is followed from Ref.~\cite{Williams:2010vh}. The term $\sigma(\vec{x}) = \bar{\sigma} / (f(\vec{x}) \int dx'$, where
$\int dx'$ is introduced as the area of the Dalitz plot, so the mean value of the denominator becomes 1. The nuisance parameter
$\bar{\sigma}$ takes a value of $0.01$ in Ref.~\cite{Williams:2010vh}. Unlike the mixed-sample test, the number of toy events should be large $(n_{\rm toy}\gg n_{\rm data})$
to avoid statistical fluctuations.
First, the test statistic $T_h$ is calculated using the full available statistics. Then, a permutation test is performed as follows.
The data and toy samples are pooled together and new samples of size $n_{\rm data}$ are randomly selected from the pooled sample.
Each new sample is then treated as the data sample, while the remaining events become the toy sample, and a new value of $T_h = T_{\rm perm}$ is calculated.
The $p$-value of the test is then simply the fraction of times that $T_{\rm perm} > T_h$.
This can then be repeated with additional toy samples to build up a distribution of $p$-values.
-\subsection{Speed}
-\label{sec:performance:speed}
\subsection{Examples}
\label{sec:performance:examples}
The \laura\ package has been used for numerous publications by several experimental collaborations and groups of phenomenologists.
Below, several examples that demonstrate the features of the package are discussed.
In addition, \laura\ has also been used for various other studies of three-body charmless \B meson decays by the \babar\ collaboration~\cite{Aubert:2007xb,Aubert:2008rr,Aubert:2008aw,delAmoSanchez:2010ur,BABAR:2011aaa}, studies of charm decays by the LHCb collaboration~\cite{Aaij:2014afa}, unpublished studies by several collaborations (for example Refs.~\cite{delAmoSanchez:2010ad,Kohl:48803}), and investigations into the phenomenology of different three-body decays~\cite{Latham:2008zs,Gershon:2014yma,Nogueira:2016mqf}.
Studies of charmless three-body \B\ meson decays provide interesting opportunities to investigate the dynamics of hadronic \B\ decays including potential \CP\ violation effects.
The $\Bp \to \pip\pip\pim$~\cite{Aubert:2005sk,Aubert:2009av} and $\Bp \to \Kp\pip\pim$~\cite{Aubert:2005ce,Aubert:2008bj} decays have been investigated by the \babar\ collaboration using the \laura\ package.
In the most recent amplitude analysis of $\Bp \to \pip\pip\pim$ decays~\cite{Aubert:2009av}, the amplitude model includes contributions from the $\rho(770)^0$, $\rho(1450)^0$, $f_2(1270)$, $f_0(1370)$ resonances and a nonresonant component.
In the most recent amplitude analysis of $\Bp \to \Kp\pip\pim$ decays~\cite{Aubert:2008bj}, the amplitude model includes the $K^*(892)^0$, $K_2^*(1430)^0$, $\rho(770)^0$, $\omega(782)$, $f_0(980)$, $f_2(1270)$, $f_X(1300)$ and $\chi_{c0}$ resonances together with $K\pi$ S-wave and nonresonant component.
In both cases, \CP violation is allowed in the amplitudes.
Projections of the fit results around the $\rho(770)^0$ resonance are shown in Fig.~\ref{fig:3pi-Kpipi-examples}.
The $\Bp \to \pip\pip\pim$ data are consistent with \CP\ conservation while there is evidence for \CP\ violation in $\Bp \to \rho(770)^0\Kp$ decays, which becomes more evident when inspecting the data in different regions of the $\pip\pim$ helicity angle, $\theta_{\pip\pim}$.
As model-independent analyses of larger data samples of these decays by the LHCb collaboration~\cite{Aaij:2013sfa,Aaij:2013bla,Aaij:2014iva} have revealed large \CP\ violation effects that vary significantly across the Dalitz plot, there is strong motivation for updated amplitude analyses on larger samples.
Projections of the data and fit results onto the $\pip\pim$ invariant mass in the $\rho(770^0)$ region, for (left) $\Bp \to \pip\pip\pim$~\cite{Aubert:2009av} and (right) $\Bp \to \Kp\pip\pim$~\cite{Aubert:2008bj} candidates observed by the \babar\ collaboration.
