aainstitutetext: School of Nuclear Science and Technology, University of
Chinese Academy of Sciences,
Beijing, P.R.China 100049bbinstitutetext: School of Physical Sciences,
University of Chinese Academy of Sciences,
Beijing, P.R.China 100049

# Spectra of glueballs and oddballs and the equation of state from holographic
QCD

Lin Zhang b Chutian Chen a Yidian Chen a and Mei Huang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the spectra of two-gluon glueballs and three-gluon oddballs and
corresponding equation of state in $5$-dimensional deformed holographic QCD
models in the graviton-dilaton system, where the metric, the dilaton field and
dilaton potential are self-consistently solved from each other through the
Einstein field equations and the equation of motion of the dilaton field. We
compare the models by inputting the dilaton field, inputting the deformed
metric and inputting the dilaton potential, and find that with only 2
parameters, the $5$-dimensional holographic QCD model predictions on
glueballs/oddballs spectra in general are in good agreement with lattice
results except two oddballs $0^{+-}$ and $2^{+-}$. From the results of
glueballs/oddballs spectra at zero temperature and the equation of state at
finite temperature, we observe that the model with quadratic dilaton field can
simultaneously describe glueballs/oddballs spectra as well as equation of
state of pure gluon system. The model with quadratic $A_{E}(z)$ can describe
glueballs/oddballs spectra, but its corresponding equation of state behaves
more like $N_{f}=2+1$ quark matter. These are consistent with dimension
analysis at UV boundary.

###### Keywords:

Glueballs, Oddballs, Equation of state, Holographic QCD

##  1 Introduction

Glueball is one of the most crucial predictions from quantum chromodynamics
(QCD), whose non-Abelian feature makes it possible to form bound states of
gauge bosons, i.e. glueballs made of two/three gluons (gg, ggg, etc.) Gell-
Mann:2015noa . The gauge field plays a more important dynamical role in
glueballs than that in the standard hadrons, therefore studying particles like
glueballs offers a good opportunity of understanding non-perturbative aspects
of QCD. The glueball spectra has attracted much attention for four decades
Gell-Mann:2015noa , and it has been widely investigated by using various non-
perturbative methods. For example, glueballs have been studied by using
lattice QCD Morningstar:1999rf ; Lucini:2001ej ; Meyer:2004gx ; Chen:2005mg ;
Gregory:2012hu ; Bennett:2017kga ; Bennett:2020hqd ; Athenodorou:2020ani ;
Bennett:2020qtj ; Athenodorou:2021qvs , by using effective models like flux
tube model Isgur:1984bm  and MIT bag model Jaffe:1975fd ; Barnes:1981kq ;
Barnes:1981kp ; Carlson:1982er ; Chanowitz:1982qj , by using QCD sum rules
Dominguez:1986td ; Dominguez:1986zv ; Latorre:1987wt ; Narison:1988ts ;
Narison:1996fm ; Narison:1997nw ; Huang:1998wj ; Narison:2008nj ; Qiao:2014vva
; Tang:2015twt ; Pimikov:2016pag ; Pimikov:2017xap ; Qiao:2017jxc ;
Pimikov:2017bkk ; Chen:2021cjr  as well as by using relativistic many-body
approach Szczepaniak:1995cw ; Llanes-Estrada:2000ozq ; Llanes-Estrada:2005bii
. There are also some other analyses of glueballs in Refs. Bugg:2000zy ;
Zhao:2005ip ; Cheng:2006hu ; Li:2009rk ; He:2009sb ; Cheng:2009zk ;
Janowski:2014ppa ; Eshraim:2015cia ; Sarantsev:2021ein . For more information,
please refer to review papers Mathieu:2008me ; Klempt:2007cp ; Amsler:2004ps .

On the other hand, the spin and mass of the glueball can be constrained from
high energy scattering data. Regge trajectories
$\alpha(t)=\alpha_{0}+\alpha^{\prime}t$ of the glueball have been used to fit
high energy pp and $p\bar{p}$ scattering cross-section. The C-even glueball,
Pomeron exchange gives the lightest $J=2^{++}$ glueball mass
$M=\sqrt{t}=1.92~{}GeV$. Analogy with the "Pomeron", C parity odd "Odderon"
contributing to large odd amplitude was proposed in 1970s in describing the
high energy $pp$ and $p{\bar{p}}$ scatteringBraun:1998fs ; Lukaszuk:1973nt .
The Odderon was regarded as three-gluon state:

$O_{abc}^{\mu\nu\sigma}(k1,k2,k3)=d_{abc}G_{a}^{\mu}(k_{1})G_{b}^{\nu}(k2)G_{c}^{\nu}(k3)$
(1)

where the lower indices refer to color and the upper ones refer to the Lorentz
structure, and $d_{abc}$ is the fundamental symmetric tensor in SU(3). The
evidence for the identification of the odderon has been debated for a
longtime. Recently, the D0 and TOTEM Collaborations announced the evidence of
a t-channel exchanged C-odd odderons in $pp$ and $p{\bar{p}}$ scattering
Abazov:2012qb ; Abazov:2020rus . Especially the odderon’s contribution at the
dip-bump region is very essential. The mass of $3^{--}$ odderon
$M_{3^{--}}=3.001\mathrm{GeV}$ and dacay width
$\Gamma_{3^{--}}=2.984\mathrm{GeV}$ are extracted by using the dipole (DP)
Regge model to fit the scattering data Szanyi:2019kkn ; Csorgo:2019ewn ;
Csorgo:2020wmw ; Bence:2018ain .

In Ref. Chen:2021cjr , the oddball spectra has been calculated by using the
QCD sum rule. In this work, we are going to investigate the glueball spectra
in the framework of holographic QCD, which is based on the gravity/gauge
duality, or anti-de Sitter/conformal field theory (AdS/CFT) correspondence
Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj . AdS/CFT correspondence
offers a new possibility to tackle the difficulty of strongly coupled gauge
theories Aharony:1999ti ; Aharony:2002up ; Zaffaroni:2005ty ; Erdmenger:2007cm
. Many efforts from both top-down and bottom-up approaches have been paid on
examining the non-perturbative properties of QCD Kovtun:2004de , e.g., QCD
equation of state, phase transitions, fluid properties of quark-gluon plasma,
meson spectra Erlich:2005qh ; Karch:2006pv ; Sakai:2004cn ; Sakai:2005yt ;
deTeramond:2005su ; DaRold:2005mxj ; Ghoroku:2005vt ; Andreev:2007vn ;
Andreev:2006ct ; Kruczenski:2004me ; Kuperstein:2004yf ; Forkel:2007cm ,
baryon spectra Hong:2006ta ; Nawa:2006gv ; Hong:2007kx , as well as the
glueball sector Csaki:1998qr ; deMelloKoch:1998vqw ; Zyskin:1998tg ;
Minahan:1998tm ; Csaki:1998cb ; Csaki:1999uw ; Brower:2000rp ;
BoschiFilho:2002vd ; BoschiFilho:2002ta ; Apreda:2003sy ; BoschiFilho:2005yh ;
Colangelo:2007pt ; Forkel:2007ru ; FolcoCapossoli:2013eao ;
Bellantuono:2015fia ; FolcoCapossoli:2015jnm ; FolcoCapossoli:2016fzj ;
FolcoCapossoli:2016uns ; Rodrigues:2016cdb ; Rodrigues:2016kez ;
FolcoCapossoli:2019imm . In Refs. Elander:2009bm ; Elander:2010wd , by
linearizing the fluctuations around a classical $\sigma$-model coupled to
gravity in $d+1$ dimensions, a gauge invariant (diffeomorphism invariant)
formalism for calculating the spectra of scalar glueballs and tensor glueballs
was developed, which was initially proposed in Refs. Bianchi:2003ug ;
Berg:2005pd ; Berg:2006xy . This algorithmic formalism was tested and some
non-trivial applications were given in Refs. Elander:2012yh ; Elander:2017cle
; Elander:2017hyr ; Elander:2018gte ; Elander:2018aub ; Elander:2020csd ;
Elander:2020ial ; Elander:2020nyd ; Elander:2020fmv ; Elander:2021wkc . The
glueball mass spectra and decay rate in the Sakai-Sugimoto model have been
investigated in Refs. Hashimoto:2007ze ; Brunner:2015oqa ; Brunner:2015yha .
Glueballs and oddballs spectra have also been widely studied by using the
bottom-up approach, where some studies are based on hard-wall Erlich:2005qh
and soft-wall holographic QCD models Karch:2006pv  with the conformal
$AdS_{5}$ background metric.

A realistic non-conformal holographic QCD model should reveal both the
spontaneous chiral symmetry breaking and color charge confinement or linear
confinement, which are two main features of QCD in the low energy regime. In
the top-down approach, the Sakai-Sugimoto (SS) model or $D_{4}-D_{8}$ brane
system Sakai:2004cn ; Sakai:2005yt  is one of the most successful non-
conformal holographic QCD models. In the bottom-up approach, the dynamical
holographic QCD (DhQCD) model constructed in Refs. Li:2012ay ; Li:2013oda ;
Li:2013xpa  can simultaneously describe both chiral symmetry breaking and
linear confinement, where the gluon dynamics background is solved by the
coupling between the graviton and the dilaton field $\Phi(z)$, which is
responsible for the gluon condensate and confinement, and the scalar field
$X(z)$ is introduced to mimic chiral dynamics. Evolution of the dilaton field
and scalar field in $5$-dimensional space-time resemble the renormalization
group from ultraviolet (UV) to infrared (IR). This dynamical holographic QCD
model describes the scalar glueball spectra and the light meson spectral quite
well Li:2012ay ; Li:2013oda ; Li:2013xpa . Further studies Li:2014hja ;
Li:2014dsa ; Chelabi:2015cwn  show that this dynamical holographic QCD model
can also describe QCD phase transition, equation of state of QCD matter and
temperature dependent transport properties, including shear viscosity, bulk
viscosity, electric conductivity as well as jet quenching parameter. Except
the dynamical holographic QCD model, there are several other non-conformal
holographic QCD models in the same graviton-dilaton system which can well
describe non-perturbative QCD propertities, e.g. the Gubser model
Gubser:2008ny ; Gubser:2008yx ; DeWolfe:2010he  and the improved holographic
QCD model Gursoy:2007cb ; Gursoy:2007er ; Gursoy:2010fj  with inputing of a
dilaton potential, and the refined model Yang:2014bqa  and Dudal model
Dudal:2017max  with inputting of a deformed metric.

In the graviton-dilaton system, the metric, the dilaton field and the dilaton
potential are self-consistently solved from each other through the Einstein
field equations and the equation of motion of the dilaton field. In principle,
the three types of models, A) inputting the form of the dilaton field, B)
inputting the deformed metric, and C) inputting the dilaton potential, should
be equivalent to describe the background at zero temperature and zero density.
We will compare the glueball including (scalar, vector as well as tensor
glueballs and their excitations) and oddball spectra, and compare
thermodynamical properties with lattice QCD results for pure gluon system
and/or $2+1$ flavors system in these three types of models.

The paper is organized as following: we introduce the general Einstein-
Maxwell-dilaton framework in section 2. Then in section 3 we introduce five
different models in the graviton-dilaton system. In section 4 we introduce the
glueball and oddball operator and calculate the mass spectra in these models
and we compare the results of mass spectra with lattice results, results from
QCD sum rule and results extracted from high energy scattering data. In
section 5 we compare thermodynamical properties of these models with lattice
results. Finally, a short summary is given in section 6.

##  2 The general Einstein-Maxwell-dilaton system

To keep the self-consistency of investigating the gluebal spectra as well as
further studies on QCD matter at finite temperature and finite chemical
potential, we firstly introduce the general framework of the Einstein-Maxwell-
dilaton (EMD) system, which comes back to the graviton-dilaton coupling system
at zero chemical potential. The total action of $5$-dimensional holographic
QCD model including glueball/oddball excitations takes the following form:

$\displaystyle S_{\text{total }}^{s}=S_{b}^{s}+S_{g}^{s},$ (2)

where $S_{b}^{s}$ is the action for the background in the string frame, and
$S_{g}^{s}$ is the action describing glueballs in the string frame.

