\subsection{The CLAS-g12 Data Set}
This section summarizes the experimental conditions of the g12 data set. The data for this experiment were
taken between April 1st and June 9th, 2008. The data set was further divided into ten different groups of runs
according to different trigger configurations.\par\smallskip
Table~\ref{Table:TriggerConfigurations} shows the different g12 trigger configurations. We used only Period~2
(starting from run 56520) for our analyses at FSU. For these data, the trigger required either (at least) three charged
tracks with no restrictions on the photon energy or only two tracks with the additional requirement of having at
least one photon detected with an energy above 3.6~GeV. Since our primary motivation initially was to extract
the $\omega$ (and $\pi^+\pi^-$) cross sections with high quality, we decided not to mix trigger configurations
and thus, avoided the prescaled data and those using an Electromagnetic Calorimeter (EC)-based photon or lepton
trigger (Period 3\,-\,8). Period~1 suffered from lower statistics and using it would not have significantly improved
the statistical uncertainties of our results. Moreover, this period switched from a three-track requirement to a
two-track requirement at a different energy and also used a different beam current.\par\smallskip
\begin{table}[tb!]
\addtolength{\extrarowheight}{4pt}
\begin{center}
\begin{tabular}{ c | l | c}
\hline
Period & Runs & Trigger Configuration\\\hline
1 & 56519 and earlier & not prescaled, trigger change at 4.4~GeV\\
2 & \bl{56520 - 56594, 56608 - 56646} & not prescaled, trigger change at 3.6~GeV\\
3 & 56601 - 56604, 56648 - 56660 & prescaled\\
4 & 56665 - 56667 & prescaled\\
5 & 56605, 56607, 56647 & prescaled\\
6 & 56668 - 56670 & prescaled\\
7 & 56897 and later & prescaled\\
8 & 57094 and later & prescaled\\\hline
9 & 56585, 56619, 56637 & single-sector, not prescaled\\
10 & 56663 and later & single-sector, not prescaled\\\hline
\end{tabular}
\caption{\it The different trigger configurations used in g12 (from the g12
wiki and Ref.~\cite{CLAS-NOTE_2017-002}).}
\label{Table:TriggerConfigurations}
\end{center}
\end{table}
The information included in the raw data consisted of QDC (Charge to Digital Convertor) and TDC (Time
to Digital Converter) channel IDs and values. In a first step, the data had to undergo reconstruction, or be
{\it cooked.} This process converted the data into physical quantities like particle IDs, positions, angles,
energies, and momenta. The data calibration was carried out independently for each detector component
of CLAS. After the detectors had been calibrated and the particle tracks had been reconstructed, the data
were made available for physics analysis. Each event had its information organized in
\href{http://clasweb.jlab.org/bos/browsebos.php?bank=gpid&build=64bit/STABL}{\underline{CLAS data
banks}}\,\footnote{http://clasweb.jlab.org/bos/browsebos.php?bank=gpid\&build=64bit/STABL}. These
data banks contained not only the properties of the particles involved in a reaction but also information
about detector hits.\par\smallskip
\clearpage
\noindent
Here we list the most relevant data banks that we used in our g12 analyses:
\begin{enumerate}
\item {\bf PART} -- This bank contained most of the details about the detected particles, such as the
particle IDs, 4-vectors, vertex of each particle, and other information from various detectors.
\item {\bf TAGR} -- In this bank, information about all incident photons was stored, e.g. the energy of
the photon(s), the time of the photon(s) after the reconstruction in the Tagger, the time of the photon(s) after
the RF correction, the status of the photon(s) (used to identify those which were not reconstructed properly),
and the E\,- and T\,-\,counter ID information of the corresponding scattered electron.
%\item {\bf MVRT} - The MVRT bank contains event-vertex information. It was obtained by using tracking
% information (from the drift chambers and the time-of-flight scintillators) of each detected particle that
% constituted the event.
\item {\bf TBER} -- Time-based tracking error bank containing fit parameters and the covariance matrix.
\item {\bf TBID} -- Bank containing information on time-based particle ID (including
$\beta$ ($=\frac{v}{c}$) values).
\item {\bf TGBI} -- Trigger bank; it also stored polarization information, e.g. the helicity bit.
\item \bl{Any other bank of importance?}
\end{enumerate}
% ----------------------------------------------------------------------------------
\subsection{Reaction Channels and General Event Selection}
\label{Subsection:GeneralEventSelection}
% ----------------------------------------------------------------------------------
The final states of interest in this analysis are $\gamma p \to p\,\pi^{+}\pi^{-}$ and $\gamma p\to p\,\pi^+
\pi^-\,(\pi^0)$. These three-track channels were broken up into different topologies as shown in
Table~\ref{Table:Topologies}. A topology is defined according to the detected particles in the final state:
the two-particle final states (Topologies 1\,-\,3) and the three-particle final states (Topologies 4\,-\,6).
A particle which was not detected in a given topology could be identified through the missing-mass technique.
For this method, the Lorentz vectors of the incoming beam and the target were used. The four-momentum of
a missing particle in the reaction was then determined from the measured three-momenta and the particle
energies. The missing four-momentum was given by:
\begin{equation}
x^{\mu} \,=\, k^{\mu} \,+\, P^{\mu} \,-\, \sum_{i=1}^{2,3} p^{\mu}_{i}\,,
\label{Equation:missing-momentum}
\end{equation}
where $k^{\mu}$ and $P^{\mu}$ are the initial photon and target-proton four-momenta and $p^{\mu}_{i}$ are the
four-momenta of the two or three detected final-state particles. The missing mass $m_{X}$ was defined as:
\begin{equation}
m^{2}_{X} \,=\, x^{\mu} x_{\mu}\,.
\end{equation}
\begin{table}[b!]
\addtolength{\extrarowheight}{4pt}
\begin{center}
\begin{tabular}{ c | c | c | c c c | c }
\hline
\hline
& & \multicolumn{4}{c|}{Reconstructed Particles} \\
\cline{3-6}
\rb{Reaction} & \rb{Topology} & Total & p & $\pi^{+}\,(K^+)$ & $\pi^{-}\,(K^-)$ & \rb{Missing Particle of Interest}\\
\hline
$\gamma p \to p \,\pi^{+}\, (\pi^{-})$ & 1 & 2 & 1 & 1 & 0 & $m_{\pi^{-}}$ \\
$\gamma p \to p \,\pi^{-}\, (\pi^{+})$ & 2 & 2 & 1 & 0 & 1 & $m_{\pi^{+}}$ \\
$\gamma p \to (p) \,\pi^{+} \pi^{-}$ & 3 & 2 & 0 & 1 & 1 & $m_{p}$ \\
$\gamma p \to p \,\pi^{+} \pi^{-}$ & 4 & 3 & 1 & 1 & 1 & 0 \\
$\gamma p \to p \,\pi^{+} \pi^{-} \,(\pi^{0})$ & 5 & 3 & 1 & 1 & 1 & $m_{\pi^{0}}$ \\
$\gamma p \to p \,K^{+} K^{-}$ & 6 & 3 & 1 & 0\,(1) & 0\,(1) & 0 \\
\hline
\hline
\end{tabular}
\caption{\it Classification of the reactions, $\gamma p \to p\,\pi^{+} \pi^{-}$, $\gamma p \to p\,K^{+} K^{-}$,
and $\gamma p \to p \,\pi^{+}\pi^{-}\, (\pi^{0})$, using different topologies. Reconstructed particles
were identified by their PID information from the TBID~bank. Note that we did not analyze
Topologies 1\,-\,3 because of the dominant three-track trigger condition in our data set.}
\label{Table:Topologies}
\end{center}
\end{table}
\noindent
The missing-mass distribution was used for a data quality check after all corrections and cuts had been
applied. The four-momentum vector $x^{\mu}$ in Equation~\ref{Equation:missing-momentum} was used to
complete the set of four-vectors for Topology~5 (Table~\ref{Table:Topologies}). The other final states with
a missing particle (Topology 1-3) were more complicated to analyze owing to the special trigger configuration
in Period~2.\par\smallskip
Events were pre-selected based on the particles' identification number (PID), which was determined during
the cooking process. Events that did not meet this requirement (Table~\ref{Table:Topologies}) were ignored
and subsequently omitted from the analysis. The calculation of the detected particles' masses, which was
necessary to determine the PIDs, used two independently-measured quantities, the momentum, $p$, and
the velocity as a fraction of the speed of light, $\beta$. The magnitude of a particle's momentum was determined
with an uncertainty of $<$ $1\,\%$ using information from the CLAS Drift Chambers (DC)~\cite{Mestayer:2000we}.
