# New hadron configuration: The double-gluon hybrid state

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

This is the first study on the double-gluon hybrid, which consists of one
valence quark and one valence antiquark together with two valence gluons. We
concentrate on the one with the exotic quantum number $J^{PC}=2^{+-}$ that
conventional $\bar{q}q$ mesons can not reach. We apply QCD sum rule method to
evaluate its mass to be $2.26^{+0.20}_{-0.25}$ GeV, and study its possible
decay patterns. Especially, its three-meson decay patterns are generally not
suppressed severely compared to two-meson decay patterns, so the $S$-wave
three-meson decay channels $f_{1}\omega\pi/f_{1}\rho\pi$ can be useful in
identifying its nature, which is of particular importance to the direct test
of QCD in the low energy sector.

hybrid state, exotic hadron, QCD sum rules

Introduction.—A hybrid state consists of one valence quark and one valence
antiquark together with some valence gluons. Its experimental confirmation is
a direct test of Quantum Chromodynamics (QCD) in the low energy sector.
Especially, the hybrid states with $J^{PC}=0^{--}/0^{+-}/1^{-+}/2^{+-}/\cdots$
are of particular interests, since these exotic quantum numbers arise from the
manifest gluon degree of freedom and can not be accessed by conventional
$\bar{q}q$ mesons. In the past half century there have been a lot of
experimental and theoretical investigations, but their nature remains elusive
pdg ; Klempt:2007cp ; Meyer:2015eta ; Chen:2016qju .

Up to now there are three candidates observed in experiments with the exotic
quantum number $I^{G}J^{PC}=1^{-}1^{-+}$, i.e., the $\pi_{1}(1400)$
alde:1988iqi , $\pi_{1}(1600)$ E852:2001ikk ; COMPASS:2009xrl , and
$\pi_{1}(2015)$ E852:2004gpn . They are possible single-gluon hybrids, which
consists of one quark-antiquark pair together with only one valence gluon.
Within the flux tube model Isgur:1984bm ; Page:1998gz ; Burns:2006wz  the
gluon degree of freedom is modeled as a semi-classical flux tube, and the
hybrid with $J^{PC}=1^{-+}$ was found to be around 1.9 GeV. The mass
calculated using the constituent gluon model is also around 1.9 GeV
Iddir:2007dq , and those extracted from quenched Lattice QCD simulations range
from 1.7 GeV to 2.0 GeV McNeile:1998cp ; Lacock:1998be ; Bernard:2003jd ;
Hedditch:2005zf ; Dudek:2009qf ; Dudek:2010wm ; Briceno:2017max . We refer to
Refs. Govaerts:1984bk ; Chetyrkin:2000tj ; Jin:2002rw ; Chen:2010ic ;
Huang:2016upt ; Xu:2018cor  for various studies on them using QCD sum rules
and Dyson-Schwinger equation. However, the above three
$I^{G}J^{PC}=1^{-}1^{-+}$ structures may also be explained as tetraquark
states Chung:2002fz ; Chen:2008qw ; Chen:2008ne ; Narison:2009vj . It is not
easy to differentiate the hybrid and tetraquark pictures so far, and this
tough problem needs to be solved by experimentalists and theorists together in
future.

In this letter we further investigate the double-gluon hybrid, which consists
of one quark-antiquark pair together with two valence gluons. This is
motivated by the recent D0 and TOTEM experiments observing the evidence of a
$C$-odd three-gluon glueball TOTEM:2020zzr , given that two-gluon glueballs
have been detailedly studied but still difficult to be identified
unambiguously pdg ; Klempt:2007cp ; Meyer:2015eta ; Chen:2016qju . We
construct twelve double-gluon hybrid currents and use them to perform QCD sum
rule analyses. As the first study, we concentrate on the one with the exotic
quantum number $J^{PC}=2^{+-}$, which makes it doubly interesting. Among the
twelve currents, we find it to be the only one with the mass predicted to be
smaller than 3.0 GeV, that is $M_{2^{+-}}=2.26^{+0.20}_{-0.25}$ GeV. This mass
value is accessible in the BESIII, GlueX, LHC, and PANDA experiments.