The Einstein-Maxwell-dilaton action $S_{b}^{s}$ for the background in the
string frame takes the form of:

$\displaystyle
S_{b}^{s}=\frac{1}{2\kappa_{5}^{2}}\int\mathrm{d}^{5}x\sqrt{-g^{s}}e^{-2\Phi}$
$\displaystyle\Big{[}$ $\displaystyle
R^{s}+4{g^{s}}^{MN}\partial_{M}\Phi\partial_{N}\Phi-V^{s}(\Phi)$
$\displaystyle-\frac{h(\Phi)}{4}e^{\frac{4\Phi}{3}}{g^{s}}^{M\widetilde{M}}{g^{s}}^{N\widetilde{N}}F_{MN}F_{\widetilde{M}\widetilde{N}}\Big{]},$
(3)

where $s$ denotes the string frame, $\kappa_{5}^{2}=8\pi G_{5}$, the $G_{5}$
is the $5$-dimensional Newton constant. The $g^{s}$ is the determinant of the
metric in the string frame: $g^{s}=\det\left(g_{MN}\right)$, and the metric
tensor in the string frame is extracted from

$\displaystyle
ds^{2}=\frac{L^{2}e^{2A_{s}(z)}}{z^{2}}\left(-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+dy_{1}^{2}+dy_{3}^{2}+dy_{3}^{2}\right),$
(4)

where $L$ is the curvature radius of the asymptotic $AdS_{5}$ space-time. For
simplicity, we set $L=1$ in the following calculations. The $R^{s}$ is the
Ricci curvature scalar in the string frame. The scalar field $\Phi(z)$ is the
dilaton field which depends only on the coordinate $z$, $F_{MN}$ is the field
strength of the $U(1)$ gauge field $A_{M}$:

$\displaystyle F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}.$ (5)

The $5$-dimensional field $A_{M}$ is dual to baryon number current. $h(\Phi)$
describes the coupling strength of $A_{M}$ in the theory, $V^{s}(\Phi)$
represents the potential of the dilaton field in the string frame. $h(\Phi)$
and $V^{s}(\Phi)$ are the functions that depends only on the value of $\Phi$.

###  2.1 The Einstein-Maxwell-dialton system in the Einstein frame

As discussed in Ref. Li:2011hp , it is convenient to calculate the vacuum
expectation value of the loop operator in the string frame, and it is more
convenient to work out the gravity solution and to study equation of state in
the Einstein frame. So we apply the Weyl transformation weyl1921raum ;
weyl1993space

$\displaystyle g_{MN}^{s}=\mathrm{e}^{\frac{4}{3}\Phi}g_{MN}^{E}$ (6)

to Eq. (3). Here $g_{MN}^{E}$ is the metric tensor in the Einstein frame, the
capital letter ’E’ denotes the Einstein frame. Then, Eq. (3) can be written as

$\displaystyle
S^{E}=\frac{1}{2\kappa_{5}^{2}}\int\mathrm{d}^{5}x\sqrt{-g^{E}}$
$\displaystyle\Big{[}$ $\displaystyle
R^{E}-\frac{4}{3}{g^{E}}^{MN}\partial_{M}\Phi\partial_{N}\Phi-V^{E}(\Phi)$
$\displaystyle-\frac{h(\Phi)}{4}{g^{E}}^{M\widetilde{M}}{g^{E}}^{N\widetilde{N}}F_{MN}F_{\widetilde{M}\widetilde{N}}\Big{]},$
(7)

with $V^{E}=\mathrm{e}^{\frac{4}{3}\Phi}V^{s}$.

Then we define a new dilaton field $\phi$:

$\displaystyle\phi=\sqrt{\frac{8}{3}}\Phi.$ (8)

Now Eq. (7) becomes

$\displaystyle
S^{E}=\frac{1}{2\kappa_{5}^{2}}\int\mathrm{d}^{5}x\sqrt{-g^{E}}$
$\displaystyle\Big{[}$ $\displaystyle
R^{E}-\frac{1}{2}{g^{E}}^{MN}\left(\partial_{M}\phi\right)\left(\partial_{N}\phi\right)-V_{\phi}(\phi)$
$\displaystyle-\frac{h_{\phi}(\phi)}{4}{g^{E}}^{M\widetilde{M}}{g^{E}}^{N\widetilde{N}}F_{MN}F_{\widetilde{M}\widetilde{N}}\Big{]},$
(9)

where $V_{\phi}(\phi)=V^{E}(\Phi)$, and $h_{\phi}(\phi)=h(\Phi)$. According to
Eqs. (4), (6) and (8), we can derive the line element in Einstein frame:

$\displaystyle
ds^{2}=\frac{L^{2}e^{2A_{E}(z)}}{z^{2}}\left(-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+dy_{1}^{2}+dy_{3}^{2}+dy_{3}^{2}\right),$
(10)

where

$\displaystyle A_{E}(z)=A_{s}(z)-\sqrt{\frac{1}{6}}\phi(z).$ (11)

After applying variation to Eq. (9), we can derive the Einstein field
equations and the equations of motion of $A_{M}$ and $\phi$ as follows

$\displaystyle R_{MN}^{E}-\frac{1}{2}g_{MN}^{E}R^{E}-T_{MN}=0,$
$\displaystyle\nabla_{M}\left[h_{\phi}(\phi)F^{MN}\right]=0,$
$\displaystyle\partial_{M}\left[\sqrt{-g}\partial^{M}\phi\right]-\sqrt{-g}\left(\frac{\mathrm{d}V_{\phi}(\phi)}{\mathrm{d}\phi}+\frac{F^{2}}{4}\frac{\mathrm{d}h_{\phi}(\phi)}{\mathrm{d}\phi}\right)=0,$
(12)

with the energy-momentum tensor $T_{MN}$

$\displaystyle T_{MN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(\partial_{M}\phi\right)\left(\partial_{N}\phi\right)-\frac{1}{2}g_{MN}^{E}{g^{E}}^{P\widetilde{P}}\left(\partial_{P}\phi\right)\left(\partial_{\widetilde{P}}\phi\right)-g_{MN}^{E}V_{\phi}(\phi)\right]$
$\displaystyle+\frac{h_{\phi}(\phi)}{2}\left({g^{E}}^{P\widetilde{P}}F_{MP}F_{N\widetilde{P}}-\frac{1}{4}g_{MN}^{E}{g^{E}}^{P\widetilde{P}}{g^{E}}^{Q\widetilde{Q}}F_{PQ}F_{\widetilde{P}\widetilde{Q}}\right).$
(13)

We can safely suppose all the components of $A_{M}(z)$ are zero except
$A_{t}(z)$. Substituting Eq. (10) into the EOMs Eq. (12), we then derive the
EOMs for the components:

$\displaystyle
A_{t}^{\prime\prime}+A_{t}^{\prime}\left(-\frac{1}{z}+\frac{{h_{\phi}}^{\prime}}{h_{\phi}}+{A_{E}}^{\prime}\right)=0,$
(14) $\displaystyle
f^{\prime\prime}+f^{\prime}\left(-\frac{3}{z}+3{A_{E}}^{\prime}\right)-\frac{e^{-2{A_{E}}}A_{t}^{\prime
2}z^{2}h_{\phi}}{L^{2}}=0,$ (15) $\displaystyle
A_{E}^{\prime\prime}+\frac{f^{\prime\prime}}{6f}+A_{E}^{\prime}\left(-\frac{6}{z}+\frac{3f^{\prime}}{2f}\right)-\frac{1}{z}\left(-\frac{4}{z}+\frac{3f^{\prime}}{2f}\right)+3{A_{E}}^{\prime
2}+\frac{L^{2}e^{2{A_{E}}}V_{\phi}}{3z^{2}f}=0,$ (16) $\displaystyle
A_{E}^{\prime\prime}-A_{E}^{\prime}\left(-\frac{2}{z}+A_{E}^{\prime}\right)+\frac{\phi^{\prime
2}}{6}=0,$ (17)
$\displaystyle\phi^{\prime\prime}+\phi^{\prime}\left(-\frac{3}{z}+\frac{f^{\prime}}{f}+3A_{E}^{\prime}\right)-\frac{L^{2}e^{2{A_{E}}}}{z^{2}f}\frac{\mathrm{d}V_{\phi}(\phi)}{\mathrm{d}\phi}+\frac{z^{2}e^{-2{A_{E}}}A_{t}^{\prime
2}}{2L^{2}f}\frac{\mathrm{d}h_{\phi}(\phi)}{\mathrm{d}\phi}=0.$ (18)

In the above 5 equations, only 4 of them are independent. Thus we can choose
Eq. (18) as a constraint, which can be used to check the solutions.

##  3 Five different models in the EMD system

In the dilaton-graviton system, the metric, the dilaton field and the dilaton
potential can be self-consistently solved from each other through the Einstein
field equations and the equation of motion of the dilaton field. At zero
temperature and zero chemical potential, the function $f(z)=1$ and
$A_{t}(z)=0$, then Eq. (14) to Eq. (18) can be simplified:

$\displaystyle
A_{E}^{\prime\prime}-\frac{6}{z}A_{E}^{\prime}+\frac{4}{z^{2}}+3{A_{E}}^{\prime
2}+\frac{L^{2}e^{2{A_{E}}}V_{\phi}}{3z^{2}}=0,$ (19) $\displaystyle
A_{E}^{\prime\prime}-A_{E}^{\prime}\left(-\frac{2}{z}+A_{E}^{\prime}\right)+\frac{\phi^{\prime
2}}{6}=0,$ (20)
$\displaystyle\phi^{\prime\prime}+\phi^{\prime}\left(-\frac{3}{z}+3A_{E}^{\prime}\right)-\frac{L^{2}e^{2{A_{E}}}}{z^{2}}\frac{\mathrm{d}V_{\phi}(\phi)}{\mathrm{d}\phi}=0,$
(21)

where Eq. (21) is the constraint. Under the condition that we have proper
boundary conditions, if we input A) the form of the dilaton field $\phi(z)$,
or B) the function $A_{E}(z)$, or C) the dilaton potential $V_{\phi}(\phi)$,
we can solve the other two. In principle, these three types of models of EMD
system are totally equivalent to describe the background in the vacuum.
However, at finite temperature and finite chemical potential, the situation
will become different. If we input $V_{\phi}(\phi)$, the form of
$V_{\phi}(\phi)$ is independent of the temperature/chemical potential, from
Eq. (14) and Eq. (18), we can solve different functions $A_{E}(z)$ and
$\phi(z)$ at different temperature/chemical potential, which can be denoted as
${A_{E}}_{\,T,\,\mu}(T,\mu,z)$ and $\phi_{T,\,\mu}(T,\mu,z)$. On the other
hand, if we input $A_{E}(z)$ (or $\phi(z)$), whose form is independent of
temperature/chemical potential, we can derive $V_{\phi}(\phi)$ with
temperature/chemical potential dependence, which can be denoted as
${V_{\phi}}_{\,T,\,\mu}(T,\mu,\phi)$. The two descriptions, that are
equivalent at vacuum, now become distinct from each other at finite
temperature/chemical potential. From now on, we call fixing $V_{\phi}(\phi)$
"description A", fixing $A_{E}(z)$ or $\phi(z)$ is denoted by "description B".

It is more convenient to solve the system in the Einstein frame from Eqs. (19)
$\sim$ (21). In the following we list two sets of vacuum solutions of
$V_{\phi}(\phi)$, $A_{E}(z)$ and $\phi(z)$ that satisfy the EOMs.