The quantity~$\beta$ of a detected final-state particle was determined with an uncertainty of up to
$5\,\%$~\cite{Smith:1999ii} using a combination of the Start Counter (SC), the Time-of-Flight (TOF) spectrometer,
and the particle's trajectory through CLAS. The detected particle's mass could then be calculated by:
\begin{equation}
m^{2}_{{\rm particle}~X} \,=\, \frac{p^2\,(1\,-\,\beta^{2})}{\beta^2}\,.
\end{equation}
\par\smallskip
After the particle's mass had been calculated, it was compared to the masses of known particles (hadrons and
leptons). If this calculated mass matched that of a known particle (within resolution), the PID associated with
that mass was assigned to the final-state particle. This value could then be used to select certain final-states
for analysis. In this analysis, the physical properties of the final-state particles (e.g. their 4-vectors, vertex
information, etc.) were extracted from the PART data banks. Photon and final-state particle selection was further
improved by applying cuts and corrections (see Section~\ref{Subsection:PhotonParticleIdentification}). We also
used kinematic fitting (see Section~\ref{Subsection:KinematicFitting}) to fine-tune the initial- and final-state
momenta by imposing energy- and momentum conservation. Finally, to separate signal events from the
remaining background, we used an event-based $Q$-factor method which is discussed in more details in
Section~\ref{Subsection:Q-factor_method}.\par\smallskip
In a short summary, listed below are the cuts and (in the right order) corrections that were applied to the g12~data
in these FSU analyses.
\subsubsection*{General g12 Corrections}
\begin{itemize}
\item Tagger-sag corrections (done in the cooking process).
\item ELoss corrections using the standard CLAS package~\cite{CLAS-NOTE_2007-016}.
\item Beam-energy corrections based on the CLAS-approved run-group approach~\cite{CLAS-NOTE_2017-002}.
\item Momentum corrections based on the CLAS-approved run-group approach~\cite{CLAS-NOTE_2017-002}.
\end{itemize}
\subsubsection*{Florida State U. Cuts}
\begin{itemize}
\item Vertex cut: $-110.0 < z$-{\rm vertex} $< -70.0$~cm.
%\item Photon selection \& accidentals (PART$[\,].$ {\sc ngrf} $=$ 1 \& PART$[\,].${\sc tagrid} equal for all
% tracks)
\item \re{Photon selection \& accidentals\\ (nGammaRF() = 1 \& {\sc tagr\_id} equal for all tracks;
information from TAGR bank)}
%\item Particle ID cut\,\footnote{In the final analysis, we applied the $\Delta\beta \leq 3\sigma$ cut on
% either the proton or the $\pi^+$ (no cut on the $\pi^-$).}, $\Delta\beta \,=\, |\beta_{\rm\,c}\,-\,\beta_{\rm\,m}|
% \,\leq\, 3\sigma$, and timing cut, $|\Delta t_{\rm\,TGPB}|< 1$~ns.
\item Particle ID cut\,\footnote{In the final analysis, we applied the $\Delta\beta \leq 3\sigma$ cut on
either the proton or the $\pi^+$ (no cut on the $\pi^-$).}, $\Delta\beta \,=\, |\beta_{\rm\,c}\,-\,\beta_{\rm\,m}|
\,\leq\, 3\sigma$, \re{and timing cut, $|\Delta t_{\rm\,TBID}|< 1$~ns}\,\footnote{\re{$\Delta t_{\rm\,TBID}
= stVtime() - vtime()$ is the coincidence time between the vertex and the photon time.}}.
\item Confidence-level cut of ${\rm CL} > 0.001$ for $\gamma p \to p\,\pi^+ \pi^-\,(\pi^0$) and $\gamma p\to
p\,K^+K^-$.
\item Fiducial cuts: {\it nominal} scenario~\cite{CLAS-NOTE_2017-002}.
\end{itemize}
\noindent
The order of these applied cuts and corrections was quite flexible with the exception of a few cases. Momentum
corrections were applied after the energy-loss corrections. The following sections describe the applied cuts and
corrections in more detail.
% -------------------------------------------------------------------
\subsection{Photon and Particle Identification}
\label{Subsection:PhotonParticleIdentification}
% -------------------------------------------------------------------
\subsubsection{Initial-Photon Selection (Cuts on Timing and Accidental
Photons)\label{Subsubsection:PhotonSelection}}
% -------------------------------------------------------------------
The electrons, which were used to produce the beam of polarized photons via bremsstrahlung radiation,
were delivered from the accelerator into Hall~B in the form of 2-ns bunches. Since each bunch contained
many electrons, there were several potential photon candidates per recorded event that could have triggered
the reaction inside the target. Random electron hits could also occur from various background sources
(e.g. cosmic rays). These did not create bremsstrahlung photons but the hits were registered in the tagger
scintillators. It was important to determine the correct photon in each event (out of about five candidates
on average) because the corresponding photon energy was key to understanding the initial state of the event.
The analysis steps taken in the photon selection were as follows:
\begin{enumerate}
\item The Start Counter time per track at the interaction point, $t_{\rm track}$, was given by:
\begin{equation}
t_{\rm track} \,=\,t_{\rm\, ST} \,-\, \frac{d}{c\,\beta_{\rm calc}}\,,
\label{Equation:eventVertexTime}
\end{equation}
where $t_{\rm\, ST}$ was the time when the particle was detected by the Start Counter, $d$ was the length
of the track from the interaction point to the Start Counter, %\,\footnote{The values of $t_{\rm\, ST}$ and $d$
%could be obtained from the GPID[\,].{\sc st\_time} and GPID[\,].{\sc st\_len}, respectively.}
and $c\,\beta_{\rm calc}$
was the calculated velocity of the particle. These (track) times could be averaged to give an event time, $t_{\rm event}$.
\par\smallskip
\begin{figure}[t!]
\begin{tabular}{cc}
%\includegraphics[height=0.4\textwidth,width=0.7\textwidth]{eps/Coincidence_time_ST_TAG.eps}
\includegraphics[width=0.48\textwidth]{g12_figures/timing_beforecut.pdf} &
\includegraphics[width=0.48\textwidth]{g12_figures/timing_afterbothcut.pdf}
\end{tabular}
\caption{\it Left: Example of a coincidence-time distribution, $\Delta t_{\rm\,TGPB}$, for the inclusive
$p\,\pi^{+} \pi^{-}$~final-state topology. The 2-ns bunching of the photon beam is clearly visible in the
histogram. Right: Distribution of %$\Delta t_{\rm\,TGPB} = t_{\rm event} - t_{\gamma}$
\re{$\Delta t_{\rm\,TBID} = t_{\rm event} - t_{\gamma}$} for the selected photon
(one entry per event) after PID cuts. The event vertex time, $t_{\rm event}$, was based on
Equation~\ref{Equation:eventVertexTime}. We only considered events which had exactly one candidate
photon in the same RF bucket per track; each identified track had to be associated with the same photon.}
\label{Figure:Timing}
\end{figure}
The time at which a candidate photon arrived at the interaction point, $t_{\gamma}$, was given by:
\begin{equation}
t_{\gamma} \,=\,t_{\rm center} \,+\, \frac{d^{\,\prime}}{c}\,,
\label{Equation:photonTime}
\end{equation}
where $t_{\rm center}$ was the time at which the photon arrived at the center of the target and $d^{\,\prime}$
was the distance between the center of the target and the event vertex along the beam-axis. We did not
consider the $x$- and $y$-coordinates of the event vertex because they were comparable to the vertex
resolution. In this analysis, the $t_{\gamma}$ values were obtained from TAGR$[\,].${\sc tpho}.
\par\smallskip
Both, $t_{\gamma}$ as well as $t_{\rm event}$, describe the time of the $\gamma p$ interaction -- based on
initial- and final-state particles, respectively. To find the correct initial photon, we looked at the
corresponding time differences. The {\it coincidence time}, %$\Delta t_{\rm\,TGPB}$,
\re{$\Delta t_{\rm\,TBID}$}, was thus defined per
photon as the difference between the Tagger time and the Start Counter time at the interaction point,
$t_{\rm event} - t_{\gamma}$. Since each event had several candidate photons, several %$\Delta t_{\rm\, TGPB}$
\re{$\Delta t_{\rm\, TBID}$}
values were available, which could be obtained from %the TGPB bank.
\re{information in the TBID bank}. Figure~\ref{Figure:Timing} (left) shows
an example distribution of the coincidence time, %$\Delta t_{\rm\,TGPB}$.
