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\author{{\large F. Iddir}$\thanks{
E-mail: \emph{iddir@univ-oran.dz}}${\large \ }and{\large \ L. Semlala}$%
\thanks{
E-mail: \emph{l\_semlala@yahoo.fr}}$ \and $Laboratoire$ $de$ $Physique$ $Th%
\acute{e}orique,$ $Univ.$ $d$'$Oran$ $Es${\small -}$S\acute{e}nia$ $31100$
\and $ALGERIA$}
\title{{\LARGE Hybrid Mesons Masses in a Quark-Gluon Constituent Model}}
\date{11 November 2002 }
\maketitle

\begin{abstract}
QCD theory allows the existence of states which cannot be built by the
na\"{i}ve quark model; both theoretical arguments and experimental data
confirm the hypothesis that gluons may have freedom degrees at the
constituent level, and should be confined. In this work, we use a
phenomenological potential motivated by QCD (with some relativistic
corrections) to determine the masses and the wavefunctions of several hybrid
mesons, within the context of a constituent \textit{q\={q}g }model. We
compare our estimates of the masses with the predictions of other
theoretical models and with the observed masses of candidates.
\end{abstract}

\section{Introduction}

Quantum Chromodynamics, acknowledged as the theory of strong interactions,
allows that mesons containing constituent gluons (as \emph{q\={q}g} hybrids)
may exist. The physical existence of these ``\emph{exotic}'' particles
(beyond the quark model) is one of the objectives of experimental projects $%
^{[1]}$. These programs would contribute significantly to the future
investigation of QCD exotics, and should improve our understanding of hybrid
physics and on the role of the gluon in QCD. From experimental efforts at
IHEP $^{[2]}$, KEK $^{[3]}$, CERN $^{[4]}$ and BNL $^{[5]}$, several hybrid
candidates have been identified, essentially with exotic quantum numbers $%
J^{PC}=1^{-\text{ }+}$.

Hybrids have been studied, using the flux-tube model $^{[6]}$, the quark
model with a constituent gluon $^{[7-9]}$, the MIT bag model $^{[10]}$, the
lattice gauge theory $^{[11]}$, and QCD Sum Rules $^{[12]}$. The models of
references $^{[9-12]}$ predicted that the lightest hybrid meson will be the $%
J^{PC}=1^{-\text{ }+}$ meson, in \emph{1.5-2.1 GeV }mass range; a charmed
hybrid meson will have a mass around $4.0$ $GeV$, and a bottom hybrid meson
with a mass around $10.0$ $GeV$.

We propose estimates for the masses of the hybrid mesons considering the
five $(u,$ $d,$ $s,$ $c$ and $b)$ flavors, within the context of a
quark-gluon constituent model using a QCD-inspired potential and taking into
account some relativistic effects.

\section{The phenomenological potential model}

Although the Lagrangian of QCD is known, its theory is not able to describe
unambiguously the strong-coupling regime. In this framework, the process
requests alternative theories like flux-tube model, Bag model, QCD string
model, Lattice QCD, or phenomenological potential models.

The potential models are essentially motivated by the experiment, and its
wave functions are used to represent the states of the strong interaction
and to describe the hadrons. The most used is the harmonic oscillator
potential, which gives very simple calculations and which is qualitatively
in good agreement with the experimental data; although its expression is not
explicitly describing the QCD characteristics, namely the confinement and
the asymptotic freedom.

The most usual kinds of potential models are using non relativistic
kinematics, which is convenient to the heavy flavors systems, but cannot be
suitable to the hadrons containing light flavors. In this work, we consider
relativistic systems to adjust the situation and then extend the study to
such bound states.

We introduce a model in which the gluon is considered as massive constituent
particle ($m_{g}\simeq 800$ $MeV$). The Hamiltonian is constructed,
containing a phenomenological potential which reproduce the QCD
characteristics; its expression has the mathematical ``Coulomb + Linear''
form, and we take into account also some additional spin effects.

