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\begin{document}
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\title{Production and Decay of Normal and Exotic Mesons in GlueX\\
\small{GlueX-doc-4788.v12} \\
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\date{December 11, 2020}
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\author{Curtis A. Meyer \\
Carnegie Mellon University}
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\maketitle
\begin{abstract}
  This document summarizes the possible photoproduction mechanisms for both normal and exotic-hybrid mesons in GlueX. It then looks at the allowed possible decay modes for the exotic mesons, and summarizes what the detected final states would be. For the normal mesons, the possible photoproduction mechanisms are summarized. Of particular interest is the
differences in photoproduction mechanisms for the $L=2$ vector mesons.
\end{abstract} 
%
%\section{Introduction}
%
\section*{\label{sec:quark-model}Mesons in The Quark Model}
In the quark model, mesons are bound states of quarks and antiquarks ($q\bar{q}$). 
The quantum numbers of such fermion-antifermion systems are functions of the 
total spin, $S$, of the quark-antiquark system, and the relative orbital angular 
momentum, $L$, between them. The spin $S$ and angular momentum $L$ combine to 
yield the total spin
\begin{eqnarray}
J & = & L \oplus S \, ,
\end{eqnarray}
where $L$ and $S$ add as two angular momentums. 

Parity is the result of a mirror reflection of the wave function, taking $\vec{r}$ into $-\vec{r}$.
It can be written as
\begin{eqnarray}
P\left[\psi(\vec{r})\right] & = \psi(-\vec{r}) & = \eta_{P}\psi(\vec{r}) \, ,
\end{eqnarray}
where $\eta_{P}$ is the eigenvalue of parity. As application of parity twice must return the 
original state, $\eta_{P}=\pm 1$. In spherical coordinates, the parity operation reduces to the 
reflection of a $Y_{lm}$ function,
\begin{eqnarray}
Y_{lm}(\pi-\theta,\pi+\phi) & = & (-1)^{l}Y_{lm}(\theta,\phi) \, .
\end{eqnarray} 
From this, we conclude that $\eta_{P}=(-1)^{l}$. 

For a $q\bar{q}$ system, the intrinsic parity of the antiquark is opposite to that of the 
quark, which yields the total parity of a $q\bar{q}$ system as 
\begin{eqnarray}
P(q\bar{q}) & = & -(-1)^{L} \, .
\end{eqnarray}

Charge conjugation, $C$, is the result of a transformation that takes a particle into its antiparticle. For
a $q\bar{q}$ system, only electrically-neutral states can be eigenstates of $C$. In order to
determine the eigenvalues of $C$ ($\eta_{C}$), we need to consider a wave function that includes
both spatial and spin information
\begin{eqnarray}
\Psi(\vec{r},\vec{s}) & = & R(r) Y_{lm}(\theta,\phi) \chi(\vec{s}) \, .
\end{eqnarray}

As an example, we consider a $u\bar{u}$ system, the $C$ operator acting on this reverses the 
meaning of $u$ and $\bar{u}$. This has the effect of mapping the vector $\vec{r}$ to 
the $u$ quark into $-\vec{r}$. Thus, following the arguments for parity, the spatial part 
of $C$ yields a factor of $(-1)^{L}$. This exchange also flips the spin wave functions, and requires
separate analysis for fermions and bosons, although they both reach the same result. For bosons, we
get a factor of $(+1)$ for $S=0$ and $(-1)$ for the $S=1$ state, resulting in a $(-1)^{S}$. For fermions,
we obtain a factor of $(-1)$ for the $S=0$ case and a factor of $(+1)$ for the $S=1$ yielding $(-1)^{S+1}$.
However, interchanging fermion and anti-fermion adds an additional factor of $(-1)$, so the end result is also
$(-1)^{S}$. When combined with the $L$ factor, we get the same result for both fermions and bosons.
Combining all of this, we find that the C-parity of  (a neutral) $q\bar{q}$ system is 
\begin{eqnarray}
C(q\bar{q}) & = & (-1)^{L+S} \, .
\end{eqnarray} 

Because $C$-parity is only defined for neutral states, it is useful to extend this to the 
more general $G$-parity which can be used to describe all $q\bar{q}$ states, independent of
charge. For isovector states ($I=1$), $C$ would transform a charged member into the oppositely
charged state (\emph{e.g.} $\pi^{+}\to\pi^{-}$). In order to transform this back to the original 
charge, we would need to perform a rotation in isospin ($\pi^{-}\to\pi^{+}$). For a state of whose 
neutral member has $C$-parity $C$, and whose total isospin is $I$, the $G$-parity is defined to be 
\begin{eqnarray}
G & = & C\cdot (-1)^{I} \, , 
\end{eqnarray}
which can be generalized to 
\begin{eqnarray}
\label{eq:g-parity}
G(q\bar{q}) & = & (-1)^{L+S+I} \, .
\end{eqnarray}
The latter is valid for all of the $I=0$ and $I=1$ members of a nonet. 
This leads to mesons having well defined quantum numbers: total angular momentum, 
$J$, isospin, $I$, parity $P$, C-parity, $C$, and G-parity, $G$. These are represented as 
$(I^{G}) J^{PC}$, or simply $J^{PC}$ for short. For the case of $L=0$ and $S=0$, we have 
$J^{PC}=0^{-+}$, while for $L=0$ and $S=1$, $J^{PC}=1^{--}$. The allowed quantum numbers 
for  $L$ smaller than $5$ are given in Table~\ref{tab:qns}. 

\begin{table}[h!]\centering
\begin{tabular}{cccclcllcl} \hline\hline
\multicolumn{4}{c}{QNs} & \multicolumn{6}{c}{Names} \\ 
$L$ & $S$ & $J^{PC}$    &  $(I^{G})$   &   &  $(I^{G})$ & & & $(I)$ & \\ \hline
% 
$0$ & $0$ & $0^{-+}$ & $(1^{-})$ &  $\mathbf{\pi}$   & $(0^{+})$ &  $\mathbf{\eta}$ & $\mathbf{\eta^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K}$ \\
$0$ & $1$ & $1^{--}$  & $(1^{+})$ & $\mathbf{\rho}$ & $(0^{-})$ &  $\mathbf{\omega}$ & $\mathbf{\phi}$ & $(\frac{1}{2})$ & $\mathbf{K^{*}}$ \\ \hline
%
  $1$ & $0$ & $1^{+-}$ & $(1^{+})$ & $\mathbf{b_{1}}$   & $(0^{-})$ &  $\mathbf{h_{1}}$ & $\mathbf{h_{1}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K_{1}}$ \\
  
$1$ & $1$ & $0^{++}$ & $(1^{-})$ & $\mathbf{a_{0}}$   & $(0^{+})$ &  $\mathbf{f_{0}}$ & $\mathbf{f_{0}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K^{*}_{0}}$ \\
$1$ & $1$ & $1^{++}$ & $(1^{-})$ & $\mathbf{a_{1}}$   & $(0^{+})$ &  $\mathbf{f_{1}}$ & $\mathbf{f_{1}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K_{1}}$ \\
  $1$ & $1$ & $2^{++}$ & $(1^{-})$ & $\mathbf{a_{2}}$   & $(0^{+})$ &  $\mathbf{f_{2}}$ & $\mathbf{f_{2}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K^{*}_{2}}$ \\ \hline
%
$2$ & $0$ & $2^{-+}$ & $(1^{-})$ &  $\mathbf{\pi_{2}}$   & $(0^{+})$ &  $\mathbf{\eta_{2}}$ & $\mathbf{\eta_{2}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K_{2}}$ \\
$2$ & $1$ & $1^{--}$ & $(1^{+})$ & $\mathbf{\rho_{1}}$ & $(0^{-})$ &  $\mathbf{\omega_{1}}$ & $\mathbf{\phi_{1}}$ & $(\frac{1}{2})$ & $\mathbf{K_{1}^{*}}$ \\  
$2$ & $1$ & $2^{--}$ & $(1^{+})$ & $\mathbf{\rho_{2}}$ & $(0^{-})$ &  $\mathbf{\omega_{2}}$ & $\mathbf{\phi_{2}}$ & $(\frac{1}{2})$ & $\mathbf{K_{2}}$ \\ 
  $2$ & $1$ & $3^{--}$ & $(1^{+})$ & $\mathbf{\rho_{3}}$ & $(0^{-})$ &  $\mathbf{\omega_{3}}$ & $\mathbf{\phi_{3}}$ & $(\frac{1}{2})$ & $\mathbf{K_{3}^{*}}$ \\ \hline
  %
 $3$ & $0$ & $3^{+-}$ & $(1^{+})$ & $\mathbf{b_{3}}$   & $(0^{-})$ &  $\mathbf{h_{3}}$ & $\mathbf{h_{3}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K_{3}}$ \\
 % 
$3$ & $1$ & $2^{++}$ & $(1^{-})$ & $\mathbf{a_{2}}$   & $(0^{+})$ &  $\mathbf{f_{2}}$ & $\mathbf{f_{2}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K^{*}_{2}}$ \\
$3$ & $1$ & $3^{++}$ & $(1^{-})$ & $\mathbf{a_{3}}$   & $(0^{+})$ &  $\mathbf{f_{3}}$ & $\mathbf{f_{3}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K_{3}}$ \\
$3$ & $1$ & $4^{++}$ & $(1^{-})$ & $\mathbf{a_{4}}$   & $(0^{+})$ &  $\mathbf{f_{4}}$ & $\mathbf{f_{4}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K^{*}_{4}}$ \\ \hline
%
  $4$ & $0$ & $4^{-+}$ & $(1^{-})$ &  $\mathbf{\pi_{4}}$   & $(0^{+})$ &  $\mathbf{\eta_{4}}$ & $\mathbf{\eta_{4}^{\prime}}$ & $(\frac{1}{2})$ & $\mathbf{K_{4}}$ \\
  % 
$4$ & $1$ & $3^{--}$ & $(1^{+})$ & $\mathbf{\rho_{3}}$ & $(0^{-})$ &  $\mathbf{\omega_{3}}$ & $\mathbf{\phi_{3}}$ & $(\frac{1}{2})$ & $\mathbf{K_{3}^{*}}$ \\  
$4$ & $1$ & $4^{--}$ & $(1^{+})$ & $\mathbf{\rho_{4}}$ & $(0^{-})$ &  $\mathbf{\omega_{4}}$ & $\mathbf{\phi_{4}}$ & $(\frac{1}{2})$ & $\mathbf{K_{4}}$ \\ 
  $4$ & $1$ & $5^{--}$ & $(1^{+})$ & $\mathbf{\rho_{5}}$ & $(0^{-})$ &  $\mathbf{\omega_{5}}$ & $\mathbf{\phi_{5}}$ & $(\frac{1}{2})$ & $\mathbf{K_{5}^{*}}$ \\   \hline\hline
\end{tabular}
\caption[]{\label{tab:qns}The naming scheme for normal $q\bar{q}$ mesons in the quark model. The first
state listed for a given quantum number is the isospin one state. The second state is
the isospin zero state that is mostly $u$ and $d$ quarks ($n\bar{n}$), while the third
name is for the mostly $s\bar{s}$ isospin zero state. Note that for the kaons, the $C$-
and $G$-parity are not defined.}
\end{table}