\re{$\Delta t_{\rm\,TBID}$}. The figure clearly shows the 2-ns
bunching of the photons that arrived at the target. In each event, the information on energy and timing,
$t_{\gamma}$, was written to the event's TAGR bank for all photons. The total number of photon candidates
per event was also available. {\it The} photon selection itself was performed by the CLAS offline software
in the cooking process. However, we applied a timing cut of %$\Delta t_{\rm\,TGPB} < 1$~ns in this analysis.
\re{$\Delta t_{\rm\,TBID} < 1$~ns} in this analysis.
\item Occasionally, events could have more than one candidate photon with %$|\Delta t_{\rm\,TGPB}|< 1$~ns.
\re{$|\Delta t_{\rm\,TBID}|< 1$~ns}.
In such cases, the photon selection could not be made based on their time information. The fraction of these
events was about 13\,\% in the g12~experiment. To prevent any ambiguity, only events with exactly \underline{one}
photon candidate in the \underline{same} RF bucket for all selected tracks %(TAGR$[\,].${\sc ngrf} = 1) were
\re{(nGammaRF() = 1)} were
considered in this analysis. In addition, we also ensured that the selected photon was the same for all reconstructed
tracks %(TAGR$[\,].${\sc tagrid}
\re{({\sc tagr\_id} equal for all tracks)}. Figure~\ref{Figure:Timing} (right) shows an example of the
coincidence-time distribution for the selected initial photon (one entry per event) after PID cuts.
\end{enumerate}
% -------------------------------------------------------------------
\subsubsection{Proton and Pion Selection}
\label{Subsubsection:betaCuts}
% -------------------------------------------------------------------
\begin{figure}[t!]
\includegraphics[width=0.325\textwidth,height=0.2\textheight]{g12_figures/oneD_proton_beta.pdf}
\hfill
\includegraphics[width=0.325\textwidth,height=0.2\textheight]{g12_figures/oneD_pip_beta.pdf}
\hfill
\includegraphics[width=0.325\textwidth,height=0.2\textheight]{g12_figures/oneD_pim_beta.pdf}
\caption{\it Distributions of $\Delta\beta\,=\,\beta_{\rm\,c}\,-\,\beta_{\rm\,m}$ for protons (left) as well as
for the $\pi^+$ (middle) and for the $\pi^-$ (right) from the g12 experiment (full statistics used in our
FSU analyses, Period 2 (see Table~\ref{Table:TriggerConfigurations})). The quantity $\beta_{\rm\,c}$ was
calculated based on the particle's PDG mass~\cite{Olive:2016xmw}. Events in the center peak were
selected after applying a $|\beta_{\rm\,c} \,-\, \beta_{\rm\,m}| \leq 3\sigma$~cut. See text for more details.}
\label{Figure:betaDiff}
\end{figure}
The photon energy for each event was selected according to the procedure outlined in
Section~\ref{Subsubsection:PhotonSelection}. In the next step, the identification of the final-state particles,
proton, $\pi^+$, and $\pi^-$, was needed. As mentioned in Section~\ref{Subsection:GeneralEventSelection},
we initially used particle ID information from the PART bank and selected those events which belonged to the
topologies of our interest (Table~\ref{Table:Topologies}). For a more refined selection of the particles, we used
the information on the measured and calculated $\beta$~values of each particle. The TBID bank contained the
CLAS-measured momentum of a particle; a theoretical value, $\beta_{\rm\,c}$, for that particle could then be
calculated from this measured momentum and an assumed mass. The $\beta_{c}$~values for all possible particle
types were compared to the CLAS-measured empirical $\beta_{\rm\,m}=\frac{v}{c}$~value. Particle identification
then proceeded by choosing the calculated $\beta_{\rm\,c}$ closest to the measured $\beta_{\rm\,m}$.
Figure~\ref{Figure:betaDiff} shows the differences, $\Delta\beta = \beta_{\rm\,c} - \beta_{\rm\,m}$ for the different
final-state particles based on the full g12 statistics that we used in our FSU analyses, (Period 2, see
Table~\ref{Table:TriggerConfigurations}). Assuming a PDG~mass~$m$ for the particle~\cite{Olive:2016xmw},
$\Delta\beta$ was given by:
\begin{equation}
\Delta \beta \,=\, \beta_{\rm\,c} \,-\, \beta_{\rm\,m} \,=\, \sqrt{
\frac{p^{2}}{m^{2} \,+\, p^{2}} } \,-\, \beta_{\rm\,m}\,.
\end{equation}
%
The prominent peaks around $\Delta\beta=0$ shown in Figure~\ref{Figure:betaDiff} correspond to the particles
of interest. It can be seen in the figures that the $\Delta\beta$~distributions for the pions are slightly broader
than for the proton and long tails including a prominent enhancement on either side of the central peak are visible.
When the PART bank was created during the track reconstruction, electrons were not separated from pions. The
additional features in the $\Delta\beta$~distributions for the pions represent these electrons which need to be
filtered out. To identify the protons and pions, loose cuts on $|\beta_{\rm\,c} - \beta_{\rm\,m}|$ were applied. The
cut values were determined by fitting the main peak around $\Delta\beta=0$ with a Gaussian.
Figure~\ref{Figure:beta_mom_diff} shows the measured momentum, $p$, versus the measured~$\beta_{\rm\,m}$
for protons and pions before (left) and after (right) applying the $|\beta_{\rm\,c} - \beta_{\rm\,m}| < 3\sigma$~cut.
The bands for the pions and protons (lower band) are clearly visible.\par\smallskip
Although the $\Delta\beta$-PID cuts significantly help avoid misidentified tracks in the selected event sample,
we applied only a loose $|\Delta\beta| < 3\sigma$ cut in our final event selection on either the proton or the
$\pi^+$ (no cut on the $\pi^-$). This allowed us to retain as many signal events as possible. The remaining
background caused by misidentified tracks did not cause structures under the signal in the relevant mass
distributions and was taken care of by our background-subtraction technique (see
Section~\ref{Subsection:Q-factor_method}). The loose cuts were also in line with an earlier CLAS analysis of the
$\omega$ and $\eta$~photoproduction cross sections~\cite{Williams:2009ab,Williams:2009yj,Williams:2007thesis}.
\begin{figure}[t!]
\includegraphics[width=0.48\textwidth,height=0.35\textheight]{g12_figures/twoD_deltabeta_before.pdf}
\hfill
\includegraphics[width=0.48\textwidth,height=0.35\textheight]{g12_figures/twoD_deltabeta_after.pdf}
\caption{\it Left: The measured $\beta_{\rm\,m}$ versus momentum on a logarithmic color scale. Note a thin
horizontal line close to one for electrons, and the broad stripes for pions (top) followed by protons (bottom).
Right: The measured $\beta_{\rm\,m}$ versus momentum after applying the $3\sigma$~cut based on the difference
$\Delta\beta\,=\,\beta_{\rm\,c}\,-\,\beta_{\rm\,m}$. Clean pion and proton bands are visible. These figures were
made using the full statistics used in our FSU analyses, (Period~2, see Table~\ref{Table:TriggerConfigurations}).}
\label{Figure:beta_mom_diff}
\end{figure}
% ------------------------------------------------------------------
\subsection{Vertex Cut}
% ------------------------------------------------------------------
In the g12~experiment, the liquid hydrogen target was not located at the center of CLAS but moved 90~cm
upstream to increase the angular resolution for heavier-meson photoproduction in the forward direction.
The target itself was 40~cm long and 2~cm in diameter. Therefore, a $z$-vertex cut of
$-110 < z~{\rm vertex} < -70$~cm was applied; the full $z$-vertex distribution is shown in
Fig.~\ref{Figure:VertexDistributions}.\par\smallskip
\begin{figure}[t!]
\centering
\includegraphics[width=0.6\textwidth,height=0.35\textheight]{g12_figures/zVertex_before.pdf}
\caption{\it The $z$-vertex distribution (axis along the beam line) of all reconstructed particles
we used in our FSU analyses. The shape of the liquid hydrogen target is clearly visible. The small
enhancement at about $z = -63$~cm originates from the exit window of the vaccuum chamber.}
\label{Figure:VertexDistributions}
\end{figure}
% ------------------------------------------------------------------
\subsection{Introduction to Kinematic Fitting}
\label{Subsection:KinematicFitting}
% ------------------------------------------------------------------
The 4-vectors of the final-state particles were determined in the {\it cooking} or reconstruction phase.