The basic hypothesis is to use a relativistic Schr\"{o}dinger-type wave
equation$^{[13]}$: 
\begin{equation}
\left\{ \sum\limits_{i=1}^{N}\sqrt{\vec{p}_{i}^{\text{ }2}+m_{i}^{\text{ }2}}%
+V_{eff}\right\} \text{ }\Psi (\vec{r}_{i})\text{ }=E\text{ }\Psi (\vec{r}%
_{i}).  \tag{1}
\end{equation}

Another wave equation, more convenient for multiparticle systems, can be used%
$^{[14,\text{ }15]}$: 
\begin{equation}
\left\{ \sum\limits_{i=1}^{N}\left( \frac{\vec{p}_{i}^{\text{ }2}}{2M_{i}}+%
\frac{M_{i}}{2}+\frac{m_{i}^{2}}{2M_{i}}\right) +V_{eff}\right\} \text{ }%
\Psi (\vec{r}_{i})\text{ }=E\text{ }\Psi (\vec{r}_{i})\text{ };  \tag{2}
\end{equation}
where $M_{i}$ are some ``dynamical masses'' satisfying the conditions: 
\begin{equation}
\frac{\partial E}{\partial M_{i}}=0\text{ };  \tag{3}
\end{equation}
$V_{eff}$ is the average over the color space of chromo-spatial potential$%
^{[16]}$: 
\begin{eqnarray}
V_{eff} &=&\left\langle V\right\rangle _{color}=\left\langle
-\sum\limits_{i<j=1}^{N}\mathbf{F}_{i}\cdot \mathbf{F}_{j}\text{ }%
v(r_{ij})\right\rangle _{color}  \nonumber \\
&=&\sum\limits_{i<j=1}^{N}\alpha _{ij}v(r_{ij})\text{ };  \tag{4}
\end{eqnarray}
where $v(r_{ij})$ is the phenomenological potential term.

We have chosen a QCD-motivated potential which has the form: 
\begin{equation}
v(r_{ij})=-\frac{\alpha _{s}}{r_{ij}}+\sigma \text{ }r_{ij}+c\text{ }; 
\tag{5}
\end{equation}
the $\alpha _{s}$, $\sigma $, and $c$ may be fitted by experimental data.

Whereas, for light quarks we should add the spin-dependent correction
represented by the (smeared) hyperfine term of Breit-Fermi interaction$^{[17,%
\text{ }18]}$: 
\begin{equation}
V_{S}=\sum_{i<j=1}^{N}\alpha _{ij}\frac{8\pi \alpha _{h}}{3M_{i}M_{j}}\frac{%
\sigma _{h}^{3}}{\sqrt{\pi ^{3}}}\exp (-\sigma _{h}^{2}\text{ }r_{ij}^{2})%
\text{ }\mathbf{S}_{i}\cdot \mathbf{S}_{j};  \tag{6}
\end{equation}
we neglect the tensor and spin-orbit terms which effects are known to be
small.

\section{The hybrid mesons}

\subsection{The quantum numbers}

For the classification of hybrid mesons in a constituent model we will use
the notations of $[8]:$

\emph{l}$_{\text{g }}$\emph{\ \ }: is the relative orbital momentum of the
gluon in the \emph{q\={q}} center of mass;

\emph{l}$_{\text{\textit{q\={q}}}}$\emph{\ \ }: is the relative orbital
momentum between \emph{q} and \emph{\={q}};

\emph{S}$_{\text{\textit{q\={q}}}}$\emph{\ }: is the total quark spin;

\emph{j}$_{\text{\textit{g }}}$\emph{\ \ }: is the total gluon angular
momentum;

\emph{L \ \ }: \textit{l}$_{\text{\textit{q\={q}}}}$ + \textit{j}$_{\text{%
\textit{g}}}.$