\newpage
Similarly, in table~\ref{tab:hybrid-names} are given the quantum numbers and names of the exotic  mesons. Lattice QCD
predicts a nonet of $0^{+-}$ hybrids, two nonets of $1^{-+}$ hybrids and two nonets of $2^{+-}$ hybrids. For
completeness, the quantum numbers $0^{--}$ and $3^{-+}$ are also exotic, so we include them in the following table as well.
\begin{table}[h!]\centering
\begin{tabular}{cccccccc} \hline\hline
\multicolumn{2}{c}{QNs} & \multicolumn{6}{c}{Names} \\ 
 $J^{PC}$    &  $(I^{G})$   &   &  $(I^{G})$ & & & $(I)$ & \\ \hline
%&&&&&&& \\ \hline
%
  $\mathbf{0^{--}}$ & $\mathbf{(1^{+})}$ & $\mathbf{\rho_{0}}$ & $\mathbf{(0^{-})}$ & $\mathbf{\omega_{0}}$ & $\mathbf{\phi_{0}}$ & $(\frac{1}{2})$     &$K^{*}_{0}$ \\
%
$\mathbf{0^{+-}}$ & $\mathbf{(1^{+})}$ &  $\mathbf{b_{0}}$       & $\mathbf{(0^{-})}$&$\mathbf{h_{0}}$     & $\mathbf{h^{\prime}_{0}}$    & $(\frac{1}{2})$     &$K^{*}_{0}$ \\
  % 
  $\mathbf{1^{-+}}$ & $\mathbf{(1^{-})}$ & $\mathbf{\pi_{1}}$     & $\mathbf{(0^{+})}$ &$\mathbf{\eta_{1}}$ & $\mathbf{\eta^{\prime}_{1}}$ & $(\frac{1}{2})$   &$K^{*}_{1}$ \\
%
  $\mathbf{2^{+-}}$ & $\mathbf{(1^{+})}$ & $\mathbf{b_{2}}$        & $\mathbf{(0^{-})}$&$\mathbf{h_{2}}$     & $\mathbf{h^{\prime}_{2}}$     & $(\frac{1}{2})$    &$K^{*}_{2}$ \\
  %
  $\mathbf{3^{-+}}$ & $\mathbf{(1^{-})}$ & $\mathbf{\pi_{3}}$     & $\mathbf{(0^{+})}$ &$\mathbf{\eta_{3}}$ & $\mathbf{\eta^{\prime}_{3}}$ & $(\frac{1}{2})$   &$K^{*}_{3}$ \\  \hline\hline
\end{tabular}
\caption[]{\label{tab:hybrid-names}The naming scheme for hybrid mesons. The first
state listed for a given quantum number is the isospin one state. The second state is
the isospin zero state that is mostly $u$ and $d$ quarks ($n\bar{n}$), while the third
name is for the mostly $s\bar{s}$ isospin zero state. Note that for the kaons, the $C$-
and $G$-parity are not defined. Kaons cannot not have manifestly exotic quantum 
numbers.}
\end{table}
\section*{Isospin Relations}
Important in both the production and decay of mesons are the isospin Clebsch-Gordan coefficients. In particular, we
want to look at neutral isospin $0$ and $1$ states decaying to pairs of isospin $1$ states. For an isospin $0$ particle connecting to two isospin $1$ states,  
\begin{eqnarray*}
  < 0\,\,0 \mid 1\, +1\, ; \, 1\, -1 > & = & \sqrt{\frac{1}{3}} \\
  < 0\,\,0 \mid 1\, 0\, ; \, 1\, 0 > & = & -\sqrt{\frac{1}{3}} \\
  < 0\,\,0 \mid 1\, -1\, ; \, 1\, +1 > & = & \sqrt{\frac{1}{3}} \, .
\end{eqnarray*}
There are non-zero clebsch-gordan coefficients to all three possible charged states: $+-$, $00$ and $-+$.
For the case of isospin $1$ connecting to two isospin $1$ states,
\begin{eqnarray*}
  < 1\,\,0 \mid 1\, +1\, ; \, 1\, -1 > & = & \sqrt{\frac{1}{2}} \\
  < 1\,\,0 \mid 1\, 0\, ; \, 1\, 0 > & = & 0 \\
  < 1\,\,0 \mid 1\, -1\, ; \, 1\, +1 > & = & \sqrt{\frac{1}{2}} \, .
\end{eqnarray*}
Thus, we can only couple to the $+-$ and $-+$ states.

In photoproduction, this means production processes that involve both an incoming $\rho$ mesons and an exchanged
isospin $1$ state can only couple to isospin $0$ produced mesons.
\section*{Identical Bosons}
Identical Bosons~\cite{dudek} must be in a symmetrical state, meaning that the product of the isospin and spin wave functions combined
with the orbital angular momentum $L$ must yield an overall symmetric wave function. If we consider two isospin $1$
particles, they can combine to both $I=0$ and $I=1$, but as we noted in the previous section, two neutral $I=0$ mesons
can only couple to an $I=0$ state, which is symmetric. For a positive and negative $I=1$ mesons, we can couple to both
$I=0$ and $I=1$. As before, the $I=0$ is symmetric, but the $I=1$ is antisymmetric. If we next consider spin, two spin
$0$ particles can only combine to $S=0$, which is also symmetric. Thus for identical spin $0$ bosons such as pions,
we have the possible couplings as shown in table~\ref{tab:iso-0}.
\begin{table}[h!]\centering
  \begin{tabular}{cc|cc|l} \hline
    Particles &  L & $I^{G}$ & $J^{PC}$ & Particles \\ \hline
    $\pi^{0}\pi^{0}$  & $0$ & $0^{+}$ & $0^{++}$ & $f_{0}$, $f^{\prime}_{0}$ \\
              &  $2$ & $0^{+}$ & $2^{++}$ & $f_{2}$, $f^{\prime}_{2}$ \\
    $\pi^{+}\pi^{-}$ & $0$ & $0^{+}$ & $0^{+}$ & $f_{0}$, $f^{\prime}_{0}$ \\
              & $2$ & $0^{+}$ & $2^{+}$ & $f_{2}$, $f^{\prime}_{2}$ \\ \hline
    $\pi^{+}\pi^{-}$ &  $1$ & $1^{+}$ & $1^{-}$ & $\rho_{1}$ \\ \hline
  \end{tabular}
  \caption[]{\label{tab:iso-0}The allowed couplings of $\pi\pi$ accounting for overall symmetric wave fuunctions.}
\end{table}

For the case of spin $1$ particles suchs as the $\omega$ or $\rho$, in addition to the isospin and orbital angular momentum noted for the pions, we need to include the spin wavefunctions. Two spin $1$ particale can combine to overall spin $0$, $1$
or $2$. Both $S=0$ and $S=2$ yield symmetric wavefunctions, while $S=1$ is antisymmetric. This yields the somewhat more
complicated possibilities shown in Table~\ref{tab:omgomg} for the $\omega\omega$ case, where we can only have $I=0$. For the case of $\rho\rho$, spin, isospin and orbital angular momentum matter, ans we get the results shown in Table~\ref{tab:iso-2}.
\begin{table}[h!]\centering
  \begin{tabular}{ccc|cc|l} \hline
    Particles & S & L & $I^{G}$ & $J^{PC}$ & Particles \\ \hline
    $\omega\omega$ & $0$ & $0$ & $0^{+}$ & $0^{++}$ & $f_{0}$, $f^{\prime}_{0}$ \\
              &  $0$ & $2$ & $0^{+}$ & $2^{++}$ & $f_{2}$, $f^{\prime}_{2}$ \\
              &  $1$ & $1$ & $0^{+}$ & $0^{-+}$, $1^{-+}$, $2^{-+}$ & $\eta_{0}$, {\color{blue}$\mathbf{\eta_{1}}$}, $\eta_{2}$, $\eta^{\prime}_{0}$, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$}, $\eta^{\prime}_{2}$ \\ 
              &  $2$ & $0$ & $0^{+}$ & $2^{++}$ & $f_{2}$, $f^{\prime}_{2}$ \\ 
              &  $2$ & $2$ & $0^{+}$ & $0^{++}, 1^{++}, 2^{++}, 3^{++}, 4^{++}$ & $f_{0}$, $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$, $f^{\prime}_{0}$, $f^{\prime}_{1}$, $f^{\prime}_{2}$, $f^{\prime}_{3}$, $f^{\prime}_{4}$ \\ \hline
  \end{tabular}
\caption[]{\label{tab:omgomg}The allowed couplings $\omega\omega$ accounting for overall symmetric wave functions.}
\end{table}

\begin{table}[h!]\centering
  \begin{tabular}{ccc|cc|l} \hline
    Particles & S & L & $I^{G}$ & $J^{PC}$ & Particles \\ \hline
    $\rho^{0}\rho^{0}$ & $0$ & $0$ & $0^{+}$ & $0^{++}$ & $f_{0}$, $f^{\prime}_{0}$ \\
              &  $0$ & $2$ & $0^{+}$ & $2^{++}$ & $f_{2}$, $f^{\prime}_{2}$ \\
              &  $1$ & $1$ & $0^{+}$ & $0^{-+}$, $1^{-+}$, $2^{-+}$ & $\eta_{0}$, {\color{blue}$\mathbf{\eta_{1}}$}, $\eta_{2}$, $\eta^{\prime}_{0}$, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$}, $\eta^{\prime}_{2}$ \\ 
              &  $2$ & $0$ & $0^{+}$ & $2^{++}$ & $f_{2}$, $f^{\prime}_{2}$ \\ 
              &  $2$ & $2$ & $0^{+}$ & $0^{++}, 1^{++}, 2^{++}, 3^{++}, 4^{++}$ & $f_{0}$, $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$, $f^{\prime}_{0}$, $f^{\prime}_{1}$, $f^{\prime}_{2}$, $f^{\prime}_{3}$, $f^{\prime}_{4}$ \\ \hline
$\rho^{+}\rho^{-}$ & $0$ & $0$ & $0^{+}$ & $0^{+}$ & $f_{0}$, $f^{\prime}_{0}$ \\
              &  $0$ & $2$ & $0^{+}$ & $2^{++}$ & $f_{2}$, $f^{\prime}_{2}$ \\
              &  $1$ & $1$ & $0^{+}$ & $0^{-+}$, $1^{-+}$, $2^{-+}$ & $\eta_{0}$, {\color{blue}$\mathbf{\eta_{1}}$}, $\eta_{2}$, $\eta^{\prime}_{0}$, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$}, $\eta^{\prime}_{2}$ \\ 
              &  $2$ & $0$ & $0^{+}$ & $2^{++}$ & $f_{2}$, $f^{\prime}_{2}$ \\ 
              &  $2$ & $2$ & $0^{+}$ & $0^{++}, 1^{++}, 2^{++}, 3^{++}, 4^{++}$ & $f_{0}$, $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$, $f^{\prime}_{0}$, $f^{\prime}_{1}$, $f^{\prime}_{2}$, $f^{\prime}_{3}$, $f^{\prime}_{4}$ \\ \hline
              %
    $\rho^{+}\rho^{-}$ & $0$ & $1$ & $1^{+}$ & $1^{-}$ & $\rho_{1}$ \\
              &  $1$ & $0$ & $1^{+}$ & $1^{+}$ & $b_{1}$ \\
              &  $1$ & $2$ & $1^{+}$ & $1^{+}$, $2^{+}$, $3^{1}$ & $b_{1}$, {\color{blue}$\mathbf{b_{2}}$}, $b_{3}$ \\
              &  $2$ & $1$ & $1^{+}$ & $1^{-}$, $2^{-}$, $3^{-}$ & $\rho_{1}$, $\rho_{2}$, $\rho_{3}$ \\ \hline
    