In both cases the top row shows all candidates, the middle row shows those with $\cos \theta_{\pip\pim} > 0$ and the bottom row shows those with $\cos \theta_{\pip\pim} < 0$, while in each row the left (right) plot is for $B^-$ ($B^+$) candidates.
The data are the points with error bars, the red/dark filled histogram shows the continuum background component, the green/light filled histogram shows the background from other \B\ meson decays, and the blue unfilled histogram shows the total fit result.
}
\label{fig:3pi-Kpipi-examples}
\end{figure}
Understanding the origin of these \CP violation effects requires related modes to also be studied.
The \laura\ package has also been used by the \babar\ collaboration for a time-dependent Dalitz plot analysis of $\Bz \to \KS\pip\pim$ decays~\cite{Aubert:2009me}, as well as for an amplitude analysis of $B^{+} \to \KS \pi^{+} \pi^{0}$ decays~\cite{Lees:2015uun}.
In the latter, the modelling of the large background contribution, as well as of the smearing of the Dalitz plot position due to the limited resolution of the neutral pion momentum (self cross-feed), is particularly important.
In addition, correlations between the Dalitz plot position and the variables that are used to discriminate signal decays from background contributions are taken into account as described in Appendix~\ref{sec:pdfs-DPdep}.
The amplitude model includes components from the $K^*(892)$ resonance and $K\pi$ S-wave (both appearing in both charged and neutral channels) as well as the $\rho(770)^+$ resonance.
Projections of the fit results are shown in Fig.~\ref{fig:KSpipi0-examples}.
The analysis reveals evidence for a \CP\ asymmetry in $\Bp \to K^*(892)^+\piz$ decays.
Projections of the data and fit results onto (top) $\KS\pimp$, (middle) $\KS\piz$ and (bottom) $\pimp\piz$ invariant mass distributions for $\Bp \to \KS\pip\piz$ candidates observed by the \babar\ collaboration~\cite{Lees:2015uun}.
In each row the left (right) plot is for $B^-$ ($B^+$) candidates.
The data are the points with error bars, the (black) dash-dotted curves represent the signal contribution, the dotted (red) curves to the continuum background component, the dashed (green) curves to the total background contribution and the solid (blue) curves the total fit result.
}
\label{fig:KSpipi0-examples}
\end{figure}
The LHCb collaboration has used the \laura\ package for several studies of multibody $B$ meson decays to charmed final states, with important results for charm spectroscopy and \CP\ violation measurements.
For example, the $\Bs \to \Dzb\Km\pip$ decay was found to have a Dalitz plot structure that contains effects from overlapping spin-1 and spin-3 resonances with masses around $m(\Dzb\Km) \sim 2.86 \gevcc$~\cite{Aaij:2014xza,Aaij:2014baa}.
The neutral charm meson is reconstructed through its $\Dzb \to \Kp\pim$ decay.
The model contains contributions from the $\Kstarb(892)^0$, $\Kstarb(1410)^0$, $\Kbar{}^*_2(1430)^0$, $\Kstarb(1680)^0$ resonances as well as a $\Km\pip$ S-wave component, and $D_{s2}^*(2573)^-$, $D_{s1}^*(2700)^-$, $D_{s1}^*(2860)^-$, $D_{s3}^*(2860)^-$ resonances together with a nonresonant $\Dzb\Km$ S-wave amplitude and virtual contributions from the $D_{s\,v}^{*-}$, $D_{s0\,v}^*(2317)^-$ and $B_{v}^{*+}$ states.
The results of the analysis include the first experimental proof of the spin-2 nature of the $D_{s2}^*(2573)^-$ state, as well as world-leading measurements of the masses and widths of many of the resonances.
Projections of the Dalitz plot fit results onto the data are shown in Fig.~\ref{fig:BsDKpi-examples}.
Projections of the data and fit results onto (top left) $\Km\pip$ and (top right) $\Dzb\Kp$ invariant mass distributions for $\Bs \to \Dzb\Km\pip$ candidates observed by the LHCb collaboration~\cite{Aaij:2014xza,Aaij:2014baa}.