The parity and charge conjugation of the hybrid are given by: 
\[
\begin{array}{l}
P=\left( -\right) ^{l_{q\bar{q}}+l_{g}}; \\ 
C=\left( -\right) ^{l_{q\bar{q}}+S_{q\bar{q}}+1}.
\end{array}
\]

Let us consider the lightest $1^{-\text{ }+}$ hybrid mesons: \textit{S}$_{%
\text{\textit{q\={q}}}}=0$, \textit{l}$_{\text{\textit{q\={q}}}}=1$ and 
\textit{l}$_{\text{\textit{g}}}=0,$ which we shall refer as \emph{the
quark-excited hybrid (QE)}, and \textit{S}$_{\text{\textit{q\={q}}}}=1$, 
\textit{l}$_{\text{\textit{q\={q}}}}=0$ and \textit{l}$_{\text{\textit{g}}%
}=1,$ which we shall refer as \emph{the gluon-excited hybrid (GE).}

\subsection{The Hamiltonian and the wavefunctions}

We have to solve the wave equation relative to the Hamiltonian: 
\begin{equation}
H=\sum\limits_{i=q,\text{ }\bar{q},\text{ }g}\left( \frac{\vec{p}_{i}^{\text{
}2}}{2M_{i}}+\frac{M_{i}}{2}+\frac{m_{i}^{2}}{2M_{i}}\right) +V_{eff\text{ }%
};  \tag{7}
\end{equation}
with, for the hybrid meson: 
\[
\begin{array}{l}
\alpha _{q\bar{q}}=-\frac{1}{6}; \\ 
\alpha _{\bar{q}g}=\alpha _{qg}=\frac{3}{2}.
\end{array}
\]

We define the Jacobi coordinates: 
\[
\begin{array}{l}
\vec{\rho}=\vec{r}_{\bar{q}}-\vec{r}_{q}; \\ 
\vec{\lambda}=\vec{r}_{g}-\frac{M_{q}\vec{r}_{q}+M_{\bar{q}}\vec{r}_{\bar{q}}%
}{M_{q}+M_{\bar{q}}}.
\end{array}
\]

Then, the relative Hamiltonian is given by: 
\begin{equation}
H_{R}=\frac{\vec{p}_{\rho }^{2}}{2\mu _{\rho }}+\frac{\vec{p}_{\lambda }^{2}%
}{2\mu _{\lambda }}+V_{eff}(\vec{\rho},\vec{\lambda})+\frac{M_{q}}{2}+\frac{%
m_{q}^{2}}{2M_{q}}+\frac{M_{\bar{q}}}{2}+\frac{m_{\bar{q}}^{2}}{2M_{\bar{q}}}%
+\frac{M_{g}}{2}+\frac{m_{g}^{2}}{2M_{g}};  \tag{8}
\end{equation}
with 
\[
\begin{array}{l}
\mu _{\rho }=\left( \frac{1}{M_{q}}+\frac{1}{\text{ }M_{\bar{q}}}\right)
^{-1} \\ 
\mu _{\lambda }=\left( \frac{1}{M_{g}}+\frac{1}{M_{q}+M_{\bar{q}}}\right)
^{-1};
\end{array}
\]
and 
\begin{eqnarray}
V_{eff}(\vec{\rho},\vec{\lambda}) &=&-\alpha _{s}\left( -\frac{1}{6\rho }+%
\frac{3}{2}\frac{1}{\left| \vec{\lambda}+\frac{\vec{\rho}}{2}\right| }+\frac{%
3}{2}\frac{1}{\left| \vec{\lambda}-\frac{\vec{\rho}}{2}\right| }\right)
+\sigma \left( -\frac{1}{6}\rho +\frac{3}{2}\left| \vec{\lambda}+\frac{\vec{%
\rho}}{2}\right| +\frac{3}{2}\left| \vec{\lambda}-\frac{\vec{\rho}}{2}%
\right| \right) +  \nonumber \\
&&+\frac{17}{6}c+V_{S}\text{ }.  \tag{9}
\end{eqnarray}

