# Two- and three-gluon glueballs of $C=+$

Hua-Xing Chen1<EMAIL_ADDRESS>Wei Chen2<EMAIL_ADDRESS>Shi-Lin
Zhu3<EMAIL_ADDRESS>1School of Physics, Southeast University, Nanjing
210094, China
2School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
3School of Physics and Center of High Energy Physics, Peking University,
Beijing 100871, China

###### Abstract

We study two- and three-gluon glueballs of $C=+$ using the method of QCD sum
rules. We systematically construct their interpolating currents, and find that
all the spin-1 currents of $C=+$ vanish. This suggests that the “ground-state”
spin-1 glueballs of $C=+$ do not exist within the relativistic framework. We
calculate masses of the two-gluon glueballs with $J^{PC}=0^{\pm+}/2^{\pm+}$
and the three-gluon glueballs with $J^{PC}=0^{\pm+}/2^{\pm+}$. We propose to
search for the $J^{PC}=0^{-+}/2^{-\pm}/3^{\pm-}$ three-gluon glueballs in
their three-meson decay channels in future BESIII, GlueX, LHC, and PANDA
experiments.

glueball, pomeron, odderon, exotic hadron, QCD sum rules

##  I Introduction

Glueballs, composed of valence gluons, are important for the understanding of
non-perturbative QCD Freund:1975pn ; Fritzsch:1975tx ; Jaffe:1975fd . There
have been tremendous theoretical studies on them in the past fifty years using
various models and methods, such as the MIT bag model Chodos:1974je , the
flux-tube model Isgur:1984bm , the Coulomb Gauge model Szczepaniak:1995cw ;
LlanesEstrada:2005jf , Regge trajectories Szanyi:2019kkn , holographic QCD
Zhang:2021itx , Lattice QCD Wilson:1974sk ; Chen:2005mg ; Mathieu:2008me ;
Meyer:2004gx ; Gregory:2012hu , and QCD sum rules Novikov:1979va ;
Novikov:1981xi ; Kataev:1981aw ; Kataev:1981gr ; Narison:1984hu ;
Narison:1996fm ; Bagan:1990sy ; Latorre:1987wt ; Forkel:2003mk ; Hao:2005hu ;
Qiao:2014vva ; Qiao:2015iea ; Pimikov:2016pag ; Pimikov:2017xap ;
Pimikov:2017bkk ; Narison:2021xhc , etc. However, experimental efforts in
searching for glueballs are confronted with the difficulty of identifying them
unambiguously, and there is currently no definite experimental evidence for
their existence.

Recently the D0 and TOTEM Collaborations studied $pp$ and $p\bar{p}$
Abazov:2012qb  cross sections, which are found to be different with a
significance of $3.4\sigma$ Abazov:2020rus . Together with their previous
result Antchev:2017yns , this significance can be increased to
$5.2\sigma$–$5.7\sigma$. The above difference leads to the evidence of a
$t$-channel exchanged odderon Levin:1990gg ; Braun:1998fs ; Cudell:2002xe ;
Khoze:2017swe ; Martynov:2018sga , that is predominantly a three-gluon
glueball of $C=-$. We refer to Refs. Lukaszuk:1973nt ; Bartels:1980pe ;
Kwiecinski:1980wb ; Donnachie:1983ff ; Khoze:2018bus ; Csorgo:2018uyp ;
Goncalves:2018nsp ; Xie:2019soz ; Dumitru:2021tqp  and the review Block:2006hy
for more discussions. Due to these studies, interests in glueballs are
reviving recently. Since the above odderon evidence is still indirect, it is
crucial and important to directly study the glueball itself.

The lowest-lying two-, three-, and four-gluon glueballs have been
systematically investigated in Ref. Jaffe:1985qp , where the authors
constructed their corresponding non-relativistic low-dimension operators.
These operators have been successfully used in Lattice QCD calculations. In
this paper we systematically study two- and three-gluon glueballs of $C=+$. We
shall construct their corresponding relativistic glueball currents, and
calculate masses of these glueballs using the method of QCD sum rules. The
same approach has been applied in Ref. Chen:2021cjr  to study three-gluon
glueballs of $C=-$, so a rather complete QCD sum rule study will be done on
the lowest-lying glueballs composed of two or three valence gluons. These
studies can largely improve our understanding of the gluon degree of freedom
as well as the non-perturbative behaviors of the strong interaction at the low
energy region.

This paper is organized as follows. We systematically construct relativistic
two- and three-gluon glueball currents of $C=+$ in Sec. II. We apply them to
perform QCD sum rule analyses in Sec. III, and perform numerical analyses in
Sec. IV. The obtained results are summarized and discussed in Sec. V, which
are compared with Lattice QCD results Chen:2005mg ; Mathieu:2008me ;
Meyer:2004gx ; Gregory:2012hu .

##  II Relativistic glueball currents

In this section we systematically construct relativistic glueball currents,
including the two-gluon glueball currents and the $C=+$ three-gluon glueball
currents. We shall do this separately in the following subsections. Note that
the two-gluon glueball currents can not reach $C=-$ Yang:1950rg , and the
$C=-$ three-gluon glueball currents have been systematically constructed in
Ref. Chen:2021cjr .

###  II.1 Couplings of tensor currents

In the present study we shall use some special tensor currents to study
glueballs with non-zero spins $J\neq 0$. These currents have $2\times J$
Lorentz indices with certain symmetries, and they couple to both positive- and
negative-parity glueballs. In this subsection we briefly explain how we deal
with them.

We assume $J_{\alpha\beta}$ to be a tensor current with two antisymmetric
Lorentz indices $\mu$ and $\nu$. Taking the current
$J_{\alpha\beta}=\bar{c}\sigma_{\alpha\beta}c$ as an example, it can be
separated into ($\alpha,\beta=0,1,2,3$ and $i,j=1,2,3$):

$J_{\alpha\beta}=\bar{c}\sigma_{\alpha\beta}c\rightarrow\left\\{\begin{array}[]{l}\bar{c}\sigma_{ij}c\,,\,{P=+}\,,\\\\[2.84526pt]
\bar{c}\sigma_{0i}c\,,\,{P=-}\,.\end{array}\right.$ (1)

Accordingly, it couples to both positive- and negative-parity charmonia
through

$\displaystyle\langle 0|J_{\alpha\beta}|h_{c}(\epsilon,p)\rangle$
$\displaystyle=$ $\displaystyle
if^{T}_{h_{c}}\epsilon_{\alpha\beta\mu\nu}\epsilon^{\mu}p^{\nu}\,,$ (2)
$\displaystyle\langle 0|J_{\alpha\beta}|J/\psi(\epsilon,p)\rangle$
$\displaystyle=$ $\displaystyle
if^{T}_{J/\psi}(p_{\alpha}\epsilon_{\beta}-p_{\beta}\epsilon_{\alpha})\,,$ (3)

where $f^{T}_{h_{c}}$ and $f^{T}_{J/\psi}$ are relevant decay constants. Given
the Lorentz structures of $J/\psi$ and $h_{c}$ to be totally different, they
can be clearly separated from each other. For example, we can isolate $h_{c}$
at the hadron level by investigating the two-point correlation function
containing

$\displaystyle\langle 0|J_{\alpha\beta}|h_{c}\rangle\langle
h_{c}|J_{\alpha^{\prime}\beta^{\prime}}^{\dagger}|0\rangle$ $\displaystyle=$
$\displaystyle\left(f^{T}_{h_{c}}\right)^{2}\epsilon_{\alpha\beta\mu\nu}\epsilon^{\mu}p^{\nu}\epsilon_{\alpha^{\prime}\beta^{\prime}\mu^{\prime}\nu^{\prime}}\epsilon^{*\mu^{\prime}}p^{\nu^{\prime}}$
$\displaystyle=$
$\displaystyle-\left(f^{T}_{h_{c}}\right)^{2}~{}p^{2}~{}\left(g_{\alpha\alpha^{\prime}}g_{\beta\beta^{\prime}}-g_{\alpha\beta^{\prime}}g_{\beta\alpha^{\prime}}\right)+\cdots\,,$

since the correlation function of $J/\psi$ dose not contain the above
coefficient. It is not so easy to isolate $J/\psi$ from $J_{\alpha\beta}$ at
the hadron level. Instead, we can investigate its partner current

$\tilde{J}_{\alpha\beta}=\epsilon_{\alpha\beta\gamma\delta}\times
J^{\gamma\delta}\,,$ (5)

which couples to $J/\psi$ and $h_{c}$ just in the opposite ways:

$\displaystyle\langle 0|\tilde{J}_{\alpha\beta}|J/\psi(\epsilon,p)\rangle$
$\displaystyle=$ $\displaystyle
i\tilde{f}^{T}_{J/\psi}\epsilon_{\alpha\beta\mu\nu}\epsilon^{\mu}p^{\nu}\,,$
(6) $\displaystyle\langle 0|\tilde{J}_{\alpha\beta}|h_{c}(\epsilon,p)\rangle$
$\displaystyle=$ $\displaystyle
i\tilde{f}^{T}_{h_{c}}(p_{\alpha}\epsilon_{\beta}-p_{\beta}\epsilon_{\alpha})\,.$
(7)

Accordingly, we can use the two currents $J_{\alpha\beta}$ and
$\tilde{J}_{\alpha\beta}$ to study and well separate $J/\psi$ and $h_{c}$.

