58$
$\Delta=5$ and $\bar{m}_{\text{Hybrid meson}}$ | Gluonic Excitation
$\alpha=0.539$ and $\kappa=2151$ MeV |  $I^{G}(J^{CP})=1^{+}(1^{+-})$ $Z_{c}$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $3721.9$ | 1 | $Z_{c}(3900)$ | $3887.2\pm 2.3$ | $4.24$
2 | $4156.4$ | 2 | $Z_{c}(4200)$ | $4196^{+35}_{-32}$ | $0.94$
3 | $4513.2$ | 3 | $Z_{c}(4430)$ | $4478^{+15}_{-18}$ | $0.78$
$\Delta=7$ and $\bar{m}_{\text{Hybrid Meson}}$ | Gluonic excitation state
$\alpha=0.863$ and $\kappa=11649$ MeV |  $I^{G}(J^{CP})=1^{+}(1^{+-})$ $Z_{B}$ mesons
n |  $M_{\text{Th}}$ (MeV) | n | State |  $M_{\text{Exp}}$ (MeV) |  $\Delta\,M$ (%)
1 | $10346.7$ | 1 | $Z_{B}(10610)$ | $10607.2\pm 2$ | $2.52$
2 | $10696.6$ | 2 | $Z_{B}(10650)$ | $10652.2\pm 1.5$ | $0.42$
Table 6: Summary of results for the set of non-$q\,\bar{q}$ states considered
in this work. Experimental results are read from PDG Tanabashi et al. (2018).

All of the mesonic states with quantum numbers not allowed by the usual
$q\,\bar{q}$ model are called _exotic_. A good review of the physics of such
states can be found in Brambilla et al. (2019); Guo et al. (2018); Lebed et
al. (2017) and references therein. We will focus on the vector exotic states
in this section. At holographic level, Liu et al. (2019a) addresses the exotic
meson spectra for $Z_{c}$ and $Z_{b}$ in the context of Sakai-Sugimoto models.

Holographically, as we explain above, the hadronic identity is controlled by
the scaling dimension associated with the operator that creates hadrons. This
information is encoded into the bulk mass of the five-dimensional field use to
mimic hadrons. Equation (14) summarizes this. Therefore, if we identify the
dimension of the operators that create exotic states we can address the
associated vector mass spectrum by using the proper holographic potential,
that in our case has the specific form

$V_{\text{non-}q\,\bar{q}}(z,\kappa,\alpha,M_{5,\Delta})=V_{q\,\bar{q}}(z,\kappa,\alpha)+\frac{M_{5}^{2}(\Delta)\,R^{2}}{z^{2}},$
(17)

where $V_{q\,\bar{q}}(z,\kappa,\alpha)$ is given by the expression (15). This
potential is obtained by following the same procedure done in the massless
vector case: we start from the bulk action for massive bulk vectors fields,
write down the mode equation, and perform the Bogoliubov transformation to
obtain the Schrodinger-like equation.

Here, we will consider the exotic meson vector states organized into two
groups: _multi-quark states_ , and _gluonic excitations_. The former is
associated with tetraquarks, hadroquarkonium, and hadronic molecules.
Pentaquarks are also part of this category. For the sake of simplicity, we
will devote to multi-quark states candidates with just four constituent
quarks. The methods developed here can be extrapolated to multiquark
candidates also.

At this point, it is important to mention the gauge invariance, since now we
have massive bulk fields. Recall that the gauge invariance should be manifest
at the conformal boundary, where all of the dual fields are massless Braga et
al. (2016b). The presence of the non-zero bulk mass does not affect the gauge
$A_{z}=0$. If we pay attention to the massive e.o.m for the vector bulk
fields, i.e., for the $z$ component

$\Box\,A_{z}-\partial_{z}\left(\partial_{\mu}\,A^{\mu}\right)+M_{5}^{2}\,e^{2\,A}\,A_{z}=0,$
(18)

and the spacetime components

$\partial_{\nu}\left[e^{-B}\,\left(\partial_{\mu}\,A^{\mu}\right)+\partial_{z}\left(e^{-B}A_{z}\right)\right]-\\\
\left\\{\partial_{z}\left[e^{-B}\,\partial_{z}\,A_{\nu}\right]+e^{-B}\,\Box\,A_{\nu}-e^{-B}\,e^{-2\,A}\,M_{5}^{2}\,A_{\nu}\right\\}=0,$
(19)

we can realize that the $A_{z}=0$ gauge condition still implies
$\partial_{\mu}\,A^{\mu}=0$. Therefore, the fields at the boundary are still
transverse.