We have chosen to develop the spatial wave function as follows: 
\begin{equation}
\psi ^{l_{q\bar{q}}l_{g}}(\vec{\rho},\vec{\lambda})=\sum%
\limits_{n=1}^{N}a_{n}\varphi _{n}^{l_{q\bar{q}}l_{g}}(\vec{\rho},\vec{%
\lambda})\text{ };  \tag{10}
\end{equation}
where $\varphi _{n}^{l_{q\bar{q}}l_{g}}(\vec{\rho},\vec{\lambda})$ are the
Gaussian-type functions: 
\begin{equation}
\varphi _{n}^{l_{q\bar{q}}l_{g}}(\vec{\rho},\vec{\lambda})=\rho ^{l_{q\bar{q}%
}}\lambda ^{l_{g}}\exp \left( -\frac{1}{2}n\text{ }\beta _{N}^{2}\text{ }%
\left( \rho ^{2}+\lambda ^{2}\right) \right) \mathbf{Y}_{l_{q\bar{q}}m_{q%
\bar{q}}}(\Omega _{\rho })\mathbf{Y}_{l_{g}m_{g}}(\Omega _{\lambda }). 
\tag{11}
\end{equation}

Thus we solve the eigenvalue problem: 
\[
H_{nm}a_{m}=\epsilon _{N}\text{ }N_{nm}a_{m}\text{ }; 
\]
where:

\[
H_{nm}=\int d\vec{\rho}d\vec{\lambda}\text{ }\varphi _{n}^{l_{q\bar{q}%
}l_{g}}(\vec{\rho},\vec{\lambda})^{*}H_{R\text{ }}\varphi _{m}^{l_{q\bar{q}%
}l_{g}}(\vec{\rho},\vec{\lambda}) 
\]
\[
N_{nm}=\int d\vec{\rho}d\vec{\lambda}\text{ }\varphi _{n}^{l_{q\bar{q}%
}l_{g}}(\vec{\rho},\vec{\lambda})^{*}\varphi _{m}^{l_{q\bar{q}}l_{g}}(\vec{%
\rho},\vec{\lambda})\text{.} 
\]

This process yields $\epsilon _{N}(\beta _{N},$ $M_{q},$ $M_{\bar{q}},$ $%
M_{g})$ which is then minimized with respect to parameters $\beta _{N},$ $%
M_{q},$ $M_{\bar{q}}$ and $M_{g}.$

For the potential parameters, we must distinguish between the light and the
heavy flavors.

In Table 1 we present the parameters, fitting to the low lying isovector S-,
P- and D-wave states of the light meson spectrum (except the mass of the
strange quark obtained by fitting to the mass of Kaons)$^{[18]}.$

In Table 2 we give the parameters fitting to $J^{PC}=1^{-\text{ }-}$ \textit{%
(c\={c})}\ and \textit{(b\={b})}\ spectrum.\linebreak

\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$\sigma _{h}$ $(GeV)$ & $\alpha _{h}$ & $\alpha _{s}$ & $\sigma $ $(GeV^{%
\text{ }2})$ & $c$ $(GeV)$ & $m_{u}=m_{d}$ $(GeV)$ & $m_{s}$ $(GeV)$ \\ 
\hline
$.70$ & $.840$ & $0.857$ & $\frac{3}{4}0.151$ & $-0.4358$ & $.375\text{ }$ & 
$.650$ \\ \hline
\end{tabular}
\]
\[
\text{{\small Table\ 1}:{\small \ Light flavors potential parameters}}^{%
\text{{\scriptsize [18]}}}\text{{\small . }} 
\]