A legend describing the various contributions is also given, together with (bottom left) a zoom around $m(\Dzb\Km) \sim 2.86 \gevcc$ and (bottom right) a projection onto the cosine of the helicity angle $\theta(\Dzb\Km)$ for candidates in that region.
In the last case, projections of the results of fits with models containing either or both of the $D_{s1}^*(2860)^-$ and $D_{s3}^*(2860)^-$ resonances are shown, demonstrating the need for both to obtain a good fit to the data.
}
\label{fig:BsDKpi-examples}
\end{figure}
A similar Dalitz plot analysis with the \laura\ package has been performed by the LHCb collaboration for the $\Bz \to \Dzb\Kp\pim$ mode~\cite{Aaij:2015kqa}.
The model obtained is an essential input into a subsequent analysis of the same decay with the neutral charm meson reconstructed through $D$ decays to the \CP\ eigenstates $\Kp\Km$ and $\pip\pim$~\cite{Aaij:2016bqv}.
In the latter case, contributions from both $\Bz \to \Dz\Kp\pim$ and $\Dzb\Kp\pim$ amplitudes can interfere, giving sensitivity to the angle $\gamma$ of the CKM unitarity triangle.
A Dalitz plot analysis allowing for \CP\ violation effects provides a powerful way to determine $\gamma$ without ambiguities~\cite{Gershon:2008pe,Gershon:2009qc}.
-In this analysis, a simultaneous fit to the final states with different $D$ decays is implemented using the {\it J}{\sc fit} method~\cite{Ben-Haim:2014afa}.
+In this analysis, a simultaneous fit to the final states with different $D$ decays is implemented using the \jfit method described in Sec.~\ref{sec:jfit} and Ref.~\cite{Ben-Haim:2014afa}.
Projections of the fit result onto the data (weighted by the signal purity) are shown in Fig.~\ref{fig:B0DKpi-examples}.
No significant \CP\ violation effect is observed, and the resulting limits on $\gamma$ are not strongly constraining.
The method is, however, expected to give competitive constraints on $\gamma$ as larger data samples become available and as additional $D$ meson decay modes are included in the analysis.
Moreover, the analysis also gives results for hadronic parameters that must be known in order to interpret results from quasi-two-body analyses of $\Bz \to D\Kstar(892)^0$ decays in terms of $\gamma$.
As such, the results have an important impact in combinations of results to obtain the best knowledge of $\gamma$~\cite{Aaij:2016kjh,Amhis:2016xyh}.
Projections of the data and fit results onto $m(\Kmp\pipm)$ for (left) $\Bzb \to D\Km\pip$ and (right) $\Bz \to D\Kp\pim$ candidates observed by the LHCb collaboration~\cite{Aaij:2016bqv}.
A legend describing the various contributions is also given.
}
\label{fig:B0DKpi-examples}
\end{figure}
Other decays of the type $B \to D^{(*)}K\pi$ have sensitivity to $\gamma$, and are also important to study as possible background contributions to the two-body $B \to D^{(*)}K$ type decays that are more conventionally used for this purpose.
The LHCb collaboration has also published results on the $\Bp \to \Dm\Kp\pip$~\cite{Aaij:2015vea} and $\Dp\Kp\pim$~\cite{Aaij:2015dwl} decays, which were obtained from analyses using the \laura\ package.
The higher-yield $B \to D^{(*)}\pi\pi$ channels provide some of the most interesting possibilities to explore charm spectroscopy.
An amplitude analysis of $\Bp \to \Dm\pip\pip$~\cite{Aaij:2016fma} has been performed by the LHCb collaboration, using the \laura\ package, in which the model contains contributions from the $\Dbar{}_2^*(2460)^0$, $\Dbar{}_1^*(2760)^0$, $\Dbar{}_3^*(2760)^0$ and $\Dbar{}_2^*(3000)^0$ resonances (the last three of which are either confirmed or observed for the first time), as well as virtual contributions from the $\Dbar{}_{v}^*(2007)^-$ and $B_{v}^{*0}$ states.
In the absence of sufficiently details models for the $\Dm\pip$ S-wave, a quasi-model-independent description based on spline interpolation is used.