We apply the above process to generally investigate the current
$J^{\alpha_{1}\cdots\alpha_{N},\beta_{1}\cdots\beta_{N}}$, which has $2N=2J$
Lorentz indices with certain symmetries, e.g., the spin-2 current
$J^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ has four Lorentz indices,
satisfying

$J^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}=-J^{\beta_{1}\alpha_{2},\alpha_{1}\beta_{2}}=-J^{\alpha_{1}\beta_{2},\beta_{1}\alpha_{2}}=J^{\alpha_{2}\alpha_{1},\beta_{2}\beta_{1}}\,.$
(8)

Its coupling can be written as:

$\langle
0|J^{\alpha_{1}\cdots\alpha_{N},\beta_{1}\cdots\beta_{N}}|X\rangle=if_{X}\mathcal{S}[\epsilon^{\alpha_{i}\beta_{i}\mu_{i}\nu_{i}}p_{\nu_{i}}]^{N}\epsilon_{\mu_{1}\cdots\mu_{N}}\,,$
(9)

where $X$ is the corresponding state having the same parity as $J^{i_{1}\cdots
i_{N},j_{1}\cdots j_{N}}$ ($i_{1}\cdots j_{N}=1,2,3$); $\mathcal{S}$ denotes
symmetrization and subtracting trace terms in the two sets
$\\{\alpha_{1}\cdots\alpha_{N}\\}$ and $\\{\beta_{1}\cdots\beta_{N}\\}$
simultaneously, with

$[\cdots]^{N}=\epsilon^{\alpha_{1}\beta_{1}\mu_{1}\nu_{1}}p_{\nu_{1}}\cdots\epsilon^{\alpha_{N}\beta_{N}\mu_{N}\nu_{N}}p_{\nu_{N}}\,.$
(10)

Note that the current
$J^{\alpha_{1}\cdots\alpha_{N},\beta_{1}\cdots\beta_{N}}$ can also couple to
the other state $X^{\prime}$ having the parity opposite to $X$, but this state
$X^{\prime}$ can not be easily isolated at the hadron level, so we do not
consider it in the present study.

###  II.2 Two-gluon glueball currents

In this subsection we use the gluon field strength tensor $G^{a}_{\mu\nu}$ to
construct two-gluon glueball currents, with $a$ the color index and $\mu,\nu$
the Lorentz indices. We also need
$\tilde{G}^{a}_{\mu\nu}=G^{a,\rho\sigma}\times\epsilon_{\mu\nu\rho\sigma}/2$
to denote the dual gluon field strength tensor, and $f^{abc}$ to denote the
totally antisymmetric $SU(3)_{C}$ structure constant. In the present study we
only consider local glueball currents without explicit derivatives, although
$G^{a}_{\mu\nu}$ and $\tilde{G}^{a}_{\mu\nu}$ contain covariant derivatives
inside themselves.

In Ref. Jaffe:1985qp  the authors use the chromoelectric and chromomagnetic
fields ($i,j=1,2,3$),

$E_{i}=G_{i0}~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}B_{i}=-{1\over
2}\epsilon_{ijk}G^{jk}\,,$ (11)

to write down all the non-relativistic low-dimension two-gluon glueball
operators:

$\displaystyle 0^{++}~{}~{}~{}~{}~{}\vec{E}_{a}^{2}\pm\vec{B}_{a}^{2}\,,$
$\displaystyle 0^{-+}~{}~{}~{}~{}~{}\vec{E}_{a}\cdot\vec{B}_{a}\,,$
$\displaystyle 1^{-+}~{}~{}~{}~{}~{}\vec{E}_{a}\times\vec{B}_{a}\,,$ (12)
$\displaystyle 2^{++}~{}~{}~{}~{}~{}\mathcal{S}^{\prime}[E_{a}^{i}E_{a}^{j}\pm
B_{a}^{i}B_{a}^{j}]\,,$ $\displaystyle
2^{-+}~{}~{}~{}~{}~{}\mathcal{S}^{\prime}[E_{a}^{i}B_{a}^{j}-B_{a}^{i}E_{a}^{j}]\,,$

where $\mathcal{S}^{\prime}$ denotes symmetrization and subtracting trace
terms in the set $\\{ij\\}$.

We construct their corresponding relativistic currents in order to perform QCD
sum rule analyses:

$\displaystyle J_{0}$ $\displaystyle=$ $\displaystyle
g_{s}^{2}G_{a}^{\mu\nu}G^{a}_{\mu\nu}\,,$ (13) $\displaystyle\tilde{J}_{0}$
$\displaystyle=$ $\displaystyle
g_{s}^{2}G_{a}^{\mu\nu}\tilde{G}^{a}_{\mu\nu}\,,$ (14) $\displaystyle
J_{1}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
g_{s}^{2}G_{a}^{\alpha\mu}\tilde{G}^{a,\beta}_{\mu}-\\{\alpha\leftrightarrow\beta\\}\,,$
(15) $\displaystyle J_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$
$\displaystyle=$
$\displaystyle\mathcal{S}[g_{s}^{2}G_{a}^{\alpha_{1}\beta_{1}}G^{a,\alpha_{2}\beta_{2}}]\,,$
(16) $\displaystyle\tilde{J}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$
$\displaystyle=$
$\displaystyle\mathcal{S}[g_{s}^{2}G_{a}^{\alpha_{1}\beta_{1}}\tilde{G}^{a,\alpha_{2}\beta_{2}}]\,.$
(17)

We shall explicitly prove in Appendix A that the third current
$J_{1}^{\alpha\beta}$ vanishes, suggesting that the “ground-state” two-gluon
glueball of $J^{PC}=1^{-+}$ does not exist within the relativistic framework.

The former two currents $J_{0}$ of $J^{PC}=0^{++}$ and $\tilde{J}_{0}$ of
$J^{PC}=0^{-+}$ couple to the $J^{PC}=0^{++}$ and $0^{-+}$ two-gluon glueballs
$|{\rm GG};{J^{PC}}\rangle$, respectively:

$\displaystyle\langle 0|J_{0}|{\rm GG};{0^{++}}\rangle$ $\displaystyle=$
$\displaystyle f_{0^{++}}\,,$ (18) $\displaystyle\langle 0|\tilde{J}_{0}|{\rm
GG};{0^{-+}}\rangle$ $\displaystyle=$ $\displaystyle f_{0^{-+}}\,,$ (19)

where $f_{0^{++}}$ and $f_{0^{-+}}$ are decay constants. Besides, the current
$J_{0}$ has a partner,

$J_{0}^{\prime}=g_{s}^{2}\tilde{G}_{a}^{\mu\nu}\tilde{G}^{a}_{\mu\nu}\,,$ (20)

whose sum rule result is the same as that of $J_{0}$.

The latter two currents $J_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ and
$\tilde{J}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ couple to the
$J^{PC}=2^{++}$ and $2^{-+}$ glueballs through:

$\displaystyle\langle 0|J_{2}^{\cdots}|{\rm GG};{2^{++}}\rangle$
$\displaystyle=$ $\displaystyle
if_{2^{++}}\mathcal{S}[\epsilon^{\alpha_{i}\beta_{i}\mu_{i}\nu_{i}}p_{\nu_{i}}]^{2}\epsilon_{\mu_{1}\mu_{2}}\,,$
(21) $\displaystyle\langle 0|\tilde{J}_{2}^{\cdots}|{\rm GG};{2^{-+}}\rangle$
$\displaystyle=$ $\displaystyle
if_{2^{-+}}\mathcal{S}[\epsilon^{\alpha_{i}\beta_{i}\mu_{i}\nu_{i}}p_{\nu_{i}}]^{2}\epsilon_{\mu_{1}\mu_{2}}\,.$
(22)

The current $J_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ also has a
partner,

$J_{2}^{\prime\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}=\mathcal{S}[g_{s}^{2}\tilde{G}_{a}^{\alpha_{1}\beta_{1}}\tilde{G}^{a,\alpha_{2}\beta_{2}}]\,.$
(23)

whose sum rule result is the same as that of
$J_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$.

###  II.3 Three-gluon glueball currents of $C=+$

In this subsection we use $G^{a}_{\mu\nu}$ and $\tilde{G}^{a}_{\mu\nu}$ to
construct three-gluon glueball currents of $C=+$. Some of their corresponding
non-relativistic operators have been constructed in Ref. Jaffe:1985qp :

$\displaystyle
0^{++}~{}~{}~{}~{}~{}f^{abc}(\vec{E}_{a}\times\vec{E}_{b})\cdot\vec{B}_{c}\,,$
$\displaystyle
0^{-+}~{}~{}~{}~{}~{}f^{abc}(\vec{E}_{a}\times\vec{E}_{b})\cdot\vec{E}_{c}\,,$
$\displaystyle
1^{++}~{}~{}~{}~{}~{}f^{abc}(\vec{B}_{a}\cdot\vec{E}_{b})~{}\vec{E}_{c}\,,$
(24) $\displaystyle
1^{-+}~{}~{}~{}~{}~{}f^{abc}(\vec{B}_{a}\cdot\vec{E}_{b})~{}\vec{B}_{c}\,,$
$\displaystyle
2^{++}~{}~{}~{}~{}~{}f^{abc}\mathcal{S}^{\prime}[(\vec{B}_{a}\times\vec{B}_{b})^{i}B_{c}^{j}]+\cdots\,,$
$\displaystyle
2^{-+}~{}~{}~{}~{}~{}f^{abc}\mathcal{S}^{\prime}[(\vec{E}_{a}\times\vec{E}_{b})^{i}E_{c}^{j}]+\cdots\,.$

We further construct their corresponding relativistic currents as follows:

$\displaystyle\eta_{0}$ $\displaystyle=$ $\displaystyle
f^{abc}g_{s}^{3}G_{a}^{\mu\nu}G_{b,\nu\rho}G_{c,\mu}^{\rho}\,,$ (25)
$\displaystyle\tilde{\eta}_{0}$ $\displaystyle=$ $\displaystyle
f^{abc}g_{s}^{3}\tilde{G}_{a}^{\mu\nu}\tilde{G}_{b,\nu\rho}\tilde{G}_{c,\mu}^{\rho}\,,$
(26) $\displaystyle\eta_{1}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
f^{abc}g_{s}^{3}\tilde{G}_{a}^{\mu\nu}G_{b,\mu\nu}\tilde{G}_{c}^{\alpha\beta}\,,$
(27) $\displaystyle\tilde{\eta}_{1}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle
f^{abc}g_{s}^{3}\tilde{G}_{a}^{\mu\nu}G_{b,\mu\nu}G_{c}^{\alpha\beta}\,,$ (28)
$\displaystyle\eta_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$
$\displaystyle=$ $\displaystyle
f^{abc}\mathcal{S}[g_{s}^{3}G_{a}^{\alpha_{1}\beta_{1}}G_{b}^{\alpha_{2}\mu}G_{c,\mu}^{\beta_{2}}-\\{\alpha_{2}\leftrightarrow\beta_{2}\\}]\,,$
$\displaystyle\tilde{\eta}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$
$\displaystyle=$ $\displaystyle
f^{abc}\mathcal{S}[g_{s}^{3}\tilde{G}_{a}^{\alpha_{1}\beta_{1}}\tilde{G}_{b}^{\alpha_{2}\mu}\tilde{G}_{c,\mu}^{\beta_{2}}-\\{\alpha_{2}\leftrightarrow\beta_{2}\\}]\,.$

We shall explicitly prove in Appendix A that the third and fourth currents
$\eta_{1}^{\alpha\beta}$ and $\tilde{\eta}_{1}^{\alpha\beta}$ both vanish,
suggesting that the “ground-state” three-gluon glueballs of $J^{PC}=1^{++}$
and $1^{-+}$ do not exist within the relativistic framework.

The former two currents $\eta_{0}$ of $J^{PC}=0^{++}$ and $\tilde{\eta}_{0}$
of $J^{PC}=0^{-+}$ couple to the $J^{PC}=0^{++}$ and $0^{-+}$ three-gluon
glueballs $|{\rm GGG};{J^{PC}}\rangle$, respectively:

$\displaystyle\langle 0|\eta_{0}|{\rm GGG};{0^{++}}\rangle$ $\displaystyle=$
$\displaystyle f^{\prime}_{0^{++}}\,,$ (31) $\displaystyle\langle
0|\tilde{\eta}_{0}|{\rm GGG};{0^{-+}}\rangle$ $\displaystyle=$ $\displaystyle
f^{\prime}_{0^{-+}}\,.$ (32)

The latter two currents $\eta_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$
and $\tilde{\eta}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ couple to the
$J^{PC}=2^{++}$ and $2^{-+}$ glueballs through:

$\displaystyle\langle 0|\eta_{2}^{\cdots}|{\rm GGG};{2^{++}}\rangle$
$\displaystyle=$ $\displaystyle
if^{\prime}_{2^{++}}\mathcal{S}[\epsilon^{\alpha_{i}\beta_{i}\mu_{i}\nu_{i}}p_{\nu_{i}}]^{2}\epsilon_{\mu_{1}\mu_{2}},$
(33) $\displaystyle\langle 0|\tilde{\eta}_{2}^{\cdots}|{\rm
GGG};{2^{-+}}\rangle$ $\displaystyle=$ $\displaystyle
if^{\prime}_{2^{-+}}\mathcal{S}[\epsilon^{\alpha_{i}\beta_{i}\mu_{i}\nu_{i}}p_{\nu_{i}}]^{2}\epsilon_{\mu_{1}\mu_{2}}.$
(34)

##  III QCD sum rule analyses

In this section we use the two-gluon glueball currents $J_{0}$,
$\tilde{J}_{0}$, $J_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$, and
$\tilde{J}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ as well as the
three-gluon glueball currents $\eta_{0}$, $\tilde{\eta}_{0}$,
$\eta_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$, and
$\tilde{\eta}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ to perform QCD
sum rule analyses. This method has been widely applied in the field of hadron
phenomenology Shifman:1978bx ; Reinders:1984sr  to study various exotic
hadrons Chen:2007xr ; Chen:2016otp ; Chen:2016jxd . Especially, all the above
spin-2 currents have four Lorentz indices with certain symmetries, so that
they couple to both positive- and negative-parity glueballs simultaneously. We
refer to Ref. Chen:2021cjr  for detailed discussions.

We take the current $\tilde{J}_{0}$ defined in Eq. (14) as an example, and
calculate its two-point correlation function

$\Pi(q^{2})\equiv i\int d^{4}xe^{iqx}\langle 0|{\bf
T}[\tilde{J}_{0}(x)\tilde{J}_{0}^{\dagger}(0)]|0\rangle\,,$ (35)

separately at hadron and quark-gluon levels.

At the hadron level we express Eq. (35) using the dispersion relation as

$\Pi(q^{2})=\int^{\infty}_{0}\frac{\rho(s)}{s-q^{2}-i\varepsilon}ds\,,$ (36)

with $\rho(s)={\rm Im}\Pi(s)/\pi$ the spectral density. It is parameterized
using one pole dominance for the ground state $X$ as well as the continuum
contribution,

$\displaystyle\rho(s)$ $\displaystyle\equiv$
$\displaystyle\sum_{n}\delta(s-M^{2}_{n})\langle
0|\tilde{J}_{0}|n\rangle\langle n|\tilde{J}_{0}^{\dagger}|0\rangle$ (37)
$\displaystyle=$ $\displaystyle
f^{2}_{X}\delta(s-M^{2}_{X})+\rm{continuum}\,.$

At the quark-gluon level we insert Eq. (14) into Eq. (35), and calculate it
using the method of operator product expansion (OPE). After performing the
Borel transformation to Eq. (36) at both hadron and quark-gluon levels, we
approximate the continuum using the spectral density above a threshold value
$s_{0}$, and obtain

$\Pi(s_{0},M_{B}^{2})\equiv
f^{2}_{X}e^{-M_{X}^{2}/M_{B}^{2}}=\int^{s_{0}}_{0}e^{-s/M_{B}^{2}}\rho(s)ds\,.$
(38)

This equation can be used to further calculate the mass of $X$ through

$M^{2}_{X}(s_{0},M_{B})=\frac{\int^{s_{0}}_{0}e^{-s/M_{B}^{2}}s\rho(s)ds}{\int^{s_{0}}_{0}e^{-s/M_{B}^{2}}\rho(s)ds}\,.$
(39)

(a) ($a$)

(b) ($b$)

Figure 1: The gluon field strength tensor
$G^{a}_{\mu\nu}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g_{s}f^{abc}A_{b,\mu}A_{c,\nu}$,
naturally separated into two parts ($a$) and ($b$).

Since the gluon field strength tensor $G^{a}_{\mu\nu}$ is defined as

$G^{a}_{\mu\nu}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g_{s}f^{abc}A_{b,\mu}A_{c,\nu}\,,$
(40)

it can be naturally separated into two parts. As shown in Fig. 1, we depict
the former two terms using the single-gluon-line, and the latter one term
using the double-gluon-line with a red vertex (see also the diagram Fig.
2$(c-3)$). Here $A_{\mu}^{a}$ is the gluon field, whose propagator is
Govaerts:1984bk :

$\displaystyle\langle 0|{\bf T}[A_{\mu}^{a}(x)A_{\nu}^{b}(y)]|0\rangle$
$\displaystyle=$ $\displaystyle{\delta^{ab}g_{\mu\nu}\over 4\pi^{2}(x-y)^{2}}$
$\displaystyle+$ $\displaystyle{g_{s}\ln(-(x-y)^{2})\over
8\pi^{2}}f^{abc}G_{c,\mu\nu}(0)$ $\displaystyle-$
$\displaystyle{g_{s}g_{\mu\nu}x^{\alpha}y^{\beta}\over
8\pi^{2}(x-y)^{2}}f^{abc}G_{c,\alpha\beta}(0)\,.$

We work in the fixed-point gauge so that

$A_{\mu}^{a}(x)\approx-{1\over 2}x^{\nu}G^{a}_{\mu\nu}(0)\,.$ (42)

(a) ($a$)

(b) ($b{\rm-}1$)

(c) ($b{\rm-}2$)

(d) ($c{\rm-}1$)

(e) ($c{\rm-}2$)

(f) ($c{\rm-}3$)

(g) ($c{\rm-}4$)

(h) ($c{\rm-}5$)

(i) ($d$)

Figure 2: Feynman diagrams for three-gluon glueball currents, including the
perturbative term, the two-gluon condensate $\langle g_{s}^{2}GG\rangle$, the
three-gluon condensate $\langle g_{s}^{3}G^{3}\rangle$, and the $D=8$
condensate $\langle g_{s}^{2}GG\rangle^{2}$. The diagrams $(a)$ and $(b-i)$
are proportional to $\alpha_{s}^{3}\times g_{s}^{0}$, the diagrams $(c-i)$ are
proportional to $\alpha_{s}^{3}\times g_{s}^{1}$, and the diagram $(d)$ is
proportional to $\alpha_{s}^{3}\times g_{s}^{2}$.