###  3.1 Vacuum solutions: set I

From the experiences in Refs. Dudal:2017max ; Yang:2014bqa , we can input the
function $A_{E}(z)$ in the Einstein frame, and solve $V_{\phi}(\phi)$ and
$\phi(z)$. The simplest ansatz for the deformed metric is $A_{E}(z)=-az^{2}$,
and from Eqs. (19) $\sim$ (21) one can derive the solution as following:

$\displaystyle A_{E}(z)=-az^{2},$ (22) $\displaystyle
V_{\phi}(\phi)=-\frac{6}{L^{2}}\mathrm{e}^{2\left(k(\phi)\right)^{2}}\left(6\left(k(\phi)\right)^{4}+5\left(k(\phi)\right)^{2}+2\right),$
(23)
$\displaystyle\phi(z)=z\sqrt{3a\left(3+2az^{2}\right)}+\frac{3}{2}\sqrt{6}\operatorname{arcsinh}\left[\sqrt{\frac{2a}{3}}z\right],$
(24)

where the auxiliary function $k(\varphi)$ is defined as the inverse function
of

$\displaystyle\varphi(\mathfrak{z})={\mathfrak{z}}\sqrt{3\left(3+2{\mathfrak{z}}^{2}\right)}+\frac{3}{2}\sqrt{6}\operatorname{arcsinh}\left[\sqrt{\frac{2}{3}}{\mathfrak{z}}\right],$
(25)

which means $k(\varphi(\mathfrak{z}))=\mathfrak{z}$ with
$\mathfrak{z}=\sqrt{a}z$. Starting from any of the above three functions,
together with proper boundary conditions, we can solve other two functions
from Eq. (19) and Eq. (20).

From Eq. (24) we know that $\phi(z=0)=0$,
$\lim_{z\to+\infty}\phi(z)\to+\infty$. At UV boundary ($z\to 0$), the
asymptotic forms of $V_{\phi}(\phi)$ and $\phi$ are given below:

$\displaystyle L^{2}V_{\phi}(\phi\to
0)=-12-\frac{3}{2}\phi^{2}-\frac{1}{12}\phi^{4}-\frac{377}{174960}\phi^{6}-\frac{977}{33067440}\phi^{8}-\frac{53483}{214277011200}\phi^{10}$
$\displaystyle\qquad\quad-\frac{1564351}{1145524901875200}\phi^{12}+\mathcal{O}\left({\phi}^{14}\right),$
(26) $\displaystyle\phi(z\to
0)=6\sqrt{a}z+\frac{2}{3}a^{\frac{3}{2}}z^{3}-\frac{1}{15}a^{\frac{5}{2}}z^{5}+\frac{1}{63}a^{\frac{7}{2}}z^{7}-\frac{5}{972}a^{\frac{9}{2}}z^{9}+\frac{7}{3564}a^{\frac{11}{2}}z^{11}$
$\displaystyle\qquad\quad-\frac{7}{8424}a^{\frac{13}{2}}z^{13}+\mathcal{O}\left(z^{15}\right).$
(27)

From the UV asymptotic form of $V_{\phi}(\phi)$, we can extract the
$5$-dimensional mass square of $\phi$

$\displaystyle M_{\phi}^{2}=-3.$ (28)

According to the mass-dimension relationship $M^{2}=(\Delta-p)(\Delta+p-4)$
and $p=0$, the dimension

$\displaystyle{\Delta_{\phi}}_{-}=1,\quad{\Delta_{\phi}}_{+}=3.$ (29)

At IR boundary ($z\to+\infty$), $V_{\phi}(\phi)$ and $\phi(z)$ behave as

$\displaystyle
L^{2}V_{\phi}(\phi\to+\infty)=-\frac{27}{4}\left(\frac{3}{8}\right)^{\frac{1}{4}}\mathrm{e}^{-\frac{3}{2}}\mathrm{e}^{\frac{\sqrt{6}}{3}\phi+\cdots}\left(\phi^{\frac{1}{2}}+\cdots\right),$
(30)
$\displaystyle\phi(z\to+\infty)=\sqrt{6}\bigg{[}az^{2}+\frac{3}{4}\left(1+\ln\left(\frac{8}{3}\right)+\ln\left(az^{2}\right)\right)+\frac{9}{32}\frac{1}{az^{2}}-\frac{27}{256}\frac{1}{a^{2}z^{4}}+\frac{135}{2048}\frac{1}{a^{3}z^{6}}$
$\displaystyle\qquad\qquad\qquad-\frac{1701}{32768}\frac{1}{a^{4}z^{8}}+\frac{15309}{327680}\frac{1}{a^{5}z^{10}}-\frac{24057}{524288}\frac{1}{a^{6}z^{12}}+\mathcal{O}\left(\frac{1}{z^{14}}\right)\bigg{]}.$
(31)

Eq. (23) lead to the masses of glueballs $m_{n}$ behave as

$\displaystyle{m}_{n}\sim{n}^{\frac{1}{2}},\qquad\text{when}\quad
n\to+\infty.$ (32)

which shows the linear Regge behavior along $n$.

###  3.2 Vacuum solutions: set II

As for another set of solution, we start from the form of $\phi(z)$. One
simple but nontrivial ansatz is to take the quadratic form of $\phi(z)$:
$\phi(z)=bz^{2}$. As discussed in Refs. Li:2011hp ; Li:2012ay ; Li:2013oda ;
Li:2013xpa , the quadratic form of the dilaton field is dual to a dimension-2
gluon condensation operator, which is responsible for the linear confinement
of the gluon system. Then the solution $V_{\phi}(\phi)$, $A_{E}(z)$ and
$\phi(z)$ take the form of

$\displaystyle\phi(z)=bz^{2},$ (33) $\displaystyle
V_{\phi}(\phi)=\frac{1}{L^{2}}2\times 2^{\frac{3}{4}}\times
3^{\frac{1}{4}}\phi^{\frac{3}{2}}\left[\Gamma\left(\frac{5}{4}\right)\right]^{2}\left\\{\left[I_{\frac{1}{4}}\left(\frac{\phi}{\sqrt{6}}\right)\right]^{2}-4\left[I_{-\frac{3}{4}}\left(\frac{\phi}{\sqrt{6}}\right)\right]^{2}\right\\},$
(34) $\displaystyle A_{E}(z)=-\ln\left[\frac{2^{\frac{3}{8}}\times
3^{\frac{1}{8}}\Gamma\left(\frac{5}{4}\right)I_{\frac{1}{4}}\left(\frac{bz^{2}}{\sqrt{6}}\right)}{{b^{\frac{1}{4}}\sqrt{z}}}\right],$
(35)

where $\Gamma(z)$ is the Euler gamma function, $I_{n}(z)$ is the modified
Bessel function of the first kind.

From Eq. (33) we know that $\phi(z=0)=0$,
$\lim_{z\to+\infty}\phi(z)\to+\infty$. At UV boundary ($z\to 0$), the
asymptotic form are

$\displaystyle L^{2}V_{\phi}(\phi\to
0)=-12-2{\phi}^{2}-\frac{4}{15}{\phi}^{4}-\frac{49}{6075}{\phi}^{6}-\frac{11}{94770}{\phi}^{8}-\frac{11}{11153700}{\phi}^{10}$
$\displaystyle\qquad\qquad\qquad\quad-\frac{38}{6851160225}{\phi}^{12}+\mathcal{O}\left({\phi}^{14}\right),$
(36) $\displaystyle A_{E}(z\to
0)=-\frac{1}{30}b^{2}z^{4}+\frac{1}{4050}b^{4}z^{8}-\frac{4}{1184625}b^{6}z^{12}+\mathcal{O}\left(z^{16}\right).$
(37)

From the UV asymptotic form of $V_{\phi}(\phi)$, we can extract the
$5$-dimensional mass square of $\phi$

$\displaystyle M_{\phi}^{2}=-4.$ (38)

According to the mass-dimension relationship $M^{2}=(\Delta-p)(\Delta+p-4)$
and $p=0$, the dimension

$\displaystyle{\Delta_{\phi}}_{-}={\Delta_{\phi}}_{+}=2.$ (39)

At IR boundary ($z\to+\infty$), $V_{\phi}(\phi)$ and $A_{E}(z)$ behave as

$\displaystyle
L^{2}V_{\phi}(\phi\to\infty)=-\frac{1}{\pi}{2}^{\frac{5}{4}}\times{3}^{\frac{7}{4}}\left[\Gamma\left(\frac{5}{4}\right)\right]^{2}\mathrm{e}^{\frac{\sqrt{6}}{3}\phi}\phi^{\frac{1}{2}}\bigg{[}1-\frac{23\sqrt{6}}{48}\frac{1}{\phi}-\frac{277}{256}\frac{1}{{\phi}^{2}}-\frac{4365\sqrt{6}}{4096}\frac{1}{{\phi}^{3}}$
$\displaystyle\qquad\qquad\quad-\frac{1271565}{131072}\frac{1}{{\phi}^{4}}-\frac{41182155\sqrt{6}}{2097152}\frac{1}{{\phi}^{5}}-\frac{9973379745}{33554432}\frac{1}{{\phi}^{6}}-\frac{481731815565\sqrt{6}}{536870912}\frac{1}{{\phi}^{7}}$
$\displaystyle\qquad\qquad\quad-\frac{650211981544125}{34359738368}\frac{1}{{\phi}^{8}}-\frac{41734532955290175\sqrt{6}}{549755813888}\frac{1}{{\phi}^{9}}$
$\displaystyle\qquad\qquad\quad-\frac{18065483595471987675}{8796093022208}\frac{1}{{\phi}^{10}}-\frac{1447278481564158318075\sqrt{6}}{140737488355328}\frac{1}{{\phi}^{11}}$
$\displaystyle\qquad\qquad\quad-\frac{1529878341359963470300425}{4503599627370496}\frac{1}{{\phi}^{12}}-\frac{146978223450520872139104375\sqrt{6}}{72057594037927936}\frac{1}{{\phi}^{13}}$
$\displaystyle\qquad\qquad\quad+\mathcal{O}\left(\frac{1}{{\phi}^{14}}\right)\bigg{]}\left\\{1+\frac{5\sqrt{2}}{6}{\mathrm{e}}^{-\frac{\sqrt{6}}{3}\phi}\left[1+\mathcal{O}\left(\frac{1}{\phi}\right)\right]\right\\},$
(40) $\displaystyle
A_{E}(z\to\infty)=-\frac{\sqrt{6}}{6}bz^{2}+\frac{3}{2}\ln\left(\sqrt{b}z\right)+\ln\left[\frac{{\pi}^{\frac{1}{2}}}{2^{\frac{1}{8}}3^{\frac{3}{8}}\Gamma\left(\frac{5}{4}\right)}\right]-\frac{3\sqrt{6}}{32}\frac{1}{bz^{2}}-\frac{9}{32}\frac{1}{b^{2}z^{4}}$
$\displaystyle\qquad\qquad\quad-\frac{297\sqrt{6}}{1024}\frac{1}{b^{3}z^{6}}-\frac{1377}{512}\frac{1}{b^{4}z^{8}}-\frac{451251\sqrt{6}}{81920}\frac{1}{b^{5}z^{10}}-\frac{172287}{2048}\frac{1}{b^{6}z^{12}}+\mathcal{O}\left(\frac{1}{z^{14}}\right)$
$\displaystyle\qquad\qquad\quad+\mathcal{O}\left({\mathrm{e}}^{-\frac{\sqrt{6}}{3}bz^{2}}\right)+\mathcal{O}\left(\frac{1}{z^{2}}{\mathrm{e}}^{-\frac{\sqrt{6}}{3}bz^{2}}\right).$
(41)

Again, from the asymptotic expansion of $V_{\phi}(\phi)$ at IR boundary, we
can conclude that linear Regge behavior of the masses of glueballs $m_{n}$:

$\displaystyle{m}_{n}^{2}\sim n,\qquad\text{when}\qquad n\to+\infty,$ (42)

which shows the linear Regge behavior along $n$.