Kinematic fitting~\cite{CLAS-NOTE_2003-017} slightly modified these {\it raw} 4-vectors by imposing
energy-momentum conservation on the event as a physical constraint. In a brief summary, all measured
components of the Lorentz 4-vectors (the magnitude of the momentum as well as the two angles used
in the drift-chamber reconstruction -- $p,~\lambda,~\phi$, respectively) in addition to the initial photon
energy were modified within their given uncertainties until the event satisfied energy-momentum conservation
exactly. The determination of the correct uncertainties (or covariance matrix) was important in this fitting
procedure. The kinematically-fitted event had then several quantities which could be used to inspect the
quality of the kinematic fitting: a pull value for each measured quantity and an overall $\chi^2$~value. The
latter could be converted to a confidence-level (CL) value to judge the goodness-of-fit. The pull distributions
were used to evaluate the initial uncertainty estimation and to study systematics. It turned out that kinematic
fitting provided an effective tool to verify kinematic corrections, e.\,g. momentum corrections.
% -------------------------------------------------------------------------
\subsubsection{Confidence Level}
% -------------------------------------------------------------------------
To check the {\it goodness-of-fit} or the agreement between the fit hypothesis and the data, the fit
$\chi^2$~value was used. The corresponding CL~value was defined as:
\begin{equation}
CL \,=\, \int_{\chi^2}^{\infty} \,f(z;n) \,dz\,,
\label{Equation:ConfidenceLevel}
\end{equation}
where $f(z;n)$ was the $\chi^2$~probability density function with $n$~degrees of freedom. It denoted the
probability distribution for certain external constraints, e.\,g. energy-momentum conservation or also a
missing-particle constraint. In the ideal case where all events satisfied the fit hypothesis and the measured
quantities were all independent and had only statistical uncertainties, the confidence-level distribution would
be flat from (0,\,1]. However, the real data had a confidence-level distribution which showed a peak near
zero (Fig.~\ref{Figure:CL_Pull}, left side). This peak contained events which did not satisfy the imposed
constraints. These events could be hadronic background events, poorly-reconstructed events with significant
systematic uncertainties, or events with misidentified particles. A cut on small CL~values eliminated the
majority of these background events while only a relatively small amount of good data was lost.
% ------------------------------------------------------------------------
\subsubsection{Pulls}
% ------------------------------------------------------------------------
A {\it pull value} is a measure of how much and in what direction the kinematic fitter has to alter a measured
parameter -- or to {\it pull} at it -- in order to make the event fulfill the imposed constraint. All three fit
parameters for every detected final-state particle had pull distributions. The pull value for the $i^{\rm \,th}$
fit parameter was given by:
\begin{equation}
z_{i} \,=\, \frac{\epsilon_{i}}{\sigma ( \epsilon_{i} ) }\,,
\end{equation}
where $\epsilon_{i}\, =\, \eta_{i}\, -\, y_{i}$ was the difference between the fitted value, $\eta_{i}$, and the
measured value, $y_{i}$. The quantity $\sigma$ represents the standard deviation of the parameter~$\epsilon_{i}$.
Therefore, the $i^{\rm \,th}$ pull can be written as:
\begin{equation}
z_{i} \,=\, \frac{\eta_{i} - y_{i}}{\sqrt{\sigma^{2}(\eta_{i}) - \sigma^{2}(y_{i})}}\,.
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{eps/cl_eloss_momc_Sit4.pdf}\hfill
\includegraphics[width=0.48\textwidth,height=0.29\textwidth]{eps/pull_eloss_momc_Sit4.pdf}
\caption{\it Example of results from kinematic fitting. Energy and momentum conservation was imposed
on Topology~4 in $\gamma p \to p\,\pi^+\pi^-$. Left: A confidence-level distribution. It peaks
toward zero but flattens out toward one. Right: Pull distribution of the incoming photon energy.
Ideally, such a distribution is Gaussian in shape, centered at the origin ($\mu = 0$) with a width of
one ($\sigma = 1$).}
\label{Figure:CL_Pull}
\end{figure}
The reaction $\gamma p\to p\,\pi^{+}\pi^{-}$ (using Topology 4, see Table~\ref{Table:Topologies}) had
three detected final-state particles: proton, $\pi^{+}$, and $\pi^{-}$. Since the reconstruction of each particle
was based on three parameters, this topology had ten pull distributions including a pull for the initial photon
energy. In the ideal case that the error matrix of these parameters was correctly determined and all remaining
systematic uncertainties were negligible, the pull distributions would be Gaussian in shape with a width of one
($\sigma = 1$) and centered at zero ($\mu = 0$); such an example is shown in Figure~\ref{Figure:CL_Pull}
(right side). A systematic problem with the data in the quantity $\eta_{i}$ would be observed as an overall shift
away from zero. Similarly, if the uncertainties of $\eta_{i}$ were consistently (overestimated) underestimated, then
the corresponding pull distribution would be too (narrow) broad, and the slope of the CL~distribution toward
$CL = 1$ would be (positive) negative. The uncertainties of the measured parameters could be corrected from
the pull distributions in an iterative procedure.\par\smallskip
In our analysis, kinematic fitting served as an effective tool to double-check the final-state corrections
approved in Ref.~\cite{CLAS-NOTE_2017-002}. We used Topology 4 (all final-state particles detected) for this.
The final mean and $\sigma$ values of Gaussian fits to our g12 pull distributions (after all corrections) are
shown in Table~\ref{Table:pull-ELoss-MCor}. The distributions themselves are presented in
Fig.~\ref{Figure:pull-ELoss-MCor-Top4},~\ref{Figure:pull-ELoss-MCor-Top5},
and~\ref{Figure:pull-ELoss-MCor-Top6}.
% ------------------------------------------------------------------
\subsection{Kinematic Corrections}
\label{Subsection:Corrections}
% ------------------------------------------------------------------
The following subsections briefly summarize some of the standard CLAS corrections. We only give a brief
description here (in the order of application) without showing the actual effect on the data. The latter was
discussed in Ref.~\cite{CLAS-NOTE_2017-002} and has been approved by the collaboration.
\begin{table}[t!]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ | c | r | r | r || r | r | r || r | r | r || r | }
\hline
& \multicolumn{3}{c||}{proton} & \multicolumn{3}{c||}{$\pi^{+}~(K^+)$}
& \multicolumn{3}{c||}{$\pi^{-}~(K^-)$} & \multicolumn{1}{c|}{$\gamma$}\\\cline{2-11}
& mom. & \multicolumn{1}{c|}{$\lambda$} & \multicolumn{1}{c||}{$\phi$}
& mom. & \multicolumn{1}{c|}{$\lambda$} & \multicolumn{1}{c||}{$\phi$}
& mom. & \multicolumn{1}{c|}{$\lambda$} & \multicolumn{1}{c||}{$\phi$} & \multicolumn{1}{c|}{E}\\\hline
\multicolumn{11}{c}{}\\
\multicolumn{11}{c}{\rb{CLAS-g12: $\gamma p\to p\,\pi^+\pi^-$}}\\\hline
$\bar{x}$ & 0.090 & $-0.044$ & $-0.001$ & 0.060 & $-0.001$ & $-0.016$ &
$-0.014$ & $-0.016$ & $-0.048$ & $-0.062$\\\hline
$\sigma$ & 1.159 & 0.970 & 1.136 & 1.048 & 1.009 & 1.089 &
1.057 & 1.013 & 1.118 & 1.136\\\hline
\multicolumn{11}{c}{}\\
\multicolumn{11}{c}{\rb{CLAS-g12: $\gamma p\to p\,\pi^+\pi^-\,(\pi^0)$}}\\\hline
$\bar{x}$ & 0.140 & 0.001 & $-0.211$ & $-0.150$ & $-0.023$ & $-0.192$ &
$-0.194$ & $-0.029$ & $-0.164$ & 0.190\\\hline
$\sigma$ & 1.167 & 1.182 & 1.173 & 1.193 & 1.178 & 1.161 &
1.194 & 1.179 & 1.143 & 1.209\\\hline
\multicolumn{11}{c}{}\\
\multicolumn{11}{c}{\rb{CLAS-g12: $\gamma p\to p\,K^+K^-$}}\\\hline
%$\bar{x}$ & $-0.003$ & $-0.046$ & $-0.041$ & $-0.048$ & 0.016 & $-0.045$ &
% $-0.070$ & $-0.005$ & $-0.043$ & 0.055\\\hline
$\sigma$ & 1.071 & 0.947 & 1.019 & 0.938 & 0.941 & 0.955 &
1.026 & 0.960 & 0.967 & 1.060\\\hline
\end{tabular}
\caption{\it Final mean, $\bar{x}$, and $\sigma$ values of Gaussian fits to our g12 pull distributions
after applying all corrections. Note that the values for $p\,\pi^+\pi^-\,(\pi^0)$ are based on distributions
which could not be perfect Gaussians owing to the missing-particle hypothesis.}
\label{Table:pull-ELoss-MCor}
\end{center}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\textwidth]{g12_figures/pull_2pion.pdf}
\caption{\it The g12 pull and CL distributions for the exclusive reaction $\gamma p\to p\,\pi^+\pi^-$
(full statistics of Period~2). A summary of the mean and $\sigma$ values of the fits can
also be found in Table~\ref{Table:pull-ELoss-MCor}.}
\label{Figure:pull-ELoss-MCor-Top4}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\textwidth]{g12_figures/pull_3pion.pdf}
\caption{\it The g12 pull and CL distributions for the reaction $\gamma p\to p\,\pi^+\pi^-\,(\pi^0)$
(full statistics of Period~2). Note that the pull distributions are not Gaussian over the full range owing
to the missing-particle hypothesis. The confidence-level distribution looks nicely flat, though. A summary
of the mean and $\sigma$~values of these fits can also be found in Table~\ref{Table:pull-ELoss-MCor}.