In the present study we consider the Feynman diagrams depicted in Fig. 2 (for
three-gluon glueballs), and calculate OPEs up to the dimension eight ($D=8$)
condensates. We take into account the perturbative term, the two-gluon
condensate $\langle g_{s}^{2}GG\rangle$, the three-gluon condensate $\langle
g_{s}^{3}G^{3}\rangle$, and the $D=8$ condensate $\langle
g_{s}^{2}GG\rangle^{2}$:

$\displaystyle\Pi_{|{\rm GG};0^{++}\rangle}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}32\alpha_{s}^{2}s^{2}+60\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle\Big{)}e^{-s/M_{B}^{2}}ds$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+24\pi\alpha_{s}\langle
g_{s}^{3}G^{3}\rangle,$ $\displaystyle\Pi_{|{\rm
GG};2^{++}\rangle}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}{2\alpha_{s}^{2}\over
15}s^{2}-{5\alpha_{s}^{2}\langle g_{s}^{2}GG\rangle\over
24}\Big{)}e^{-s/M_{B}^{2}}ds$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+{\pi\alpha_{s}\langle
g_{s}^{3}G^{3}\rangle\over 3},$ $\displaystyle\Pi_{|{\rm
GG};0^{-+}\rangle}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}32\alpha_{s}^{2}s^{2}e^{-s/M_{B}^{2}}ds-40\pi\alpha_{s}\langle
g_{s}^{3}G^{3}\rangle,$ $\displaystyle\Pi_{|{\rm
GG};2^{-+}\rangle}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}{2\alpha_{s}^{2}\over
5}s^{2}+{\alpha_{s}^{2}\langle g_{s}^{2}GG\rangle\over
12}\Big{)}e^{-s/M_{B}^{2}}ds$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-{\pi\alpha_{s}\langle
g_{s}^{3}G^{3}\rangle\over 2},$ $\displaystyle\Pi_{|{\rm
GGG};0^{++}\rangle}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}{3\alpha_{s}^{3}\over
10\pi}s^{4}+{135\alpha_{s}^{3}\langle g_{s}^{2}GG\rangle\over 32\pi}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-{81\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle\over 2}s\Big{)}e^{-s/M_{B}^{2}}ds,$
$\displaystyle\Pi_{|{\rm GGG};2^{++}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$ $\displaystyle\int_{0}^{s_{0}}\Big{(}{2\alpha_{s}^{3}\over
315\pi}s^{4}+{\alpha_{s}^{2}\langle g_{s}^{2}GG\rangle\over 15}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}+{53\alpha_{s}^{3}\langle
g_{s}^{2}GG\rangle\over 320\pi}s^{2}+{\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle\over 3}s\Big{)}e^{-s/M_{B}^{2}}ds,$
$\displaystyle\Pi_{|{\rm GGG};0^{-+}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$ $\displaystyle\int_{0}^{s_{0}}\Big{(}{3\alpha_{s}^{3}\over
10\pi}s^{4}+{135\alpha_{s}^{3}\langle g_{s}^{2}GG\rangle\over 32\pi}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+{27\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle\over 2}s\Big{)}e^{-s/M_{B}^{2}}ds,$
$\displaystyle\Pi_{|{\rm GGG};2^{-+}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$ $\displaystyle\int_{0}^{s_{0}}\Big{(}{2\alpha_{s}^{3}\over
315\pi}s^{4}-{\alpha_{s}^{2}\langle g_{s}^{2}GG\rangle\over 15}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}+{57\alpha_{s}^{3}\langle g_{s}^{2}GG\rangle\over
320\pi}s^{2}+{5\alpha_{s}^{2}\langle g_{s}^{3}G^{3}\rangle\over
4}s\Big{)}e^{-s/M_{B}^{2}}ds.$

In the calculations we have considered all the diagrams proportional to
$\alpha_{s}^{n}\times g_{s}^{0}$ and $\alpha_{s}^{n}\times g_{s}^{1}$ ($n=2$
for two-gluon glueballs and $n=3$ for three-gluon glueballs); however, there
are so many diagrams proportional to $\alpha_{s}^{n}\times g_{s}^{2}$, so we
have only taken into account one of them. Especially, we find all the $D=8$
terms proportional to $\langle g_{s}^{2}GG\rangle^{2}$ vanish, so the
convergence of the above OPE series are quite good.

In Ref. Latorre:1987wt  the authors studied $J^{PC}=0^{++}$ three-gluon
glueballs using the current $\eta_{0}$ defined in Eq. (25), where they
calculated the Feynman diagrams depicted in Figs. 2($a,b-i,c-1,c-2$). In Ref.
Hao:2005hu  the authors studied $J^{PC}=0^{-+}$ three-gluon glueballs using
the current $\tilde{\eta}_{0}$ defined in Eq. (26), where they calculated the
diagrams depicted in Figs. 2($a,b-i,c-i$). Their calculations are done
(mainly) by hand. In the present study we use the software Mathematica with
the package FeynCalc, and we can obtain exactly the same results for these
diagrams. In Refs. Novikov:1981xi ; Bagan:1990sy ; Forkel:2003mk  the authors
studied $J^{PC}=0^{++}$ and $0^{-+}$ two-gluon glueballs using the currents
$J_{0}$ and $\tilde{J}_{0}$ defined in Eqs. (13) and (14), where they
calculated more diagrams than those calculated in the present study. However,
such calculations are too complicated to be applied to three-gluon glueballs,
and we still calculate similar diagrams as those depicted in Fig. 2 for two-
gluon glueballs to make the present study unified as a whole.

For completeness, we also investigate the following three-gluon glueball
currents of $C=-$:

$\displaystyle\xi_{1}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
d^{abc}g_{s}^{3}G_{a}^{\mu\nu}G_{b,\mu\nu}G_{c}^{\alpha\beta}\,,$ (51)
$\displaystyle\tilde{\xi}_{1}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
d^{abc}g_{s}^{3}G_{a}^{\mu\nu}G_{b,\mu\nu}\tilde{G}_{c}^{\alpha\beta}\,,$ (52)
$\displaystyle\xi_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$
$\displaystyle=$ $\displaystyle
d^{abc}\mathcal{S}[g_{s}^{3}\tilde{G}_{a}^{\alpha_{1}\beta_{1}}G_{b}^{\alpha_{2}\mu}\tilde{G}_{c,\mu}^{\beta_{2}}-\\{\alpha_{2}\leftrightarrow\beta_{2}\\}],$
$\displaystyle\tilde{\xi}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$
$\displaystyle=$ $\displaystyle
d^{abc}\mathcal{S}[g_{s}^{3}G_{a}^{\alpha_{1}\beta_{1}}\tilde{G}_{b}^{\alpha_{2}\mu}G_{c,\mu}^{\beta_{2}}-\\{\alpha_{2}\leftrightarrow\beta_{2}\\}],$
$\displaystyle\xi_{3}^{\cdots}$ $\displaystyle=$ $\displaystyle
d^{abc}\mathcal{S}[g_{s}^{3}G_{a}^{\alpha_{1}\beta_{1}}G_{b}^{\alpha_{2}\beta_{2}}G_{c}^{\alpha_{3}\beta_{3}}]\,,$
(55) $\displaystyle\tilde{\xi}_{3}^{\cdots}$ $\displaystyle=$ $\displaystyle
d^{abc}\mathcal{S}[g_{s}^{3}\tilde{G}_{a}^{\alpha_{1}\beta_{1}}\tilde{G}_{b}^{\alpha_{2}\beta_{2}}\tilde{G}_{c}^{\alpha_{3}\beta_{3}}]\,,$
(56)

where $d^{abc}$ is the totally symmetric $SU(3)_{C}$ structure constant. Their
sum rule equations are:

$\displaystyle\Pi_{|{\rm GGG};1^{+-}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}\frac{4\alpha_{s}^{3}}{81\pi}s^{4}+\frac{10\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle}{9}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{35\alpha_{s}^{3}\langle
g_{s}^{2}GG\rangle}{36\pi}s^{2}+\frac{5\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle}{27}s\Big{)}e^{-s/M_{B}^{2}}ds,$
$\displaystyle\Pi_{|{\rm GGG};2^{+-}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}\frac{\alpha_{s}^{3}}{324\pi}s^{4}-\frac{5\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle}{108}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{15\alpha_{s}^{3}\langle
g_{s}^{2}GG\rangle}{128\pi}s^{2}+\frac{65\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle}{216}s\Big{)}e^{-s/M_{B}^{2}}ds,$
$\displaystyle\Pi_{|{\rm GGG};3^{+-}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}\frac{5\alpha_{s}^{3}}{2016\pi}s^{4}+\frac{\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle}{16}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}-\frac{59\alpha_{s}^{3}\langle
g_{s}^{2}GG\rangle}{512\pi}s^{2}-\frac{\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle}{2}s\Big{)}e^{-s/M_{B}^{2}}ds,$ $\displaystyle\Pi_{|{\rm
GGG};1^{--}\rangle}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}\frac{4\alpha_{s}^{3}}{81\pi}s^{4}-\frac{10\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle}{9}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{25\alpha_{s}^{3}\langle
g_{s}^{2}GG\rangle}{36\pi}s^{2}+\frac{35\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle}{27}s\Big{)}e^{-s/M_{B}^{2}}ds,$
$\displaystyle\Pi_{|{\rm GGG};2^{--}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}\frac{\alpha_{s}^{3}}{324\pi}s^{4}+\frac{5\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle}{108}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{15\alpha_{s}^{3}\langle
g_{s}^{2}GG\rangle}{128\pi}s^{2}+\frac{5\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle}{24}s\Big{)}e^{-s/M_{B}^{2}}ds,$
$\displaystyle\Pi_{|{\rm GGG};3^{--}\rangle}(s_{0},M_{B}^{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\Big{(}\frac{5\alpha_{s}^{3}}{2016\pi}s^{4}-\frac{\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle}{16}s^{2}$
$\displaystyle~{}~{}~{}~{}~{}-\frac{49\alpha_{s}^{3}\langle
g_{s}^{2}GG\rangle}{1536\pi}s^{2}-\frac{11\alpha_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle}{432}s\Big{)}e^{-s/M_{B}^{2}}ds.$

The above three-gluon glueball currents of $C=-$ have been systematically
studied in Ref. Chen:2021cjr , but there we did not calculate the Feynman
diagrams depicted in Figs. 2($c-3,c-4,c-5$). Similar to Eqs. (III)-(III), we
find all the $D=8$ terms proportional to $\langle g_{s}^{2}GG\rangle^{2}$
vanish, so the convergence of the above OPE series are also quite good.