\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$\alpha _{s}$ & $\sigma $ $(GeV^{\text{ }2})$ & $c$ $(GeV)$ & $m_{c}$ $(GeV)$
& $m_{b}$ $(GeV)$ \\ \hline
$0.36$ & $0.144$ & $-0.45$ & $1.70$ & $5.05\text{ }$ \\ \hline
\end{tabular}
\]
\[
\text{{\small Table\ 2}:{\small \ Heavy flavors potential parameters.}} 
\]

\section{Hybrid mesons masses}

We present in Table 3 our estimates of hybrid mesons masses for different
flavors without spin effects; we take $800$ $MeV$ for the mass of the gluon.
We compare our results with the predictions by other models.

\[
\begin{tabular}{|c|c|c|c|c|}
\hline
model & $u,d$ & $s$ & $c$ & $b$ \\ \hline
\begin{tabular}{c}
{\small Coul. + Lin.} \\ 
{\small (Our results)}
\end{tabular}
$\ 
\begin{array}{c}
l_{g}=0;\text{ }l_{q\bar{q}}=1\text{; {\small QE}} \\ 
l_{g}=1;\text{ }l_{q\bar{q}}=0\text{ ; {\small GE}}
\end{array}
$ & $
\begin{array}{c}
1.31 \\ 
1.70
\end{array}
$ & $
\begin{array}{c}
1.57 \\ 
2.00
\end{array}
$ & $
\begin{array}{c}
4.09 \\ 
4.45
\end{array}
$ & $
\begin{array}{c}
10.34 \\ 
10.81
\end{array}
$ \\ \hline
{\small QCD Sum Rules} & 
\begin{tabular}{l}
$1.5$%
\end{tabular}
& 
\begin{tabular}{l}
$1.6-1.7$%
\end{tabular}
& 
\begin{tabular}{l}
$4.1-5.3$%
\end{tabular}
& 
\begin{tabular}{l}
$10.6-11.2$%
\end{tabular}
\\ \hline
{\small Lattice QCD} & 
\begin{tabular}{l}
$1.5-1.8$%
\end{tabular}
& 
\begin{tabular}{l}
$2.1$%
\end{tabular}
& 
\begin{tabular}{l}
$4.19$%
\end{tabular}
& 
\begin{tabular}{l}
$10.81$%
\end{tabular}
\\ \hline
{\small Flux Tube Model} & 
\begin{tabular}{l}
$1.8-2.0$%
\end{tabular}
& 
\begin{tabular}{l}
$2.1-2.2$%
\end{tabular}
& 
\begin{tabular}{l}
$4.2-4.5$%
\end{tabular}
& 
\begin{tabular}{l}
$10.8-11.1$%
\end{tabular}
\\ \hline
{\small Bag Model} & 
\begin{tabular}{l}
$1.3-1.8$%
\end{tabular}
& 
\begin{tabular}{l}
$2.5$%
\end{tabular}
& 
\begin{tabular}{l}
$3.9$%
\end{tabular}
& 
\begin{tabular}{l}
$10.49$%
\end{tabular}
\\ \hline
{\small Massless} {\small Const. Gluon Model} & 
\begin{tabular}{l}
$1.7$%
\end{tabular}
& 
\begin{tabular}{l}
$2.0$%
\end{tabular}
& 
\begin{tabular}{l}
$4.1$%
\end{tabular}
& 
\begin{tabular}{l}
$10.64$%
\end{tabular}
\\ \hline
\end{tabular}
\]
\[
\text{{\small Table 3: Predicted hybrid mesons masses} {\small (in} }{\small %
GeV}\text{{\small )}}
\]

\[
\begin{tabular}{|c|c|c|c|}
\hline
$l_{q\bar{q}}$ & $l_{g}$ & $P$ & $J^{C}$ \\ \hline
$0$ & $0$ & $+$ & $0^{+},1^{\pm },2^{+}$ \\ \hline
$1$ & $0$ & $-$ & $0^{\pm },1^{\pm },2^{\pm },3^{-}$ \\ \hline
$0$ & $1$ & $-$ & $0^{\pm },1^{\pm },2^{\pm },3^{+}$ \\ \hline
\end{tabular}
\]
\[
\text{{\small Table 4:} {\small Hybrid meson quantum numbers.}} 
\]

We find the masses of the hybrid mesons larger in the \emph{GE} mode than
the \emph{QE} mode. Indeed, the strong force being proportional to the color
charge, the exchange of a color octet does require an important energy. Then
the GE hybrid meson will be heavier than the QE one, which is lower
attractive.