\bl{Resolution.}}
\label{Figure:pull-ELoss-MCor-Top5}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\textwidth,height=0.8\textheight]{g12_figures/pull_2kaon_data.pdf}
\caption{\it The g12 pull and CL distributions for the exclusive reaction $\gamma p\to p\,K^+K^-$ (full
statistics of Period~2). A summary of the mean and $\sigma$~values of the fits can also be found in
Table~\ref{Table:pull-ELoss-MCor}. \bl{The CL distribution is still rising toward one (positive slope)
which is correlated with several $\sigma$~values that are smaller than one. We will go through
an additional iteration of pull adjustments for the second round of this review.}}
\label{Figure:pull-ELoss-MCor-Top6}
\end{figure}
% ------------------------------------------------------------------------
\subsubsection{Tagger-Sag Correction}
% ------------------------------------------------------------------------
The energy of the incoming photons was determined by the Hall-B tagging system. It was observed in previous
experiments that a physical sagging of the holding structure supporting the E-counter scintillator bars could
be attributed to gravitational forces~\cite{Stepanyan:2006vv}. The consequence of this time-dependent
sagging was a misalignment of the scintillator bars which led to a small shift of the scattered electron's
energy~\cite{CLAS-NOTE_2009-030}. In the CLAS-g12 experiment, the tagger sag was taken into account
and corrected in the offline reconstruction code. No further photon energy correction was applied.
% ------------------------------------------------------------------
\subsubsection{Enery-Loss (ELoss) Correction}
\label{Subsubsection:Energy_Loss_Correction}
% ------------------------------------------------------------------
As charged particles traveled from the production vertex to the active components of the CLAS spectrometer,
they lost energy through inelastic scattering, atomic excitation or ionization when interacting with the target,
target walls, support structures, beam pipe, Start Counter, and the air gap between the Start Counter and the
Region~1 Drift Chambers. Therefore, the momentum reconstructed from the drift chambers was smaller than
the momentum of the particle at the production vertex. To account and correct for this, the 4-vectors of the
final-state particles were modified event-by-event using the ``ELoss'' package, which was developed for
charged particles moving through CLAS~\cite{CLAS-NOTE_2007-016}. This ELoss package determined the
lost momentum of each particle in the materials it had interacted with. In this procedure, the particle's
4-momentum -- as measured by the Region 1 Drift Chambers -- was used to track the particle back to the
reaction vertex in the target cell. The energy loss was then calculated based on the distance and the materials
it traversed. The corresponding 4-vector was corrected by multiplying an ELoss correction factor to the magnitude
of the momentum:
\begin{align}
\begin{split}
P_{\:{\rm particle}\,({\rm ELoss})} & \,=\, \eta_{\:\rm particle}\,\cdot P_{\:{\rm particle}\,({\rm CLAS})} \\
%P_{(\, \pi^{+}\, , \,{\rm ELoss}\,)} & \,=\, \eta_{\pi^+}\,\cdot P_{(\, \pi^+\, , \,{\rm CLAS}\,)}\, \\
%P_{(\, \pi^{-}\, , \,{\rm ELoss}\,)} & \,=\, \eta_{\pi^-}\,\cdot P_{(\, \pi^-\, , \,{\rm CLAS}\,)}\,, \\
\end{split}
\label{Equation:pim_mom_ELoss}
\end{align}
where $P_{\,x\,({\rm ELoss})}$ is the momentum of the particle $x$ after applying the energy-loss correction,
$P_{\,x\,({\rm CLAS})}$ is the raw momentum measured in CLAS and $x$ is either the proton, $\pi^{+}$, or $\pi^{-}$.
The parameters $\eta_{\,p}$,~$\eta_{\,\pi^{+}}$, and~$\eta_{\,\pi^{-}}$ are the ELoss correction factors which modified
the momentum by a few MeV, on average.
% ------------------------------------------------------------------------
\subsubsection{Momentum Corrections}
\label{Subsubsection:mom_cor}
% ------------------------------------------------------------------------
The CLAS-g12 experimental setup was not absolutely perfect. For this reason, corrections of a few~MeV
had to be determined and applied to the final-state particles' momenta to account for unknown variations
in the CLAS magnetic field (Torus Magnet) as well as inefficiencies and misalignments of the drift chambers.
As a matter of fact, the momenta of the tracks as measured by the drift chambers exhibited a systematic shift
within each sector as a function of the azimuthal angle $\phi$ of one of the tracks~\cite{CLAS-NOTE_2017-002}.
In our FSU analyses, we have followed the CLAS-approved procedure outlined in Ref.~\cite{CLAS-NOTE_2017-002}.
\clearpage
%-------------------------------------------------------------------------
\subsubsection{Bad or Malfunctioning Time-of-Flight Paddles}
%-------------------------------------------------------------------------
Some TOF paddles of the CLAS spectrometer were dead or malfunctioning during the g12 experiment. The
timing resolution of each paddle was investigated on a run-by-run basis to determine the stability throughout
the experiment. Reference~\cite{CLAS-NOTE_2017-002} contains the results of an extensive study on bad TOF
paddles in CLAS-g12. The list of identified bad paddles recommended to knock out was taken directly from
Table~19 of Ref.~\cite{CLAS-NOTE_2017-002} and is also given in Table~\ref{Table:bad_paddles} for convenience.
\par\medskip
\begin{table}[h!]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ | c | c |}
\hline
Sector Number & Bad TOF Paddles in CLAS-g12\\\hline
1 & 6, 25, 26, 35, 40, 41, 50, 56\\
2 & 2, 8, 18, 25, 27, 34, 35, 41, 44, 50, 54, 56\\
3 & 1, 11, 18, 32, 35, 40, 41, 56\\
4 & 8, 19, 41, 48\\
5 & 48\\
6 & 1, 5, 24, 33, 56\\\hline
\end{tabular}
\caption{\it The list of bad time-of-flight paddles recommended to knock out~\cite{CLAS-NOTE_2017-002}.}
\label{Table:bad_paddles}
\end{center}
\end{table}
% --------------------------------------------------------------------------
\subsection{Monte Carlo Simulations\label{Subsection:MC}}
% --------------------------------------------------------------------------
To extract the differential cross sections for the reactions (1) $\gamma p\to p\,\omega$, (2) $\gamma
p\to p\,\eta$, (3) $\gamma p\to K^0\,\Sigma^+$, and (4) $\gamma p\to p\,\phi$, we needed to apply
detector-acceptance corrections, where the latter accounted for the probability that an event of certain
kinematics would be detected and recorded (also called efficiency corrections). The performance of the
detector was simulated in GEANT3\,-\,based Monte-Carlo studies. We followed the steps outlined in
Ref.~\cite{CLAS-NOTE_2017-002} for generating events, digitization and smearing, as well as reconstruction.
\par\smallskip
The generated raw events were processed by {\sc{gsim}} to simulate the detector acceptance for each
propagated track from the event vertex through the GEANT3-modeled CLAS detector. The CLAS smearing
package known as {\sc{gpp}} then processed the output to reflect the resolution of the detector. Finally,
the {\sc{a1c}}~package was used to perform the {\it cooking}. We generated a total of 175~million
$\gamma p\to p\,\omega\to p\,\pi^+\pi^-\pi^0$ phase-space events for the whole range of incident-photon
energies, i.e. $1.1 < E_\gamma < 5.4$~GeV. We have also generated 11~million $\gamma p\to p\,\eta\to
p\,\pi^+\pi^-\pi^0$ and 40~million $\gamma p\to K^0\,\Sigma^+\to p\,\pi^+\pi^-\pi^0$ Monte Carlo
events. To guarantee phase-space (generated) events which are flat in cos\,$\theta_{\rm c.m.}^{\rm \,meson}$, we
chose a $t$-slope of {\it zero}.\par\smallskip
In this section, we show the quality of the simulated events by comparing various data distributions with
Monte Carlo events:
\begin{enumerate}
\item In the CLAS-g12 experiment, the 40-cm-long liquid-hydrogen target was pulled upstream by 90~cm
from the center of the CLAS detector. Figure~\ref{Figure:vertex_mc} compares the $z$-vertex distribution
for data and Monte Carlo events after applying our cut of $-110 < z~{\rm vertex} < -70$~cm:
$\gamma p\to p\,\omega$ (left) and $\gamma p\to K_S\,\Sigma^+$ (right). This figure shows that the
vertex distribution is very well modeled.
\begin{figure}[b!]