We shall use the above sum rule equations to perform numerical analyses in the
next section.

##  IV Numerical analyses

In this section we perform numerical analyses using the sum rules given in
Eqs. (III)-(III) and Eqs. (III)-(III). The glueball mass $M_{X}$ depends
significantly on the gluon condensates $\langle g_{s}^{2}GG\rangle$ and
$\langle g_{s}^{3}G^{3}\rangle$, both of which are still not precisely known.
In the present study we use the following values for these parameters
Narison:2011xe ; Narison:2018dcr :

$\displaystyle\langle\alpha_{s}GG\rangle$ $\displaystyle=$
$\displaystyle(6.35\pm 0.35)\times 10^{-2}\mbox{ GeV}^{4}\,,$ (63)
$\displaystyle\langle g_{s}^{3}G^{3}\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha_{s}GG\rangle\times(8.2\pm 1.0)\mbox{ GeV}^{2}\,.$

Besides, we use the following value for the strong coupling constant at the
QCD scale $\Lambda_{\rm QCD}=300$ MeV pdg :

$\alpha_{s}(Q^{2})={4\pi\over 11\ln(Q^{2}/\Lambda_{\rm QCD}^{2})}\,.$ (64)

Figure 3: CVGA (short-dashed curve, defined in Eq. (65)), CVGB (long-dashed
curve, defined in Eq. (66)), and PC (solid curve, defined in Eq. (69)) as
functions of the Borel mass $M_{B}$. The current $\tilde{J}_{0}$ is used here
when setting $s_{0}=9.0$ GeV2.

We still take the current $\tilde{J}_{0}$ as an example, and use Eq. (39) to
calculate the mass of $|{\rm GG};0^{-+}\rangle$. It depends on two free
parameters, the Borel mass $M_{B}$ and the threshold value $s_{0}$. We use two
criteria to determine the Borel window. The first criterion is to insure the
convergence of OPE by requiring a) the $\alpha_{s}^{2}\times g_{s}^{2}$ term
$\alpha_{s}^{2}\langle g_{s}^{2}GG\rangle$ to be less than 5%, and b) the
$D=6$ term $\alpha_{s}\langle g_{s}^{3}G^{3}\rangle$ to be less than 10%:

$\displaystyle\mbox{CVG}_{A}$ $\displaystyle\equiv$
$\displaystyle\left|\frac{\Pi^{g_{s}^{n=6}}(s_{0},M_{B}^{2})}{\Pi(s_{0},M_{B}^{2})}\right|\leq
5\%\,,$ (65) $\displaystyle\mbox{CVG}_{B}$ $\displaystyle\equiv$
$\displaystyle\left|\frac{\Pi^{{\rm
D=6}}(s_{0},M_{B}^{2})}{\Pi(s_{0},M_{B}^{2})}\right|\leq 10\%\,.$ (66)

As shown in Fig. 3 using the dashed curves, we determine the lower limit of
$M_{B}$ to be $M_{B}^{2}\geq 3.28$ GeV2 when setting $s_{0}=9.0$ GeV2.

The above condition is the cornerstone of a reliable sum rule analysis, where
we have taken into account two terms because the OPE is expanded in two
directions: the dimension of condensates and the coupling constant $g_{s}$.
Eqs. (65) and (66) are for two-gluon glueball currents, and the conditions for
three-gluon glueball currents are

$\displaystyle\mbox{CVG}^{\prime}_{A}$ $\displaystyle\equiv$
$\displaystyle\left|\frac{\Pi^{g_{s}^{n=8}}(s_{0},M_{B}^{2})}{\Pi(s_{0},M_{B}^{2})}\right|\leq
5\%\,,$ (67) $\displaystyle\mbox{CVG}^{\prime}_{B}$ $\displaystyle\equiv$
$\displaystyle\left|\frac{\Pi^{{\rm
D=6}}(s_{0},M_{B}^{2})}{\Pi(s_{0},M_{B}^{2})}\right|\leq 10\%\,.$ (68)

The second criterion is to insure the one-pole-dominance assumption by
requiring the pole contribution (PC) to be larger than 40%:

$\mbox{PC}\equiv\left|\frac{\Pi(s_{0},M_{B}^{2})}{\Pi(\infty,M_{B}^{2})}\right|\geq
40\%\,.$ (69)

As shown in Fig. 3 using the solid curve, we determine the upper limit of
$M_{B}$ to be $M_{B}^{2}\leq 3.70$ GeV2 when setting $s_{0}=9.0$ GeV2.

Figure 4:  Mass of the two-gluon glueball $|{\rm GG};0^{-+}\rangle$ as a
function of the Borel mass $M_{B}$ (left) and the threshold value $s_{0}$
(right), calculated using the current $\tilde{J}_{0}$. In the left panel the
short-dashed/solid/long-dashed curves are obtained by setting
$s_{0}=8.0/9.0/10.0$ GeV2, respectively. In the right panel the short-
dashed/solid/long-dashed curves are obtained by setting
$M_{B}^{2}=3.28/3.49/3.70$ GeV2, respectively.

Altogether we determine the Borel window to be $3.28$ GeV${}^{2}\leq
M_{B}^{2}\leq 3.70$ GeV2 for the fixed threshold value $s_{0}=9.0$ GeV2. Then
we redo the same procedures by changing $s_{0}$, and find that there exist
non-vanishing Borel windows as long as $s_{0}\geq s^{\rm min}_{0}=8.2$ GeV2.
Accordingly, we choose $s_{0}$ to be slightly larger, and determine our
working regions to be $8.0$ GeV${}^{2}\leq s_{0}\leq 10.0$ GeV2 and $3.28$
GeV${}^{2}\leq M_{B}^{2}\leq 3.70$ GeV2, where we calculate the mass of $|{\rm
GG};0^{-+}\rangle$ to be

$M_{|{\rm GG};0^{-+}\rangle}=2.17\pm 0.11{\rm~{}GeV}\,.$ (70)

Its central value corresponds to $M_{B}^{2}=3.49$ GeV2 and $s_{0}=9.0$ GeV2,
and its uncertainty comes from the threshold value $s_{0}$, the Borel mass
$M_{B}$, and the gluon condensates listed in Eqs. (63).

We show $M_{|{\rm GG};0^{-+}\rangle}$ in the left panel of Fig. 4 as a
function of the Borel mass $M_{B}$, and find it quite stable inside the Borel
window $3.28$ GeV${}^{2}\leq M_{B}^{2}\leq 3.70$ GeV2. We also show it in the
right panel of Fig. 4 as a function of the threshold value $s_{0}$. We find
there exists a mass minimum around $s_{0}\sim 5$ GeV2, and the $s_{0}$
dependence is weak and acceptable inside the working region $8.0$
GeV${}^{2}\leq s_{0}\leq 10.0$ GeV2.