The gluonic excitations cathegory classifies glueballs and hybrid mesons. In
this paper we will focus on vector hybrid mesons only, consisting of a quark-
antiquark pair with a finite number of active gluons.

####  V.3.1 Multi-quark states

In the case of multi-quark states, a degeneracy appears when the conformal
dimension is defined. Furthermore, since the conformal dimension is counting
indirectly the number of constituent quarks, this dimension does not
distinguish between four quarks in the diquark antidiquark pair, the
hadroquarkonium or hadronic molecule configurations.

This degeneracy can be removed if we consider the constituent mass of each
multi-quark configuration as a collection of $N$ constituents, quarks or
mesons, given by

$\bar{m}_{\text{multi-
quark}}=\sum_{i=1}^{N}({P^{\text{quark}}_{i}\,\bar{m}_{q_{i}}+P^{\text{meson}}_{i}\,m_{\text{meson}_{i}}}),$
(20)

with the condition that

$\sum_{i=1}^{N}\,(P^{\text{quark}}_{i}+P^{\text{meson}}_{i})=1.$ (21)

Notice that each weight $P^{\text{quark(meson)}}_{i}$ measures the
contribution of a given constituent (quark or meson) with mass
$m_{\text{quark(meson)}}$. Each multi-quark state has a different mass
configuration, used to calculated the parameters $\kappa$ and $\alpha$ in the
respective calibration curves, as we did in the heavy-light mesons case.

_Diquark constituent model._ Tetraquark states can be considered as hadronic
states consisting of a pair of _diquark_ and an _anti-diquark_ interacting
between them. A diquark is a non-colored singlet object used as essential
building blocks forming tetraquark mesons and pentaquark baryons. These
fundamental blocks either a color anti-triplet or a color sextet in the SU(3)
color representation Jaffe (2005). These diquarks are bounded by spin-spin
interactions. The constituent diquark approach is useful to describe the
spectroscopy and decays of multiquark states. It is expected that these
diquark composed candidates appear as poles in the $S$-matrix, described by
narrow widths.

Theoretical approaches are done in the QCD sum rules Bracco et al. (2009);
Kleiv et al. (2013), potential models Monemzadeh et al. (2015) framework, and
lattice QCD Junnarkar et al. (2019), where they approach the diquark-
antidiquark. Experimentally, charmonium and bottomonium tetraquark states can
be identified because they decay into open-flavor states instead of a
quarkonium with a light meson due to the _spin-spin interaction dominance_
(See Brambilla et al. (2019)).

Following PDG Tanabashi et al. (2018), the charmonium $Z_{c}$ states, with
quantum numbers $I^{G}(J^{CP})=1^{+}(1^{+-})$, are candidates to be vector
tetraquarks. For these states we can use the charm quark constituent mass
given in section II to find the values of $\kappa$ and $\alpha$ for these
states. Following Bambrilla, we consider the $Z_{c}$ states as a single
trajectory. Table 6 summarizes the experimental candidates.

Other studies, as Wang (2018), suggest $\psi(4260)$ with $0^{+}(1^{--})$ as a
vector tetraquark instead of $Z_{c}$. As we will prove later, at least at the
holographic level, $\psi(4260)$ seems to be consistent with the hypothesis
that it is a hadrocharmonium state.

In this case, we can organize the diquark and antidiquark as
$(q\bar{q})_{1}\,(q\bar{q})_{2}$, with conformal dimension $\Delta=6$. This
implies that the bulk mass is $M_{5}^{2}\,R^{2}=15$ for these states. The
parameter $\bar{m}$ can be set as a sort of _holographic threshold_ that will
allow us to distinguish between multiquark state descriptions.

In the case of the diquark constituent model, the threshold in this
charmonium-like diquark-antidiquark case is set as

$\bar{m}_{\text{diquark-Antidiquark}}=\bar{m}_{c},$ (22)

implying that $P^{\text{quark}}_{i}=1/4$ and $P^{\text{meson}}_{i}=0$ for this
configuration. With this definition we can set the proper values for $\kappa$
and $\alpha$. Numerical results for this approximation are shown in the first
left panel of table 6.