For the orders of magnitude of the masses, our results are in good agreement
with the masses obtained by QCD Sum Rules$^{[12]}$, Lattice QCD $^{[11]}$,
Flux-Tube Model$^{[6]}$, Bag Model$^{[10]}$, (massless) Const. Gluon Model$%
^{[9]}$ and with the observed masses of candidates (namely $J^{PC}=1^{-\text{
}+}$ at $1.4$ and $1.6$ $GeV$\ ).

\section{The mixed-hybrid states and spin effects}

The followed expansion representing the hybrid wave function in the cluster
approximation: 
\begin{eqnarray*}
\Psi _{JM}(\vec{\rho},\vec{\lambda}) &=&\sum\limits_{n,\text{ }l_{q\bar{q}},%
\text{ }l_{g}}a_{n}^{l_{q\bar{q}}l_{g}}\sum\limits_{j_{g},\text{ }L,\text{ (}%
m),\text{ (}\mu )}\varphi _{n}^{l_{q\bar{q}}l_{g}}(\vec{\rho},\vec{\lambda})%
\text{\textbf{e}}^{\mu _{g}}\chi _{_{S_{_{q\bar{q}}}}}^{^{\mu _{q\bar{q}%
}}}\left\langle l_{g}m_{g}1\mu _{g}\mid J_{g}M_{g}\right\rangle \\
&&\times \left\langle l_{q\bar{q}}m_{q\bar{q}}J_{g}M_{g}\mid Lm\right\rangle
\left\langle LmS_{q\bar{q}}\mu _{q\bar{q}}\mid JM\right\rangle \text{ .}
\end{eqnarray*}

For the $J^{PC}=1^{-\text{ }+}$ states, restricting ourselves to the first
orbital excitations ( $l_{q\bar{q}}$ and $l_{g}\leqslant 1$\ ) we can expand
a mixing of the two modes (QE and GE): 
\[
\Psi _{1^{-\text{ }+}}(\vec{\rho},\vec{\lambda})\simeq
\sum\limits_{n=1}^{N}a_{n}^{QE}\varphi _{n}^{QE}(\vec{\rho},\vec{\lambda}%
)+\sum\limits_{n=1}^{N}a_{n}^{GE}\varphi _{n}^{GE}(\vec{\rho},\vec{\lambda}%
). 
\]

For the spin states we choused $\left\{ \left| S_{q\bar{q}},\text{ }s_{g};%
\text{ }S\right\rangle \right\} $ ( $s_{g}=1$ et $\mathbf{S}=\mathbf{S}_{q%
\bar{q}}+\mathbf{s}_{g}$ ).\linebreak

\underline{\emph{Numerical results}}

Our numerical results show that the QE-hybrid and the GE-hybrid mix very
weakly: 
\[
\begin{array}{ll}
\left| 1^{-\text{ }+}(u\bar{u}g)\right\rangle \simeq -.999\left|
QE\right\rangle +.040\left| GE\right\rangle ; & E\simeq 1.34\text{ }GeV \\ 
\left| 1^{-\text{ }+}(u\bar{u}g)\right\rangle \simeq -\left| GE\right\rangle
; & E\simeq 1.72\text{ }GeV
\end{array}
\]