###  3.3 Five different models

The two sets of vacuum solutions listed above have linear confinement and can
produce glueball bound state. Not all models can show such feature. According
to Refs. Gursoy:2010fj ; Gursoy:2008za , if we require that the theory is
confined and bad singularities are absent, the asymptotic behavior of
$V_{\phi}(\phi)$ at IR boundary should be

$\displaystyle
L^{2}V_{\phi}(\phi\to+\infty)=c_{V}\,\mathrm{e}^{c_{\phi,1}\phi+\cdots}\left(\phi^{c_{\phi,2}}+\cdots\right),$
$\displaystyle\left\\{\begin{aligned}
&\frac{\sqrt{6}}{3}<c_{\phi,1}<\frac{2\sqrt{3}}{3},\quad
c_{\phi,2}\,\,\text{is real number},\\\ &c_{\phi,1}=\frac{\sqrt{6}}{3},\quad
c_{\phi,2}\geqslant 0,\end{aligned}\right.$ (43)

where $c_{V}$ is constant. When $c_{\phi,1}=\frac{\sqrt{6}}{3}$ and
$c_{\phi,2}>0$, there are asymptotically linear glueball spectra:

$\displaystyle m_{n}\sim n^{c_{\phi,2}}\qquad\text{when}\quad n\to+\infty.$
(44)

For comparison, we plot three different dilaton potentials $V_{\phi}(\phi)$ in
Fig. 1. One of them is the Gubser model taken from Ref. DeWolfe:2010he :

$\displaystyle
V_{\phi}(\phi)=\frac{-12\cosh{\left(0.606\phi\right)}+2.057{\phi}^{2}}{L^{2}},$
(45)

the others two are Eq. (23) and Eq. (34). Here we set $L=1$. The dashed black
line is ${\mathrm{e}}^{\frac{\sqrt{6}}{3}\phi}$. According to the conclusion
in subsection 3.1, if the potential is more gradual than this line, such as
the blue line that represents the Gubser model in Eq. (45), the theory is
gapless and non-confining.

Figure 1: These are three different dilaton potentials $V_{\phi}(\phi)$. The
longitudinal axis is the value of $-V_{\phi}(\phi)$ in logarithm coordinate.
The horizontal axis is the value of $\phi$. The dashed black line is
${\mathrm{e}}^{\frac{\sqrt{6}}{3}\phi}$. The blue line, orange line and green
line represent the potential in Eq. (45), Eq. (23) and Eq. (34) respectively.
The meaning of "model II" and "model IV" will be explained later. The dashed
black line is ${\mathrm{e}}^{\frac{\sqrt{6}}{3}\phi}$. The bound is given by
Eq.(44) from Refs. Gursoy:2010fj ; Gursoy:2008za . If the potential is more
gradual than this bound, the theory is gapless and non-confining.

As we stated below Eqs. (19) $\sim$ (21), there are two different descriptions
of the input of EMD system. Combining with the two different sets of vacuum
solutions Eqs. (23) $\sim$ (24) and Eq. (34) $\sim$ (33), we consider five
models in this article.

####  3.3.1 Model I and II

In model I, we use description-B and input $A_{E}(z)$ as Eq. (22):

$\displaystyle A_{E}(z)=-az^{2}.$ (46)

Note that the dimension of the parameter $a$ is ${\left[E\right]}^{2}$ and its
value decides the energy scale of the EMD system. At vacuum, the boundary
condition is chosen as

$\displaystyle\phi(z=0)=0.$ (47)

Combining the boundary condition Eq. (47) with the EOMs Eq. (20) and Eq. (19),
we can solve $\phi(z)$ and $V_{\phi}(\phi)$. The results are Eq. (24) and Eq.
(23).

As for finite temperature and finite chemical potential, the EOMs are Eq. (14)
$\sim$ (17). There may exist the black hole 111 Actually, the black hole may
not exist for all values of temperature and chemical potential. Under some
situations, it only exists above a certain temperature $T_{\text{min}}$. The
$AdS$ thermal gas is the only solution below $T_{\text{min}}$. However, under
other situations, we always have the black hole solution.  in space-time
manifold, the metric of which in conformal coordinate $z$ is Eq. (10). The
boundary conditions are given as

$\displaystyle A_{t}(z=0)=\mu,$ $\displaystyle A_{t}(z=z_{h})=0,$
$\displaystyle f(z=0)=1,$ $\displaystyle f(z=z_{h})=0,$
$\displaystyle\phi(z=0)=0,$ (48)

where $z=z_{h}$ is the location of the event horizon of black hole on the
coordinate $z$, $\mu$ is the chemical potential. Besides the boundary
condition Eq. (48), the form of $h_{\phi}(\phi)$ are also needed to solve the
EOMs. However, we consider the $\mu=0$ case, which means $A_{t}(z)=0$ through
the article. Thus our calculations and results are independent on
$h_{\phi}(\phi)$.

In model II, we use description-A and input $V_{\phi}(\phi)$ as Eq. (23):

$\displaystyle
V_{\phi}(\phi)=-\frac{6}{L^{2}}\mathrm{e}^{2\left(k(\phi)\right)^{2}}\left(6\left(k(\phi)\right)^{4}+5\left(k(\phi)\right)^{2}+2\right).$
(49)

At vacuum, the boundary conditions are chosen as

$\displaystyle A_{E}(z=0)=0,$ (50) $\displaystyle\phi(z=0)=0,$ (51)
$\displaystyle\left.\frac{\mathrm{d}\phi(z)}{\mathrm{d}z}\right|_{z=0}=6\sqrt{a}.$
(52)

Eq. (50) guarantees the space-time is asymptotic $AdS_{5}$ at UV boundary. Eq.
(52) contains a parameter $a$, the dimension of which is
${\left[E\right]}^{2}$ and the value of which decides the energy scale of the
EMD system. Given by these boundary conditions and set the value of $a$ in Eq.
(52) to be the same with that in Eq. (46) , we can solve the EOMs at vacuum,
then it will be found that the solutions are totally equivalent to those in
model I at vacuum.

As for finite temperature and finite chemical potential, the EOMs are Eq. (14)
$\sim$ (17). The boundary conditions are given as

$\displaystyle A_{t}(z=0)=\mu,$ $\displaystyle A_{t}(z=z_{h})=0,$
$\displaystyle f(z=0)=1,$ $\displaystyle f(z=z_{h})=0,$ $\displaystyle
A_{E}(z=0)=0,$ $\displaystyle\phi(z=0)=0,$
$\displaystyle\left.\frac{\mathrm{d}\phi(z)}{\mathrm{d}z}\right|_{z=0}=6\sqrt{a},$
(53)

where $z=z_{h}$ is the location of the event horizon of black hole on the
coordinate $z$, $\mu$ is the chemical potential. We should emphasize here that
at finite temperature or finite chemical potential case, the solutions here
are different from those in model I.

####  3.3.2 Model III and IV

In model III, we use description B and input $\phi(z)$ as Eq. (33):

$\displaystyle\phi(z)=bz^{2}.$ (54)

Note that the dimension of the parameter $b$ is ${\left[E\right]}^{2}$ and its
value decides the energy scale of the EMD system. Substituting Eq. (54) into
Eq. (20), we can solve a general solution for $A_{E}(z)$ with two integration
constants. However, the value of this general solution is usually complex. If
we force the reality of $A_{E}(z)$ and consider the boundary condition

$\displaystyle A_{E}(z=0)=0,$ (55)

we can solve $A_{E}(z)$ and $V_{\phi}(\phi)$. The results are Eq. (35) and Eq.
(33).

As for finite temperature and finite chemical potential, the EOMs are Eqs.
(14) $\sim$ (17). The boundary conditions are imposed as

$\displaystyle A_{t}(z=0)=\mu,$ $\displaystyle A_{t}(z=z_{h})=0,$
$\displaystyle f(z=0)=1,$ $\displaystyle f(z=z_{h})=0,$ (56)

where $z=z_{h}$ is the location of the event horizon of black hole on the
coordinate $z$, $\mu$ is the chemical potential. Collecting these boundary
conditions and Eq. (54), Eq. (35), we can then solve the EMD system.

In model IV, we use description A and input $V_{\phi}(\phi)$ as Eq. (34):

$\displaystyle V_{\phi}(\phi)=\frac{1}{L^{2}}2\times 2^{\frac{3}{4}}\times
3^{\frac{1}{4}}\phi^{\frac{3}{2}}\left[\Gamma\left(\frac{5}{4}\right)\right]^{2}\left\\{\left[I_{\frac{1}{4}}\left(\frac{\phi}{\sqrt{6}}\right)\right]^{2}-4\left[I_{-\frac{3}{4}}\left(\frac{\phi}{\sqrt{6}}\right)\right]^{2}\right\\}.$
(57)

At vacuum, the boundary conditions are chosen as

$\displaystyle A_{E}(z=0)=0,$ (58) $\displaystyle\lim_{z\to
0}\frac{\phi(z)}{z^{2}}=b,$ (59)

Again, we force that $A_{E}(z)$ is real. The Eq. (58) guarantees the space-
time is asymptotic $AdS_{5}$ at UV boundary. Eq. (59) contains a parameter
$b$, the dimension of which is ${\left[E\right]}^{2}$ and the value of which
decides the energy scale of the EMD system. Given by these boundary conditions
and set the value of $b$ in Eq. (59) to be the same with that in Eq. (54), we
can solve the EOMs at vacuum, then it will be found that the solutions are
totally equivalent to those in model III at vacuum.

As for finite temperature and finite chemical potential, the EOMs are Eqs.
(14) $\sim$ (17). The boundary conditions are given as

$\displaystyle A_{t}(z=0)=\mu,$ $\displaystyle A_{t}(z=z_{h})=0,$
$\displaystyle f(z=0)=1,$ $\displaystyle f(z=z_{h})=0,$ $\displaystyle
A_{E}(z=0)=0,$ $\displaystyle\phi(z=0)=0.$ (60)

Given by these boundary conditions, we still have the freedom to choose the
energy scale of the EMD system. We should emphasis here that at finite
temperature or finite chemical potential case, the solutions here are
different from those in model III.

####  3.3.3 Model V

In model V, we input $\phi(z)$ as

$\displaystyle\phi(z)=\frac{2\sqrt{6}}{3}z\sqrt{3d\left(3+2dz^{2}\right)}+6\operatorname{arcsinh}\left[\sqrt{\frac{2d}{3}}z\right].$
(61)

Note that the dimension of the parameter $d$ is ${\left[E\right]}^{2}$ and its
value decides the energy scale of the EMD system. Substituting Eq. (61) into
Eq. (20), we force the reality of $A_{E}(z)$ and consider the boundary
condition

$\displaystyle A_{E}(z=0)=0,$ (62)

we can solve $A_{E}(z)$ and $V_{\phi}(\phi)$ numerically. Although we can’t
get the analytical form of $A_{E}(z)$, we can still can derive its asymptotic
expansions:

$\displaystyle A_{E}(z\to
0)=-\frac{8}{3}dz^{2}+\frac{8}{9}d^{2}z^{4}-\frac{512}{567}d^{3}z^{6}+\frac{1664}{1701}d^{4}z^{8}-\frac{311296}{280665}d^{5}z^{10}$
$\displaystyle\qquad\qquad\qquad+\frac{19972096}{15324309}d^{6}z^{12}+\mathcal{O}\left(z^{14}\right),$
(63)

As for finite temperature and finite chemical potential, the EOMs are Eqs.
(14) $\sim$ (17). The boundary conditions are imposed as

$\displaystyle A_{t}(z=0)=\mu,$ $\displaystyle A_{t}(z=z_{h})=0,$
$\displaystyle f(z=0)=1,$ $\displaystyle f(z=z_{h})=0,$ (64)

where $z=z_{h}$ is the location of the event horizon of black hole on the
coordinate $z$, $\mu$ is the chemical potential. Collecting these boundary
conditions, the Eq. (61), and the numerical solution of $A_{E}(z)$, we can
then solve the EMD system.