  \end{tabular}
  \caption[]{\label{tab:iso-2}The allowed couplings $\rho\rho$ accounting for overall symmetric wave functions.}
\end{table}



\clearpage
\section*{Photoproduction of Exotic Hybrid Mesons}
In the photoproduction of mesons, the usual assumption is to assume vector meson dominance (VMD) where the
incident photon fluctuates into a vector meson ($\rho$, $\omega$ or $\phi$), which then undergoes a $t$-channel
exchange with the proton target to produce the outgoing meson. For production to happen, the reaction must
conserve $J$, $P$, $C$, $G$ and $I$. In the following tables, exotic quantum number states are shown in {\color{blue}blue}, the the allowed normal meson states are shown in black.
\subsubsection*{$\pi^{0}$ Exchange}
\begin{table}[h!]\centering
  \begin{tabular}{cccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{3}{c}{$J^{PC}$} & Exotics \\ \hline
 $\rho$ & $\pi^{0}$   &$(0)^{-}$ & $L=0$ & {\color{OliveGreen}$1^{+-}$} & & & $h_{1}$\, , \, $h^{\prime}_{1}$ \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{--}}$} & $1^{--}$ & $2^{--}$ & {\color{blue}$\mathbf{\omega_{0}}$} \, , \,  {\color{blue}$\mathbf{\phi_{0}}$} \, , \, $\omega_{1}$\, , \, $\phi_{1}$ \, , \, $\omega_{2}$ \, , \, $\phi_{2}$ \\
           & &  & $L=2$ & $1^{+-}$ & {\color{blue}$\mathbf{2^{+-}}$} & $3^{+-}$ & $h_{1}$\, , \, $h^{\prime}_{1}$ \, , \, {\color{blue}$h_{2}$} \, , \, {\color{blue}$h_{2}^{\prime}$}\, , \, $h_{3}$\, , \, $h^{\prime}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $\pi^{0}$   &$(1)^{+}$ & $L=0$ & $1^{+-}$ & & & $b^{0}_{1}$ \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{--}}$} & $1^{--}$ & $2^{--}$ & {\color{blue}$\mathbf{\rho^{0}_{0}}$} \, , \, $\rho^{0}_{1}$ \, , \, $\rho^{0}_{2}$\\
           & &  & $L=2$ & $1^{+-}$ & $\mathbf{2^{+-}}$ & $3^{+-}$ & $b^{0}_{1}$ \,  , \, {\color{blue}$\mathbf{b^{0}_{2}}$} \,  , \, $b^{0}_{3}$ \\ \hline
   \end{tabular}
 \end{table}
 \vspace{-0.5cm}
 \subsubsection*{$\pi^{\pm}$ Exchange}
\begin{table}[h!]\centering
  \begin{tabular}{cccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{3}{c}{$J^{P}$} & Exotics \\ \hline
 $\rho$ & $\pi^{\pm}$   &$(1)^{-}$ & $L=0$ & $1^{+}$ & & & $a^{\pm}_{1}$ \\
   & &  & $L=1$ & $0^{-}$ & {\color{blue}$\mathbf{1^{-}}$} & $2^{-}$ & $\pi^{\pm}$ \, , \,  {\color{blue}$\pi_{1}^{\pm}$} \, , \, $\pi^{\pm}_{2}$ \\
           & &  & $L=2$ & $1^{+}$ & $2^{+}$ & $3^{+}$ &  $a^{\pm}_{1}$ \, , \, $a^{\pm}_{2}$ \, , \, $a^{\pm}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $\pi^{\pm}$   &$(1)^{+}$ & $L=0$ & $1^{+}$ & & & $b^{\pm}_{1}$ \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{-}}$} & $1^{-}$ & $2^{-}$ & {\color{blue}$\mathbf{\rho^{\pm}_{0}}$} \, , \, $\rho^{\pm}_{1}$ \, , \, $\rho^{\pm}_{2}$ \\
           & &  & $L=2$ & $1^{+}$ & {\color{blue}$\mathbf{2^{+}}$} & $3^{+}$ & $b^{\pm}_{1}$ \, , \, {\color{blue}$b^{\pm}_{2}$} \, , \, $b^{\pm}_{3}$ \\ \hline
   \end{tabular}
\end{table}
%
\vspace{-0.5cm}
\subsubsection*{$\eta$ and $\eta^{\prime}$ Exchange}
\begin{table}[h]\centering
  \begin{tabular}{cccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{3}{c}{$J^{PC}$} & Exotics \\ \hline
 $\rho$ & $\eta$, $\eta^{\prime}$   &$(1)^{+}$ & $L=0$ & $1^{+-}$ & & & $b^{0}_{1}$ \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{--}}$} & $1^{--}$ & $2^{--}$ & {\color{blue}$\mathbf{\rho^{0}_{0}}$} \, , \,  $\rho^{0}_{1}$ \, , \, $\rho^{0}_{2}$ \\
           & &  & $L=2$ & $1^{+-}$ & {\color{blue}$\mathbf{2^{+-}}$} & $3^{+-}$ & $b^{0}_{1}$ \, , \, {\color{blue}$\mathbf{b_{2}^{0}}$} \, , \, $b^{0}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $\eta$, $\eta^{\prime}$   &$(0)^{-}$ & $L=0$ & $1^{+-}$ & & & $h_{1}$ \, , \, $h^{\prime}_{1}$ \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{--}}$} & $1^{--}$ & $2^{--}$ & {\color{blue}$\mathbf{\omega_{0}}$} \, , \, {\color{blue}$\mathbf{\phi_{0}}$} \, , \, $\omega_{1}$ \, , \, $\phi_{1}$ \, , \, $\omega_{2}$ \, , \, $\phi_{2}$ \\
           & &  & $L=2$ & $1^{+-}$ & {\color{blue}$\mathbf{2^{+-}}$} & $3^{+-}$ & $h_{1}$ \, , \, $h_{1}^{\prime}$ \, , \, {\color{blue}$\mathbf{h_{2}}$}\, , \, {\color{blue}$\mathbf{h_{2}^{\prime}}$} \, , \, $h_{3}$\, , \, $h_{3}^{\prime}$\\ \hline
   \end{tabular}
\end{table}
\vspace{-0.5cm}
\subsubsection*{Pomeron Exchange}
\begin{table}[h!]\centering
  \begin{tabular}{cccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{3}{c}{$J^{PC}$} & Exotics \\ \hline
 $\rho$ & $\cal{P}$   &$(1)^{+}$ & $L=0$ & $1^{--}$ &  &  & $\rho^{0}_{1}$ \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{+-}}$} & $1^{+-}$ & {\color{blue}$\mathbf{2^{+-}}$} &   {\color{blue}$\mathbf{b_{0}^{0}}$} \, , \, $b_{1}^{0}$ \, , \, {\color{blue}$\mathbf{b_{2}^{0}}$} \\
           & &  & $L=2$ & $1^{--}$ & $2^{--}$ & $3^{--}$ &  $\rho^{0}_{1}$ \, , \, $\rho^{0}_{2}$ \, , \, $\rho^{0}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $\cal{P}$   & $(0)^{-}$ & $L=0$ & $1^{--}$ &  &  & $\omega_{1}$ \, , \, $\phi_{1}$ \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{+-}}$} & $1^{+-}$ & {\color{blue}$\mathbf{2^{+-}}$} &  {\color{blue}$\mathbf{h_{0}}$} \, , \, {\color{blue}$\mathbf{h_{0}^{\prime}}$} \, , \, $h_{1}$\, , \, $h_{1}^{\prime}$ \, , \, {\color{blue}$\mathbf{h_{2}}$}\, , \, {\color{blue}$\mathbf{h_{2}^{\prime}}$} \\
           & &  & $L=2$ & $1^{--}$ & $2^{--}$ & $3^{--}$ & $\omega_{1}$ \, , \, $\phi_{1}$ \, , \, $\omega_{2}$ \, , \, $\phi_{2}$ \, , \, $\omega_{3}$ \, , \, $\phi_{3}$ \\ \hline
   \end{tabular}
\end{table}
\clearpage
%
\subsubsection*{$\rho^{0}$ Exchange}
%Recall that two identical bosons ($\rho\rho$) need to be in a symmetric state,  This means that the $L=1$
% $\rho\rho$ production of $\eta_{1}$ and $\eta_{3}$ is forbidden.
\begin{table}[h]\centering
  \begin{tabular}{ccccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{4}{c}{$J^{PC}$} & Exotics \\ \hline
    $\rho$ & $\rho^{0}$   &$(0)^{+}$ & $L=0$ & $0^{++}$ & $2^{++}$ & & & $f_{0}$ \, , \, $f^{\prime}_{0}$ \, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \\
   & &  & $L=1$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & {\color{blue}$\mathbf{3^{-+}}$} &  {\color{blue}$\mathbf{\eta_{1}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$} \, , \,$\eta_{2}$ \, , \,$\eta^{\prime}_{2}$ \, , \, {\color{blue}$\mathbf{\eta_{3}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{3}}$} \\
           & &  & $L=2$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & $3^{++}$ &  $f_{1}$ \, , \, $f^{\prime}_{1}$ \, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \, , \, $f_{3}$ \, , \, $f^{\prime}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $\rho^{0}$   &$(1)^{-}$ & $L=0$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & & $a^{0}_{0}$ \, , \, $a^{0}_{1}$ \, , \, $a^{0}_{2}$ \\
   & &  & $L=1$ & $0^{-+}$ &  {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & {\color{blue}$\mathbf{3^{-+}}$} & $\pi^{0}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{0}}$}\, , \, $\pi^{0}_{2}$ \, , \, {\color{blue}$\mathbf{\pi_{3}^{0}}$} \\
           & &  & $L=2$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & $3^{++}$ & $a^{0}_{0}$ \, , \, $a^{0}_{1}$ \, , \, $a^{0}_{2}$ \, , \, $a^{0}_{3}$ \\ \hline
   \end{tabular}
 \end{table}
 \vspace{-0.5cm}
 \subsubsection*{$\rho^{\pm}$ Exchange}
\begin{table}[h!]\centering