We study possible decay patterns of the double-gluon hybrids with
$I^{G}J^{PC}=1^{+}2^{+-}$ and $0^{-}2^{+-}$, separately for two- and three-
meson final states. We propose to search for the one of
$I^{G}J^{PC}=1^{+}2^{+-}$ in its decay channels $\rho
f_{0}(980)/\omega\pi/K^{*}\bar{K}/f_{1}\omega\pi/\rho\pi\pi/\cdots$, and the
one of $I^{G}J^{PC}=0^{-}2^{+-}$ in its decay channels $\rho
a_{0}(980)/\rho\pi/K^{*}\bar{K}/f_{1}\rho\pi/\omega\pi\pi/\cdots$, both of
which are worthy to be searched for in the decay process
$J/\psi\to\pi/\pi\pi/\eta+X(\to K^{*}\bar{K}^{*}/K^{*}\bar{K}\pi/\rho
K\bar{K}\to K\bar{K}\pi\pi)$. Especially, their three-meson decay patterns are
generally not suppressed severely compared to two-meson decay patterns, since
they are both at the $\mathcal{O}(\alpha_{s})$ order. Accordingly, the
$S$-wave three-meson decay channels $f_{1}\omega\pi/f_{1}\rho\pi$ can be
useful in distinguishing their nature from the tetraquark picture, therefore,
of particular importance to the direct test of QCD in the low energy sector.

Double-gluon hybrid currents.—As the first step, we use the light $up/down$
quark field $q_{a}(x)$ and the gluon field strength tensor $G^{n}_{\mu\nu}(x)$
to construct double-gluon hybrid currents. Here $a=1\cdots 3$ and $n=1\cdots
8$ are color indices; $\mu$ and $\nu$ are Lorentz indices. Besides, we need
the antiquark field $\bar{q}_{a}(x)$ and the dual gluon field strength tensor
$\tilde{G}^{n}_{\mu\nu}=G^{n,\rho\sigma}\times\epsilon_{\mu\nu\rho\sigma}/2$.

Generally speaking, one can construct many double-gluon hybrid currents by
combining the color-octet quark-antiquark fields,

$\displaystyle\bar{q}_{a}\lambda_{n}^{ab}q_{b}\,,\,\bar{q}_{a}\lambda_{n}^{ab}\gamma_{5}q_{b}\,,$
(1)
$\displaystyle\bar{q}_{a}\lambda_{n}^{ab}\gamma_{\mu}q_{b}\,,\,\bar{q}_{a}\lambda_{n}^{ab}\gamma_{\mu}\gamma_{5}q_{b}\,,\,\bar{q}_{a}\lambda_{n}^{ab}\sigma_{\mu\nu}q_{b}\,,$

and the relativistic color-octet double-gluon fields,

$d^{npq}G_{p}^{\alpha\beta}G_{q}^{\gamma\delta}\,,\,f^{npq}G_{p}^{\alpha\beta}G_{q}^{\gamma\delta}\,,$
(2)

with suitable Lorentz matrices $\Gamma^{\mu\nu\cdots\alpha\beta\gamma\delta}$.
Here $d^{npq}$ and $f^{npq}$ are totally symmetric and antisymmetric $SU(3)$
structure constants, respectively.

As the first study on the double-gluon hybrid, in the present study we shall
investigate the following double-gluon hybrid currents:

$\displaystyle J_{0^{-+}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}d^{npq}~{}g_{s}^{2}G_{p}^{\mu\nu}G_{q,\mu\nu}\,,$
(3) $\displaystyle J_{0^{--}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}f^{npq}~{}g_{s}^{2}G_{p}^{\mu\nu}G_{q,\mu\nu}\,,$
(4) $\displaystyle J_{0^{++}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}d^{npq}~{}g_{s}^{2}G_{p}^{\mu\nu}\tilde{G}_{q,\mu\nu}\,,$
(5) $\displaystyle J_{0^{+-}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}f^{npq}~{}g_{s}^{2}G_{p}^{\mu\nu}\tilde{G}_{q,\mu\nu}\,,$
(6) $\displaystyle J^{\alpha\beta}_{1^{-+}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}d^{npq}~{}g_{s}^{2}G_{p}^{\alpha\mu}G_{q,\mu}^{\beta}-\\{\alpha\leftrightarrow\beta\\}\,,$
$\displaystyle J^{\alpha\beta}_{1^{--}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}f^{npq}~{}g_{s}^{2}G_{p}^{\alpha\mu}G_{q,\mu}^{\beta}-\\{\alpha\leftrightarrow\beta\\}\,,$
$\displaystyle J^{\alpha\beta}_{1^{++}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}d^{npq}~{}g_{s}^{2}G_{p}^{\alpha\mu}\tilde{G}_{q,\mu}^{\beta}-\\{\alpha\leftrightarrow\beta\\}\,,$
$\displaystyle J^{\alpha\beta}_{1^{+-}}$ $\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}f^{npq}~{}g_{s}^{2}G_{p}^{\alpha\mu}\tilde{G}_{q,\mu}^{\beta}-\\{\alpha\leftrightarrow\beta\\}\,,$
$\displaystyle J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{-+}}$
$\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}d^{npq}~{}\mathcal{S}[g_{s}^{2}G_{p}^{\alpha_{1}\beta_{1}}G_{q}^{\alpha_{2}\beta_{2}}]\,,$
(11) $\displaystyle J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{--}}$
$\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}f^{npq}~{}\mathcal{S}[g_{s}^{2}G_{p}^{\alpha_{1}\beta_{1}}G_{q}^{\alpha_{2}\beta_{2}}]\,,$
(12) $\displaystyle J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{++}}$
$\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}d^{npq}~{}\mathcal{S}[g_{s}^{2}G_{p}^{\alpha_{1}\beta_{1}}\tilde{G}_{q}^{\alpha_{2}\beta_{2}}]\,,$
(13) $\displaystyle J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{+-}}$
$\displaystyle=$
$\displaystyle\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}~{}f^{npq}~{}\mathcal{S}[g_{s}^{2}G_{p}^{\alpha_{1}\beta_{1}}\tilde{G}_{q}^{\alpha_{2}\beta_{2}}]\,,$
(14)