##  4 Spectra of glueballs and oddballs

In this section, we discuss the spectra of glueballs and oddballs. In our
treatment, the $5$-dimensional fields dual to glueballs/oddballs are excited
from the background. The action describing scalar, vector, and tensor
glueballs/oddballs in the string frame is

$\displaystyle
S_{g}^{s}=-c_{g}\int\mathrm{d}^{5}x\sqrt{-g_{s}}\mathrm{e}^{-p\Phi}\Bigg{\\{}$
$\displaystyle\bigg{[}\frac{1}{2}{g^{s}}^{MN}\partial_{M}\mathscr{S}(z)\partial_{N}\mathscr{S}+\frac{1}{2}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{S},5}^{2}\mathscr{S}^{2}\bigg{]}$
$\displaystyle+\bigg{[}\frac{1}{4}{g^{s}}^{M\widetilde{M}}{g^{s}}^{N\widetilde{N}}\left(\partial_{M}\mathscr{V}_{N}-\partial_{N}\mathscr{V}_{M}\right)\left(\partial_{\widetilde{M}}\mathscr{V}_{\widetilde{N}}-\partial_{\widetilde{N}}\mathscr{V}_{\widetilde{M}}\right)$
$\displaystyle\qquad+\frac{1}{2}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{V},5}^{2}\mathscr{V}^{2}\bigg{]}$
$\displaystyle+\bigg{[}\frac{1}{2}\nabla_{L}\mathscr{T}_{MN}\nabla^{L}\mathscr{T}^{MN}-\nabla_{L}\mathscr{T}^{LM}\nabla^{N}\mathscr{T}_{NM}$
$\displaystyle\qquad+\nabla_{M}\mathscr{T}^{MN}\nabla_{N}\mathscr{T}-\frac{1}{2}\nabla_{M}\mathscr{T}\nabla^{M}\mathscr{T}$
$\displaystyle\qquad+\frac{1}{2}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{T},5}^{2}\left(\mathscr{T}^{MN}h_{MN}-\mathscr{T}^{2}\right)\bigg{]}$
$\displaystyle+\text{terms for high spin fields (spin $S\geqslant
3$)}\Bigg{\\}},$ (65)

where $s$ denotes the string frame, $c_{g}$ describes the coupling strength of
glueballs part in the whole theory. The fields $\mathscr{S}$,
$\mathscr{V}_{M}$, and $\mathscr{T}_{MN}$ are $5$-dimensional fields that are
dual to scalar glueball, vector glueball, and spin-2 glueball operators
respectively. $\mathscr{T}={g^{s}}^{MN}{\mathscr{T}}_{MN}$, and
${\mathscr{T}}_{MN}$ satisfies the following constraints

$\displaystyle\nabla_{M}{\mathscr{T}}^{MN}=0,\quad{\mathscr{T}}=0,\quad{\mathscr{T}}_{\mu\nu}=\frac{1}{z^{2}}\mathrm{e}^{2A_{s}}\mathscr{T}_{\mu\nu},\quad{\mathscr{T}}_{Mz}=0.$
(66)

As in Ref. Chen:2015zhh , the parameter $p$ is introduced to make a
distinction between glueballs (oddballs) with different P-parity:

$\displaystyle\left\\{\begin{aligned} &p=1,\quad\,\,\,\,\,\text{for even
parity,}\\\ &p=-1,\quad\text{for odd parity.}\end{aligned}\right.$ (67)

Also we introduce a $z$ dependent modified $5$-dimensional mass:

$\displaystyle M_{5}^{2}(z)=\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{5}^{2},$ (68)

where $c_{\text{r.m.}}$ is a constant. The $M_{5}^{2}$ is listed in Table 1
given by the $\text{AdS}_{5}/\text{CFT}_{4}$ correspondence dictionary. The
$\text{AdS}_{5}/\text{CFT}_{4}$ duality gives one-to-one correspondence
between $4$-dimensional operators in the ${\cal N}=4$ Super Yang-Mills theory
and the spectrum of the type IIB string theory on $AdS_{5}\times S^{5}$. Based
on the AdS/CFT dictionary, the conformal dimension of a $p$-form operator at
the ultraviolet (UV) boundary is related to the $5$-dimensional mass square
$M_{5}^{2}$ of its dual field in the bulk as follows Maldacena:1997re ;
Gubser:1998bc ; Witten:1998qj  :

$M_{5}^{2}=(\Delta-p)(\Delta+p-4).$ (69)

###  4.1 Glueballs and oddballs

In the bottom-up holographic QCD models, one can expect a more general
correspondence, i.e. each $4$-dimensional operator ${\cal O}(x)$ corresponds
to a $5$-dimensional field $O(x,z)$ in the bulk theory. To investigate the
glueball spectra, we consider the lowest dimension operators with the
corresponding quantum numbers defined in the field theory living on the
$4$-dimensional boundary. We show the C-even/odd glueball and oddball
operators and their corresponding $5$-dimensional mass square in Table 1.

$J^{PC}$ |  $4$-dimensional operator: $\mathscr{O}(x)$ | $\Delta$ | $p$ | $M_{5}^{2}$
---|---|---|---|---
$0^{++}$ | $Tr(G^{2})=\vec{E}^{a}\cdot\vec{E}^{a}-\vec{B}^{a}\cdot\vec{B}^{a}$ | 4 | 0 | 0
$0^{-+}$ | $Tr(G\tilde{G})=\vec{E}^{a}\cdot\vec{B}^{a}$ | 4 | 0 | 0
$0^{+-}$ | $\operatorname{Tr}\left(\left\\{\left(D_{\tau}G_{\mu\nu}\right),\left(D_{\tau}G_{\rho\nu}\right)\right\\}\left(D_{\mu}G_{\rho\alpha}\right)\right)$ | 9 | 0 | 45
$0^{--}$ | $\operatorname{Tr}\left(\left\\{\left(D_{\tau}G_{\mu\nu}\right),\left(D_{\tau}G_{\rho\nu}\right)\right\\}\left(D_{\mu}\tilde{G}_{\rho\alpha}\right)\right)$ | 9 | 0 | 45
$1^{-+}$ |  $f^{abc}\partial_{\mu}\left[G_{\mu\nu}^{a}\right]\left[G_{v\rho}^{b}\right]\left[G_{\rho\alpha}^{c}\right]$, $f^{abc}\partial_{\mu}\left[G_{\mu\nu}^{a}\right]\left[\tilde{G}_{v\rho}^{b}\right]\left[\tilde{G}_{\rho\alpha}^{c}\right]$, | 7 | 1 | 24
|  $f^{abc}\partial_{\mu}\left[\tilde{G}_{\mu\nu}^{a}\right]\left[G_{v\rho}^{b}\right]\left[\tilde{G}_{\rho\alpha}^{c}\right]$, $f^{abc}\partial_{\mu}\left[\tilde{G}_{\mu\nu}^{a}\right]\left[\tilde{G}_{v\rho}^{b}\right]\left[G_{\rho\alpha}^{c}\right]$ |  |  |
$1^{+-}$ | $d^{abc}\left(\vec{E}_{a}\cdot\vec{E}_{b}\right)\vec{B}_{c}$ | 6 | 1 | 15
$1^{--}$ | $d^{abc}\left(\vec{E}_{a}\cdot\vec{E}_{b}\right)\vec{E}_{c}$ | 6 | 1 | 15
$2^{++}$ | $E_{i}^{a}E_{j}^{a}-B_{i}^{a}B_{j}^{a}-trace$ | 4 | 2 | 4
$2^{-+}$ | $E_{i}^{a}B_{j}^{a}+B_{i}^{a}E_{j}^{a}-trace$ | 4 | 2 | 4
$2^{+-}$ | $d^{abc}\mathcal{S}\left[E_{a}^{i}\left(\vec{E}_{b}\times\vec{B}_{c}\right)^{j}\right]$ | 6 | 2 | 16
$2^{--}$ | $d^{abc}\mathcal{S}\left[B_{a}^{i}\left(\vec{E}_{b}\times\vec{B}_{c}\right)^{j}\right]$ | 6 | 2 | 16
$3^{+-}$ | $d^{abc}\mathcal{S}\left[B_{a}^{i}B_{b}^{j}B_{c}^{k}\right]$ | 6 | 3 | 15
$3^{--}$ | $d^{abc}\mathcal{S}\left[E_{a}^{i}E_{b}^{j}E_{c}^{k}\right]$ | 6 | 3 | 15
Table 1: $5$-dimensional mass square of C-even glueballs and C-odd oddballs.
The operators are taken from Refs. Brower:2000rp ; Pimikov:2017bkk ;
Tang:2015twt ; Chen:2021cjr .

The lowest dimension gauge invariant three-gluon currents that couple to the
exotic $0^{+-}$ and $0^{--}$ glueballs are constructed in Ref. Pimikov:2017bkk
:

$\displaystyle
j_{\alpha}^{0^{+-}}(x)=g_{s}^{3}\operatorname{Tr}\left(\left\\{\left(D_{\tau}G_{\mu\nu}(x)\right),\left(D_{\tau}G_{\rho\nu}(x)\right)\right\\}\left(D_{\mu}G_{\rho\alpha}(x)\right)\right),$
(70) $\displaystyle
j_{\alpha}^{0^{--}}(x)=g_{s}^{3}\operatorname{Tr}\left(\left\\{\left(D_{\tau}G_{\mu\nu}(x)\right),\left(D_{\tau}G_{\rho\nu}(x)\right)\right\\}\left(D_{\mu}\tilde{G}_{\rho\alpha}(x)\right)\right).$
(71)

For trigluon glueball $1^{-+}$, and $2^{+-}$, the currents that match the
unconventional quantum number and satisfy the constraints of the gauge
invariance are given in Refs. Tang:2015twt :

$\displaystyle
j_{\alpha}^{1^{-+},A}(x)=g_{s}^{3}f^{abc}\partial_{\mu}\left[G_{\mu\nu}^{a}(x)\right]\left[G_{v\rho}^{b}(x)\right]\left[G_{\rho\alpha}^{c}(x)\right],$
$\displaystyle
j_{\alpha}^{1^{-+},B}(x)=g_{s}^{3}f^{abc}\partial_{\mu}\left[G_{\mu\nu}^{a}(x)\right]\left[\tilde{G}_{v\rho}^{b}(x)\right]\left[\tilde{G}_{\rho\alpha}^{c}(x)\right],$
$\displaystyle
j_{\alpha}^{1^{-+}},C_{(x)}=g_{s}^{3}f^{abc}\partial_{\mu}\left[\tilde{G}_{\mu\nu}^{a}(x)\right]\left[G_{v\rho}^{b}(x)\right]\left[\tilde{G}_{\rho\alpha}^{c}(x)\right],$
$\displaystyle
j_{\alpha}^{1^{-+},D}(x)=g_{s}^{3}f^{abc}\partial_{\mu}\left[\tilde{G}_{\mu\nu}^{a}(x)\right]\left[\tilde{G}_{v\rho}^{b}(x)\right]\left[G_{\rho\alpha}^{c}(x)\right],$
(72)

and

$\displaystyle
j_{\mu\alpha}^{2^{+-},A}(x)=g_{s}^{3}d^{abc}\left[G_{\mu\nu}^{a}(x)\right]\left[G_{v\rho}^{b}(x)\right]\left[G_{\rho\alpha}^{c}(x)\right],$
$\displaystyle
j_{\mu\alpha}^{2^{+-},B}(x)=g_{s}^{3}d^{abc}\left[G_{\mu\nu}^{a}(x)\right]\left[\tilde{G}_{v\rho}^{b}(x)\right]\left[\tilde{G}_{\rho\alpha}^{c}(x)\right],$
$\displaystyle
j_{\mu\alpha}^{2^{+-},C}(x)=g_{s}^{3}d^{abc}\left[\tilde{G}_{\mu\nu}^{a}(x)\right]\left[G_{v\rho}^{b}(x)\right]\left[\tilde{G}_{\rho\alpha}^{c}(x)\right],$
$\displaystyle
j_{\mu\alpha}^{2^{+-},D}(x)=g_{s}^{3}d^{abc}\left[\tilde{G}_{\mu\nu}^{a}(x)\right]\left[\tilde{G}_{v\rho}^{b}(x)\right]\left[G_{\rho\alpha}^{c}(x)\right],$
(73)

where $d^{abc}$ stands for the totally symmetric $SU_{c}(3)$ structure
constant and
$g^{t}_{\alpha\beta}=g_{\alpha\beta}-\partial_{\alpha}\partial_{\beta}/\partial^{2}$.