\centering
\includegraphics[width=0.49\textwidth,height=0.3\textheight]{./g12_figures/vertex_omega.pdf}\hfill
\includegraphics[width=0.49\textwidth,height=0.3\textheight]{./g12_figures/vertex_Kaon.pdf}
\caption{\it Left: The z-vertex distribution of $\gamma p\to p\,\omega$ events. The black line denotes the
data, the read line denotes the Monte Carlo distribution; good agreement is observed. These figures were
made using the full data statistics of 4.4~million events and an equal amount of Monte Carlo events after
applying our z-vertex cut of $-110 < z < -70$~cm. Right: The z-vertex distribution of
$\gamma p\to K_S\,\Sigma^+$~events.}
\label{Figure:vertex_mc}
\end{figure}
\begin{figure}[t!]
\subfloat{\includegraphics[width=0.50\textwidth,height=0.29\textheight]{./g12_figures/proton_theta.pdf}}
\subfloat{\includegraphics[width=0.50\textwidth,height=0.29\textheight]{./g12_figures/proton_phi.pdf}}\hfill
\subfloat{\includegraphics[width=0.50\textwidth,height=0.29\textheight]{./g12_figures/pion_theta.pdf}}
\subfloat{\includegraphics[width=0.50\textwidth,height=0.29\textheight]{./g12_figures/pion_phi.pdf}}
\caption{\it The polar ($\theta$) and azimuthal ($\phi$) angle distributions of the proton (top row) and of the
$\pi^-$ (bottom row) in the reaction $\gamma p\to p\,\omega$ for data (black line) and Monte Carlo events
(red line). These figures were made using the full data statistics of 4.4 million events and the same number of
Monte Carlo events. The $\theta_{\pi^-},~\phi_{\pi^-} $ and $\phi_p$ distributions are in very good agreement.}
\label{Figure:angle_distribution}
\end{figure}
\item Figure~\ref{Figure:angle_distribution} shows the distributions of $\theta$~(polar angle) and $\phi$~(azimuthal
angle) for the proton (top) and for the $\pi^-$ (bottom). The data and Monte Carlo distributions match well
for the azimuthal angles of the proton and the $\pi^-$ as well as for the polar angle of the pion. However,
the MC polar angle of the proton, $\theta_p$, does not agree very well with the data. This is reasonable
because our Monte Carlo events do not contain any reaction dynamics (simple generation of phase space
events), but the distribution covers the same polar-angle range.
\item We also checked all the signal distributions (peaks for $\omega,~\eta$, and~$K_S$) to see if our Monte Carlo
mass resolution matches the real detector resolution. Figure~\ref{Figure:signal_mass} shows invariant-mass
distributions for both data (black line) and Monte Carlo (red line) events. Since the mass resolution is slightly
energy dependent, we compare data and Monte Carlo for $E_\gamma < 3$~GeV (left) and $E_\gamma > 3$~GeV
(right). It is observed in this figure that the MC resolution is in reasonable agreement with the actual detector
resolution.\par\smallskip
{\addtolength{\extrarowheight}{5pt}
\begin{center}
\begin{tabular}{| l | r | r | c | c |}
\hline
& \multicolumn{4}{c|}{Resolution (Gaussian $\sigma$ in [\,MeV\,])}\\\cline{2-5}
Reaction & \multicolumn{2}{c|}{Low Energy} & \multicolumn{2}{c|}{High Energy}\\\cline{2-5}
& Data & MC & Data & MC\\\hline
$\gamma p\to p\,\omega$ & 7.68 & 7.98 & 12.0 & 12.0\\
$\gamma p\to p\,\eta$ & 6.5 & 6.9 & 7.2 & 7.1\\
$\gamma p\to p\,\phi$ & & & &\\\hline
& \multicolumn{2}{c|}{$K_S$ Peak} & \multicolumn{2}{c|}{$\Sigma$ Peak}\\\cline{2-5}
\rb{$\gamma p\to K_S\,\Sigma^+$} & 5.4 & 4.4 & 6.5 & 6.2\\\hline
\end{tabular}
\end{center}}
\item Figure~\ref{Figure:cospimz} shows the distributions for the cos\,$\theta^{\,\pi^-}_{\rm c.m.}$ versus $z$~vertex
for $\gamma p\to p\,\omega$ data and Monte Carlo events; the distributions are almost identical. In the
very backward region of the target, an angle range of only about $-0.6 < cos\,\theta^{\pi^-}_{c.m.} < 0.8$ is
covered, whereas $-0.8 < cos\,\theta^{\pi^-}_{c.m.} < 0.8$ is covered in the very forward region.\par\smallskip
% Figure~\ref{Figure:xyVertex} shows the distribution of $x$- vs. y-vertex with our cut superimposed. The very
%good agreement provides confidence in our detector simulations.
\begin{table}[t!]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ | c | r | r | r || r | r | r || r | r | r || r | }
\hline
& \multicolumn{3}{c||}{proton} & \multicolumn{3}{c||}{$\pi^{+}$}
& \multicolumn{3}{c||}{$\pi^{-}$} & \multicolumn{1}{c|}{$\gamma$}\\\cline{2-11}
& mom. & \multicolumn{1}{c|}{$\lambda$} & \multicolumn{1}{c||}{$\phi$}
& mom. & \multicolumn{1}{c|}{$\lambda$} & \multicolumn{1}{c||}{$\phi$}
& mom. & \multicolumn{1}{c|}{$\lambda$} & \multicolumn{1}{c||}{$\phi$} & \multicolumn{1}{c|}{E}\\\hline
\multicolumn{11}{c}{}\\
\multicolumn{11}{c}{\rb{Monte Carlo: $\gamma p\to p\,\pi^+\pi^-$}}\\\hline
$\bar{x}$ & 0.023 & 0.003 & 0.042 & 0.053 & $-0.002$ & 0.041 &
0.053 & 0.004 & 0.040 & $-0.056$\\\hline
$\sigma$ & 1.117 & 1.045 & 1.010 & 1.017 & 1.028 & 0.997 &
1.018 & 1.048 & 0.994 & 1.102\\\hline
\multicolumn{11}{c}{}\\
\multicolumn{11}{c}{\rb{Monte Carlo: $\gamma p\to p\,\pi^+\pi^-\,(\pi^0)$}}\\\hline
$\bar{x}$ & 0.040 & 0.018 & 0.024 & 0.027 & 0.000 & 0.024 &
0.022 & 0.004 & 0.030 & $-0.052$\\\hline
$\sigma$ & 1.078 & 1.054 & 1.081 & 1.045 & 1.056 & 1.015 &
1.055 & 1.056 & 1.004 & 1.086\\\hline
\multicolumn{11}{c}{}\\
\multicolumn{11}{c}{\rb{Monte Carlo: $\gamma p\to p\,K^+K^-$}}\\\hline
$\bar{x}$ & $-0.002$ & 0.005 & $-0.121$ & 0.077 & 0.017 & $-0.128$ &
0.084 & 0.002 & $-0.148$ & $-0.112$\\\hline
$\sigma$ & & & & & & & & & & \\\hline
\end{tabular}
\caption{\it Final mean ($\bar{x}$) and $\sigma$ values of Gaussian fits to our g12 pull distributions
after applying all corrections. Note that the values for $p\,\pi^+\pi^-\,(\pi^0)$ are based on distributions
which cannot be perfect Gaussians owing to the missing-particle hypothesis. \bl{Similar to the data, the
CL~distribution for the $\gamma p\to p\,K^+K^-$~hypothesis is still rising toward one (positive slope)
which is correlated with all MC $\sigma$~values currently being smaller than one. We will go through
an additional iteration of pull adjustments for the second round of this review.}}
\label{Table:pull-ELoss-MCor-MC}
\end{center}
\end{table}
\item The quality of the kinematic fitting for the Monte Carlo events is shown in the pull and confidence-level (CL)
distributions for the reaction $\gamma p\to p\pi^+\pi^-$~(Fig.~\ref{Figure:pull_cl_2pi}) and for the reaction
$\gamma p\to p\,\omega\to p\pi^+\pi^-\pi^0$~(Fig.~\ref{Figure:pull_cl_3pi}). A summary of the mean and
$\sigma$~values is given in Table~\ref{Table:pull-ELoss-MCor-MC}. Recall that each of these distributions
should have a mean of {\it zero} and width of {\it one.} The agreement of the extracted values with these ideal
values is very good. The CL distributions are flat toward {\it one}.\par\smallskip
To further check the quality of the confidence
level in all kinematic regions, we considered the normalized slope of the distribution:
\begin{equation}
\bar{a}\,=\,\frac{a}{a/2\,+\,b}\,,
\end{equation}
where $a$ is the slope and $b$ is the intercept obtained by fitting the confidence-level distribution to a linear
function. Figure~\ref{Figure:NormalizedSlopesExamples} shows examples of confidence-level distributions and
their respective normalized slopes. If the errors are overestimated (underestimated), then the confidence-level
distribution will have a positive (negative) slope. In line with the procedure outlined in
Ref.~\cite{Williams:2007thesis}, we would consider the covariance matrix to be aceptable if all kinematic
regions yielded normalized slopes in the range $[-0.5,0.5]$. Figure~\ref{Figure:NormalizedSlopes} shows
the normalized slopes extracted in $(p,\,{\rm cos}\,\theta)$~bins for the proton and the $\pi^-$. Notice
that all kinematic regions (excluding edge bins with low statistics) have $|\bar{a}| < 0.5$. Thus, we conclude
that the covariance matrix is acceptable.