Table 1: QCD sum rule results of two- and three-gluon glueballs. Glueball |  Current |  $s_{0}^{min}~{}[{\rm GeV}^{2}]$ | Working Regions |  Pole [%] |  Mass [GeV]
---|---|---|---|---|---
$s_{0}~{}[{\rm GeV}^{2}]$ |  $M_{B}^{2}~{}[{\rm GeV}^{2}]$
$|{\rm GG};0^{++}\rangle$ | $J_{0}$ | 7.8 | $9.0\pm 1.0$ |  $3.70$–$4.19$ |  $40$–$48$ | $1.78^{+0.14}_{-0.17}$
$|{\rm GG};2^{++}\rangle$ | $J_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ | 8.5 | $10.0\pm 1.0$ |  $3.99$–$4.60$ |  $40$–$50$ | $1.86^{+0.14}_{-0.17}$
$|{\rm GG};0^{-+}\rangle$ | $\tilde{J}_{0}$ | 8.2 | $9.0\pm 1.0$ |  $3.28$–$3.70$ |  $40$–$47$ | $2.17^{+0.11}_{-0.11}$
$|{\rm GG};2^{-+}\rangle$ | $\tilde{J}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ | 8.1 | $10.0\pm 1.0$ |  $3.27$–$4.20$ |  $40$–$55$ | $2.24^{+0.11}_{-0.11}$
$|{\rm GGG};0^{++}\rangle$ | $\eta_{0}$ | 31.6 | $33.0\pm 3.0$ |  $7.25$–$7.61$ |  $40$–$44$ | $4.46^{+0.17}_{-0.19}$
$|{\rm GGG};2^{++}\rangle$ | $\eta_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ | 16.0 | $35.0\pm 3.0$ |  $4.77$–$9.04$ |  $40$–$90$ | $4.18^{+0.19}_{-0.42}$
$|{\rm GGG};0^{-+}\rangle$ | $\tilde{\eta}_{0}$ | 17.0 | $33.0\pm 3.0$ |  $4.48$–$8.13$ |  $40$–$88$ | $4.13^{+0.18}_{-0.36}$
$|{\rm GGG};2^{-+}\rangle$ | $\tilde{\eta}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ | 33.1 | $35.0\pm 3.0$ |  $8.10$–$8.53$ |  $40$–$44$ | $4.29^{+0.20}_{-0.22}$
$|{\rm GGG};1^{+-}\rangle$ | $\xi_{1}^{\alpha\beta}$ | 9.0 | $34.0\pm 4.0$ |  $3.16$–$9.09$ |  $40$–$99$ | $4.01^{+0.26}_{-0.95}$
$|{\rm GGG};2^{+-}\rangle$ | $\xi_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ | 32.7 | $35.0\pm 4.0$ |  $7.53$–$8.09$ |  $40$–$46$ | $4.42^{+0.24}_{-0.29}$
$|{\rm GGG};3^{+-}\rangle$ | $\xi_{3}^{\alpha_{1}\alpha_{2}\alpha_{3},\beta_{1}\beta_{2}\beta_{3}}$ | 30.2 | $33.0\pm 4.0$ |  $7.69$–$8.40$ |  $40$–$47$ | $4.30^{+0.23}_{-0.26}$
$|{\rm GGG};1^{--}\rangle$ | $\tilde{\xi}_{1}^{\alpha\beta}$ | 31.2 | $34.0\pm 4.0$ |  $5.81$–$6.77$ |  $40$–$51$ | $4.91^{+0.20}_{-0.18}$
$|{\rm GGG};2^{--}\rangle$ | $\tilde{\xi}_{2}^{\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}}$ | 19.7 | $36.0\pm 4.0$ |  $5.80$–$9.47$ |  $40$–$81$ | $4.25^{+0.22}_{-0.33}$
$|{\rm GGG};3^{--}\rangle$ | $\tilde{\xi}_{3}^{\alpha_{1}\alpha_{2}\alpha_{3},\beta_{1}\beta_{2}\beta_{3}}$ | 35.8 | $38.0\pm 4.0$ |  $6.15$–$7.22$ |  $40$–$49$ | $5.59^{+0.33}_{-0.22}$

Similarly, we use the sum rules given in Eqs. (III)-(III) and Eqs. (III)-(III)
to perform numerical analyses, and calculate masses of two- and three-gluon
glueballs systematically. The obtained results are summarized in Table 1,
where we choose threshold values $s_{0}$ for two-gluon glueballs to be around
$9\sim 10$ GeV2, and those for three-gluon glueballs to be around $33\sim 38$
GeV2.

##  V Summary and Discussions

In this paper we study two- and three-gluon glueballs of $C=+$ using the
method of QCD sum rules, including

  * •

the two-gluon glueballs with the quantum numbers $J^{PC}=0^{\pm+}$, $1^{-+}$,
and $2^{\pm+}$;

  * •

the three-gluon glueballs with the quantum numbers $J^{PC}=0^{\pm+}$,
$1^{\pm+}$, and $2^{\pm+}$.

We systematically construct their interpolating currents, and find that all
the spin-1 currents of $C=+$ vanish, suggesting that the “ground-state” spin-1
glueballs of $C=+$ do not exist within the relativistic framework. This
behavior is consistent with Lattice QCD calculations Chen:2005mg ;
Mathieu:2008me ; Meyer:2004gx ; Gregory:2012hu .

We use spin-0 and spin-2 glueball currents to perform QCD sum rule analyses,
and calculate masses of their corresponding spin-0 and spin-2 glueballs. All
these spin-2 currents have four Lorentz indices with certain symmetries, so
that they couple to both positive- and negative-parity glueballs, which need
to be further separated at the hadron level. We refer to Ref. Chen:2021cjr
for detailed discussions.

Table 2: Masses of two- and three-gluon glueballs, in units of GeV. Our QCD sum rule results are listed in the 2nd column. Lattice QCD results are listed in the 3rd-6th columns, taken from Refs. Chen:2005mg ; Mathieu:2008me ; Meyer:2004gx  (quenched) and Ref. Gregory:2012hu  (unquenched). Glueball |  QCD sum rules |  Ref. Chen:2005mg  |  Ref. Mathieu:2008me  |  Ref. Meyer:2004gx  |  Ref. Gregory:2012hu
---|---|---|---|---|---
$|{\rm GG};0^{++}\rangle$ | $1.78^{+0.14}_{-0.17}$ | $1.71\pm 0.05\pm 0.08$ | $1.73\pm 0.05\pm 0.08$ | $1.48\pm 0.03\pm 0.07$ | $1.80\pm 0.06$
$|{\rm GG};2^{++}\rangle$ | $1.86^{+0.14}_{-0.17}$ | $2.39\pm 0.03\pm 0.12$ | $2.40\pm 0.03\pm 0.12$ | $2.15\pm 0.03\pm 0.10$ | $2.62\pm 0.05$
$|{\rm GG};0^{-+}\rangle$ | $2.17^{+0.11}_{-0.11}$ | $2.56\pm 0.04\pm 0.12$ | $2.59\pm 0.04\pm 0.13$ | $2.25\pm 0.06\pm 0.10$ | –
$|{\rm GG};2^{-+}\rangle$ | $2.24^{+0.11}_{-0.11}$ | $3.04\pm 0.04\pm 0.15$ | $3.10\pm 0.03\pm 0.15$ | $2.78\pm 0.05\pm 0.13$ | $3.46\pm 0.32$
$|{\rm GGG};0^{++}\rangle$ | $4.46^{+0.17}_{-0.19}$ | – | $2.67\pm 0.18\pm 0.13$ | $2.76\pm 0.03\pm 0.12$ | $3.76\pm 0.24$
$|{\rm GGG};2^{++}\rangle$ | $4.18^{+0.19}_{-0.42}$ | – | – | $2.88\pm 0.10\pm 0.13$ | –
$|{\rm GGG};0^{-+}\rangle$ | $4.13^{+0.18}_{-0.36}$ | – | $3.64\pm 0.06\pm 0.18$ | $3.37\pm 0.15\pm 0.15$ | $4.49\pm 0.59$
$|{\rm GGG};2^{-+}\rangle$ | $4.29^{+0.20}_{-0.22}$ | – | – | $3.48\pm 0.14\pm 0.16$ | –
$|{\rm GGG};1^{+-}\rangle$ | $4.01^{+0.26}_{-0.95}$ | $2.98\pm 0.03\pm 0.14$ | $2.94\pm 0.03\pm 0.14$ | $2.67\pm 0.07\pm 0.12$ | $3.27\pm 0.34$
$|{\rm GGG};2^{+-}\rangle$ | $4.42^{+0.24}_{-0.29}$ | $4.23\pm 0.05\pm 0.20$ | $4.14\pm 0.05\pm 0.20$ | – | –
$|{\rm GGG};3^{+-}\rangle$ | $4.30^{+0.23}_{-0.26}$ | $3.60\pm 0.04\pm 0.17$ | $3.55\pm 0.04\pm 0.17$ | $3.27\pm 0.09\pm 0.15$ | $3.85\pm 0.35$
$|{\rm GGG};1^{--}\rangle$ | $4.91^{+0.20}_{-0.18}$ | $3.83\pm 0.04\pm 0.19$ | $3.85\pm 0.05\pm 0.19$ | $3.24\pm 0.33\pm 0.15$ | –
$|{\rm GGG};2^{--}\rangle$ | $4.25^{+0.22}_{-0.33}$ | $4.01\pm 0.05\pm 0.20$ | $3.93\pm 0.04\pm 0.19$ | $3.66\pm 0.13\pm 0.17$ | $4.59\pm 0.74$
$|{\rm GGG};3^{--}\rangle$ | $5.59^{+0.33}_{-0.22}$ | $4.20\pm 0.05\pm 0.20$ | $4.13\pm 0.09\pm 0.20$ | $4.33\pm 0.26\pm 0.20$ | –

We summarize the obtained results in Table 2, which are compared with the
Lattice QCD results obtained using non-relativistic glueball operators
Chen:2005mg ; Mathieu:2008me ; Meyer:2004gx ; Gregory:2012hu . For
completeness, we also reanalysis the results of $C=-$ three-gluon glueballs
(also called as odderons), which have been previously studied in Ref.
Chen:2021cjr . Therefore, a rather complete QCD sum rule study have been done
on the lowest-lying glueballs composed of two or three valence gluons. We find
that our QCD sum rule results are generally consistent with unquenched Lattice
QCD results Gregory:2012hu .