###  4.2 Equation of motion for scalar, vector and tensor glueballs/oddballs

From the $5$-dimensional action for the glueball/oddball in the string frame
Eq. (65), we can derive the equation of motion for the glueballs. The equation
of motion for the scalar glueballs $\mathscr{S}$ is given as:

$\displaystyle-z^{3}{\mathrm{e}}^{-(3A_{s}-p\Phi)}\partial_{z}\left[\frac{1}{z^{3}}{\mathrm{e}}^{3A_{s}-p\Phi}\partial_{z}{\mathscr{S}_{n}}\right]$
$\displaystyle+\frac{1}{z^{2}}{\mathrm{e}}^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{S},5}^{2}{\mathscr{S}_{n}}=m_{\mathscr{S},n}^{2}{\mathscr{S}_{n}}.$
(74)

Via the substitution

$\displaystyle\mathscr{S}_{n}\rightarrow
z^{\frac{3}{2}}\mathrm{e}^{-\frac{1}{2}(3A_{s}-p\Phi)}\mathscr{S}_{n},$ (75)

the equation can be brought into Schrödinger-like equation

$\displaystyle-{\mathscr{S}_{n}}^{{}^{\prime\prime}}+V_{\mathscr{S}}{\mathscr{S}_{n}}=m_{\mathscr{G},n}^{2}{\mathscr{S}_{n}},$
(76)

with the $5$-dimensional effective Schrödinger potential

$V_{\mathscr{S}}=\frac{3A_{s}^{{}^{\prime\prime}}+\frac{3}{z^{2}}-p\Phi^{{}^{\prime\prime}}}{2}+\frac{\left[3A_{s}^{{}^{\prime}}-\frac{3}{z}-p\Phi^{{}^{\prime}}\right]^{2}}{4}+\frac{1}{z^{2}}e^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{S},5}^{2}.$
(77)

The equation of motion for the vector glueballs $\mathscr{V}_{M}$ is given as:

$\displaystyle-z\mathrm{e}^{-(A_{s}-p\Phi)}\partial_{z}\left[\frac{1}{z}\mathrm{e}^{A_{s}-p\Phi}\partial_{z}\mathscr{V}_{n}\right]+\frac{1}{z^{2}}\mathrm{e}^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{V},5}^{2}\mathscr{V}_{n}=m_{\mathscr{V},n}^{2}\mathscr{V}_{n}.$
(78)

Via the substitution

$\displaystyle\mathscr{V}_{n}\rightarrow
z^{\frac{1}{2}}e^{-\frac{1}{2}(A_{s}-p\Phi)}\mathscr{V}_{n},$ (79)

the equation can be brought into Schrödinger-like equation

$\displaystyle-\mathscr{V}_{n}^{{}^{\prime\prime}}+V_{\mathscr{V}}\mathscr{V}_{n}=m_{\mathscr{V},n}^{2}\mathscr{V}_{n},$
(80)

with the $5$-dimensional effective Schrödinger potential

$V_{\mathscr{V}}=\frac{A_{s}^{{}^{\prime\prime}}+\frac{1}{z^{2}}-p\Phi^{{}^{\prime\prime}}}{2}+\frac{\left[A_{s}^{{}^{\prime}}-\frac{1}{z}-p\Phi^{{}^{\prime}}\right]^{2}}{4}+\frac{1}{z^{2}}\mathrm{e}^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{V},5}^{2}.$
(81)

The equation of motion for the spin-2 glueballs $\mathscr{T}_{MN}$ is given as

$\displaystyle-z^{3}\mathrm{e}^{-(3A_{s}-p\Phi)}\partial_{z}\left[\frac{1}{z^{3}}\mathrm{e}^{3A_{s}-p\Phi}\partial_{z}\mathscr{T}_{n}\right]+\frac{1}{z^{2}}\mathrm{e}^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{T},5}^{2}\mathscr{T}_{n}=m_{\mathscr{T},n}^{2}\mathscr{T}_{n}.$
(82)

Via the substitution

$\displaystyle\mathscr{T}_{n}\rightarrow
z^{\frac{3}{2}}\mathrm{e}^{-\frac{1}{2}(3A_{s}-p\Phi)}\mathscr{T}_{n},$ (83)

the equation can be brought into Schrödinger-like equation

$\displaystyle-\mathscr{T}_{n}^{{}^{\prime\prime}}+V_{\mathscr{T}}\mathscr{T}_{n}=m_{\mathscr{T},n}^{2}\mathscr{T}_{n},$
(84)

with the $5$-dimensional effective Schrödinger potential

$V_{\mathscr{T}}=\frac{3A_{s}^{{}^{\prime\prime}}+\frac{3}{z^{2}}-p\Phi^{{}^{\prime\prime}}}{2}+\frac{\left[3A_{s}^{{}^{\prime}}-\frac{3}{z}-p\Phi^{{}^{\prime}}\right]^{2}}{4}+\frac{1}{z^{2}}\mathrm{e}^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{T},5}^{2}.$
(85)

According to Ref. Karch:2006pv , the equation of motion for the high spin
glueballs $\mathscr{H}_{M_{1}M_{2}\cdots M_{S}}$, the spin $S$ of which are
larger than $2$, is given as

$\displaystyle-z^{2S-1}\mathrm{e}^{-\left[\left(2S-1\right)A_{s}-p\Phi\right]}\partial_{z}\left[\frac{1}{z^{2S-1}}\mathrm{e}^{\left(2S-1\right)A_{s}-p\Phi}\partial_{z}\mathscr{H}_{n}\right]$
$\displaystyle+\frac{1}{z^{2}}\mathrm{e}^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{H},5}^{2}\mathscr{H}_{n}=m_{\mathscr{H},n}^{2}\mathscr{H}_{n},$
(86)

where $S\geqslant 3$. Via the substitution

$\displaystyle\mathscr{H}_{n}\rightarrow
z^{\frac{2S-1}{2}}\mathrm{e}^{-\frac{1}{2}\left[\left(2S-1\right)A_{s}-p\Phi\right]}\mathscr{H}_{n},$
(87)

the equation can be brought into Schrödinger-like equation

$\displaystyle-\mathscr{H}_{n}^{{}^{\prime\prime}}+V_{\mathscr{H}}\mathscr{H}_{n}=m_{\mathscr{H},n}^{2}\mathscr{H}_{n},$
(88)

with the $5$-dimensional effective Schrödinger potential

$V_{\mathscr{H}}=\frac{\left(2S-1\right)A_{s}^{{}^{\prime\prime}}+\frac{2S-1}{z^{2}}-p\Phi^{{}^{\prime\prime}}}{2}+\frac{\left[\left(2S-1\right)A_{s}^{{}^{\prime}}-\frac{2S-1}{z}-p\Phi^{{}^{\prime}}\right]^{2}}{4}+\frac{1}{z^{2}}\mathrm{e}^{2A_{s}}\mathrm{e}^{-c_{\text{r.m.}}\Phi}M_{\mathscr{H},5}^{2}.$
(89)

###  4.3 Numerical results of glueballs/oddballs spectra

We calculate the glueballs spectra using five different holographic models
defined in last section. We list the parameters used for calculating the
glueballs spectra below.

####  4.3.1 Model I and II

In model I and model II, we choose the parameter $a=0.6032\mathrm{GeV}^{2}$.
Firstly, we don’t consider the distinction between glueballs (oddballs) with
different P-parity and don’t introduce $z$ dependent modified $5$-dimensional
masses, that means $p=1$ for even and odd parity, and $c_{\text{r.m.}}=0$.
Then we calculate the glueballs/oddballs mass spectra in model I and II, which
is denoted by "Model I,II(O)" in Tab. 2. We find the calculation results of
the masses of glueballs/oddballs, of which the $5$D mass square $M_{5}^{2}$ in
Tab. 1 are large, are much heavier than the lattice data, as we mentioned in
subsubsection 4.3.1. That’s why we introduce a $z$-dependent modified
$5$-dimensional mass of glueball/oddball fields in Eq. (68). The value of the
constant $c_{\text{r.m.}}$ in Eq. (68) is chosen as $\frac{3}{5}$, which means

$\displaystyle
M_{5}^{2}(z)={\mathrm{e}}^{-\frac{3}{5}\Phi}M_{5}^{2},~{}~{}~{}\text{model I,
and II}.$ (90)

The results of glueballs/oddballs spectra are denoted by "Model I,II" in Tab.
3.

Note that the $5$-dimensional field $\Phi$ and $\phi$ are different, the
relationship between them is Eq. (8):

$\displaystyle\phi=\sqrt{\frac{8}{3}}\Phi.$

####  4.3.2 Model III and IV

In model III and model IV, we choose the parameter $b=1.760\mathrm{GeV}^{2}$.
The value of the constant $c_{\text{r.m.}}$ in Eq. (68) is chosen as
$\frac{1}{2}$, which means

$\displaystyle
M_{5}(z)^{2}={\mathrm{e}}^{-\frac{1}{2}\Phi}M_{5}^{2},~{}~{}~{}\text{model III
and IV}.$ (91)

The results of glueballs/oddballs spectra is denoted as "Model III,IV(1)" in
Tab. 3.

In Ref. Chen:2015zhh , the authors also use model III to calculate the
glueballs spectra. There they use the parameter
$b=\frac{2\sqrt{6}}{3}\mathrm{GeV}^{2}$ 222 Please remember Eq. (8). This
value of $b$ means  $\Phi(z)=\left(1\mathrm{GeV}^{2}\right)z^{2}.$ (92) and
$c_{\text{r.m.}}=\frac{2}{3}$. We also calculate the glueballs spectra using
these parameters and list the results denoted by "Model III,IV(2)" in the Tab.
3.

####  4.3.3 Model V

In model V, we choose the parameter $d=0.2\mathrm{GeV}^{2}$. The value of the
constant $c_{\text{r.m.}}$ in Eq. (68) is chosen as $\frac{1}{3}$, which means

$M_{5}(z)^{2}={\mathrm{e}}^{-\frac{1}{3}\Phi}M_{5}^{2},~{}~{}~{}\text{model
V}.$ (93)

The results are denoted by "Model V" in Tab. 3.

The corresponding results for glueballs and oddballs spectra are also shown in
Fig. 2 and Fig. 3, respectively.

####  4.3.4 Compare results with lattice QCD, QCD sum rule and $pp$ high
energy scattering

We summarize our holographic results of glueballs/oddballs spectra and then
compare them with the results from lattice simulation and QCD sum rule in Tab.
3. To explicitly see the difference between results from holographic QCD
models and those from lattice simulation, we also list results in Fig. 2 for
C-even glueballs, and in Fig. 3 for C-odd oddballs.