Vector hybrid meson | $P_{q}$ | $P_{\bar{q}}$ | $P_{G}$
---|---|---|---
$\pi_{1}$ | $0.497$ | $0.497$ | $6\times 10^{-3}$
$Z_{c}$ | $0.49$ | $0.49$ | $0.02$
$Z_{b}$ | $0.495$ | $0.495$ | $0.01$
Table 7: Summary of coefficients fixed for each hybrid meson candidate. In the
case of $Z_{b}$ we are considering two flux tubes instead one.

_Hadroquarkonium model._ The hadroquarkonium states can be constructed by
considering a vector meson core with a _cloud_ of two quarks Liu et al.
(2019b). From the experiments, it was observed that most of the candidates to
be heavy exotic states appear as final states composed by heavy quarkonium and
light quarks. This motivated the idea that these states were made of a compact
heavy quarkonium core surrounded by a light quark cloud Voloshin (2008). This
quarkonium core interacts with the light quark cloud through a _colored Van
der Waals-like force_ (similar as the one in molecular physics), allowing the
decay of these states into the observed quarkonium core and the light quarks
Brambilla et al. (2017).

Following Brambilla et al. (2017), we will suppose that the states
$\psi(4260)$, $\psi(4360)$ and $\psi(4660)$ with $0^{+}(1^{--})$ are possible
hadrocharmonium states, forming a single vector trajectory. Holographically,
the operator that creates these states has dimension six, i.e., $\Delta=6$,
implying that the bulk mass is $M_{5}^{2}\,R^{2}=15$, as in the case of the
diquark-antidiquark pair configuration. The difference will be the definition
of the holographic threshold used to set the parameters $\alpha$ and $\kappa$.

In this case, we will consider a charmonium ($J/\psi$ meson) core
characterized by its mass plus the light quark cloud, consisting of a pair of
$u$ and $d$ quarks. Therefore, the holographic threshold is defined as

$\bar{m}_{\text{Hadrocharmonium}}=\frac{1}{2}m_{J/\psi}+\frac{1}{4}\,(\bar{m}_{u}+\bar{m}_{d}).$
(23)

With this criterion, we can extrapolate the model parameter and compute the
mass spectra for these exotic states. The summary of these results is shown in
the third left panel of the table 6. Another possible candidates to be vector
hadrocharmonium are the pair of states $\chi_{c1}(3872)$ and $\chi_{c1}(4140)$
Cleven et al. (2015), with quantum numbers given by $0^{+}(1^{++})$. In our
case, the model developed here is not sensitive to such difference between the
quantum numbers, i.e., the transition $CP=++\rightarrow--$ is not described by
this non-quadratic dilaton (11), implying that for us these states are
degenerate. A similar situation occurs in other bottom-up models, such as
Folco Capossoli et al. (2019), where is not possible to distinguish between
mesonic states with different isospin since the model does not consider chiral
symmetry breaking. In our case, we need to add extra parameters to split up
these two sets of exotic states.

_Hadronic molecule model._ 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 Lebed et al. (2017). These structures can
be realized as a two heavy quarkonia interacting or one heavy quarkonium plus
a light meson. This proposal is as old as QCD itself De Rújula et al. (1977).
The first theoretical approaches are applications of the deuteron Weinberg’s
model Weinberg (1963); Weinstein and Isgur (1982). Experimental results for
$X(3872)$ and $D_{S_{0}}(2317)$ are consistent with these ideas. Other
approaches are done in the context of sum rules Wang et al. (2009) or lattice
QCD Stewart and Koniuk (1998).

In the heavy sector, Brambilla et al. (2019) and Guo et al. (2018) suggest
that $Z_{c}$ or $Y$ mesons could be possible hadronic molecule charmonium
states, containing at least one pair of $c\,\bar{c}$ in the inner core of the
molecule. The most relevant decay of these states is $J/\psi\,\pi$. Following
this, we will construct the threshold mass for the holographic $Y$ or $\psi$
mesons as

$\bar{m}_{\text{hadronic
molecule}}=\frac{1}{3}m_{J/\psi}+\frac{2}{3}m_{\rho}.$ (24)

In the case of the $Z_{c}$ mesons, we have proposed the following threshold
mass

$\bar{m}_{\text{hadronic molecule}}=0.283\,m_{J/\psi}+0.717\,m_{\rho}.$ (25)

We will extend these ideas to the bottomonium hadronic molecule candidates,
the $z_{B}$ mesons, where the expected core is the $\Upsilon(1S)$ state. The
holographic threshold in this case is

$\bar{m}_{\text{hadronic molecule}}=0.458\,m_{\Upsilon(1S)}+0.542\,m_{\rho}.$
(26)

Results for all of these fits are showed in the table 6. As in the other
multiquark cases, the conformal dimension is $\Delta=6$, implying a bulk mass
given by $M^{2}\,R^{2}=15$.