where $\mathcal{S}$ denotes symmetrization and subtracting trace terms in the
two sets $\\{\alpha_{1}\alpha_{2}\\}$ and $\\{\beta_{1}\beta_{2}\\}$
simultaneously.

The above double-gluon hybrid currents have very clear Lorentz structures,
simply because the color-octet quark-antiquark field
$\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}$ does not contain any surplus
Lorentz index. Besides, this quark-antiquark pair has the $S$-wave spin-parity
quantum number $J^{P}=0^{-}$, so these currents are capable of coupling to the
lowest-lying double-gluon hybrid states.

QCD sum rule analyses.—The method of QCD sum rules has been widely applied in
the study of hadron phenomenology Shifman:1978bx ; Reinders:1984sr , and in
this letter we apply it to study the double-gluon hybrid currents defined in
Eqs. (3–14). We find that only the double-gluon hybrid state coupled by the
current $J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{+-}}$ has the mass
smaller than 3.0 GeV. This state has the exotic quantum number $J^{PC}=2^{+-}$
that conventional $\bar{q}q$ mesons can not reach, making it doubly
interesting. Moreover, the current
$J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{+-}}$ contains the double-
gluon field with the symmetric spin $J=2$ that can not (easily) transform to
the single-gluon field, making this current significantly different from the
single-gluon hybrid current.

We briefly introduce how we use the method of QCD sum rules to study the
current $J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{+-}}$ defined in Eq.
(14). Its two-point correlation function

$\displaystyle\Pi^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2};\alpha_{1}^{\prime}\beta_{1}^{\prime},\alpha_{2}^{\prime}\beta_{2}^{\prime}}(q^{2})$
$\displaystyle\equiv$ $\displaystyle i\int d^{4}xe^{iqx}\langle 0|{\bf
T}[J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{+-}}(x)J^{\alpha_{1}^{\prime}\beta_{1}^{\prime},\alpha_{2}^{\prime}\beta_{2}^{\prime}\dagger}_{2^{+-}}(0)]|0\rangle$
$\displaystyle=$
$\displaystyle\mathcal{S}^{\prime}[g^{\alpha_{1}\alpha_{1}^{\prime}}g^{\beta_{1}\beta_{1}^{\prime}}g^{\alpha_{2}\alpha_{2}^{\prime}}g^{\beta_{2}\beta_{2}^{\prime}}]~{}\Pi(q^{2})\,,$

can be investigated at both hadron and quark-gluon levels. Here
$\mathcal{S}^{\prime}$ denotes anti-symmetrization in the four sets
$\\{\alpha_{1}\beta_{1}\\}$, $\\{\alpha_{2}\beta_{2}\\}$,
$\\{\alpha_{1}^{\prime}\beta_{1}^{\prime}\\}$, and
$\\{\alpha_{2}^{\prime}\beta_{2}^{\prime}\\}$, and symmetrization and
subtracting trace terms in the four sets $\\{\alpha_{1}\alpha_{2}\\}$,
$\\{\beta_{1}\beta_{2}\\}$, $\\{\alpha_{1}^{\prime}\alpha_{2}^{\prime}\\}$,
and $\\{\beta_{1}^{\prime}\beta_{2}^{\prime}\\}$, simultaneously.

The spectral density $\rho(s)\equiv{\rm Im}\Pi(s)/\pi$ can be extracted from
Eq. (New hadron configuration: The double-gluon hybrid state) through the
dispersion relation

$\Pi(q^{2})=\int_{0}^{\infty}\frac{\rho(s)}{s-q^{2}-i\varepsilon}ds\,.$ (16)

At the hadron level we parameterize it using one pole dominance for the
possible ground state $|X;2^{+-}\rangle$ together with the continuum
contribution:

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

At the quark-gluon level we calculate Eq. (New hadron configuration: The
double-gluon hybrid state) and extract $\rho_{\rm OPE}(s)$ using the method of
operator product expansion (OPE). After performing the Borel transformation to
Eq. (16) at both hadron and quark-gluon levels, we obtain

$\Pi(s_{0},M_{B}^{2})\equiv
f^{2}_{X}e^{-M_{X}^{2}/M_{B}^{2}}=\int_{0}^{s_{0}}\rho_{\rm
OPE}(s)e^{-s/M_{B}^{2}}ds\,,$ (18)

where we have approximated the continuum using the OPE spectral density above
the threshold value $s_{0}$.