$J^{PC}$ | LQCD1 | LQCD2 | LQCD3 | LQCD4 | QCDSR | Model I,II(O)
---|---|---|---|---|---|---
$0^{++}$ | 1.653(26) | 1.475(30)(65) | 1.710(50)(80) | 1.730 (50) (80) | $1.50\pm 0.19$ | 2.099
$0^{*++}$ | 2.842(40) | 2.755(70)(120) | – | 2.670 (180)(130) | 2.0 - 2.1 | 2.842
$0^{**++}$ | – | 3.370(100)(150) | – | – | – | 3.425
$0^{***++}$ | – | 3.990(210)(180) | – | – | – | 3.922
$2^{++}$ | 2.376(32) | 2.150(30)(100) | 2.390(30)(120) | 2.400 (25) (120) | $2.0\pm 0.1$ | 8.831
$2^{*++}$ | 3.30(5) | 2.880(100)(130) | – | – | 2.2 - 2.3 | 9.515
$0^{-+}$ | 2.561(40) | 2.250(60)(100) | 2.560(35)(120) | 2.590 (40) (130) | $2.05\pm 0.19$ | 2.099
$0^{*-+}$ | 3.54(8) | 3.370(150)(150) | – | 3.640 (60) (180) | 2.1 - 2.3 | 2.842
$1^{-+}$ | 4.12(8) | – | – | – | – | 20.675
$1^{*-+}$ | 4.16(8) | – | – | – | – | 21.405
$1^{**-+}$ | 4.20(9) | – | – | – | – | 22.093
$2^{-+}$ | 3.07(6) | 2.780(50)(130) | 3.040(40)(150) | 3.100 (30) (150) | – | 8.831
$2^{*-+}$ | 3.97(7) | 3.480(140)(160) | – | 3.890 (40) (190) | – | 9.515
$0^{+-}$ | – | – | 4.780(60)(230) | 4.740 (70) (230) | $9.2_{-1.4}^{+1.3}$ | 28.137
$1^{+-}$ | 2.944(42) | 2.670(65)(120) | 2.980(30)(140) | 2.940 (30) (140) | $2.87_{-0.20}^{+0.17}$ | 16.457
$1^{*+-}$ | 3.80(6) | – | – | – | – | 17.176
$2^{+-}$ | 4.24(8) | – | 4.230(50)(200) | 4.140 (50) (200) | $2.85_{-0.20}^{+0.16}$ | 16.996
|  |  |  |  | $6.06\pm{0.13}$ |
$3^{+-}$ | 3.53(8) | 3.270(90)(150) | 3.600(40)(170) | 3.550 (40) (170) | $2.78_{-0.23}^{+0.18}$ | 16.491
$3^{*+-}$ | – | 3.630(140)(160) | – | – | – | 17.212
$0^{--}$ | – | – | – | – | $6.8_{-1.2}^{+1.1}$ | 28.137
$1^{--}$ | 4.03(7) | 3.240(330)(150) | 3.830(40)(190) | 3.850 (50) (190) | $3.29_{-0.32}^{+1.49}$ | 16.457
$2^{--}$ | 3.92(9) | 3.660(130)(170) | 4.010(45)(200) | 3.930 (40) (190) | $3.16_{-0.23}^{+0.33}$ | 16.996
$2^{*--}$ | – | 3.740(200)(170) | – | – | – | 17.717
$3^{--}$ | – | 4.330(260)(200) | 4.200(45)(200) | 4.130 (90) (200) | $3.47_{-0.50}^{+\;\;?}$ | 16.491
Table 2: The glueballs and oddballs mass spectra in the dynamical soft-wall model I and II without making a distinction between glueballs (oddballs) with different $P$-parity and introducing $z$ dependent modified $5$-dimensional masses, compared with results from lattice QCD and QCD sum rule. The units of all the data in the table are $\mathrm{GeV}$. The lattice data in the column "LQCD1", column "LQCD2", column "LQCD3", and column "LQCD4" are taken from Ref. Athenodorou:2020ani , Ref Meyer:2004gx , Ref Chen:2005mg , and Ref Morningstar:1999rf  respectively. The QCD sum rule results are take from Refs. Narison:1996fm ; Pimikov:2017bkk ; Chen:2021cjr ; Tang:2015twt . Here we also list the data predicted by the single pole (SP) and dipole (DP) Regge model Szanyi:2019kkn : using the SP Regge model, the predicted mass for $2^{++}$ glueball is $1.747\mathrm{GeV}$; using the DP Regge model, the predicted masses for $2^{++}$ glueball and $3^{--}$ oddall are $1.758\mathrm{GeV}$ and $3.001\mathrm{GeV}$ respectively. $J^{PC}$ | LQCD1-4 | QCDSR | Model I,II | Model III,IV(1) | Model III,IV(2) | Model V
---|---|---|---|---|---|---
$0^{++}$ | 1.475(30)(65) - 1.730(50)(80) | $1.50\pm 0.19$ | 2.099 | 1.653 | 1.593 | 1.761
$0^{*++}$ | 2.670 (180)(130) - 2.842(40) | 2.0 - 2.1 | 2.842 | 2.718 | 2.618 | 2.251
$0^{**++}$ | 3.370(100)(150) | – | 3.425 | 3.437 | 3.311 | 2.653
$0^{***++}$ | 3.990(210)(180) | – | 3.922 | 4.024 | 3.877 | 3.001
$2^{++}$ | 2.150(30)(100) - 2.400(25)(120) | $2.0\pm 0.1$ | 2.633 | 2.546 | 2.203 | 2.553
$2^{*++}$ | 2.880(100)(130) - 3.30(5) | 2.2 - 2.3 | 3.292 | 3.258 | 3.006 | 2.939
$0^{-+}$ | 2.250(60)(100) - 2.590(40)(130) | $2.05\pm 0.19$ | 2.599 | 2.705 | 2.606 | 2.043
$0^{*-+}$ | 3.370(150)(150) - 3.640(60)(180) | 2.1 - 2.3 | 3.280 | 3.443 | 3.317 | 2.522
$1^{-+}$ | 4.12(8) | – | 3.585 | 4.079 | 3.588 | 3.304
$1^{*-+}$ | 4.16(8) | – | 4.180 | 4.692 | 4.221 | 3.678
$1^{**-+}$ | 4.20(9) | – | 4.667 | 5.195 | 4.730 | 4.000
$2^{-+}$ | 2.780(50)(130) - 3.100(30)(150) | – | 3.252 | 3.474 | 3.161 | 2.919
$2^{*-+}$ | 3.480(140)(160) - 3.97(7) | – | 3.806 | 3.960 | 3.703 | 3.263
$0^{+-}$ | 4.740 (70) (230) - 4.780(60)(230) | $9.2_{-1.4}^{+1.3}$ | 3.268 | 3.742 | 3.165 | 3.184
$1^{+-}$ | 2.670(65)(120) - 2.980(30)(140) | $2.87_{-0.20}^{+0.17}$ | 3.114 | 3.448 | 2.954 | 2.982
$1^{*+-}$ | 3.80(6) | – | 3.755 | 4.099 | 3.652 | 3.373
$2^{+-}$ | 4.140 (50) (200) - 4.24(8) |  $2.85_{-0.20}^{+0.16}$, | 3.002 | 3.303 | 2.786 | 2.926
|  | $6.06\pm{0.13}$ |  |  |  |
$3^{+-}$ | 3.270(90)(150) - 3.600(40)(170) | $2.78_{-0.23}^{+0.18}$ | 2.857 | 3.100 | 2.572 | 2.837
$3^{*+-}$ | 3.630(140)(160) | – | 3.542 | 3.818 | 3.369 | 3.244
$0^{--}$ | – | $6.8_{-1.2}^{+1.1}$ | 3.843 | 4.426 | 3.907 | 3.514
$1^{--}$ | 3.240(330)(150) - 4.03(7) | $3.29_{-0.32}^{+1.49}$ | 3.470 | 3.901 | 3.441 | 3.190
$2^{--}$ | 3.660(130)(170) - 4.010(45)(200) | $3.16_{-0.23}^{+0.33}$ | 3.602 | 4.073 | 3.619 | 3.273
$2^{*--}$ | 3.740(200)(170) | – | 4.177 | 4.645 | 4.211 | 3.637
$3^{--}$ | 4.130(90)(200) - 4.330(260)(200) | $3.47_{-0.50}^{+\;\;?}$ | 3.705 | 4.201 | 3.765 | 3.326
Table 3: The glueballs and oddballs mass spectra in the dynamical soft-wall
model, compared with results from lattice QCD and QCD sum rule. The units of
all the data in the table are $\mathrm{GeV}$. The lattice data are taken from
Refs. Athenodorou:2020ani ; Meyer:2004gx ; Chen:2005mg ; Morningstar:1999rf .
The QCD sum rule results are take from Refs. Narison:1996fm ; Pimikov:2017bkk
; Chen:2021cjr ; Tang:2015twt . Here we also list the data predicted by the SP
and DP Regge model Szanyi:2019kkn : using the SP Regge model, the predicted
mass for $2^{++}$ glueball is $1.747\mathrm{GeV}$; using the DP Regge model,
the predicted masses for $2^{++}$ glueball and $3^{--}$ oddall are
$1.758\mathrm{GeV}$ and $3.001\mathrm{GeV}$ respectively. Figure 2: The mass
spectra of $J^{PC}$ ($C=1$) glueballs in the dynamical soft-wall model,
compared with lattice data. This figure are split into five panels, that are
divided by black solid lines. From left to right, the mass data in these
panels belong to $0^{++}$ states, $2^{++}$ states, $0^{-+}$ states, $1^{-+}$
states, and $2^{-+}$ states respectively. In every panel, the black dashed
line split it into two parts. The left one contains lattice data taken from
Refs. Athenodorou:2020ani ; Meyer:2004gx ; Chen:2005mg ; Morningstar:1999rf .
The steel blue lines, goldenrod lines, olive drab lines, orange red lines are
lattice data taken from Ref. Athenodorou:2020ani , Ref Meyer:2004gx , Ref
Chen:2005mg , and Ref Morningstar:1999rf  respectively. The minimal value and
maximal value of a set of discrete data that belongs to the same glueball
state decide the positions of lower and upper bound of the bar in the figure
respectively. The data in the right part are calculated in our holographic
models. The medium purple lines, sienna lines, sky blue lines, and magenta
lines are results from "Model I,II", "Model III,IV(1)", "Model III,IV(2)", and
"Model V" respectively. Figure 3: The mass spectra of $J^{PC}$ ($C=-1$)
oddballs in the dynamical soft-wall model, compared with lattice data. This
figure are split into eight panels, that are divided by black solid lines.
From left to right, the mass data in these panels belong to $0^{+-}$ states,
$1^{+-}$ states, $2^{+-}$ states, $3^{+-}$ states, $0^{--}$ states, $1^{--}$
states, $2^{--}$ states, and $3^{--}$ states respectively. In every panel, the
black dashed line split it into two parts. The left one contains lattice data
taken from Refs. Athenodorou:2020ani ; Meyer:2004gx ; Chen:2005mg ;
Morningstar:1999rf . The steel blue lines, goldenrod lines, olive drab lines,
orange red lines are lattice data taken from Ref. Athenodorou:2020ani , Ref
Meyer:2004gx , Ref Chen:2005mg , and Ref Morningstar:1999rf  respectively. The
minimal value and maximal value of a set of discrete data that belongs to the
same oddball state decide the positions of lower and upper bound of the bar in
the figure respectively. The data in the right part are calculated in our
holographic models. The medium purple lines, sienna lines, sky blue lines, and
magenta lines are results from "Model I,II", "Model III,IV(1)", "Model
III,IV(2)", and "Model V" respectively.

In the framework of holography, the states $J^{PC}$ with the same angular
momentum $J$ and the same C-parity share the same operator. Thus, they have
the same dimension and $5$-dimensional mass, and the mass splitting for
different P-parity states is realized by $\mathrm{e}^{-p\Phi}$ in Eq. (65).
The states $J^{PC}$ with the same angular momentum $J$ and the same P-parity
but different C-parity have different operators. Thus, they have different
$5$-dimensional masses, which naturally induces the mass splitting for
different C-parity states. From the results in Table 3, Fig.2, and Fig. 3, we
can see that with only 2 parameters, the model predictions on
glueballs/oddballs spectra in general are in good agreement with lattice
results except two oddballs $0^{+-}$ and $2^{+-}$. Here we also would like to
mention that the data predicted by the single pole (SP) and dipole (DP) Regge
model Szanyi:2019kkn  to fit the high energy $pp$ scattering: using the SP
Regge model, the predicted mass for $2^{++}$ glueball is $1.747\mathrm{GeV}$;
using the DP Regge model, the predicted masses for $2^{++}$ glueball and
$3^{--}$ oddall are $1.758\mathrm{GeV}$ and $3.001\mathrm{GeV}$ respectively.
These predicted values are a little bit lower than the results predicted from
holography but still in reasonable regions. It might indicate that the mass
$1.747\mathrm{GeV}$/$1.758\mathrm{GeV}$ $2^{++}$ glueball and mass
$3.001\mathrm{GeV}$ $3^{--}$ oddball are hybrid glueball/oddball states mixing
with quark states.

##  5 Equation of state

With parameters used to calculate the glueballs/oddballs spectra listed in
Table 3, we check the corresponding thermodynamical properties of the system.