At this point, we can notice that, at the holographic level, $Z_{c}$ and $Y$
states are better described as hadronic molecules. When $Z_{c}$ is described
as a pair of diquark-antidiquark, the RMS error (7.5%) is bigger than in the
hadronic molecule case (2.5%). For the $Y$ mesons we observe the same: the RMS
error in the hadrocharmonium description (6.8%) is bigger than in the
molecular case (5.5%).

####  V.3.2 Gluonic excitations: Hybrid mesons

Gluonic excitations are defined as hadrons with constituent gluonic fields.
QCD confined states are naturally non-perturbative, therefore it is not
surprising to have constituent gluons inside hadrons. This kind of structure
is realized as pairs of quarks and anti-quarks joined by gluonic flux tubes.
This particular configuration allows us to introduce other sets of quantum
numbers not possible in the quark constituent model, for example, the
$J^{CP}=1^{+-}$ configuration that we will explore in this section. The
mesonic states consisting of valence quarks and constituent gluons are called
_hybrid mesons_. Another set of gluonic excitations are the glueballs, not
addressed here, characterized by the absence of quark quantum numbers. In
general, hybrid mesons have been studied using flux tube model Isgur and Paton
(1985), the MIT bag model Iddir et al. (1988), coulomb-like potentials Guo et
al. (2008), gluon constituent model Hou et al. (2001), quenched QCD Lacock et
al. (1997) or lattice QCD Dudek et al. (2009).

Experimentally speaking, it is possible to find candidates across the entire
mass range, from light mesons up to bottomonium states. In particular, we will
focus on the $\pi_{1}$, $Z_{c}$ and $Z_{b}$ states.

To build up the holographic description, we need to define the hadronic
operators creating hybrid mesons. Following the standard AdS/QFT dictionary,
the phenomenological motivation comes from the two-point functions at the
conformal boundary. These objects are defined in terms of operators that are
composites of quarks and gluons, that generally, can be defined as
$q\,\gamma_{\mu}\,\bar{q}\,G^{\mu\nu}$, where $G$ is a gluonic field on its
ground stated and $\gamma_{\mu}$ are the Dirac matrices Richard (2016). This,
in terms of the operator dimension, means that $\Delta=5$ if we consider one
single constituent gluon, or $\Delta=7$ if we consider two constituent gluons.
This information is translated in the bulk mass as $M_{5}^{2}\,R^{2}=8$ and
$M_{5}^{2}\,R^{2}=24$ respectively.

Since we want to define the holographic threshold, we need to infer a mass for
the constituent gluon. Following Hou et al. (2001), we adopt $M_{G}=700$ MeV.
Therefore, our general proposal for the holographic threshold has the form

$\bar{m}_{\text{hybrid
meson}}=P_{q}\,m_{q}+P_{\bar{q}}\,m_{\bar{q}}+P_{G}\,m_{G}.$ (27)

In table 7 we summarize the choices for the $P_{i}$ coefficients used to
describe the hybrid meson candidates in the context of the model developed
here. With these holographic thresholds, we can obtain the proper values for
$\kappa$ and $\alpha$ to fix the non-linear trajectory. Results are depicted
in the last three panels of the table 6.

The light and the charmonium were fitted supposing a single constituent gluon,
which is translated in a conformal dimension fixed as $\Delta=6$. The RMS
error in both cases is about $1\%$ in the former and $4.4\%$ in the latter. In
the case of the bottomonium hybrids, the best fits were obtained for two
constituent gluons, implying that $\Delta=7$. The RMS error, in this case, was
near to $2.55\%$.

It is important to notice that, at holographic level, in this model
constituent gluons are not so relevant for the definition of the holographic
threshold, since their associated weight in each of the three cases at hand
was almost near to zero, as we can read from the table 7.