  \begin{tabular}{ccccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{4}{c}{$J^{PC}$} & Exotics \\ \hline
 $\rho$ & $\rho^{\pm}$   &$(1)^{+}$ & $L=0$ & {\color{blue}$\mathbf{0^{+}}$} & $1^{+}$ & {\color{blue}$\mathbf{2^{+}}$} & & {\color{blue}$\mathbf{b_{0}^{\pm}}$} \, , \, $b^{\pm}_{1}$ \, , \, {\color{blue}$\mathbf{b_{2}^{\pm}}$} \\
   & &  & $L=1$ & {\color{blue}$\mathbf{0^{-}}$} & $1^{-}$ & $2^{-}$ & $3^{-}$ &  {\color{blue}$\mathbf{\rho^{\pm}_{0}}$} \, , \, $\rho^{\pm}_{1}$ \, , \, $\rho^{\pm}_{2}$ \, , \, $\rho^{\pm}_{3}$ \\
           & &  & $L=2$ & {\color{blue}$\mathbf{0^{+}}$} & $1^{+}$ & {\color{blue}$\mathbf{2^{+}}$} & $3^{+}$ &  {\color{blue}$\mathbf{b_{0}^{\pm}}$}\, , \, $b^{\pm}_{1}$ \, , \, {\color{blue}$\mathbf{b_{2}^{\pm}}$} \, , \, $b^{\pm}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $\rho^{\pm}$   &$(1)^{-}$ & $L=0$ & $0^{+}$ & $1^{+}$ & $2^{+}$ & & $a^{\pm}_{0}$ \, , \, $a^{\pm}_{1}$ \, , \, $a^{\pm}_{2}$ \\ 
   & &  & $L=1$ & $0^{-}$ &  {\color{blue}$\mathbf{1^{-}}$} & $2^{-}$ & {\color{blue}$\mathbf{3^{-}}$} & $\pi^{\pm}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{\pm}}$} \, , \, $\pi^{\pm}_{2}$ \, , \, {\color{blue}$\mathbf{\pi_{3}^{\pm}}$} \\
           & &  & $L=2$ & $0^{+}$ & $1^{+}$ & $2^{+}$ & $3^{+}$ & $a^{\pm}_{0}$ \, , \, $a^{\pm}_{1}$ \, , \, $a^{\pm}_{2}$ \, , \, $a^{\pm}_{3}$ \\ \hline
   \end{tabular}
 \end{table}
 \vspace{-0.5cm}
\subsubsection*{$\omega$ and $\phi$ Exchange}
Note that there is a slight difference between $\omega\omega$ or $\phi\phi$ and $\omega\phi$ or $\phi\omega$.
%
\begin{table}[h!]\centering
  \begin{tabular}{ccccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{4}{c}{$J^{PC}$} & Exotics \\ \hline
 $\rho$ & $\omega$,$\phi$   &$(1)^{-}$ & $L=0$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & & $a^{0}_{0}$ \, , \, $a^{0}_{1}$ \, , \, $a^{0}_{2}$ \\
   & &  & $L=1$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & {\color{blue}$\mathbf{3^{-+}}$} &  $\pi^{0}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{0}}$} \, , \, $\pi^{0}_{2}$ \, , \, {\color{blue}$\mathbf{\pi_{3}^{0}}$} \\
           & &  & $L=2$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & $3^{++}$ &  $a^{0}_{0}$ \, , \, $a^{0}_{1}$ \, , \, $a^{0}_{2}$ \, , \, $a^{0}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $\omega$, $\phi$  & $(0)^{+}$ & $L=0$ & $0^{++}$ & $2^{++}$ & & & $f_{0}$ \, , \, $f^{\prime}_{0}$ \, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \\
   & &  & $L=1$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & {\color{blue}$\mathbf{3^{-+}}$} & $\eta$\, , \, $\eta^{\prime}$ \, , \, {\color{blue}$\mathbf{\eta_{1}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$} \, , \, $\eta_{2}$ \, , \, $\eta^{\prime}_{2}$ \, , \, {\color{blue}$\mathbf{\eta_{3}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{3}}$} \\
           & &  & $L=2$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & $3^{++}$ & $f_{0}$ \, , \, $f^{\prime}_{0}$ \, , \, $f_{1}$ \, , \, $f^{\prime}_{1}$ \, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \, , \, $f_{3}$ \, , \, $f^{\prime}_{3}$ \\ \hline
$\omega$,$\phi$ & $\phi$, $\omega$  & $(0)^{+}$ & $L=0$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & & $f_{0}$ \, , \, $f^{\prime}_{0}$ \, , \, $f_{1}$ \, , \, $f^{\prime}_{1}$\, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \\
   & &  & $L=1$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & {\color{blue}$\mathbf{3^{-+}}$} & $\eta$\, , \, $\eta^{\prime}$ \, , \, {\color{blue}$\mathbf{\eta_{1}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$} \, , \, $\eta_{2}$ \, , \, $\eta^{\prime}_{2}$ \, , \, {\color{blue}$\mathbf{\eta_{3}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{3}}$} \\
           & &  & $L=2$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & $3^{++}$ & $f_{0}$ \, , \, $f^{\prime}_{0}$ \, , \, $f_{1}$ \, , \, $f^{\prime}_{1}$ \, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \, , \, $f_{3}$ \, , \, $f^{\prime}_{3}$ \\ \hline
   \end{tabular}
\end{table}
%
 \subsubsection*{$b_{1}^{0}$ Exchange}
\begin{table}[h!]\centering
  \begin{tabular}{ccccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{4}{c}{$J^{PC}$} & Exotics \\ \hline
 $\rho$ & $b_{1}^{0}$   &$(0)^{+}$ & $L=0$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & & $\eta$ \, ,\, $\eta^{\prime}$ \, , \, {\color{blue}$\mathbf{\eta_{1}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$} \, , \, $\eta_{2}$ \, , \, $\eta^{\prime}_{2}$ \\ 
           & &  & $L=1$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & $3^{++}$ &  $f_{0}$ \, , \, $f^{\prime}_{0}$ \, , \, $f_{1}$ \, , \, $f^{\prime}_{1}$ \, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \, , \, $f_{3}$ \, , \, $f^{\prime}_{3}$ \\
           & &  & $L=2$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ &  {\color{blue}$\mathbf{3^{-+}}$} & $\eta$ \, ,\, $\eta^{\prime}$ \, , \, {\color{blue}$\mathbf{\eta_{1}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$} \, , \, $\eta_{2}$ \, , \, $\eta^{\prime}_{2}$ \, , \, {\color{blue}$\mathbf{\eta_{3}}$} \,\, ,\, {\color{blue}$\mathbf{\eta^{\prime}_{3}}$}  \\ \hline
 %   
 $\omega$,$\phi$ & $b_{1}^{0}$   &$(1)^{-}$ & $L=0$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & & $\pi^{0}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{0}}$} \, , \, $\pi^{0}_{2}$ \\
           & &  & $L=1$ & $0^{++}$ &  $1^{++}$ & $2^{++}$ & $3^{++}$ & $a^{0}_{0}$ \, , \, $a^{0}_{1}$ \, , \, $a^{0}_{2}$ \, , \, $a^{0}_{3}$ \\
           & &  & $L=2$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & {\color{blue}$\mathbf{3^{-+}}$} & $\pi^{0}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{0}}$} \,  , \, $\pi^{0}_{2}$ \, , \, {\color{blue}$\mathbf{\pi_{3}^{0}}$} \\ \hline
   \end{tabular}
 \end{table}
\newpage
 \subsubsection*{$b_{1}^{\pm}$ Exchange}
\begin{table}[h!]\centering
  \begin{tabular}{ccccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{4}{c}{$J^{P}$} & Exotics \\ \hline
 $\rho$ & $b_{1}^{\pm}$   &$(1)^{+}$ & $L=0$ & {\color{blue}$\mathbf{0^{-}}$} & $1^{-}$ & $2^{-}$ & & {\color{blue}$\mathbf{\rho^{\pm}_{0}}$} \, , \, $\rho^{\pm}_{1}$ \, , \,  $\rho^{\pm}_{2}$ \\
           & &  & $L=1$ & {\color{blue}$\mathbf{0^{+}}$} & $1^{+}$ & {\color{blue}$\mathbf{2^{+}}$} & $3^{+}$ &  {\color{blue}$\mathbf{b_{0}^{\pm}}$} \, , \, $b_{1}^{\pm}$ \, , \, {\color{blue}$\mathbf{b_{2}^{\pm}}$} \, , \, $b_{3}^{\pm}$ \\
           & &  & $L=2$ & {\color{blue}$\mathbf{0^{-}}$} & $1^{-}$ & $2^{-}$ & $3^{-}$ & {\color{blue}$\mathbf{\rho^{\pm}_{0}}$} \, , \, $\rho^{\pm}_{1}$ \, , \, $\rho^{\pm}_{2}$ \, , \, $\rho^{\pm}_{3}$ \\ \hline
 %   
$\omega$,$\phi$ & $b_{1}^{\pm}$   &$(1)^{-}$ & $L=0$ & $0^{-}$ & {\color{blue}$\mathbf{1^{-}}$} & $2^{-}$ & & $\pi^{\pm}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{\pm}}$} \, , \, $\pi^{\pm}_{2}$ \\
           & &  & $L=1$ & $0^{+}$ &  $1^{+}$ & $2^{+}$ & $3^{+}$ & $a^{\pm}_{0}$ \, , \, $a^{\pm}_{1}$ \, , \, $a^{\pm}_{3}$ \, , \, $a^{\pm}_{3}$ \\
           & &  & $L=2$ & $0^{-}$ & {\color{blue}$\mathbf{1^{-}}$} & $2^{-}$ & {\color{blue}$\mathbf{3^{-}}$} & $\pi^{\pm}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{\pm}}$} \, , \, $\pi^{\pm}_{2}$ \, , \, {\color{blue}$\mathbf{\pi_{3}^{\pm}}$} \\ \hline
   \end{tabular}
 \end{table}