Finally, we calculate the mass of $|X;2^{+-}\rangle$ through

$M^{2}_{X}(s_{0},M_{B})=\frac{\int_{0}^{s_{0}}s\rho_{\rm
OPE}(s)e^{-s/M_{B}^{2}}ds}{\int_{0}^{s_{0}}\rho_{\rm
OPE}(s)e^{-s/M_{B}^{2}}ds}\,.$ (19)

(a) (a)

(b) (b–1)

(c) (b–2)

(d) (b–3)

(e) (b–4)

(f) (c–1)

(g) (c–2)

(h) (c–3)

(i) (c–4)

(j) (d–1)

(k) (d–2)

(l) (d–3)

(m) (d–4)

(n) (d–5)

(o) (d–6)

(p) (e–1)

(q) (e–2)

(r) (e–3)

Figure 1: Feynman diagrams for the double-gluon hybrid, including the
perturbative term, the quark condensate $\langle\bar{q}q\rangle$, the quark-
gluon mixed condensate $\langle\bar{g}_{s}q\sigma Gq\rangle$, the two-gluon
condensate $\langle g_{s}^{2}GG\rangle$, the three-gluon condensate $\langle
g_{s}^{3}G^{3}\rangle$, and their combinations. The diagrams (a) and (b–i) are
proportional to $\alpha_{s}^{2}\times g_{s}^{0}$, the diagrams (c–i) and (d–i)
are proportional to $\alpha_{s}^{2}\times g_{s}^{1}$, and the diagrams (e–i)
are proportional to $\alpha_{s}^{2}\times g_{s}^{2}$.

In the present study we take into account the Feynman diagrams depicted in
Fig. 1, and calculate $\rho_{\rm OPE}(s)$ up to the dimension eight ($D=8$)
condensates. The gluon field strength tensor $G^{n}_{\mu\nu}$ is defined as

$G^{n}_{\mu\nu}=\partial_{\mu}A_{\nu}^{n}-\partial_{\nu}A_{\mu}^{n}+g_{s}f^{npq}A_{p,\mu}A_{q,\nu}\,,$
(20)

so it can be naturally separated into two parts. We depict the former two
terms using the single-gluon-line, and the third term using the double-gluon-
line with a red vertex, e.g., the diagram depicted in Fig. 1(c–3).

We calculate the spectral density from the current
$J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{+-}}$ to be

$\displaystyle\rho_{\rm OPE}(s)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}^{2}s^{5}}{80640\pi^{4}}+\left(\frac{\alpha_{s}\langle
g_{s}^{2}GG\rangle}{3840\pi^{3}}+\frac{7\alpha_{s}^{2}\langle
g_{s}^{2}GG\rangle}{61440\pi^{4}}\right)s^{3}$
$\displaystyle+\left(\frac{\alpha_{s}^{2}\langle\bar{q}q\rangle^{2}}{9}-\frac{\alpha_{s}\langle
g_{s}^{3}G^{3}\rangle}{1536\pi^{3}}\right)s^{2}$
$\displaystyle+\left(-\frac{2\alpha_{s}^{2}\langle\bar{q}q\rangle\langle\bar{g}_{s}q\sigma
Gq\rangle}{9}-\frac{\alpha_{s}\langle
g_{s}^{2}GG\rangle^{2}}{18432\pi^{3}}\right)s\,,$

where we have taken into account all the diagrams proportional to
$\alpha_{s}^{2}\times g_{s}^{0}$ and $\alpha_{s}^{2}\times g_{s}^{1}$; while
there are so many diagrams proportional to $\alpha_{s}^{2}\times g_{s}^{2}$,
and we have kept only three of them, as depicted in Fig. 1(e–i).

Numerical analyses.—We study the sum rules given in Eq. (New hadron
configuration: The double-gluon hybrid state) numerically using the following
values for various QCD parameters at the renormalization scale $2$ GeV and the
QCD scale $\Lambda_{\rm QCD}=300$ MeV pdg ; Ovchinnikov:1988gk ; Jamin:2002ev
; Gimenez:2005nt ; Narison:2011xe ; Narison:2018dcr :

