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\textbf{The Flatte Parametrization of the $a_{0}(980)$} \\
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\textbf{Curtis A. Meyer} \\
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\small{GlueX-Doc: 4824} \\
December 18, 2020 \\
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In the Crystal Barrel paper on $\bar{p}p\rightarrow K_{L}K^{\pm}\pi^{\mp}$~\cite{cbar-KlKpp}, a Flatte
parametrization~\cite{flatte} is used for the $a_{0}(980)$ to accurately account for both the $\eta\pi$ and the
$K\bar{K}$ decay. That parametrization is given as
\begin{eqnarray}
  F_{0}(m) & = & \beta_{0}^{\prime} \, \frac{ \left( \begin{array}{c} g_{1} \\ g_{2} \end{array} \right) }{m^{2}_{0} - m^{2} -
  i\left( \rho_{1}(m)g^{2}_{1} + \rho_{2}(m)g^{2}_{2}\right)} \, ,
\end{eqnarray}
where $m_{0}$ is the nominal mass of the $a_{0}(980)$, $m$ is the running mass of the $a_{0}(980)$, $g_{1}$ and $g_{2}$
are the couplings to the $\eta\pi$ and $K\bar{K}$ channels. The paper and an earlier Crystal Barrel paper on the analysis of $\eta\pi^{0}\pi^{0}$~\cite{cbar-etapipi} final states finds
\begin{eqnarray*}
  g_{\eta\pi}   & = & 353\, \mathrm{MeV} \\
  g_{K\bar{K}} & = & 311\, \mathrm{MeV} \, . 
\end{eqnarray*}
That paper also finds a pole mass of the $a_{0}$ of $m_{a_{0}}=984.45$~MeV. The parameter $\beta_{0}^{\prime}$ describes the production
of the resonance from the initial $p\bar{p}$ and the $\rho_{i}(m)$s are the phase space available for the decay with an
$a_{0}$ of mass $m$, and is given as
\begin{eqnarray}
\rho_{i}(m) & = & \frac{2q(m)}{m}\, ,
\end{eqnarray}
where the decay momentum of the $a_{0}$ to daughters of mass $m_{1}$ and $m_{2}$ is given as
\begin{eqnarray}
q(m) & = & \frac{\sqrt{\left[ m^{2} - (m_{1}-m_{2})^{2}\right]\,\left[ m^{2} - (m_{1}+m_{2})^{2}\right]}}{2m} \, .
\end{eqnarray}
What will ultimately replace the Breit-Wigner for the $a_{0}(980)$ is $\rho\mid F \mid^{2}$.



For the case of $a_{0}\rightarrow K_{S}K_{S}$, we would have the decay momentum being
\begin{eqnarray*}
  q_{K\bar{K}}(m) & = & \frac{1}{2}\sqrt{ m^{2} - 4\, m^{2}_{K_{S}} } \,\,\, m > 2m_{K_{S}} \\
  q_{K\bar{K}}(m) & = & 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, m \leq 2m_{K_{S}} \, .
\end{eqnarray*}
and for the decay to $\eta\pi$, we have
\begin{eqnarray*}
  q_{\eta\pi}(m) & = & \frac{\sqrt{\left[ m^{2} - (m_{\eta}-m_{\pi})^{2}\right]\,\left[ m^{2} - (m_{\eta}+m_{\pi})^{2}\right]}}{2m} \, .
\end{eqnarray*}

From these, the phase space factors are
\begin{eqnarray*}
  \rho_{K\bar{K}}(m) & = & \sqrt{ 1 - \left(\frac{2 m_{K_{S}}}{m}\right)^{2} } \,\,\, m > 2m_{K_{S}} \\
  \rho_{K\bar{K}}(m) & = & 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, m \leq 2m_{K_{S}} \, .
\end{eqnarray*}
and
\begin{eqnarray*}
\rho_{\eta\pi}(m) & = & \frac{\sqrt{\left[ m^{2} - (m_{\eta}-m_{\pi})^{2}\right]\,\left[ m^{2} - (m_{\eta}+m_{\pi})^{2}\right]}}{m^{2}} \, .
\end{eqnarray*}

So, the Flatte formulation for the $K\bar{K}$ decay will be
\begin{eqnarray*}
  F_{K\bar{K}}(m) & = & \frac{ g_{K\bar{K}}}{m^{2}_{a_0} - m^{2} -
  i\left( \rho_{\eta\pi}(m)g^{2}_{\eta\pi} + \rho_{K\bar{K}}(m)g^{2}_{K\bar{K}}\right)} \, ,
\end{eqnarray*}
and the function to replace the Breit-Wigner form is
\begin{eqnarray}
BW(m) & = & c\, \rho_{K\bar{K}}(m) \, \mid F_{K\bar{K}}(m) \mid^{2}\, ,
\end{eqnarray}
where $c$ is a normalization constant. This function will be $0$ for $m\leq 2m_{K_{S}}$, and then turn on after
that, yielding a bump above $1$~GeV.
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\begin{thebibliography}{99}
\bibitem{cbar-KlKpp} A. Abele \emph{et al.} [Crystal Barrel], \textbf{$\mathbf{\overline{p}p}$ annihilation at rest into $\mathbf{K_{L}K^{\pm}\pi^{\mp}}$},
  Phys. Rev. D\textbf{57}, 3860 (1998).
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\bibitem{flatte} S.~M.~Flatt\'{e}, \textbf{Coupled-channel analysis of the $\mathbf{\eta\pi}$ and $\mathbf{K\overline{K}}$
systems near $\mathbf{K\overline{K}}$ threshold}, Phys. Lett. B\textbf{63}, 224 (1976).
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\bibitem{cbar-etapipi} C. Amsler \emph{et al.} [Crystal Barrel], \textbf{Observation of a new
    $\mathbf{I^{G}(J^{PC}=1^{-}(0^{++})}$ resonance at 1450 MeV}, Phys. Lett. B\textbf{333}, 277 (1994).
\end{thebibliography}  
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