 \subsubsection*{$h_{1}$ and $h_{1}^{\prime}$ Exchange}
\begin{table}[h!]\centering
  \begin{tabular}{ccccccccl}
  Incident & Exchange & $(I)^{G}$ & $L$ & \multicolumn{4}{c}{$J^{PC}$} & Exotics \\ \hline
 $\rho$ & $h_{1}$,$h_{1}^{\prime}$   &$(1)^{-}$ & $L=0$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & & $\pi^{0}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{0}}$} \, , \, $\pi^{0}_{2}$ \\
           & &  & $L=1$ & $0^{++}$ & $1^{++}$ & $2^{++}$ & $3^{++}$ & $a^{0}_{0}$ \, , \, $a^{0}_{1}$ \, , \, $a^{0}_{2}$ \, , \, $a^{0}_{3}$ \\
           & &  & $L=2$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ &  {\color{blue}$\mathbf{3^{-+}}$} & $\pi^{0}$ \, , \, {\color{blue}$\mathbf{\pi_{1}^{0}}$} \, , \, $\pi^{0}_{2}$ \, , \, {\color{blue}$\mathbf{\pi_{3}^{0}}$}  \\ \hline
 %   
 $\omega$,$\phi$ & $h_{1}$,$h_{1}^{\prime}$   &$(0)^{+}$ & $L=0$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & & $\eta$ \, ,\, $\eta^{\prime}$ \, , \, {\color{blue}$\mathbf{\eta_{1}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$} \, , \, $\eta_{2}$ \, , \, $\eta^{\prime}_{2}$ \\
           & &  & $L=1$ & $0^{++}$ &  $1^{++}$ & $2^{++}$ & $3^{++}$ & $f_{0}$ \, , \, $f^{\prime}_{0}$ \, , \, $f_{1}$ \, , \, $f^{\prime}_{1}$ \, , \, $f_{2}$ \, , \, $f^{\prime}_{2}$ \, , \, $f_{3}$ \, , \, $f^{\prime}_{3}$ \\
           & &  & $L=2$ & $0^{-+}$ & {\color{blue}$\mathbf{1^{-+}}$} & $2^{-+}$ & {\color{blue}$\mathbf{3^{-+}}$} & $\eta$ \, ,\, $\eta^{\prime}$ \, , \, {\color{blue}$\mathbf{\eta_{1}}$} \, , \, {\color{blue}$\mathbf{\eta^{\prime}_{1}}$} \, , \, $\eta_{2}$ \, , \, $\eta^{\prime}_{2}$ \, , \, {\color{blue}$\mathbf{\eta_{3}}$} \, ,\, {\color{blue}$\mathbf{\eta^{\prime}_{3}}$}  \\ \hline
   \end{tabular}
 \end{table}
%  \newpage
 \subsection*{Summary of Production Mechanisms}
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $\pi^{0}_{1}$ & $\rho^{0}$ & $\omega$,$\phi$ & $1$ \\
            & $\rho^{0}$ & $h_{1}$,$h^{\prime}_{1}$ & $0$, $2$ \\
            &$\omega$,$\phi$ & $\rho^{0}$ & $1$ \\
            & $\omega$,$\phi$ & $b_{1}^{0}$ & $0$, $2$ \\ \hline
%
     $\pi^{\pm}_{1}$ & $\rho^{0}$ & $\pi^{\pm}$ & $1$ \\
            & $\omega$,$\phi$ & $\rho^{\pm}$ & $1$ \\ 
            & $\omega$,$\phi$ & $b_{1}^{\pm}$ & $0$, $2$ \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $\eta_{1}$,$\eta^{\prime}_{1}$ & $\rho^{0}$ & $b^{0}_{1}$ & $0$, $2$ \\
            & $\omega$ & $\omega$ & $1$ \\
            & $\rho$ & $\rho$ & $1$ \\
            & $\omega$ & $\phi$ & $1$ \\
            & $\phi$ & $\phi$ & $1$ \\
            & $\phi$ & $\omega$ & $1$ \\
            & $\omega$,$\phi$ & $h_{1}$, $h^{\prime}_{1}$ & $0$, $2$ \\ \hline
   \end{tabular}
 \end{table}
 %
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $b^{0}_{2}$ & $\rho^{0}$ & $\eta$, $\eta^{\prime}$ & $2$ \\
            & $\rho^{0}$ & $\cal{P}$ & $1$ \\
            & $\omega$, $\phi$ &  $\pi^{0}$ & $2$ \\ \hline
     $b^{\pm}_{2}$ & $\rho^{0}$ & $\rho^{\pm}$ & $0$, $2$ \\
            & $\omega$,$\phi$ & $\pi^{\pm}$ & $2$  \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $h_{2}$, $h^{\prime}_{2}$ & $\omega$, $\phi$ & $\eta$, $\eta^{\prime}$ & $2$ \\
            & $\omega$, $\phi$ &  $\cal{P}$ & $1$ \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $b^{0}_{0}$ & $\rho^{0}$ & $\cal{P}$ & $1$ \\ \hline
     $b^{\pm}_{0}$ & $\rho^{0}$ & $\rho^{\pm}$ & $0$, $2$ \\
            & $\rho^{0}$ & $b^{\pm}_{1}$ & $1$ \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $h_{0}$, $h^{\prime}_{0}$ & $\omega$, $\phi$ &  $\cal{P}$ & $1$ \\ \hline
   \end{tabular}
 \end{table}
 %
  \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $\rho^{0}_{0}$ & $\rho^{0}$ & $\eta$, $\eta^{\prime}$ & $1$ \\
            & $\omega$, $\phi$ & $\pi^{0}$ & $1$ \\ \hline
     $\rho^{\pm}_{0}$ & $\rho^{0}$ & $\rho^{\pm}$ & $1$ \\
            & $\rho^{0}$ & $b^{\pm}_{1}$ & $0$, $2$ \\
            &$\omega$, $\phi$ & $\pi^{\pm}$ & $1$ \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $\omega_{0}$, $\phi_{0}$ & $\rho$ &  $\pi^{0}$ & $1$ \\
            & $\omega$, $\phi$ & $\eta$, $\eta^{\prime}$ & $1$ \\ \hline
   \end{tabular}
 \end{table}
 %
   \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $\pi^{0}_{3}$ & $\rho^{0}$ & $\omega$, $\phi$ & $1$ \\
            & $\rho^{0}$ & $h_{1}$, $h^{\prime}$ & $2$ \\
            & $\omega$, $\phi$ & $\rho^{0}$ & $1$ \\
            & $\omega$, $\phi$ & $b^{0}_{1}$ & $2$ \\
     \hline
     $\pi^{\pm}_{3}$ & $\rho^{0}$ & $\pi^{\pm}$ & $1$ \\
            & $\rho^{0}$ & $b^{\pm}_{1}$ & $0$, $2$ \\
            & $\omega$, $\phi$ & $\rho^{\pm}$ & $1$ \\
                 & $\rho^{0}$ & $b^{\pm}_{1}$ & $0$, $2$ \\
            & $\omega$, $\phi$ & $b^{\pm}_{1}$ & $2$ \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc} 
     Exotic & Beam & Exchange & L \\ \hline
     $\eta_{3}$, $\eta^{\prime}_{3}$ & $\rho^{0}$ & $b^{0}_{1}$ & $2$ \\
            & $\omega$, $\phi$ &  $\phi$, $\omega$ & $1$ \\
            & $\omega$, $\phi$ & $h_{1}$, $h^{\prime}_{1}$ & $2$ \\ \hline
   \end{tabular}
 \end{table}
\clearpage
\section*{The Decays of Exotic Hybrid Mesons}
Please note that this is not a complete list of all possible decays. In particular, the listed decays for the $\rho^{0}_{0}$,
$\rho^{\pm}_{0}$, $\omega_{0}$, $\phi_{0}$, $\pi^{0}_{3}$, $\pi^{\pm}_{3}$, $\eta_{3}$ and the $\eta^{\prime}_{3}$ are a very limited subset of the possible decays.

\subsubsection*{$\pi_{1}^{0}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{PC}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $\pi_{1}^{0}$ & $(1)^{-}1^{-+}$ & $\rho^{\pm}$ & $\pi^{\mp}$ & $1$ & $\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $\eta^{\prime}$ & $\pi^{0}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{0}$ & $\eta\pi^{0}\pi^{0}\pi^{0}$  \\
       & & $f_{1}$ & $\pi^{0}$ & $0$ & $\eta\pi^{+}\pi^{-}\pi^{0}$ & $\eta\pi^{0}\pi^{0}\pi^{0}$ \\
       & & $b_{1}^{\pm}$ & $\pi^{\mp}$ & $1$ & $\omega\pi^{+}\pi^{-}$ & \\
       & & $\rho^{0}$ & $\omega$ & $1$ & $\omega\pi^{+}\pi^{-}$ & \\
       & & $a_{1}^{0}$ & $\eta$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $b_{1}^{0}$ & $\omega$ & $0$ & $\omega\omega\pi^{0}$ \\ \hline
  \end{tabular}
\end{table}
\hspace{-0.5cm}
\subsubsection*{$\pi_{1}^{\pm}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $\pi_{1}^{\pm}$ & $(1)^{-}1^{-}$ & $\rho^{\pm}$ & $\pi^{0}$ & $1$ & $\pi^{\pm}\pi^{0}\pi^{0}$ & \\
       & & $\rho^{0}$ & $\pi^{\pm}$ & $1$ & $\pi^{\pm}\pi^{+}\pi^{-}$ & \\
       & & $\eta^{\prime}$ & $\pi^{\pm}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{\pm}$ & $\eta\pi^{0}\pi^{0}\pi^{\pm}$  \\
       & & $f_{1}$ & $\pi^{\pm}$ & $0$ & $\eta\pi^{+}\pi^{-}\pi^{\pm}$ & $\eta\pi^{0}\pi^{0}\pi^{\pm}$  \\
       & & $b_{1}^{\pm}$ & $\pi^{0}$ & $1$ & $\omega\pi^{\pm}\pi^{0}$ & \\
       & & $b_{1}^{0}$ & $\pi^{\pm}$ & $1$ & $\omega\pi^{0}\pi^{\pm}$ & \\
       & & $b_{1}^{\pm}$ & $\omega$ & $0$ & $\omega\omega\pi^{\pm}$ \\ 
       & & $\rho^{\pm}$ & $\omega$ & $1$ & $\omega\pi^{\pm}\pi^{0}$ & \\
       & & $a_{1}^{\pm}$ & $\eta$ & $0$ & $\eta\pi^{\pm}\pi^{+}\pi^{-}$ & $\eta\pi^{\pm}\pi^{0}\pi^{0}$ \\ \hline
  \end{tabular}
  \end{table}
  \hspace{-0.5cm}
  \newpage
  \subsubsection*{$\eta_{1}$ and $\eta_{1}^{\prime}$ Decays}
  \begin{table}[h!]\centering
  \begin{tabular}{ccccclll}
Exotic & $(I)^{G}J^{PC}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{3}{l}{Final States} \\ \hline
%    $\eta_{1}$ & $(0)^{+}1^{-+}$& $\pi^{\pm}$ & $\pi^{\mp}$ & $1$ & $\pi^{\pm}\pi^{\mp}$ \\
    $\eta_{1}$ & $(0)^{+}1^{-+}$  & $\eta^{\prime}$ & $\eta$ & $1$ & $\eta\eta\pi^{+}\pi^{-}$ & $\eta\eta\pi^{0}\pi^{0}$ \\
       & & $f_{1}$ & $\eta$ & $0$ & $\eta\eta\pi^{+}\pi^{-}$ & $\eta\eta\pi^{0}\pi^{0}$ \\
       & & $f_{2}$ & $\eta$ & $2$ & $\eta\pi^{+}\pi^{-}$ & $\eta\pi^{0}\pi^{0}$ \\
       & & $\omega$ & $\omega$ & $1$ & $\omega\omega$ \\
       & & $\rho^{0}$ & $\rho^{0}$ & $1$ & $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ \\
       & & $\rho^{\pm}$ & $\rho^{\mp}$ & $1$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ \\
       & & $a_{2}^{\pm}$ & $\pi^{\mp}$ & $2$ & $\eta\pi^{+}\pi^{-}$ & $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ \\
       & & $a_{2}^{0}$ & $\pi^{0}$ & $2$ & $\eta\pi^{0}\pi^{0}$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ \\
%      & & $a_{2}^{0}$ & $\pi^{0}$ & $2$ & $\eta\pi^{0}\pi^{0}$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ \\ 
       & & $b_{1}^{0}$ & $\rho^{0}$ & $0$ & $\omega\pi^{+}\pi^{-}\pi^{0}$ & \\ \hline