$\displaystyle\alpha_{s}(Q^{2})$ $\displaystyle=$ $\displaystyle{4\pi\over
11\ln(Q^{2}/\Lambda_{\rm QCD}^{2})}\,,$ $\displaystyle\langle\bar{q}q\rangle$
$\displaystyle=$ $\displaystyle-(0.240\pm 0.010)^{3}\mbox{ GeV}^{3}\,,$
$\displaystyle\langle g_{s}\bar{q}\sigma Gq\rangle$ $\displaystyle=$
$\displaystyle(0.8\pm 0.2)\times\langle\bar{q}q\rangle\mbox{ GeV}^{2}\,,$ (22)
$\displaystyle\langle\alpha_{s}GG\rangle$ $\displaystyle=$
$\displaystyle(6.35\pm 0.35)\times 10^{-2}\mbox{ GeV}^{4}\,,$
$\displaystyle\langle g_{s}^{3}G^{3}\rangle$ $\displaystyle=$
$\displaystyle(8.2\pm 1.0)\times\langle\alpha_{s}GG\rangle\mbox{ GeV}^{2}\,.$

As shown in Eq. (19), the mass of $|X;2^{+-}\rangle$ depends on the Borel mass
$M_{B}$ and the threshold value $s_{0}$. To insure the convergence of Eq. (New
hadron configuration: The double-gluon hybrid state), we require a) the
$\alpha_{s}^{2}\times g_{s}^{2}$ terms to be less than 5%, and b) the $D=8$
terms to be less than 10%:

CVG $\displaystyle\equiv$
$\displaystyle\left|\frac{\Pi^{g_{s}^{n=6}}(s_{0},M_{B}^{2})}{\Pi(s_{0},M_{B}^{2})}\right|\leq
5\%\,,$ (23) $\displaystyle\mbox{CVG}^{\prime}$ $\displaystyle\equiv$
$\displaystyle\left|\frac{\Pi^{{\rm
D=8}}(s_{0},M_{B}^{2})}{\Pi(s_{0},M_{B}^{2})}\right|\leq 10\%\,.$ (24)

To insure the one-pole-dominance assumption, we require the pole contribution
(PC) to be larger than 40%:

$\mbox{PC}\equiv\left|\frac{\Pi(s_{0},M_{B}^{2})}{\Pi(\infty,M_{B}^{2})}\right|\geq
40\%\,.$ (25)

Altogether we determine the Borel window to be $1.61$ GeV${}^{2}\leq
M_{B}^{2}\leq 1.78$ GeV2 when setting $s_{0}=7.0$ GeV2.

We redo the same procedures by changing $s_{0}$, and find that there are non-
vanishing Borel windows as long as $s_{0}\geq s^{\rm min}_{0}=6.3$ GeV2.
Accordingly, we set $s_{0}$ to be about 10% larger, and determine our working
regions to be $5.0$ GeV${}^{2}\leq s_{0}\leq 9.0$ GeV2 and $1.61$
GeV${}^{2}\leq M_{B}^{2}\leq 1.78$ GeV2. The mass of $|X;2^{+-}\rangle$ is
evaluated to be

$\displaystyle M_{|X;2^{+-}\rangle}$ $\displaystyle=$ $\displaystyle
2.26^{+0.19}_{-0.25}\pm 0.07\pm 0.03{\rm~{}GeV}$ (26) $\displaystyle=$
$\displaystyle 2.26^{+0.20}_{-0.25}{\rm~{}GeV}\,,$

whose uncertainty is due to the threshold value $s_{0}$, Borel mass $M_{B}$,
and various quark and gluon parameters listed in Eqs. (22), respectively. We
show it in Fig. 2 as a function of the Borel mass $M_{B}$ and the threshold
value $s_{0}$. This mass value is obtained for both isoscalar and isovector
states, so actually we can not differentiate them in the present QCD sum rule
study.

(a) (a) (b) (b)

Figure 2:  Mass of the double-gluon hybrid $|X;2^{+-}\rangle$ as a function of
the Borel mass $M_{B}$ (a) and the threshold value $s_{0}$ (b). In the
subfigure (a) the short-dashed/solid/long-dashed curves are obtained by
setting $s_{0}=5.0/7.0/9.0$ GeV2, respectively. In the subfigure (b) the
short-dashed/solid/long-dashed curves are obtained by setting
$M_{B}^{2}=1.61/1.70/1.78$ GeV2, respectively.

For completeness, we also use the other eleven hybrid currents defined in Eqs.
(3–13) to perform QCD sum rule analyses. We explicitly prove the four currents
$J^{\cdots}_{0^{--}/0^{+-}/1^{-+}/1^{++}}$ to be zero, while masses extracted
from the seven currents
$J^{\cdots}_{0^{-+}/0^{++}/1^{--}/1^{+-}/2^{--}/2^{-+}/2^{++}}$ are all larger
than 3.0 GeV. We leave their detailed discussions for our future studies.