\end{enumerate}
\begin{figure}[h!]
\centering
\includegraphics[width=0.49\textwidth,height=0.26\textheight]{./g12_figures/omega_mass_low.pdf}\hfill
\includegraphics[width=0.49\textwidth,height=0.26\textheight]{./g12_figures/omega_mass_high.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.26\textheight]{./g12_figures/eta_mass_low.pdf}\hfill
\includegraphics[width=0.49\textwidth,height=0.26\textheight]{./g12_figures/eta_mass_high.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.26\textheight]{./g12_figures/kaon_mass_low.pdf}\hfill
\includegraphics[width=0.49\textwidth,height=0.26\textheight]{./g12_figures/sigma_mass_low.pdf}
\caption{\it Invariant mass (signal) distributions for data (black line) and Monte Carlo (red line). The left
distributions are for $E_\gamma < 3.0$~GeV, the right distibutions are for $E_\gamma > 3.0$~GeV. Top row:
The $M_{\pi^+\pi^-\pi^0}$~distribution showing the $\omega$~meson. Middle row: The $M_{\pi^+\pi^-\pi^0}$
distribution showing the $\eta$~meson. Bottom row: The $M_{\pi^+\pi^-}$~distribution showing the $K_S$
signal (left) and the $M_{p\pi^0}$~distribution showing the $\Sigma$~signal (right). The overall agreement
between the data and Monte Carlo distributions indicates that the GEANT simulations model the resolution
of the actual detector reasonably well.}
\label{Figure:signal_mass}
\end{figure}
\begin{figure}[t!]
\subfloat{\includegraphics[width=0.48\textwidth,height=0.38\textheight]{./g12_figures/cospimz_data.pdf}}\hfill
\subfloat{\includegraphics[width=0.48\textwidth,height=0.38\textheight]{./g12_figures/cospimz_mc.pdf}}
\caption{\it The z-vertex vs. cos\,$\theta^{\pi^-}_{c.m.}$~distributions using a logarithmic color scale for data
(left) and Monte Carlo events (right); the distributions are very similar. In the very backward region of the target,
an angle range of only about $-0.6 < cos\,\theta^{\pi^-}_{c.m.} < 0.8$ is covered,
whereas $-0.8 < cos\,\theta^{\pi^-}_{c.m.} < 0.8$ is covered in the very forward region.}
\label{Figure:cospimz}
\end{figure}
\begin{figure}[t!]
\subfloat{\includegraphics[width=0.50\textwidth,height=0.34\textheight]{./g12_figures/proton_cl_fit.pdf}}
\subfloat{\includegraphics[width=0.50\textwidth,height=0.34\textheight]{./g12_figures/pion_cl_fit.pdf}}\hfill
\caption{Examples of normalized slopes from confidence-level distributions for the proton (left) and for the $\pi^-$
(right): Normalized slopes have been extracted by fitting the distributions in the range (0.5, 1) to a linear function.}
\label{Figure:NormalizedSlopesExamples}
\end{figure}
\begin{figure}[b!]
\centering
\includegraphics[width=1.0\textwidth,height=1.0\textwidth]{./g12_figures/2pi_pull_MC.pdf}
\caption{\it Monte Carlo (reaction: $\gamma p\to p\,\omega\to p\,\pi^+\pi^-$) pull and confidence-level
distributions for the four-constraint fit to $p\,\pi^+\pi^-$ (check for energy and momentum conservation,
no mass constraint) along with the mean and $\sigma$ values of the fits. A summary of the mean and
$\sigma$~values of these fits (for data and Monte Carlo) can also be found in Table~\ref{Table:pull-ELoss-MCor-MC}.}
\label{Figure:pull_cl_2pi}
\end{figure}
\begin{figure}[b!]
\centering
\includegraphics[width=1.0\textwidth,height=1.0\textwidth]{./g12_figures/pull_cl_MC.pdf}
\caption{\it Monte Carlo (reaction: $\gamma p\to p\,\omega\to p\,\pi^+\pi^-\pi^0$) pull and confidence-level
distributions for the one-constraint fit to $p\,\pi^+\pi^-\,(\pi^0)$ (no $\omega$-mass constraint) along with
the mean and $\sigma$ values of the fits. Note that the pull distributions are not Gaussian over the full range
owing to the missing-particle hypothesis. A summary of the mean and σ values of these fits (for data and
Monte Carlo) can also be found in Table~\ref{Table:pull-ELoss-MCor-MC}.}
\label{Figure:pull_cl_3pi}
\end{figure}
\begin{figure}[b!]
\subfloat{\includegraphics[width=0.50\textwidth,height=0.36\textheight]{./g12_figures/proton_slope.pdf}}\hfill
\subfloat{\includegraphics[width=0.50\textwidth,height=0.36\textheight]{./g12_figures/proton_slope_mc.pdf}}\\
\subfloat{\includegraphics[width=0.50\textwidth,height=0.36\textheight]{./g12_figures/pim_slope.pdf}}\hfill
\subfloat{\includegraphics[width=0.50\textwidth,height=0.36\textheight]{./g12_figures/pim_slope_mc.pdf}}
\caption{\it Confidence-Level Checks. Normalized confidence-level slopes presented in cos\,$\theta$ versus
p [GeV/c] distributions for the proton (top row) and for the $\pi^-$ (bottom row). The results for the g12-data
are shown on the left and for Monte Carlo on the right. Notice that - excluding edge bins with low statistics
- all kinematic regions have $|\bar{a}| < 0.5$.}
\label{Figure:NormalizedSlopes}
\end{figure}
\clearpage
% -------------------------------------------------------------------------
\subsubsection{Trigger Simulation}
% -------------------------------------------------------------------------
To simulate the trigger conditions for our g12~data, we used the same technique that was developed for the
measurement of the $\omega$~and $\eta$~cross sections in the g11a
experiment~\cite{Williams:2009ab,Williams:2009yj}. The procedure is outlined in Ref.~\cite{CLAS-NOTE_2006-017}.
This technique used the \textit{trigger word} or \textit{trigger bit}, which was written into the BOS data during the
cooking. Reminder: The trigger conditions for the data that we used are described on Page 5 of
Ref.~\cite{CLAS-NOTE_2017-002}. In summary, the recorded events had:
\begin{enumerate}
\item Either three charged time-of-flight hits in three different sectors (three-sector events),
\item Or two hits in different sectors (two-sector events), in combination with at least one photon in the
beam bucket whose energy was above 3.6~GeV. The term ``beam bucket'' refers to all photons that were
detected during the life time of the trigger (detector).
\end{enumerate}
Therefore, to simulate the trigger conditions in the Monte Carlo events, two pieces of information were needed:
\begin{enumerate}
\item The efficiency of the trigger as a function of particle type, momentum, and detector position (trigger
efficiency map).
\item The probability for having at least one photon with $E_{\gamma} > 3.6$~GeV (two-sector events).
\end{enumerate}
\begin{figure}[b!]