It is worthy to mention that, as a holographic prediction, the $Z_{c}$ mesons
are better described as a holographic hadronic molecule (R.M.S near to 2.48 %)
than a holographic hybrid meson (R.M.S. near to 4.41 %).

##  VI Discussions and conclusions

In the model AdS/QCD with dilaton, the usual approach considers quadratic
dilatons at large $z$, because configuration produces linear Regge
trajectories. But it is important to notice that this sort of Regge
trajectories is a good description only in the light sector. For this reason
we propose a new shape for dilaton field, (namely $\phi(z)=(\kappa
z)^{2-\alpha}$), breaking the conformal invariance and producing trajectories
with the generic form $M_{n}^{2}=a(n+b)^{\nu}$. This set of trajectories
reproduce, in a satisfactory form, masses for vector mesons with different
constituent quarks, catching linearity in the massless quark case and
exhibiting how this linearity starts to cease when constituent quark masses
increased.

We consider that $\alpha$ and $\nu$ depend on the average of constituent quark
masses for the mesons considered. Also we proposed a explicit shape for
$\alpha(\bar{m})$ and $\nu(\bar{m})$ in order to built a model that produce a
good spectrum for vector mesons with different constituents.

Nowadays in literature, it is possible to found some models AdS/QCD applied to
charmonium or bottomonium Braga et al. (2016a, b, c); Braga and Ferreira
(2017a); Braga et al. (2017), but spectra are no so good enough in these
models, although other observables (as the melting temperature) have the
proper qualitative behavior. Therefore, these ideas, as we discussed here, can
be useful in this kind of application.

Regarding the chiral symmetry, even though the model describes the spectra for
$\phi$ and $\omega$ mesons, it does not reproduce a proper chiral symmetry
breaking picture, as most of the static soft wall-like models developed. The
main drawback is the impossibility to distinguish between the explicit and the
spontaneous breaking since the quark condensate $\sigma_{q}$ and the quark
mass $m_{q}$ are not independent. In the case of the dilaton proposed here,
although its UV behavior is different from the static quadratic one, this does
not guarantee the independence between $m_{q}$ and $\sigma_{q}$. The best
advances are done in the frame of dynamical AdS/QCD models, as Gherghetta et
al. (2009); Li et al. (2013), or direct modifications of the bulk _vev_ by
changing the bulk mass, as it was done in Vega and Schmidt (2010).

In this direction, linear trajectories are associated with low constituent
quark mass, as the results in tables 2 and 3 are demonstrating. Therefore, we
conclude that the soft wall model is set before the chiral symmetry breaking
scenario. Furthermore, the meson spectra obtained is degenerate: there is no
form to distinguish $\rho$, $\omega$ and $a_{1}$ vector mesons the using
quadratic dilaton only. It is necessary to do explicitly the chiral symmetry
breaking by using SU(2) bulk vector fields and a tachyonic _vev_ to address
this, even though soft wall-like models do not represent a QCD-like chiral
symmetry breaking. See for example the analisys done in Ballon-Bayona and
Mamani (2020).

By introducing the $\nu$ exponent in the radial Regge trajectory we can
explore the effect of the quark mass. As it was pointed out in Afonin and
Pusenkov (2014); Chen (2018b, a), increasing the quark mass should deviate the
trajectory from the linear case. In this holographic approach, such behavior
was observed. Therefore, despite the fact we do not deal with the meson
structure directly, we can capture information about it in the non-linearity
behavior. Translated to the bulk side, this information is captured in the
$\alpha$ parameter, which measures the deviation from the quadratic form in
the dilaton. As we observed in graphic 3, increasing constituent quark mass
implies a strong deviation from the quadratic dilaton.