    $\eta^{\prime}_{1}$  & $(0)^{+}1^{-+}$ & $\omega$ & $\phi$ & $1$ & $\omega\phi$ \\
       & & $\phi$ & $\phi$ & $1$ & $\phi\phi$ \\
       & & $(K^{*})^{\pm}$ & $K^{\mp}$ & $1$ & $K^{+}K^{-}\pi^{0}$ & \\
       & & $(K^{*})^{0}$ & $K_{S}$ & $0$ & $K^{+}K_{S}\pi^{-}$ & $K^{-}K_{S}\pi^{+}$ \\
       & & $K_{1}^{\pm}$ & $K^{\mp}$ & $0$ & $K^{+}K^{-}\pi^{+}\pi^{-}$ & \\
       & & $K_{1}^{0}$ & $K_{S}$ & $1$ & $K^{+}K_{S}\pi^{-}\pi^{0}$ & $K^{-}K_{S}\pi^{+}\pi^{0}$\\ \hline
  \end{tabular}
\end{table}
%\clearpage
\subsubsection*{$b_{2}^{0}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{ccccclll}
Exotic & $(I)^{G}J^{PC}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{3}{l}{Final States} \\ \hline
%    $b_{2}^{0}$ & $(1)^{+}2^{+-}$& $\pi^{\pm}$ & $\pi^{\mp}$ & $2$ & $\pi^{\pm}\pi^{\mp}$ \\
   $b_{2}^{0}$ & $(1)^{+}2^{+-}$  & $\omega$ & $\pi^{0}$ & $2$ & $\omega\pi^{0}$ & &\\
       & & $\rho^{0}$ & $\eta$ & $2$ & $\eta\pi^{+}\pi^{-}$ & & \\
       & & $a_{2}^{\pm}$ & $\pi^{\mp}$ & $1$ & $\eta\pi^{+}\pi^{-}$ & $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ \\
       & & $\rho^{\pm}$ & $\rho^{\mp}$ & $1$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ \\
       & & $b_{1}^{0}$ & $\eta$ & $1$ & $\omega\eta\pi^{0}$ & & \\
       & & $f_{1}$ & $\rho^{0}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\eta\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ & \\ 
       & & $a_{1}^{\pm}$ & $\pi^{\mp}$ & $1$ & $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ & \\ \hline
  \end{tabular}
\end{table}
\hspace{-0.5cm}
\subsubsection*{$b_{2}^{\pm}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{ccccclll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{3}{l}{Final States} \\ \hline
%    $b_{2}^{\pm}$ & $(1)^{+}2^{+}$& $\pi^{\pm}$ & $\pi^{0}$ & $2$ & $\pi^{\pm}\pi^{0}$ \\
    $b_{2}^{\pm}$ & $(1)^{+}2^{+}$  & $\omega$ & $\pi^{\pm}$ & $2$ & $\omega\pi^{\pm}$ & &\\
       & & $\rho^{\pm}$ & $\eta$ & $2$ & $\eta\pi^{\pm}\pi^{0}$ & &\\
       & & $a_{2}^{\pm}$ & $\pi^{0}$ & $1$ & $\eta\pi^{\pm}\pi^{0}$ & $\pi^{\pm}\pi^{+}\pi^{-}\pi^{0}$ & $\pi^{\pm}\pi^{0}\pi^{0}\pi^{0}$ \\
       & & $a_{2}^{0}$ & $\pi^{\pm}$ & $1$ & $\pi^{\pm}\pi^{+}\pi^{-}\pi^{0}$ & & \\
       & & $b_{1}^{\pm}$ & $\eta$ & $1$ & $\omega\eta\pi^{\pm}$ & & \\
       & & $b_{1}^{0}$ & $\rho^{\pm}$ & $1$ & $\omega\pi^{0}\pi^{0}\pi^{\pm}$ & & \\
       & & $b_{1}^{\pm}$ & $\rho^{0}$ & $1$ & $\omega\pi^{+}\pi^{-}\pi^{\pm}$ & & \\
       & & $f_{1}$ & $\rho^{\pm}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{0}\pi^{\pm}$ & $\eta\pi^{0}\pi^{0}\pi^{0}\pi^{\pm}$ & \\
       & & $a_{1}^{\pm}$ & $\pi^{0}$ & $1$ & $\pi^{\pm}\pi^{+}\pi^{-}\pi^{0}$ & $\pi^{\pm}\pi^{0}\pi^{0}\pi^{0}$ & \\ 
       & & $a_{1}^{0}$ & $\pi^{\mp}$ & $1$ & $\pi^{\pm}\pi^{+}\pi^{-}\pi^{0}$ & & \\ \hline
  \end{tabular}
\end{table}
\hspace{-0.5cm}
\subsubsection*{$h_{2}$ and $h_{2}^{\prime}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{PC}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $h_{2}$ & $(0)^{-}2^{+-}$ & $\rho^{0}$ & $\pi^{0}$ & $2$ & $\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $\omega$ & $\eta$ & $2$ & $\omega\eta$ & \\
       & & $b_{1}^{\pm}$ & $\pi^{\mp}$ & $1$ & $\omega\pi^{+}\pi^{-}$ & \\
       & & $b_{1}^{0}$ & $\pi^{0}$ & $1$ & $\omega\pi^{0}\pi^{0}$ & \\
       & & $f_{1}$ & $\omega$ & $1$ & $\omega\eta\pi^{+}\pi^{-}$ & $\omega\eta\pi^{0}\pi^{0}$ \\ \hline
    $h^{\prime}_{2}$   & $(0)^{-}2^{+-}$ & $\phi$ & $\eta$ & $2$ & $\phi\eta$ & \\
       & & $f_{1}$ & $\phi$ & $1$ & $\phi\eta\pi^{+}\pi^{-}$ & $\phi\eta\pi^{0}\pi^{0}$ \\
       & & $K_{1}^{\pm}$ & $K^{\mp}$ & $1$ & $K^{+}K^{-}\pi^{+}\pi^{-}$ & \\
       & & $K_{1}^{0}$ & $K_{S}$ & $2$ & $K^{+}K_{S}\pi^{-}$ & $ K^{-}K_{S}\pi^{+}$ \\
       & & $(K_{2}^{*})^{\pm}$ & $K^{\mp}$ & $1$ & $K^{+}K^{-}\pi^{0}$ & $K^{+}K^{-}\pi^{+}\pi^{-}$ \\
       & & $(K_{2}^{*})^{0}$ & $K_{S}$ & $0,2$ & $K^{\pm}K_{S}\pi^{\mp}$ & $K^{\pm}K_{S}\pi^{\mp}\pi^{0}$ \\ \hline
  \end{tabular}
\end{table}