Decay analyses.—The double-gluon hybrid can decay after exciting two
$\bar{q}q/\bar{s}s$ ($q=u/d$) pairs from two gluons, followed by recombining
three color-octet $\bar{q}q/\bar{s}s$ pairs into two color-singlet mesons or
three mesons, as depicted in Fig. 3. These two possible decay processes are
both at the $\mathcal{O}(\alpha_{s})$ order, so three-meson decay patterns are
generally not suppressed severely compared to two-meson decay patterns, or
even enhanced due to the quark-antiquark annihilation during the two-meson
decay process. This behavior can be useful in identifying the nature of the
double-gluon hybrid.

(a) (a)

(b) (b)

Figure 3: Possible decay processes of the double-gluon hybrid.

To investigate decay properties of the double-gluon hybrid, we assume its
final quark content to be either

$\displaystyle(\bar{q}q)_{\mathbf{8}_{C}}\times(\bar{q}q+\bar{s}s)_{\mathbf{8}_{C}}$
$\displaystyle\rightarrow$
$\displaystyle(\bar{q}q)_{\mathbf{1}_{C}}(\bar{q}q)_{\mathbf{1}_{C}}$
$\displaystyle+$
$\displaystyle(\bar{q}s)_{\mathbf{1}_{C}}(\bar{s}q)_{\mathbf{1}_{C}}\,,$

or

$\displaystyle(\bar{q}q)_{\mathbf{8}_{C}}\times(\bar{q}q+\bar{s}s)^{2}_{\mathbf{8}_{C}}$
$\displaystyle\rightarrow$
$\displaystyle(\bar{q}q)_{\mathbf{1}_{C}}(\bar{q}q)_{\mathbf{1}_{C}}(\bar{q}q)_{\mathbf{1}_{C}}$
$\displaystyle+$
$\displaystyle(\bar{q}q)_{\mathbf{1}_{C}}(\bar{q}s)_{\mathbf{1}_{C}}(\bar{s}q)_{\mathbf{1}_{C}}$
$\displaystyle+$
$\displaystyle(\bar{s}s)_{\mathbf{1}_{C}}(\bar{q}s)_{\mathbf{1}_{C}}(\bar{s}q)_{\mathbf{1}_{C}}\,.$

Accordingly, we list some possible decay patterns of the double-gluon hybrids
with the exotic quantum numbers $I^{G}J^{PC}=1^{+}2^{+-}$ and $0^{-}2^{+-}$ in
Table 1, separately for two- and three-meson decay processes. The one of
$I^{G}J^{PC}=1^{+}2^{+-}$ may be observed in its two-meson decay channels
$\rho f_{0}(980)/\omega\pi/K^{*}\bar{K}/\cdots$ and three-meson decay channels
$f_{1}\omega\pi/\rho\pi\pi/\cdots$; the one of $I^{G}J^{PC}=0^{-}2^{+-}$ may
be observed in its two-meson decay channels $\rho
a_{0}(980)/\rho\pi/K^{*}\bar{K}/\cdots$ and three-meson decay channels
$f_{1}\rho\pi/\omega\pi\pi/\cdots$. Especially, both of them are worthy to be
searched for in the decay process $J/\psi\to\pi/\pi\pi/\eta+X(\to
K^{*}\bar{K}^{*}/K^{*}\bar{K}\pi/\rho K\bar{K}\to K\bar{K}\pi\pi)$, and the
$S$-wave three-meson decay channels $f_{1}\omega\pi/f_{1}\rho\pi$ can be
useful in identifying their nature.

Table 1: Some possible two- and three-meson decay patterns of the double-gluon hybrids with the exotic quantum numbers $I^{G}J^{PC}=1^{+}2^{+-}$ and $0^{-}2^{+-}$. Two-Meson |  $1^{+}2^{+-}$ |  $0^{-}2^{+-}$
---|---|---
S-wave | $K_{2}^{*}\bar{K}_{0}^{*}$
P-wave | $h_{1}\pi,a_{1}\pi,a_{2}\pi,b_{1}\eta,\rho f_{0}$ | $b_{1}\pi,h_{1}\eta,\rho a_{0}$
$K_{1}\bar{K},K_{2}^{*}\bar{K},K^{*}\bar{K}_{0}^{*}$
D-wave | $\rho^{+}\rho^{-},\omega\pi,\rho\eta,\rho\eta^{\prime}$ | $\rho\pi,\omega\eta,\omega\eta^{\prime}$
$K^{*}\bar{K},K^{*}\bar{K}^{*}$
Three-Meson |  $1^{+}2^{+-}$ |  $0^{-}2^{+-}$
S-wave | $f_{1}\omega\pi,a_{1}\rho\pi$ | $f_{1}\rho\pi,a_{1}\omega\pi$
P-wave | $\rho\pi\pi,\omega\eta\pi,\rho\eta\eta$ | $\omega\pi\pi,\rho\eta\pi$
$K^{*}\bar{K}\pi,\rho K\bar{K},\omega K\bar{K}$