In the case of the energy scale $\kappa$, it is important to notice that its
value is near to the constituent quark mass for each meson considered.
Furthermore, in linear soft wall model, $\kappa$ defines the vector Regge
slope (string tension), i.e, $4\kappa^{2}=a$, where the linear trajectory is
defined as $M^{2}=a(n+b)$. In the holographic non-linear case, where the
trajectory is defined as $M^{2}=a(n+b)^{\nu}$, the energy scale $\kappa$ also
increases with the quark mass, indicating that it is connected indirectly with
the meson structure. In tables 3, 4 and 5 we see that each $\kappa$ increases
with the constituent quark mass. In the case of the non-linear trajectory, $a$
should be proportional to $\kappa^{2}$, and also carries information about the
string tension and the quark constituent mass in each mesonic family. Recall
that in this case, the meson is modeled as the usual flux tube with two
massive quarks at the ends, thus it is expected that the quark mass
information should appear in the slope. Thus, the factor $a$ in the non-linear
trajectory should be a function of $\alpha$ and $\nu$. Moreover, from the data
reported in table 3 we can infer the fitted form for $a$ as:

$a(\kappa^{2},\alpha,\nu)=(11.304\,e^{-0.4141\,\alpha}-7.3054\,e^{-0.00348\,\nu})\,\kappa^{2},$
(28)

where the correlation coefficient for this fit is $R^{2}=1$. Notice that in
the case where $\alpha=0$ (implying $\nu=0$) and $\kappa=388$ MeV, we obtain
$a=0.6022$ which is consistent with the usual soft wall model Karch et al.
(2006). This expression could be useful to construct general non-linear
trajectories just by using as inputs the parameters $\kappa$, $\alpha$ and
$\nu$.

It is worthy to say that this dilaton is not capturing the expected low $z$
behavior in the eigenmodes. Notice that the meson ground states are not well
fitted in the light sector as long as the $\nu$ exponent. This can be inferred
by the fact that trajectories are not exactly fitted. Therefore, this proposed
dilaton should be interpolated with other low $z$ proposals, as Braga and
Ferreira (2017b, a). But, on the other hand, it was possible to fit heavy
light vector mesons and to test, at holographic level, possibles candidates to
be non-$q\,\bar{q}$ states, just by considering how the hadronic operators at
the boundary change their conformal dimension, that has information about the
meson constituent indirectly, altogether with the holographic threshold
$\bar{m}$, that parameterizes the structure of the state at hand. The change
in the conformal dimension is translated into a modification of the bulk mass
term that appears in the holographic potential (17), whilst $\bar{m}$ fixes in
the calibration curves (13) and (12) the values for $\kappa$ and $\alpha$. The
same methodology was used to do the holographic fit for the heavy-light mesons
and the $K^{*}$ vector states.

As a final comment is important to recall the predictability of the
holographic picture discussed in this manuscript. A good criterion is given by
the RMS error for estimating $N$ quantities using $N_{p}$ parameters, that can
be defined as

$\delta_{\text{RMS}}=\sqrt{\frac{1}{N-N_{p}}\sum_{i}^{N}\left(\frac{\delta\,O_{i}}{O_{i}}\right)},$
(29)

where $O_{i}$ is a given experimental measure with $\delta\,O_{i}$ defining
the deviation of the theoretical value from the experimental one. Although the
ground states for light mesons were not well fitted (errors near to 20%), the
RMS for the model with 27 mesonic states (we do not consider the
non-$q\,\bar{q}$ states since they are holographic predictions) with fifteen
holographic parameters listed as follows:

  * •

Two parameters, $\kappa$ and $\alpha$, for each isovector meson family, i.e.,
$\omega$, $\phi$, $J/\Psi$ and $\Upsilon$, implying eight in total.

  * •

One threshold mass $\bar{m}$ for the vector kaon $K^{*}$ system.

  * •

Six threshold masses $\bar{m}$ for each heavy-light vector meson considered,
i.e., $D^{*0}$, $D^{+0}$, $D^{*0}_{s}$, $B^{*}$, $B^{*0}$ and $B^{0*}_{s}$.

This parameter fixing implies an RMS error near to $12.61\%$. It is important
to remark that we fit four families of isovector mesons, with constituent
masses going from the light to the heavy sector. Also, we have fitted heavy-
light resonances with parameters interpolated from the isovector matches for
$\kappa$ and $\alpha$, namely equations (13) and (12), making the model self-
consistent. We have also approached the spectra of some non-$q\,\bar{q}$
candidates with the same interpolations. Therefore an RMS error around 12% is
reasonable for this model, considering the simplicity of the proposal done and
the complexity of the QCD physics at a strong regime.

###### Acknowledgements.

We wish to acknowledge the financial support provided by FONDECYT (Chile)
under Grants No. 1180753 (A. V.) and No. 3180592 (M. A. M. C.).

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