where $m_0$ is the pole mass and $\Gamma_0$ is the resonance width.
The more commonly used relativistic Breit--Wigner lineshape is described in Sec.~\ref{sec:lineshapes}.
We note here that the relativistic Breit--Wigner lineshape can also describe so-called virtual contributions, from resonances with masses outside the kinematically accessible region of the Dalitz plot, with one modification:
in the calculation of the momenta, the mass $m_0$ is set to a value $m_0^{\rm{eff}}$ within the kinematically allowed range.
This is accomplished with the {\it ad-hoc} formula
Here the fractional coefficients come from isospin conservation,
$m_{i,j}$ denotes the invariant mass of the daughter particle $j$ (1--4) in
channel $i$ (1--2),
and $g_1$ and $g_2$ are coupling constants whose values are assumed to be those
provided in Table~\ref{tab:flattepars}.
The Clebsch-Gordan coefficients in Eqs.~\ref{eq:Gamma1} and~\ref{eq:Gamma2} are not guaranteed to be correct for every possible resonance that could be modelled with the Flatt\'e lineshape, but are appropriate for every case considered in Table~\ref{tab:flattepars}.
The expressions for the widths are continued analytically ($\Gamma \ra i |\Gamma|$) when $m$ is below any of the specific channel thresholds,
contributing to the real part of the amplitude, while the Adler-zero
term $f_A = (m^2 - s_A)/(m^2_0 - s_A)$ can be used to suppress false kinematic
singularities when $m$ goes below threshold~\cite{Bugg:2003kj}
(otherwise $f_A$ is set to unity).
Variants of the Flatt\'e lineshape have been used in the literature.
In some cases, \eg\ Refs.~\cite{Ablikim:2004wn,Abele:1998qd}, the constant
$m_0$ that multiplies the widths in the denominator of Eq.~(\ref{eq:Flatte})
is absorbed into the couplings.
As a consequence the couplings have dimensions of mass-squared, and are
sometimes denoted as $g_i$~\cite{Ablikim:2004wn} and sometimes as $g_i^2$~\cite{Abele:1998qd}.
In Table~\ref{tab:flattepars} all values have been converted to be consistent
with Eqs.~(\ref{eq:Flatte})--(\ref{eq:Gamma2}).
In \laura\ it is only possible to use the Flatt\'e lineshape for the systems
specified in Table~\ref{tab:flattepars}.
At construction time the resonance name is checked and the corresponding
parameter values are set; these can be modified by the user if desired.
\begin{table}[!htb]
\caption{
The four daughter particles used for each channel term $m_{ij}$, as well as the coupling ($g_1$, $g_2$) and Adler-zero ($s_A$) constants for the Flatt\'e lineshapes.
Units of \gev\ for $g_{1,2}$ (or $\gev^2$ for $m_0g_{1,2}$ when taken from Refs.~\cite{Ablikim:2004wn,Abele:1998qd}) and \gevgevcccc\ for $s_A$ are implied.
}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{lllllll}
\hline
Resonance & Channel 1 & Channel 2 & $g_1$ or $m_0g_1$ & $g_2$ or $m_0g_2$ & $s_A$ & Reference \\