Summary.—As the first study on the double-gluon hybrid, we systematically
construct twelve double-gluon hybrid currents and use them to perform QCD sum
rule analyses. These currents are constructed by using the $S$-wave color-
octet quark-antiquark field $\bar{q}_{a}\gamma_{5}\lambda_{n}^{ab}q_{b}$
together with the relativistic color-octet double-gluon fields. Among them, we
find that only the double-gluon hybrid state coupled by the current
$J^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2}}_{2^{+-}}$ has the mass smaller
than 3.0 GeV, that is

$M_{|X;2^{+-}\rangle}=2.26^{+0.20}_{-0.25}{\rm~{}GeV}\,,$ (29)

which is accessible in the BESIII, GlueX, LHC, and PANDA experiments.
Moreover, this state has the exotic quantum number $J^{PC}=2^{+-}$ that
conventional $\bar{q}q$ mesons can not reach, making it doubly interesting.

We study its possible decay patterns separately for two- and three-meson final
states. We propose to search for the one of $I^{G}J^{PC}=1^{+}2^{+-}$ in its
decay channels $\rho
f_{0}(980)/\omega\pi/K^{*}\bar{K}/f_{1}\omega\pi/\rho\pi\pi/\cdots$, and the
one of $I^{G}J^{PC}=0^{-}2^{+-}$ in its decay channels $\rho
a_{0}(980)/\rho\pi/K^{*}\bar{K}/f_{1}\rho\pi/\omega\pi\pi/\cdots$, both of
which are worthy to be searched for in the decay process
$J/\psi\to\pi/\pi\pi/\eta+X(\to K^{*}\bar{K}^{*}/K^{*}\bar{K}\pi/\rho
K\bar{K}\to K\bar{K}\pi\pi)$. Especially, their three-meson decay patterns are
generally not suppressed severely compared to two-meson decay patterns, since
they are both at the $\mathcal{O}(\alpha_{s})$ order. Accordingly, the
$S$-wave three-meson decay channels $f_{1}\omega\pi/f_{1}\rho\pi$ can be
useful in identifying their nature, therefore, of particular importance to the
direct test of QCD in the low energy sector.

###### Acknowledgements.

This project is supported by the National Natural Science Foundation of China
under Grant No. 11975033, No. 12075019, No. 12175318, 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.