###  5.1 Model I and II

In Model I and Model II, we choose the parameter $a=0.6032\mathrm{GeV}^{2}$,
the $5$-dimensional Newtown constant $G_{5}=1$. Then we numerically calculate
the thermodynamical propertities for Model I and II. In model II, we utilize
the numerical method in Refs. DeWolfe:2010he ; Critelli:2017oub  to
investigate the thermodynamical propertities. The results are different for
these two models, as we emphasized in subsubsection 3.3.1. The deconfined
temperature $T_{c}=537.960\mathrm{MeV}$ for model I with inputting $A_{E}(z)$
and $T_{c}=521.147\mathrm{MeV}$ for model II with inputting $V_{\phi}(\phi)$.
We plot the thermodynamical quantities in Fig. 4. The red points with error
bar is lattice simulation of $SU(3)$ Yang-Mills results in Ref.
Caselle:2018kap .

Figure 4: The results of equation of state from model I and model II with
$a=0.6032\mathrm{GeV}^{2}$ and $G_{5}=1$. The results of the entropy density
over cubic temperature (upper left panel), the pressure density over quartic
temperature (upper right panel), the energy density over quartic temperature
(lower left panel) and the trace anomaly over quartic temperature (lower right
panel) as functions of the scaled temperature $T/T_{c}$ in model I and model
II, respectively. The blue line is the result for model I with inputting
$A_{E}(z)$, the orange line is result for model II with inputting
$V_{\phi}(\phi)$. The red points are $SU(3)$ lattice data taken from Ref.
Caselle:2018kap .

It is noticed that even though Model I and Model II can describe
glueballs/oddballs spectra, the corresponding thermodynamical properties shown
in Fig. 4 are not in good agreement with lattice results Caselle:2018kap  for
the pure gluon system. From the asymptotic analysis of the dilaton field at UV
boundary Eq. (27) in subsection 3.1, we can see that the leading order of the
$5$-dimensional dilaton field is a term proportional to $z$, and the sub-
leading order is a term proportional to $z^{3}$. So we expect the
thermodynamical properties of model I and II behaves more like quark matter.
We fix the value of the parameter $a$, and tune the value of $G_{5}=1$ to
$G_{5}=0.42$ to meet the degrees of freedom of quark matter. In this case, the
critical temperatures remain unchanged. It is found that the equation of state
calculated in model I and II are qualitatively consistent with the $2+1$
flavors lattice results in Ref. Borsanyi:2013bia . We plot the equation of
state in Fig. 5. The red points with error bar is lattice simulation of
$SU(3)$ equation of state taken from Ref. Caselle:2018kap  for pure gluon
system. The purple points with error bar is lattice simulation of $N_{f}=2+1$
QCD equation of state taken from Ref. Borsanyi:2013bia .

Figure 5: The results of equation of state from model I and model II with
$a=0.6032\mathrm{GeV}^{2}$ and $G_{5}=0.42$. Upper left panel: The ratio of
entropy density over cubic temperature as function of scaled temperature
$T/T_{c}$. Upper right panel: The ratio of pressure density over quartic
temperature as function of scaled temperature $T/T_{c}$. Lower left panel: The
energy density over quartic temperature as function of scaled temperature
$T/T_{c}$. Lower right panel: The trace anomaly over quartic temperature as
function of scaled temperature $T/T_{c}$. The blue line is for model I with
inputting $A_{E}(z)$. The orange line is for model II with inputting
$V_{\phi}(\phi)$. The red points are $SU(3)$ lattice data taken from Ref.
Caselle:2018kap , and the purple points are $N_{f}=2+1$ lattice data taken
from Ref. Borsanyi:2013bia .

###  5.2 Model III and IV

We also check the correponding thermodynamical properties of Model III and IV.
In model III, we choose two sets of parameters. The parameters I are
$b=1.760\mathrm{GeV}^{2}$, and the $5$-dimensional Newtown constant
$G_{5}=1.35$; the parameters II are $b=\frac{2\sqrt{6}}{3}\mathrm{GeV}^{2}$ as
in Chen:2015zhh , as we mention in subsubsection 4.3.2, the $5$-dimensional
Newtown constant $G_{5}=1.35$. Again, we employ the numerical method in Refs.
DeWolfe:2010he ; Critelli:2017oub  to investigate the thermodynamical
propertities for model IV. Then we fix the values of the characteristic energy
scale 333 As explained in Ref. Critelli:2017oub , the characteristic energy
scale $\Lambda$, the energy dimension of which is 1, is introduced to express
dimensionful observables in physical unit. When an gauge/gravity observable
with dimension ${\left[E\right]}^{q}$ is expressed in physical unit, it should
be multiplied by ${\Lambda}^{q}$.  of the EMD system $\Lambda$ and the
$5$-dimensional Newtown constant $G_{5}$: $\Lambda=1\mathrm{GeV}$, and
$G_{5}=1.35$. Then we numerically calculate the equation of state for these
two models respectively. The results are actually different for the two
models, as we emphasized in subsubsection 3.3.2. In model III, the deconfined
temperature $T_{c}=367.597\mathrm{MeV}$ for parameters I with
$b=1.760\mathrm{GeV}^{2}$ and $T_{c}=354.131\mathrm{MeV}$ for parameters II
with $b=\frac{2\sqrt{6}}{3}\mathrm{GeV}^{2}$. The deconfined temperature
$T_{c}=269.371\mathrm{MeV}$ in model IV with $\Lambda=1\mathrm{GeV}$. We plot
the equation of state in Fig. 6. The red points with error bar is lattice
simulation of $SU(3)$ equation of state for pure gluon system in Ref.
Caselle:2018kap .

Figure 6: The results of equation of state from model III and model IV. Upper
left panel: The ratio of entropy density over cubic temperature as function of
scaled temperature $T/T_{c}$. Upper right panel: The ratio of pressure density
over quartic temperature as function of scaled temperature $T/T_{c}$. Lower
left panel: The energy density over quartic temperature as function of scaled
temperature $T/T_{c}$. Lower right panel: The trace anomaly over over quartic
temperature as function of scaled temperature $T/T_{c}$. The blue line is for
parameters I: $b=1.760\mathrm{GeV}^{2}$, and $G_{5}=1.35$ in model III, in
which we input $\phi(z)$. The green line is for parameters II:
$b=\frac{2\sqrt{6}}{3}\mathrm{GeV}^{2}$, and $G_{5}=1.35$ in model III. The
orange line is for model IV, in which we input $V_{\phi}(\phi)$ and the
parameters are $\Lambda=1\mathrm{GeV}$, and $G_{5}=1.35$. The red points is
$SU(3)$ lattice data taken from Ref. Caselle:2018kap  for pure gluon sysytem.
The positions of the blue line and the green line are totally the same in each
panel.

We can see from the Fig. 6 that the lines from parameters I and parameters II
in model III are totally the same with each other. That is not surprising
because all the quantities are dimensionless in this plot.

###  5.3 Model V

In Model V, we take the parameter $d=0.2\mathrm{GeV}^{2}$, and the
$5$-dimensional Newtown constant $G_{5}=\frac{10}{11}$. Then we numerically
calculate the equation of state. The deconfined temperature
$T_{c}=470.833\mathrm{MeV}$. We plot the equation of state in Fig. 7. The red
points with error bar is lattice simulation of $SU(3)$ equation of state from
Ref. Caselle:2018kap .

Figure 7: The results of equation of state from model V. Upper left panel: The
ratio of entropy density over cubic temperature as function of scaled
temperature $T/T_{c}$. Upper right panel: The ratio of pressure density over
quartic temperature as function of scaled temperature $T/T_{c}$. Lower left
panel: The energy density over quartic temperature as function of scaled
temperature $T/T_{c}$. Lower right panel: The trace anomaly over over quartic
temperature as function of scaled temperature $T/T_{c}$. The blue line is for
Model V, in which we input $\phi(z)$ and the parameters are
$d=0.2\mathrm{GeV}^{2}$, and $G_{5}=\frac{10}{11}$. The red points are $SU(3)$
lattice data taken from Ref. Caselle:2018kap  for pure gluon system.

If we fix the value of the parameter $d$ and tune the value of $G_{5}$ to
$G_{5}=0.39$, the critical temperature remains unchanged. However, the
equation of state in Model V will be qualitatively consistent with the $2+1$
flavors lattice results, which is taken from Ref. Borsanyi:2013bia . We plot
the equation of state in Fig. 8. The red points with error bar is lattice
simulation of $SU(3)$ equation of state taken from Ref. Caselle:2018kap . The
purple points with error bar is lattice simulation of $N_{f}=2+1$ QCD equation
of state taken from Ref. Borsanyi:2013bia .

Figure 8: The results of equation of state from model V. Upper left panel: The
ratio of entropy density over cubic temperature as function of scaled
temperature $T/T_{c}$. Upper right panel: The ratio of pressure density over
quartic temperature as function of scaled temperature $T/T_{c}$. Lower left
panel: The energy density over quartic temperature as function of scaled
temperature $T/T_{c}$. Lower right panel: The trace anomaly over over quartic
temperature as function of scaled temperature $T/T_{c}$. The blue line is for
Model V, in which we input $\phi(z)$. The blue line is theoretical result from
model V, in which we input $\phi(z)$ and the parameters are
$d=0.2\mathrm{GeV}^{2}$, and $G_{5}=0.39$. The red points are $SU(3)$ lattice
data taken from Ref. Caselle:2018kap  for pure gluon system, and the purple
points are $N_{f}=2+1$ lattice data taken from Ref. Borsanyi:2013bia .

##  6 Conclusion and discussion

In this work, we study scalar, vector and tensor glueballs/oddballs spectra in
the framework of $5$-dimensional dynamical holographic QCD model, where the
metric structure is deformed self-consistently by the dilaton field. In the
framework of holography, the states $J^{PC}$ with the same angular momentum
$J$ and the same C-parity share the same operator thus have the same dimension
and $5$-dimensional mass, and the mass splitting for different P-parity states
is realized by $\mathrm{e}^{-p\Phi}$ in Eq. (65). The states $J^{PC}$ with the
same angular momentum $J$ and the same P-parity but different C-parity have
different operators. Thus, they have different $5$-dimensional masses, which
naturally induces the mass splitting for different C-parity states.

From the results in Table 3, Fig. 2, and Fig. 3, we can see that with only 2
parameters, the model predictions on glueballs/oddballs spectra in general are
in good agreement with lattice results except two oddballs $0^{+-}$ and
$2^{+-}$. Here we also would like to mention that the data predicted by the SP
and DP Regge model Szanyi:2019kkn  to fit the high energy $pp$ scattering:
using the SP Regge model, the predicted mass for $2^{++}$ glueball is
$1.747\mathrm{GeV}$; using the DP Regge model, the predicted masses for
$2^{++}$ glueball and $3^{--}$ oddall are $1.758\mathrm{GeV}$ and
$3.001\mathrm{GeV}$ respectively. These predicted values are a little bit
lower than the results predicted from holography but still in reasonable
regions. It might indicate that the mass
$1.747\mathrm{GeV}$/$1.758\mathrm{GeV}$ $2^{++}$ glueball and mass
$3.001\mathrm{GeV}$ $3^{--}$ oddball are hybrid glueball/oddball states mixing
with quark states.

From the results of glueballs/oddballs spectra at zero temperature and zero
density and the equation of state at finite temperature, we obtain the
following conclusions. 1) For the same set of vacuum solutions for the
Einstein field equations and the equation of motion of the dilaton field
$\phi(z)$, inputting the function $A_{E}(z)$ and inputting the dilaton
potential $V_{\phi}(\phi)$ give different equation of state indeed. The
difference between Model III and IV are much less than that between Model I
and II. 2) The model with quadratic dilaton field $\phi(z)$ can simultaneously
describe glueballs/oddballs spectra as well as equation of state of pure gluon
system. The model with quadratic $A_{E}(z)$ can describe glueballs/oddballs
spectra, but its corresponding equation of state behaves more like $N_{f}=2+1$
quark matter. These are consistent with dimension analysis at UV boundary.

Acknowledgement

We thank Danning Li and Cong-Feng Qiao for helpful discussions. This work is
supported in part by the National Natural Science Foundation of China (NSFC)
Grant Nos. 11735007, 11725523, and Chinese Academy of Sciences under Grant No.
XDPB09, the start-up funding from University of Chinese Academy of
Sciences(UCAS), and the Fundamental Research Funds for the Central
Universities.

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