To end this paper, we briefly discuss possible decay patterns of two- and
three-gluon glueballs. The two-gluon glueballs can decay after exciting two
quark-antiquark pairs, and recombine into two mesons. However, it is rather
difficult to differentiate them from standard $q\bar{q}$ states. The three-
gluon glueballs can decay after exciting three quark-antiquark pairs, and
recombine into three mesons. Their possible decay patterns are:

$\displaystyle 0^{-+}$ $\displaystyle\to$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\,VVP,VVV~{}~{}~{}~{}~{}~{}~{}~{}~{}(S\mbox{-wave})\,,$
$\displaystyle 0^{++}$ $\displaystyle\to$
$\displaystyle~{}~{}~{}~{}\,VPP,VVP,VVV~{}~{}~{}~{}~{}(P\mbox{-wave})\,,$
$\displaystyle 1^{--}$ $\displaystyle\to$
$\displaystyle~{}~{}~{}~{}\,VPP,VVP,VVV~{}~{}~{}~{}~{}(S\mbox{-wave})\,,$
$\displaystyle 1^{+-}$ $\displaystyle\to$ $\displaystyle
PPP,VPP,VVP,VVV~{}(P\mbox{-wave})\,,$ $\displaystyle 2^{-\pm}$
$\displaystyle\to$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\,VVP,VVV~{}~{}~{}~{}~{}~{}~{}~{}~{}(S\mbox{-wave})\,,$
$\displaystyle 2^{+\pm}$ $\displaystyle\to$
$\displaystyle~{}~{}~{}~{}VPP,VVP,VVV~{}~{}~{}~{}~{}(P\mbox{-wave})\,,$
$\displaystyle 3^{--}$ $\displaystyle\to$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,VVV~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(S\mbox{-wave})\,,$
$\displaystyle 3^{+-}$ $\displaystyle\to$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}VVP,VVV~{}~{}~{}~{}~{}~{}~{}~{}~{}(P\mbox{-wave})\,,$

where $P$ and $V$ denote light vector and pseudoscalar mesons, respectively.
Considering their limited decay patterns, the
$J^{PC}=0^{-+}/2^{-\pm}/3^{\pm-}$ three-gluon glueballs may have relatively
smaller widths, and we propose to search for them in their $VVV$ and $VVP$
decay channels in future BESIII, GlueX, LHC, and PANDA experiments.

## Acknowledgments

We thank Zi-Yang Lin very much for helping us prove that all the spin-1 two-
and three-gluon glueball currents of $C=+$ vanish. We thank the anonymous PRD
referee very much for checking our formulae and correcting one critical
mistake. This project is supported by the National Natural Science Foundation
of China under Grant No. 11722540, No. 11975033, No. 12075019, and No.
12070131001, the National Key R$\&$D Program of China under Contracts No.
2020YFA0406400, the Jiangsu Provincial Double-Innovation Program under Grant
No. JSSCRC2021488, and the Fundamental Research Funds for the Central
Universities.

##  Appendix A Spin-1 currents of $C=+$

In this appendix we prove that the three spin-1 currents
$J_{1}^{\alpha\beta}$, $\eta_{1}^{\alpha\beta}$, and
$\tilde{\eta}_{1}^{\alpha\beta}$ all vanish. Their definitions are given in
Eqs. (15), (27), and (28), respectively. For simplicity, we shall not
differentiate the superscript and subscript in the following calculations.

Firstly, we investigate the current $J_{1}^{\alpha\beta}$. Due to the Lorentz
invariant, we simply assume $\alpha=0$ and $\beta=1\cdots 3$; besides, we need
the Lorentz indices $\mu=0/i$, $\rho=0/k$, and $\sigma=0/l$, with
$i/k/l=1\cdots 3$. We obtain:

$\displaystyle 2J_{1}^{0\beta}$ $\displaystyle=$ $\displaystyle
2G_{a}^{0\mu}\tilde{G}_{a}^{\beta\mu}-\\{0\leftrightarrow\beta\\}\,,$
$\displaystyle=$ $\displaystyle
G_{a}^{0\mu}G_{a}^{\rho\sigma}\epsilon^{\beta\mu\rho\sigma}-G_{a}^{\beta\mu}G_{a}^{\rho\sigma}\epsilon^{0\mu\rho\sigma}$
$\displaystyle=$ $\displaystyle G_{a}^{0i}G_{a}^{k0}\epsilon^{\beta
ik0}+G_{a}^{0i}G_{a}^{0l}\epsilon^{\beta i0l}-G_{a}^{\beta
i}G_{a}^{kl}\epsilon^{0ikl}\,.$

After interchanging $i\leftrightarrow k$, the first term turns out to be zero:

$G_{a}^{0i}G_{a}^{k0}\epsilon^{\beta ik0}=G_{a}^{0k}G_{a}^{i0}\epsilon^{\beta
ki0}=G_{a}^{0i}G_{a}^{k0}\epsilon^{\beta ki0}\rightarrow 0\,.$ (72)

So does the second term. The third term is non-zero when $\beta=k$ or
$\beta=l$. However, for the case $\beta=k$, we can interchange
$i\leftrightarrow l$ and obtain (not sum over $\beta$ here):

$G_{a}^{\beta i}G_{a}^{\beta l}\epsilon^{0i\beta l}=G_{a}^{\beta
l}G_{a}^{\beta i}\epsilon^{0l\beta i}=G_{a}^{\beta i}G_{a}^{\beta
l}\epsilon^{0l\beta i}\rightarrow 0\,.$ (73)

So does the case $\beta=l$. Therefore, the third term is also zero, and the
current $J_{1}^{\alpha\beta}$ vanishes.

Secondly, we investigate the current $\eta_{1}^{\alpha\beta}$:

$\displaystyle 2\eta_{1}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
2f_{abc}\tilde{G}_{a}^{\mu\nu}G_{b}^{\mu\nu}G_{c}^{\alpha\beta}$
$\displaystyle=$ $\displaystyle
f_{abc}\epsilon^{\mu\nu\rho\sigma}G_{a}^{\rho\sigma}G_{b}^{\mu\nu}G_{c}^{\alpha\beta}$
$\displaystyle=$ $\displaystyle
f_{abc}\epsilon^{\mu\nu\rho\sigma}G_{a}^{\mu\nu}G_{b}^{\rho\sigma}G_{c}^{\alpha\beta}$
$\displaystyle=$ $\displaystyle-
f_{abc}\epsilon^{\mu\nu\rho\sigma}G_{b}^{\mu\nu}G_{a}^{\rho\sigma}G_{c}^{\alpha\beta}$
$\displaystyle\rightarrow$ $\displaystyle 0\,.$

In the above expressions, we have consequently interchanged
$\mu\nu\leftrightarrow\rho\sigma$ and $a\leftrightarrow b$. Similarly, we can
prove the current $\tilde{\eta}_{1}^{\alpha\beta}$ to be zero.

One can construct more spin-1 three-gluon glueball currents of $C=+$, such as:

$\displaystyle\eta_{1}^{\prime\alpha\beta}$ $\displaystyle=$ $\displaystyle
f_{abc}G_{a}^{\alpha\mu}G_{b}^{\mu\nu}G_{c}^{\nu\beta}-\\{\alpha\leftrightarrow\beta\\}\,,$
(75) $\displaystyle\tilde{\eta}_{1}^{\prime\alpha\beta}$ $\displaystyle=$
$\displaystyle
f_{abc}G_{a}^{\alpha\mu}G_{b}^{\mu\nu}\tilde{G}_{c}^{\nu\beta}-\\{\alpha\leftrightarrow\beta\\}\,.$
(76)

It is straightforward to prove the former current
$\eta_{1}^{\prime\alpha\beta}$ to be zero:

$\displaystyle\eta_{1}^{\prime\alpha\beta}$ $\displaystyle=$ $\displaystyle
f_{abc}G_{a}^{\alpha\mu}G_{b}^{\mu\nu}G_{c}^{\nu\beta}-\\{\alpha\leftrightarrow\beta\\}$
$\displaystyle=$ $\displaystyle
f_{abc}G_{a}^{\alpha\nu}G_{b}^{\nu\mu}G_{c}^{\mu\beta}-\\{\alpha\leftrightarrow\beta\\}$
$\displaystyle=$ $\displaystyle-
f_{abc}G_{a}^{\beta\nu}G_{b}^{\nu\mu}G_{c}^{\mu\alpha}+\\{\alpha\leftrightarrow\beta\\}$
$\displaystyle=$ $\displaystyle
f_{abc}G_{c}^{\beta\nu}G_{b}^{\nu\mu}G_{a}^{\mu\alpha}-\\{\alpha\leftrightarrow\beta\\}$
$\displaystyle=$ $\displaystyle-
f_{abc}G_{c}^{\nu\beta}G_{b}^{\mu\nu}G_{a}^{\alpha\mu}-\\{\alpha\leftrightarrow\beta\\}$
$\displaystyle\rightarrow$ $\displaystyle 0\,.$

It is a bit tricky but one can still prove the latter current
$\tilde{\eta}_{1}^{\prime\alpha\beta}$ to be zero, after explicitly writing
out all its Lorentz indices. We have done this using the software Mathematica.