%\clearpage
\subsubsection*{$b_{0}^{0}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{PC}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
 %   $b_{0}^{0}$ & $(1)^{+}0^{+-}$& $\pi^{\pm}$ & $\pi^{\mp}$ & $0$ & $\pi^{\pm}\pi^{\mp}$ \\
       $b_{0}^{0}$ & $(1)^{+}0^{+-}$ & $f_{1}$ & $\rho^{0}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\eta\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ \\
       & & $b_{1}^{0}$ & $\eta$ & $1$ & $\omega\eta\pi^{0}$ & \\
       & & $h_{1}$ & $\pi^{0}$ & $1$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ & \\ \hline
  \end{tabular}
\end{table}
\hspace{-0.5cm}
\subsubsection*{$b_{0}^{\pm}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $b_{0}^{\pm}$ & $(1)^{+}0^{+}$ & $\pi^{\pm}$ & $\pi^{0}$ & $0$ & $\pi^{\pm}\pi^{0}$ \\
       & & $f_{1}$ & $\rho^{\pm}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{0}\pi^{\pm}$ & $\eta\pi^{0}\pi^{0}\pi^{0}\pi^{\pm}$ \\
       & & $b_{1}^{\pm}$ & $\eta$ & $1$ & $\omega\eta\pi^{\pm}$ & \\
       & & $b_{1}^{0}$ & $\rho^{\pm}$ & $1$ & $\omega\pi^{0}\pi^{0}\pi^{\pm}$ & \\
       & & $b_{1}^{\pm}$ & $\rho^{0}$ & $1$ & $\omega\pi^{+}\pi^{-}\pi^{\pm}$ & \\
       & & $h_{1}$ & $\pi^{\pm}$ & $1$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{\pm}$ & \\ \hline
  \end{tabular}
\end{table}
\hspace{-0.5cm}
\subsubsection*{$h_{0}$ and $h_{0}^{\prime}$ Decays}
%
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{PC}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $h_{0}$ & $(0)^{-}0^{+-}$ & $b_{1}^{\pm}$ & $\pi^{\mp}$ & $1$ & $\omega\pi^{+}\pi^{-}$ &  \\
       & & $b_{1}^{0}$ & $\pi^{0}$ & $1$ & $\omega\pi^{0}\pi^{0}$ &  \\
       & & $h_{1}$ & $\eta$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{0}$  & \\ \hline
   $h^{\prime}_{0}$  & $(0)^{-}0^{+-}$ & $K_{1}^{\pm}$ & $K^{\mp}$ & $1$ & $K^{+}K^{-}\pi^{+}\pi^{-}$ & \\
       & & $K(1460)^{\pm}$ & $K^{\mp}$ & $0$ & $K^{+}K^{-}\pi^{+}\pi^{-}$ & $K^{+}K^{-}\pi^{0}\pi^{0}$ \\ \hline
  \end{tabular}
\end{table}
%
\newpage
\subsubsection*{$\rho_{0}^{0}$ Decays}
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $\rho_{0}^{0}$ & $(1)^{+}0^{--}$ & $\omega$ & $\pi^{0}$ & $1$ & $\omega\pi^{0}$ & \\
       &  & $\phi$       & $\pi^{0}$ & $1$ & $\phi\pi^{0}$           & \\
       &  & $\rho^{0}$ & $\eta$     & $1$ & $\eta\pi^{+}\pi^{-}$ & \\
       &  & $\rho^{0}$ & $\eta^{\prime}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\eta\pi^{0}\pi^{0}\pi^{+}\pi^{-}$  \\
       &  & $\rho^{0}$ & $f_{1}$ & $0$ & $\eta\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\eta\pi^{0}\pi^{0}\pi^{+}\pi^{-}$  \\
       &  & $\rho^{0}$ & $f_{2}$ & $2$ & $pi^{+}\pi^{-}\pi^{+}\pi^{-}$ & $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$  \\
       & & $K^{*\pm}$ & $K^{\mp}$ & $1$ & $K^{+}K^{-}\pi^{0}$ & $K^{\pm}K_{S}\pi^{\pm}$ \\ \hline
  \end{tabular}
\end{table}
%\clearpage
\subsubsection*{$\rho_{0}^{\pm}$ Decays}
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $\rho_{0}^{\pm}$ & $(1)^{+}0^{-}$ & $\rho^{0}$ & $\rho^{\pm}$ & $1$ & $\pi^{\pm}\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $\rho^{0}$ & $b^{\pm}_{1}$ & $0$, $2$ & $\omega\pi^{\pm}\pi^{+}\pi^{-}$ & \\
       & & $\rho^{\pm}$ & $b^{0}_{1}$ & $0$, $2$ & $\omega\pi^{\pm}\pi^{0}\pi^{0}$ & \\
       & & $\omega$ & $\pi^{\pm}$ & $1$ & $\omega\pi^{\pm}$ & \\
       &  & $\phi$       & $\pi^{\pm}$ & $1$ & $\phi\pi^{\pm}$           & \\
       &  & $\rho^{\pm}$ & $\eta$     & $1$ & $\eta\pi^{\pm}\pi^{0}$ & \\
       &  & $\rho^{\pm}$ & $\eta^{\prime}$ & $1$ & $\eta\pi^{+}\pi^{-}\pi^{\pm}\pi^{0}$ & $\eta\pi^{0}\pi^{0}\pi^{\pm}\pi^{0}$  \\
       &  & $\rho^{\pm}$ & $f_{1}$ & $0$ & $\eta\pi^{+}\pi^{-}\pi^{\pm}\pi^{0}$ & $\eta\pi^{0}\pi^{0}\pi^{\pm}\pi^{0}$  \\
       &  & $\rho^{\pm}$ & $f_{2}$ & $2$ & $pi^{\pm}\pi^{0}\pi^{+}\pi^{-}$ & $\pi^{\pm}\pi^{0}\pi^{0}\pi^{0}$  \\ \hline
  \end{tabular}
\end{table}


\subsubsection*{$\omega_{0}$ and $\phi_{0}$ Decays}
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $\omega_{0}^{0}$ & $(0)^{-}0^{--}$ & $\rho^{0}$ & $\pi^{0}$ & $1$ & $\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $\omega$ & $\eta$ & $1$ & $\omega\eta$ \\ \hline
    $\phi_{0}$ & $(0^{-})0^{--}$ & $\phi$ & $\eta$ & $\phi\eta$ &  \\
       & & $\omega$ & $\eta\prime$ & $\omega\eta\pi^{+}\pi^{-}$ & $\omega\eta\pi^{0}\pi^{0}$ \\
       & & $\phi$ & $\eta\prime$ & $\phi\eta\pi^{+}\pi^{-}$ & $\phi\eta\pi^{0}\pi^{0}$ \\
       & & $\rho^{0}$ & $a^{0}_{1}$ & $0$ & $\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $\rho^{0}$ & $a^{0}_{2}$ & $2$ & $\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}$ & $\eta\pi^{+}\pi^{-}\pi^{0}$ \\
       & & $K^{*\pm}$ & $K^{\mp}$ & $1$ & $K^{+}K^{-}\pi^{0}$ & $K^{\pm}K_{S}\pi^{\pm}$ \\ \hline
  \end{tabular}
\end{table}
%
\subsubsection*{$\pi^{0}_{3}$ Decays}
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $\pi^{0}_{3}$ & $(1^{-})3^{-+}$ & $\rho^{0}$ & $\omega$ & $1$ & $\omega\pi^{+}\pi^{-}$ & \\
       & & $\rho^{0}$ & $\phi$ & $1$ & $\phi\pi^{+}\pi^{-}$ & \\
       & & $\rho^{0}$ & $h_{1}$ & $2$ & $\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $\omega$ & $b^{0}_{1}$ & $2$ & $\omega\omega\pi^{0}$ & \\
       & & $\phi$ & $b^{0}_{1}$ & $2$ & $\phi\omega\pi^{0}$ & \\ \hline
  \end{tabular}
\end{table}
\subsubsection*{$\pi^{\pm}_{3}$ Decays}
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
    $\pi^{\pm}_{3}$ & $(1^{-})3^{-+}$ & $\rho^{\pm}$ & $\omega$ & $1$ & $\omega\pi^{\pm}\pi^{0}$ & \\
       & & $\rho^{\pm}$ & $\phi$ & $1$ & $\phi\pi^{\pm}\pi^{0}$ & \\
       & & $\rho^{\pm}$ & $h_{1}$ & $2$ & $\pi^{\pm}\pi^{0}\pi^{+}\pi^{-}\pi^{0}$ & \\
       & & $\omega$ & $b^{\pm}_{1}$ & $2$ & $\omega\omega\pi^{\pm}$ & \\
       & & $\phi$ & $b^{\pm}_{1}$ & $2$ & $\phi\omega\pi^{\pm}$ & \\ \hline
  \end{tabular}
\end{table}
%\clearpage
\subsubsection*{$\eta_{3}$ and $\eta^{\prime}_{3}$ Decays}
\begin{table}[h!]\centering
  \begin{tabular}{cccccll}
Exotic & $(I)^{G}J^{P}$ & \multicolumn{2}{c}{Daughters} & $L$ &\multicolumn{2}{l}{Final States} \\ \hline
$\eta_{3}$ & $(0^{+})3^{-+}$ & $\omega$ & $\phi$ & $1$ & $\omega\phi$ & \\
       & & $\rho^{0}$ & $b^{0}_{1}$ & $2$ & $\omega\pi^{0}\pi^{+}\pi^{-}$ & \\
       & & $\omega$ & $h_{1}$ & $2$ & $\omega\pi^{+}\pi^{-}\pi^{0}$ & \\ \hline
$\eta^{\prime}_{3}$ & $(0^{+})3^{-+}$ & $\phi$ & $h_{1}$ & $1$ & $\phi\pi^{+}\pi^{-}\pi^{0}$ & \\ \hline
  \end{tabular}
\end{table}
%
%\clearpage
%
\section*{Photoproduction of Normal Mesons}
From the photoproduction tables above, it appears that there are photoproduction mechanisms that, in principle,
could produce all normal mesons up to $L=2$ in Table~\ref{tab:qns}. In this section, we summarize those mechanisms.
It is interesting to note that for the $L=2$ vector mesons, the $3^{--}$ states have a much more limited set of production
mechanisms compared to the $1^{--}$ and $2^{--}$ states.
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L & Meson & Beam & Exchange & L  \\ \hline
     $\pi^{0}$ & $\rho^{0}$ & $\omega$ & $1$ & $\eta$, $\eta^{\prime}$ & $\omega$ & $\phi$ & $1$ \\
            & $\omega$ & $\rho^{0}$ & $1$ &  & $\phi$ & $\omega$ & $1$ \\
            & $\rho^{0}$ & $h_{1}$ & $0$, $2$ &  & $\rho^{0}$ & $b^{0}_{1}$ & $0$, $2$ \\
            & $\omega$ & $b^{0}_{1}$ & $0$, $2$ & & $\omega$ & $h_{1}$ & $0$, $2$  \\ \hline
   \end{tabular}
 \end{table}
 %
  \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $\rho^{0} $ & $\rho^{0}$ & $\eta$ & $1$ & $\omega$ & $\rho^{0}$ & $\pi^{0}$ & $1$ \\
            & $\rho^{0}$ & $\cal{P}$ & $0$, $2$ & & $\omega$ & $\eta$ & $1$ \\
            & $\omega$ & $\pi^{0}$ & $1$ & & $\omega$ & $\cal{P}$ & $0$, $2$ \\
           & & & & $\phi$ & $\rho$ & $\pi^{0}$ & $1$ \\
           & & & & & $\phi$ & $\eta$ & $1$ \\
           & & & & & $\phi$ & $\cal{P}$ & $0$, $2$ \\\hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $\pi^{\pm} $ & $\rho^{0}$ & $\pi^{\pm}$ & $1$ & $\rho^{\pm}$ & $\rho^{0}$ & $\rho^{\pm}$ & $1$ \\