## References

  * (1) P. A. Zyla et al. (Particle Data Group), Review of Particle Physics, PTEP 2020, 083C01 (2020).
  * (2) E. Klempt and A. Zaitsev, Glueballs, hybrids, multiquarks: Experimental facts versus QCD inspired concepts, Phys. Rept. 454, 1-202 (2007).
  * (3) C. A. Meyer and E. S. Swanson, Hybrid mesons, Prog. Part. Nucl. Phys. 82, 21-58 (2015).
  * (4) H. X. Chen, W. Chen, X. Liu, and S. L. Zhu, The hidden-charm pentaquark and tetraquark states, Phys. Rept. 639, 1-121 (2016).
  * (5) D. Alde et al. (LAPP Collaboration), Evidence for a $1^{-+}$ exotic meson, Phys. Lett. B 205, 397-400 (1988).
  * (6) E. I. Ivanov et al. (E852 Collaboration), Observation of Exotic Meson Production in the Reaction $\pi^{-}p\to\eta^{\prime}\pi^{-}p$ at 18 GeV$/c$, Phys. Rev. Lett. 86, 3977 (2001).
  * (7) M. Alekseev et al. (COMPASS Collaboration), Observation of a $J^{PC}=1^{-+}$ Exotic Resonance in Diffractive Dissociation of 190 GeV$/c$ $\pi^{-}$ into $\pi^{-}\pi^{-}\pi^{+}$, Phys. Rev. Lett. 104, 241803 (2010).
  * (8) J. Kuhn et al. (E852 Collaboration), Exotic meson production in the $f_{1}(1285)\pi^{-}$ system observed in the reaction $\pi^{-}p\to\eta\pi^{+}\pi^{-}\pi^{-}p$ at 18 GeV$/c$, Phys. Lett. B 595, 109-117 (2004).
  * (9) N. Isgur and J. E. Paton, Flux-tube model for hadrons in QCD, Phys. Rev. D 31, 2910 (1985).
  * (10) P. R. Page, E. S. Swanson, and A. P. Szczepaniak, Hybrid meson decay phenomenology, Phys. Rev. D 59, 034016 (1999).
  * (11) T. Burns and F. E. Close, Hybrid-meson properties in lattice QCD and flux-tube models, Phys. Rev. D 74, 034003 (2006).
  * (12) F. Iddir and L. Semlala, Hybrid states from constituent glue model, Int. J. Mod. Phys. A 23, 5229-5250 (2008).
  * (13) C. McNeile, C. W. Bernard, T. A. DeGrand, C. E. DeTar, S. A. Gottlieb, U. M. Heller, J. Hetrick, R. Sugar, and D. Toussaint, Exotic meson spectroscopy from the clover action at $\beta=5.85$ and $6.15$, Nucl. Phys. B Proc. Suppl. 73, 264-266 (1999).
  * (14) P. Lacock et al. (TXL Collaboration), Hybrid and orbitally excited mesons in full QCD, Nucl. Phys. B Proc. Suppl. 73, 261-263 (1999).
  * (15) C. Bernard, T. Burch, E. B. Gregory, D. Toussaint, C. E. DeTar, J. Osborn, S. A. Gottlieb, U. M. Heller, and R. Sugar, Lattice calculation of $1^{-+}$ hybrid mesons with improved Kogut-Susskind fermions, Phys. Rev. D 68, 074505 (2003).
  * (16) J. N. Hedditch, W. Kamleh, B. G. Lasscock, D. B. Leinweber, A. G. Williams, and J. M. Zanotti, $1^{-+}$ exotic meson at light quark masses, Phys. Rev. D 72, 114507 (2005).
  * (17) J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas, Highly Excited and Exotic Meson Spectrum from Dynamical Lattice QCD, Phys. Rev. Lett. 103, 262001 (2009).
  * (18) J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas, Toward the excited meson spectrum of dynamical QCD, Phys. Rev. D 82, 034508 (2010).
  * (19) R. A. Briceno, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018).
  * (20) J. Govaerts, F. de Viron, D. Gusbin, and J. Weyers, QCD sum rules and hybrid mesons, Nucl. Phys. B 248, 1-18 (1984).
  * (21) K. G. Chetyrkin and S. Narison, Light hybrid mesons in QCD, Phys. Lett. B 485, 145-150 (2000).
  * (22) H. Y. Jin, J. G. Körner, and T. G. Steele, Improved determination of the mass of the $1^{-+}$ light hybrid meson from QCD sum rules, Phys. Rev. D 67, 014025 (2003).
  * (23) H. X. Chen, Z. X. Cai, P. Z. Huang, and S. L. Zhu, Decay properties of the $1^{-+}$ hybrid state, Phys. Rev. D 83, 014006 (2011).
  * (24) Z. R. Huang, H. Y. Jin, T. G. Steele and Z. F. Zhang, Revisiting the $b_{1}\pi$ and $\rho\pi$ decay modes of the 1-+ light hybrid state with light-cone QCD sum rules, Phys. Rev. D 94, 054037 (2016).
  * (25) S. S. Xu, Z. F. Cui, L. Chang, J. Papavassiliou, C. D. Roberts, and H. S. Zong, New perspective on hybrid mesons, Eur. Phys. J. A 55, 113 (2019).
  * (26) S. U. Chung, E. Klempt, and J. G. Körner, SU(3) classification of $p$-wave $\eta\pi$ and $\eta^{\prime}\pi$ systems, Eur. Phys. J. A 15, 539-542 (2002).
  * (27) H. X. Chen, A. Hosaka, and S. L. Zhu, $I^{G}J^{PC}=1^{-}1^{-+}$ tetraquark states, Phys. Rev. D 78, 054017 (2008).
  * (28) H. X. Chen, A. Hosaka and S. L. Zhu, $I^{G}J^{PC}=0^{+}1^{-+}$ tetraquark state, Phys. Rev. D 78, 117502 (2008).
  * (29) S. Narison, $1^{-+}$ light exotic mesons in QCD, Phys. Lett. B 675, 319-325 (2009).
  * (30) V. M. Abazov et al. (TOTEM and D0 Collaborations), Odderon Exchange from Elastic Scattering Differences between $pp$ and $p\bar{p}$ Data at 1.96 TeV and from pp Forward Scattering Measurements, Phys. Rev. Lett. 127, 062003 (2021).
  * (31) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, QCD and resonance physics. theoretical foundations, Nucl. Phys. B 147, 385-447 (1979).
  * (32) L. J. Reinders, H. Rubinstein, and S. Yazaki, Hadron properties from QCD sum rules, Phys. Rept. 127, 1-97 (1985).
  * (33) A. A. Ovchinnikov and A. A. Pivovarov, QCD Sum Rule Calculation Of The Quark Gluon Condensate, Sov. J. Nucl. Phys. 48, 721 (1988) [Yad. Fiz. 48, 1135 (1988)].
  * (34) M. Jamin, Flavour-symmetry breaking of the quark condensate and chiral corrections to the Gell-Mann-Oakes-Renner relation, Phys. Lett. B 538, 71-76 (2002).
  * (35) V. Gimenez, V. Lubicz, F. Mescia, V. Porretti, and J. Reyes, Operator product expansion and quark condensate from lattice QCD in coordinate space, Eur. Phys. J. C 41, 535-544 (2005).
  * (36) 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, 412-422 (2012).
  * (37) 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)].