## References

  * (1) P. G. O. Freund and Y. Nambu, Dynamics in the Zweig-Iizuka Rule and a New Vector Meson Below 2-GeV/c2, Phys. Rev. Lett. 34, 1645 (1975).
  * (2) H. Fritzsch and P. Minkowski, Psi Resonances, Gluons and the Zweig Rule, Nuovo Cim. A 30, 393 (1975).
  * (3) R. L. Jaffe and K. Johnson, Unconventional States of Confined Quarks and Gluons, Phys. Lett. 60B (1976) 201.
  * (4) A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, New extended model of hadrons, Phys. Rev. D 9, 3471 (1974).
  * (5) N. Isgur and J. E. Paton, Flux-tube model for hadrons in QCD, Phys. Rev. D 31, 2910 (1985).
  * (6) A. Szczepaniak, E. S. Swanson, C. R. Ji and S. R. Cotanch, Glueball Spectroscopy in a Relativistic Many-Body Approach to Hadron Structure, Phys. Rev. Lett. 76, 2011 (1996).
  * (7) F. J. Llanes-Estrada, P. Bicudo and S. R. Cotanch, $J^{--}$ Glueballs and a Low Odderon Intercept, Phys. Rev. Lett. 96, 081601 (2006).
  * (8) I. Szanyi, L. Jenkovszky, R. Schicker and V. Svintozelskyi, Pomeron/glueball and odderon/oddball trajectories, Nucl. Phys. A 998, 121728 (2020).
  * (9) L. Zhang, C. Chen, Y. Chen and M. Huang, Spectra of glueballs and oddballs and the equation of state from holographic QCD, [arXiv:2106.10748 [hep-ph]].
  * (10) K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974).
  * (11) Y. Chen et al., Glueball spectrum and matrix elements on anisotropic lattices, Phys. Rev. D 73, 014516 (2006).
  * (12) V. Mathieu, N. Kochelev and V. Vento, The physics of glueballs, Int. J. Mod. Phys. E 18, 1 (2009).
  * (13) H. B. Meyer, Glueball regge trajectories, hep-lat/0508002.
  * (14) E. Gregory, A. Irving, B. Lucini, C. McNeile, A. Rago, C. Richards and E. Rinaldi, Towards the glueball spectrum from unquenched lattice QCD, JHEP 1210, 170 (2012).
  * (15) V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, In search of scalar gluonium, Nucl. Phys. B 165, 67 (1980).
  * (16) V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Are All Hadrons Alike? Nucl. Phys. B 191 (1981), 301-369.
  * (17) A. L. Kataev, N. V. Krasnikov and A. A. Pivovarov, The Connection Between the Scales of the Gluon and Quark Worlds in Perturbative QCD, Phys. Lett. B 107, 115-118 (1981).
  * (18) A. L. Kataev, N. V. Krasnikov and A. A. Pivovarov, Two Loop Calculations for the Propagators of Gluonic Currents, Nucl. Phys. B 198, 508-518 (1982) [erratum: Nucl. Phys. B 490, 505-507 (1997)].
  * (19) S. Narison, Spectral function sum rules for gluonic currents, Z. Phys. C 26, 209 (1984).
  * (20) S. Narison, Masses, decays and mixings of gluonia in QCD, Nucl. Phys. B 509, 312 (1998).
  * (21) E. Bagan and T. G. Steele, Mass of the scalar glueball. Higher-loop effects in the QCD sum rules, Phys. Lett. B 243, 413 (1990).
  * (22) J. I. Latorre, S. Narison and S. Paban, $0^{++}$ trigluonium sum rules, Phys. Lett. B 191, 437 (1987).
  * (23) H. Forkel, Direct instantons, topological charge screening and QCD glueball sum rules, Phys. Rev. D 71 (2005), 054008.
  * (24) G. Hao, C. F. Qiao and A. L. Zhang, $0^{-+}$ trigluon glueball and its implication for a recent BES observation, Phys. Lett. B 642, 53 (2006).
  * (25) C. F. Qiao and L. Tang, Finding the $0^{--}$ Glueball, Phys. Rev. Lett. 113, 221601 (2014).
  * (26) L. Tang and C. F. Qiao, Mass spectra of $0^{+-}$, $1^{-+}$, and $2^{+-}$ exotic glueballs, Nucl. Phys. B 904, 282 (2016).
  * (27) A. Pimikov, H. J. Lee, N. Kochelev and P. Zhang, Is the exotic $0^{--}$ glueball a pure gluon state? Phys. Rev. D 95, 071501(R) (2017).
  * (28) A. Pimikov, H. J. Lee and N. Kochelev, Comment on ”Finding the $0^{--}$ Glueball”, Phys. Rev. Lett. 119, no.7, 079101 (2017).
  * (29) A. Pimikov, H. J. Lee, N. Kochelev, P. Zhang and V. Khandramai, Exotic glueball $0^{\pm-}$ states in QCD sum rules, Phys. Rev. D 96, no.11, 114024 (2017).
  * (30) S. Narison, Di-Gluonium Sum Rules, Conformal Charge and I = 0 Scalar Mesons, arXiv:2108.13089 [hep-ph].
  * (31) V. M. Abazov et al. [D0 Collaboration], Measurement of the differential cross section $d\sigma/dt$ in elastic $p\bar{p}$ scattering at $\sqrt{s}=1.96$ TeV, Phys. Rev. D 86, 012009 (2012).
  * (32) V. M. Abazov et al. [D0 and TOTEM Collaborations], Comparison of $pp$ and $p\bar{p}$ differential elastic cross sections and observation of the exchange of a colorless $C$-odd gluonic compound, arXiv:2012.03981 [hep-ex].
  * (33) G. Antchev et al. [TOTEM Collaboration], First determination of the ${\rho}$ parameter at ${\sqrt{s}=13}$ TeV: probing the existence of a colourless $C$-odd three-gluon compound state, Eur. Phys. J. C 79, 785 (2019).
  * (34) E. M. Levin and M. G. Ryskin, High-energy hadron collisions in QCD, Phys. Rept. 189, 267 (1990).
  * (35) M. A. Braun, Odderon and QCD, hep-ph/9805394.
  * (36) J. R. Cudell et al. [COMPETE Collaboration], Benchmarks for the Forward Observables at RHIC, the Tevatron-Run II and the LHC, Phys. Rev. Lett. 89, 201801 (2002).
  * (37) V. A. Khoze, A. D. Martin and M. G. Ryskin, Elastic proton-proton scattering at 13 TeV, Phys. Rev. D 97, 034019 (2018).
  * (38) E. Martynov and B. Nicolescu, Odderon effects in the differential cross-sections at Tevatron and LHC energies, Eur. Phys. J. C 79, 461 (2019).
  * (39) V. A. Khoze, A. D. Martin and M. G. Ryskin, Black disk, maximal Odderon and unitarity, Phys. Lett. B 780, 352 (2018).
  * (40) T. Csörgő, R. Pasechnik and A. Ster, Odderon and proton substructure from a model-independent Lévy imaging of elastic $pp$ and $p\bar{p}$ collisions, Eur. Phys. J. C 79, 62 (2019).
  * (41) V. P. Gonçalves and P. V. R. G. Silva, The Phillips-Barger model for the elastic cross section and the Odderon, Eur. Phys. J. C 79, 237 (2019).
  * (42) W. Xie, A. Watanabe and M. Huang, Elastic proton-proton scattering at LHC energies in holographic QCD, JHEP 1910, 053 (2019).
  * (43) L. Lukaszuk and B. Nicolescu, A possible interpretation of $pp$ rising total cross-sections, Lett. Nuovo Cim. 8, 405 (1973).
  * (44) J. Bartels, High-energy behavior in a nonabelian gauge theory (II). First corrections to $T_{n\to m}$ beyond the leading $\ln s$ approximation, Nucl. Phys. B 175, 365 (1980).
  * (45) J. Kwiecinski and M. Praszalowicz, Three gluon integral equation and odd C singlet Regge singularities in QCD, Phys. Lett. 94B, 413 (1980).
  * (46) A. Donnachie and P. V. Landshoff, Multi-gluon exchange in $pp$ elastic scattering, Phys. Lett. 123B, 345 (1983).
  * (47) A. Dumitru and R. Paatelainen, Cubic color charge correlator in a proton made of three quarks and a gluon, [arXiv:2106.12623 [hep-ph]].
  * (48) M. M. Block, Hadronic forward scattering: Predictions for the Large Hadron Collider and cosmic rays, Phys. Rept. 436, 71 (2006).
  * (49) R. L. Jaffe, K. Johnson and Z. Ryzak, Qualitative features of the glueball spectrum, Annals Phys. 168, 344 (1986).
  * (50) H. X. Chen, W. Chen and S. L. Zhu, Toward the existence of the odderon as a three-gluon bound state, Phys. Rev. D 103, L091503 (2021).
  * (51) C. N. Yang, Selection Rules for the Dematerialization of a Particle Into Two Photons, Phys. Rev. 77, 242 (1950).
  * (52) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCD and resonance physics. theoretical foundations, Nucl. Phys. B 147, 385 (1979).
  * (53) L. J. Reinders, H. Rubinstein and S. Yazaki, Hadron properties from QCD sum rules, Phys. Rept. 127, 1 (1985).
  * (54) H. X. Chen, A. Hosaka and S. L. Zhu, Light Scalar Tetraquark Mesons in the QCD Sum Rule, Phys. Rev. D 76, 094025 (2007).
  * (55) H. X. Chen, E. L. Cui, W. Chen, X. Liu, T. G. Steele and S. L. Zhu, QCD sum rule study of hidden-charm pentaquarks, Eur. Phys. J. C 76, 572 (2016).
  * (56) W. Chen, H. X. Chen, X. Liu, T. G. Steele and S. L. Zhu, Hunting for exotic doubly hidden-charm/bottom tetraquark states, Phys. Lett. B 773, 247-251 (2017).
  * (57) J. Govaerts, F. de Viron, D. Gusbin and J. Weyers, QCD Sum Rules and Hybrid Mesons, Nucl. Phys. B 248, 1-18 (1984).
  * (58) S. Narison, QCD parameter correlations from heavy quarkonia, Int. J. Mod. Phys. A 33, 1850045 (2018) Addendum: [Int. J. Mod. Phys. A 33, 1892004 (2018)].
  * (59) S. Narison, Gluon condensates and precise $\overline{m}_{c,b}$ from QCD-moments and their ratios to order $\alpha_{s}^{3}$ and $\langle G^{4}\rangle$, Phys. Lett. B 706, 412 (2012).
  * (60) P. A. Zyla et al. [Particle Data Group], Review of Particle Physics, PTEP 2020, 083C01 (2020).