           & $\omega$ & $\rho^{\pm}$ & $1$ & & $\rho^{0}$ & $b^{\pm}_{1}$ & $0$, $2$ \\
           & $\omega$ & $b^{\pm}_{1}$ & $0$, $2$&  & $\omega$ & $\pi^{\pm}$ & $1$ \\ \hline
   \end{tabular}
 \end{table}
 %
   \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $b^{0}_{1} $ & $\rho^{0}$ & $\eta$ & $0$, $2$ & $h_{1}$ & $\rho^{0}$ & $\pi^{0}$ & $0$, $2$ \\
           & $\rho^{0}$ & $\cal{P}$ & $0$, $2$ & & $\omega$ & $\eta$ & $0$, $2$ \\
           & $\omega$ & $\pi^{0}$ & $0$, $2$ & & $\omega$ & $\cal{P}$ & $1$ \\
           & & & & $h^{\prime}_{1}$ & $\phi$ & $\eta$ & $0$, $2$ \\
           & & & & & $\phi$ & $\cal{P}$ & $1$ \\ \hline
    \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $a^{0}_{0}$, $a^{0}_{1}$, $a^{0}_{2}$ & $\rho^{0}$ & $\omega$ & $0$, $2$ & $f_{0}$, $f_{1}$, $f_{2}$& $\rho^{0}$ & $\rho^{0}$ & $0$, $2$ \\
           & $\omega$ & $\rho^{0}$ & $0$, $2$ & & $\omega$ & $\omega$ & $0$, $2$ \\
           & $\rho^{0}$ & $h_{1}$   & $1$ & & $\rho$ & $b^{0}_{1}$ & $1$ \\
           & $\omega$ & $b^{0}_{1}$ & $1$ &  & $\omega$ & $h_{1}$ & $1$ \\
           & & & &  $f^{\prime}_{0}$, $f^{\prime}_{1}$, $f^{\prime}_{2}$ & $\omega$ & $\phi$ & $0$, $2$ \\
           & & & &  & $\phi$ & $\phi$ & $0$, $2$ \\ \hline
    \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $b_{1}^{\pm} $ & $\rho^{0}$ & $\rho^{\pm}$ & $0$, $2$ & $a^{\pm}_{0}$ & $\omega$ & $\rho^{\pm}$ & $0$, $2$ \\
           & $\rho^{0}$ & $b^{\pm}_{1}$ & $1$ & & $\omega$ & $b^{\pm}_{1}$ & $1$\\
           & $\omega$ & $\pi^{\pm}$ & $0$, $2$&  & &  & \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $a_{1}^{\pm} $ & $\rho^{0}$ & $\pi^{\pm}$ & $0$, $2$ & $a^{\pm}_{2}$ & $\rho^{0}$ & $\pi^{\pm}$ & $2$ \\
           & $\omega$ & $\rho^{\pm}$ & $0$, $2$ & & $\omega$ & $rho^{\pm}$ & $0$, $2$ \\
           & $\omega$ & $b^{\pm}_{1}$ & $1$        &  & $\omega$ & $b^{\pm}_{1}$ & $1$ \\ \hline
   \end{tabular}
 \end{table}
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L & Meson & Beam & Exchange & L  \\ \hline
     $\pi^{0}_{2}$ & $\rho^{0}$ & $\omega$ & $1$ & $\eta_{2}$, $\eta_{2}^{\prime}$ & $\omega$ & $\phi$ & $1$ \\
            & $\omega$ & $\rho^{0}$ & $1$ &  & $\phi$ & $\omega$ & $1$ \\
            & $\rho^{0}$ & $h_{1}$ & $0$, $2$ &  & $\rho^{0}$ & $b^{0}_{1}$ & $0$, $2$ \\
            & $\omega$ & $b^{0}_{1}$ & $0$, $2$ & & $\omega$ & $h_{1}$ & $0$, $2$  \\ \hline
   \end{tabular}
 \end{table}
 %
%  \begin{table}[h!]\centering
%   \begin{tabular}{cccc|cccc} 
%     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
%     $\rho^{0}_{1} $ & $\rho^{0}$ & $\eta$ & $1$ & $\omega_{1}$ & $\rho^{0}$ & $\pi^{0}$ & $1$ \\
%            & $\rho^{0}$ & $\cal{P}$ & $0$, $2$ & & $\omega$ & $\eta$ & $1$ \\
%            & $\omega$ & $\pi^{0}$ & $1$         & & $\omega$ & $\cal{P}$ & $0$, $2$ \\
%           & & & & $\phi_{1}$ & $\omega$ & $\eta^{\prime}$ & $1$ \\
%           & & & & & $\phi$ & $\eta$ & $1$ \\
%           & & & & & $\phi$ & $\cal{P}$ & $2$ \\ \hline
%   \end{tabular}
% \end{table}
%
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $\rho^{0}_{2} $ & $\rho^{0}$ & $\eta$ & $1$ & $\omega_{2}$ & $\rho^{0}$ & $\pi^{0}$ & $1$ \\
           & $\rho^{0}$ & $\cal{P}$ & $2$ & & $\omega$ & $\eta$ & $1$ \\
           & $\omega$ & $\pi^{0}$ & $1$ & & $\omega$ & $\cal{P}$ & $2$ \\
           & & & & $\phi_{2}$ & $\omega$ & $\eta^{\prime}$ & $1$ \\
           & & & & & $\phi$ & $\eta$ & $1$ \\
           & & & & & $\phi$ & $\cal{P}$ & $2$ \\ \hline
   \end{tabular}
 \end{table}
 %
    \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $\rho^{0}_{3} $ & $\rho^{0}$ & $\cal{P}$ & $2$ & $\omega_{3}$ & $\omega^{0}$ & $\cal{P}$ & $2$ \\
           & & & & $\phi_{3}$ & $\phi$ & $\cal{P}$ & $2$ \\ \hline
   \end{tabular}
 \end{table}
 %
 % Charged States
 %
 \begin{table}[h!]\centering
   \begin{tabular}{cccc|cccc} 
     Meson & Beam & Exchange & L &  Meson & Beam & Exchange & L \\ \hline
     $\rho_{2}^{\pm} $ & $\rho^{0}$ & $\rho^{\pm}$ & $1$ & $\rho^{\pm}_{3}$ & $\rho^{0}$ & $\rho^{\pm}$ & $1$ \\
           & $\rho^{0}$ & $b^{\pm}_{1}$ & $0$, $2$ & & $\rho^{0}$ & $b^{\pm}_{1}$ & $2$\\
           & $\omega$ & $\pi^{\pm}$ & $1$ &  & &  & \\ \hline
   \end{tabular}
 \end{table}
 %
 \clearpage
\section*{Lattice QCD}
Recent lattice QCD calculations~\cite{antoni-20} predict the dominant decay modes of the $\pi_{1}(1600)$. These
are reproduced from Table VIII in that reference in Table~\ref{tab:lqcd} below. The partial width $\Gamma_{i}$ is related
to the physical coupling $c^{phys}_{i}$ through equation~\ref{eq:gamma}
\begin{eqnarray}
  \label{eq:gamma}
  \Gamma_{i} & = & \frac{\mid c^{phys}_{i}\mid^{2}}{m^{phys}}\, \rho_{i}(m^{phys})  \, ,
\end{eqnarray}
where $m^{phys}$ is the physical mass of the $\pi_{1}$ and $\rho_{i}$ is the phase space available for a particle
of mass $m^{phys}$ to the particular final states. An overview of the current experimental
situation can be found in references~\cite{meyer,swanson}. The $\pi_{1}(1600)$ has been reported in decays to
$\eta^{\prime}\pi$, $\rho\pi$, $b_{1}\pi$ and $f_{1}\pi$. Lattice QCD clearly favors the $b_{1}\pi$ mode to be the
dominant decay of the $\pi_{1}$, with the $\eta^{\prime}\pi$, $\rho\pi$ and $f_{1}(1285)\pi$ being good places to look.
As for production, we note that the charged states $pi_{1}^{\pm}$, $\pi$, $\rho$ and $b_{1}$ exchange are possible
mechanisms, while for the neutral state $\pi_{1}^{0}$, $\rho$, $\omega$, $h_{1}$ and $b_{1}$ exchange are possible.
Based on this, the photoproduction of $\pi_{1}^{\pm}$ may be stronger than that of the $\pi_{1}^{0}$, particularly if
pion exchange is important.

\begin{table}[h!]\centering
  \begin{tabular}{r|c|c|c}
    Decay & Threshold & $c^{phys}_{i}$ & $\Gamma_{i}$ \\ \hline
    %
    $\eta\pi$            &  $688$~MeV   & $0\rightarrow 43$~MeV         & $0\rightarrow 1$~MeV \\
    $\rho\pi$            &  $910$~MeV   & $0\rightarrow 203$~MeV       & $0\rightarrow 20$~MeV \\
    $\eta^{\prime}\pi$  &  $1098$~MeV & $0\rightarrow 173$~MeV       & $0\rightarrow 12$~MeV \\
    $b_{1}\pi$            &  $1375$~MeV & $799\rightarrow 1599$~MeV & $139\rightarrow 529$~MeV \\
    $K^{*}\bar{K}$      &  $1386$~MeV & $0\rightarrow 87$~MeV   & $0\rightarrow 2$~MeV \\
    $f_{1}(1285)\pi$   &  $1425$~MeV & $0\rightarrow 363$~MeV & $0\rightarrow 24$~MeV \\
    $\rho\omega(^{1}P_{1})$  &  $1552$~MeV & $\lesssim 19$~MeV & $\lesssim 0.03$~MeV \\
    $\rho\omega(^{3}P_{1})$  &  $1552$~MeV & $\lesssim 32$~MeV & $\lesssim 0.09$~MeV \\
    $\rho\omega(^{5}P_{1})$  &  $1552$~MeV & $\lesssim 19$~MeV & $\lesssim 0.03$~MeV \\
    $f_{1}(1420)\pi$   &  $1560$~MeV & $0\rightarrow 245$~MeV & $0\rightarrow 2$~MeV \\ \hline
\multicolumn{4}{r}{$\Gamma=\Sigma_{i}\Gamma_{i}=139\rightarrow 590$~MeV} \\ \hline
  \end{tabular}
  \caption[]{\label{tab:lqcd}The lattice QCD predictions for the decay of the $\pi_{1}$ exotic~\cite{antoni-20}. Given are
    threshold for the channel, the coupling from lattice QCD ($c{i}^{phys}$, and the partial width $\Gamma_{i}$.}
\end{table}
%\clearpage
 % 
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibliography{gluex-sdme.bib}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{antoni-20}   J.~W.~Antoni, J.~J.~Dudek, R.~G.~Edwards, C.~E.~Thomas, and D.~J.~WILSON,
\textbf{Decays of an exotic $1^{-+}$ hybrid meson resonance in QCD},
Submitted for publication (2020).
\href{https://arxiv.org/abs/2009.10034}{arXiv:2009.10034[hep-lat]}
  %
\bibitem{meyer} C. A. Meyer and Y. van Haarlem, 
  \textbf{The Status of Exotic-quantum-number Mesons} 
Phys. Rev. C\textbf{82}, 025208 (2010). 
\href{https://doi.org/10.1103/PhysRevC.82.025208}{DOI: 10.1103/PhysRevC.82.025208}.
%
\bibitem{swanson} C.~A.~Meyer and E.~S.~Swanson, 
\textbf{Hybrid Mesons},
Progress in Particle and Nuclear Physics B\textbf{82}, 21, (2015). 
\href{http://dx.doi.org/10.1016/j.ppnp.2015.03.001}{DOI: 10.1016/j.ppnp.2015.03.001}
%
\bibitem{dudek} J.~J.Dudek, private communication.
%
\end{thebibliography}

\end{document}