\[
\begin{array}{ll}
\left| 1^{-\text{ }+}(s\bar{s}g)\right\rangle \simeq -.999\left|
QE\right\rangle +.050\left| GE\right\rangle ; & E\simeq 1.60\text{ }GeV \\ 
\left| 1^{-\text{ }+}(s\bar{s}g)\right\rangle \simeq -\left| GE\right\rangle
; & E\simeq 2.02\text{ }GeV
\end{array}
\]

\[
\begin{array}{ll}
\left| 1^{-\text{ }+}(c\bar{c}g)\right\rangle \simeq -.999\left|
QE\right\rangle -.040\left| GE\right\rangle ; & E\simeq 4.10\text{ }GeV \\ 
\left| 1^{-\text{ }+}(c\bar{c}g)\right\rangle \simeq -.031\left|
QE\right\rangle -.999\left| GE\right\rangle ; & E\simeq 4.45\text{ }GeV
\end{array}
\]
\linebreak

In Table 5 we present the $1^{-\text{ }+}$ light hybrid mesons masses
calculated within spin-spin corrections (6). 
\[
\begin{tabular}{|c|c|c|c|c|}
\hline
& $u\bar{u}g$ & $u\bar{s}g$ & $s\bar{s}g$ &  \\ \hline
\begin{tabular}{l}
$S=1$%
\end{tabular}
& 
\begin{tabular}{l}
$1.32$%
\end{tabular}
& 
\begin{tabular}{l}
$1.45$%
\end{tabular}
& 
\begin{tabular}{l}
$1.58$%
\end{tabular}
& 
\begin{tabular}{l}
QE Mode ($l_{q\bar{q}}=1;$ $l_{g}=0$ and $S_{q\bar{q}}=0$)
\end{tabular}
\\ \hline
\begin{tabular}{l}
$S=0$%
\end{tabular}
& 
\begin{tabular}{l}
$1.56$%
\end{tabular}
& 
\begin{tabular}{l}
$1.72$%
\end{tabular}
& 
\begin{tabular}{l}
$1.87$%
\end{tabular}
&  \\ \cline{1-4}
$S=1$ & $1.69$ & $1.84$ & $1.99$ & 
\begin{tabular}{l}
GE Mode ($l_{q\bar{q}}=0;$ $l_{g}=1$ and $S_{q\bar{q}}=1$)
\end{tabular}
\\ \cline{1-4}
$S=2$ & $1.75$ & $1.89$ & $2.04$ &  \\ \hline
\end{tabular}
\]
\[
\text{{\small Table5:} }1^{-\text{ }+}\text{ Light hybrid mesons masses (}GeV%
\text{)};\text{ }2N=6. 
\]

\section{Conclusion}

In this work we use a QCD-motivated potential (Coulombic plus Linear) to
estimate masses of both light and heavy hybrid mesons, in the context of a
constituent quark-gluon model, taking into account the spin-spin interaction
effects for light hybrids. Our results are in a good agreement with the
other methods, like Lattice QCD, QCD Sum Rules, Bag Model, ... and with the
experimental candidates ($1^{-\text{ }+}$ $1400$ and $1600$ and $2000$ $MeV$%
). We find also that $1^{-\text{ }+}$ hybrid mesons may exist in two
weakly-mixed modes: \emph{QE} ($l_{q\bar{q}}=1,l_{g}=0$ and $S_{q\bar{q}}=0$%
) and \emph{GE} ($l_{q\bar{q}}=0,l_{g}=1$ and $S_{q\bar{q}}=1$), the later
being much heavier.

\paragraph{{\protect\Large Acknowledgments}\newline
}

We are grateful to O. P\`{e}ne \emph{(Laboratoire de Physique Th\'{e}orique,
Univ. of Paris-sud)} for extremely useful discussions. We would like to
thank the Abdus Salam International Center for Theoretical Physics, in
Trieste (Italy), for their Fellowship to visit the Center by the
Associateship scheme, where a part of this work was done.

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\end{document}