Projections of the results of the fit onto the data are shown in Fig.~\ref{fig:BDpipi-examples}.
Projections of the data and fit results onto $m(\Dm\pip)_{\rm min}$ for $\Bp \to \Dm\pip\pip$ candidates observed by the LHCb collaboration~\cite{Aaij:2016fma} on (top left) linear and (bottom right) logarithmic $y$-axis scales.
Here, $m(\Dm\pip)_{\rm min}$ is the smaller of the two values of $m(\Dm\pip)$ for each $\Bp \to \Dm\pip\pip$ candidate.
A legend describing the various contributions is also given.
The (bottom right) Argand diagram of the $\Dm\pip$ S-wave amplitude shows the expected phase motion corresponding to the $\Dbar{}_0^*(2400)^0$ resonance.
}
\label{fig:BDpipi-examples}
\end{figure}
+
+
+\subsection{Speed}
+\label{sec:performance:speed}
+
+In this section some performance benchmarks for the \laura code are provided.
+More specifically, each of the examples that are provided with the package
+(several of which are based on analyses presented in the previous section)
+are run out of the box on the same machine (\textbf{provide here the
+specification of the machine}) and in each case average timings for both
+toy generation and for fitting are provided in Table~\ref{tab:examples-speed}.
+
+\begin{table}[!htb]
+\caption{Speed of execution of each of the examples provided with the \laura package. The times given are for 500 toy experiments.}
+\label{tab:examples-speed}
+\centering
+\begin{tabular}{llcc}
+\hline
+Example & Description & Execution time of toy MC generation & Execution time of fit \\
+\hline
+\texttt{GenFit3pi.cc} & 1500 $\Bp\to\pip\pip\pim$ signal with 1250 uniform background & $?.?\sec$ & $?.?\sec$ \\
+\hline
+\end{tabular}
+\end{table}
+
+In addition, the $\Bp\to\Kp\Kp\Km$ example is run with:
+\begin{itemize}
+\item no floating resonance parameters
+\item only the mass of the $\phi(1020)$ floating
+\item the mass and width of the $\phi(1020)$ floating
+\item the mass and width of both the $\phi(1020)$ and $f^{\prime}_2(1525)$ floating
+\end{itemize}
+and the timings for each of those scenarios are provided in Table~\ref{tab:respars-speed}.
+
+\begin{table}[!htb]
+\caption{Speed of execution of the \texttt{GenFit3K.cc} example provided with the \laura package with different sets of floating resonance parameters. The times given are for a single toy experiment.}
+\label{tab:respars-speed}
+\centering
+\begin{tabular}{lc}
+\hline
+Floated parameters & Execution time of fit \\
+\hline
+None & $?.?\sec$ \\
+Mass of $\phi(1020)$ & $?.?\sec$ \\
+Mass and width of $\phi(1020)$ & $?.?\sec$ \\
+Mass and width of $\phi(1020)$ and $f^{\prime}_2(1525)$ & $?.?\sec$ \\
This section provides the complete set of available resonances, indicating the
name, mass $m_0$, width $\Gamma_0$, spin, charge and Blatt--Weisskopf barrier radius $r_{\rm BW}$.
Table~\ref{tab:resNames1} contains information for light meson resonances,
Table~\ref{tab:resNames2} for charm, charmonium, strange--charm, beauty and strange--beauty resonances,
Table~\ref{tab:resNames3} for $K^*$ resonances and
Table~\ref{tab:resNames4} for non--resonant terms.
The tables list the information contained in the information records for both neutral and positively-charged resonances.
Negatively-charged resonance records are implemented as charge-conjugates of
the positively charged ones; the plus sign in the name just needs to be
replaced with a minus sign.
In case a user wishes to modify the values of the parameters from those given in the tables, the {\tt LauAbsResonance::changeResonance} function, which takes the mass, width and spin as arguments, can be used.
The Blatt--Weisskopf barrier radius can be changed with the {\tt LauAbsResonance::changeBWBarrierRadii} function, and other parameters specific to particular lineshapes can be changed with the {\tt LauAbsResonance::setResonanceParameter} function.
The same approach can be used to include a resonance that is not available in these tables, by using any of the existing states of appropriate charge and redefining its properties.