\includegraphics[width=0.49\textwidth]{./g12_figures/two_sector.pdf}\hfill
\includegraphics[width=0.49\textwidth]{./g12_figures/three_sector.pdf}
\caption{\it The distribution of events as a function of beam energy. The left figure shows the distribution
for two-sector events. It clearly shows a discontinuity at 3.6~GeV. The right figure shows the distribution
for three-sector events. The distribution is smooth since there is no additional photon-energy requirement.}
\label{Figure:sector_study}
\end{figure}
(1) The trigger efficiency map was derived using $\gamma p\to p\,\pi^+\pi^-$ events and required all three
tracks to be detected in three different sectors. If the trigger were 100\,\% efficient, then all three detected
particles would also be recorded in the trigger word, i.e. contributed to the trigger decision. However, if the
trigger were not 100\,\%, an event with three charged tracks would still be reconstructed, although one of the
tracks would not fire the trigger (and be recorded in the trigger word). Therefore, it was possible to build a map
of the trigger efficiency for each particle type as a function of sector, time-of-flight paddle, and azimuthal angle.
The trigger efficiency map for the $\pi^-,~\pi^+$, and the proton are shown in Fig.~\ref{Figure:trigger_map_pim},
\ref{Figure:trigger_map_pip}, and~\ref{Figure:trigger_map_proton}.\par\smallskip
\begin{figure}[b!]
\centering
\includegraphics[width=0.9\textwidth,height=0.47\textheight]{./g12_figures/two_sector_ratio.pdf}
\caption{\it The ratio of two-sector and three-sector events. The discontinuity at 3.6~GeV is
an effect of the trigger condition. The ratios are flat as expected because the physics of using an unpolarized
beam must be independent of the azimuthal angle. By fitting these ratio distributions below and above 3.6~GeV,
we studied the probability for tow-sector events of having at least one photon in the beam bucket with
$E_{\gamma} > 3.6$~GeV. This probability was determined to be 0.51.}
\label{Figure:sector_study_ratio}
\end{figure}
\begin{figure}[t!]
\begin{tabular}{cc}
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pim_1.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pim_2.pdf}\\
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pim_3.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pim_4.pdf}\\
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pim_5.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pim_6.pdf}
\end{tabular}
\caption{\it Trigger efficiency map for the $\pi^-$ as a function of sector, tof-paddle number,
and azimuthal angle.}
\label{Figure:trigger_map_pim}
\end{figure}
\begin{figure}[t!]
\begin{tabular}{cc}
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pip_1.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pip_2.pdf}\\
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pip_3.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pip_4.pdf}\\
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pip_5.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/pip_6.pdf}
\end{tabular}
\caption{\it Trigger efficiency map for the $\pi^+$ as a function of sector, tof-paddle number,
and azimuthal angle.}
\label{Figure:trigger_map_pip}
\end{figure}
\begin{figure}[t!]
\begin{tabular}{cc}
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/proton_1.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/proton_2.pdf}\\
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/proton_3.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/proton_4.pdf}\\
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/proton_5.pdf} &
\includegraphics[width=0.48\textwidth,height=0.3\textheight]{./g12_figures/proton_6.pdf}
\end{tabular}
\caption{\it Trigger efficiency map for the proton as a function of sector, tof-paddle number,
and azimuthal angle.}
\label{Figure:trigger_map_proton}
\end{figure}
(2) The probability for two-sector events of having at least one photon with $E_{\gamma} > 3.6$~GeV in the beam
bucket could be determined by comparing energy-dependent intensity distributions of two-sector and three-sector
events. These distributions are shown in Fig.~\ref{Figure:sector_study}. A discontinuity at about 3.6~GeV is clearly
observed in the top distribution (two-sector events) due to the additional photon-energy requirement. Further
structures can be seen around 3~GeV and 4.4~GeV owing to broken tagger scintillators. On the other hand, the
bottom figure shows a smooth distribution for three-sector events because events were recorded independent
of their photon energy.\par\smallskip
\noindent
Figure~\ref{Figure:sector_study_ratio} shows the ratio of two-sector events and three-sector events. Since the
physics for using an unpolarized beam is independent of the azimuthal angle, we expect the ratio to be flat.
And we clearly see two flat distributions that disconnect at about 3.6~GeV. By fitting the two plateaus using a
zeroth-order polynomial below and above 3.6~GeV, we concluded that the probability for two-sector events
of having at least one photon with an energy above 3.6~GeV is about 0.51.\par\smallskip
After building the efficiency map and determining the probability for having at least one photon with an energy
above 3.6~GeV for two-sector events, we simulated the Monte Carlo events using the following steps:
\begin{enumerate}
\item The efficiency map was based on events that had all three particles in different sectors. Therefore, we cut
out events if two particles ended up in the same sector (for both data and MC events).
\item For each event, we generated three random numbers between 0 and 1 for the three final-state particles,
denoted by $R\,_p$, $R\,_{\pi^+}$, and $R\,_{\pi^-}$.
\begin{enumerate}
\item Denoting the trigger efficiency for each particle $P\,_p$, $P\,_{\pi^+}$, and $P_{\pi^-}$, then the particles
were considered to fire the trigger if the generated random number were smaller than the efficiencies. For
example, we considered the proton to fire the trigger if $R_p < P_p$.
\item If all particles fired the trigger, we kept the Monte Carlo event no matter what the photon energy was.
\item If only two particles fired the trigger and the photon energy was above 3.6~GeV, we kept the event.
\item If only two particles fired the trigger and the photon energy was below 3.6~GeV, then we generated
another random number, $R_{\,\rm tagger}$ . If $R_{\,\rm tagger} < 0.51$, then we kept the event. Otherwise,
if If $R_{\,\rm tagger} > 0.51$, the Monte Carlo event was discarded.
\end{enumerate}
\item If no particle or only one particle fired the trigger, then the Monte Carlo event was discarded.
\end{enumerate}
\bl{Anything else about the trigger simulation?}
\clearpage
% --------------------------------------------------------------------------
\subsection{\texorpdfstring{Angular Distribution of the Undetected $\pi^0$}
{Angular Distribution of the Undetected Neutral pion}}
% --------------------------------------------------------------------------
\subsubsection*{\texorpdfstring{The cos\,$\theta^{\,\pi^0}_{\rm c.m.}$ Distribution}
{The Angular Distribution}}
The channel $\gamma p\to p\,\pi^+\pi^-$ has a significantly larger cross section than
$\gamma p\to p\,\pi^+\pi^-\,(\pi^0)$. This fact, coupled with the relatively small difference in the missing
masses of the two channels, makes $p\,\pi^+\pi^-$~leakage into the $p\,\pi^+\pi^-\,(\pi^0)$ sample a cause
for concern. In this section, we consider the possibility of $p\,\pi^+\pi^-$~leakage resulting from selecting
the wrong photon.\par\smallskip
If the incorrect photon has a higher energy than the correct one, the extra energy will create a fake $\pi^0$ that
will move along the beam direction. Consider a $\gamma p\to p\,\pi^+\pi^-$~event that was produced in the
detector. Our analysis procedure will attempt to reconstruct a $\pi^0$ from the missing momentum,
$\vec{p}_{\rm \,miss}$. Since the event produced was actually a $p\,\pi^+\pi^-$~event, the missing transverse
momentum measured should be approximately zero, regardless of whether the correct photon has been found.
Thus, the momentum vector of the reconstructed~$\pi^0$ must point (approximately) along the beam direction:
$\vec{p}_{\rm \,miss} \approx \pm |\vec{p}_{\rm \,miss}|\,\hat{z}$.\par\smallskip
Therefore, we expect any leakage from the $\gamma p\to p\,\pi^+\pi^-$ channel, due to an incorrect photon
selection, to result in an excess of events in the very forward direction with cos\,$\theta^{\,\pi^0}_{\rm c.m.} \approx
+1$. Figure~\ref{Figure:cospi0} clearly shows a pronounced excess of events in the very forward direction. Therefore,
we cut out all events with cos\,$\theta^{\,\pi^0}_{\rm c.m.} > 0.99$.
\begin{figure}[t!]
\subfloat{\includegraphics[width=0.47\textwidth,height=0.25\textheight]{g12_figures/cospi0.pdf}}\hfill
\subfloat{\includegraphics[width=0.47\textwidth,height=0.25\textheight]{g12_figures/cospi0_v2.pdf}}
\caption{\it Left: The cos\,$\theta^{\,\pi^0}_{\rm c.m.}$ distribution of all 18~million $\gamma p\to
p\,\pi^+\pi^-\,(\pi^0)$~events which passed a $p > 0.001$~CL cut. This figure shows an excess of events
in the very forward region. Right: The same figure zoomed in on the forward region.}
\label{Figure:cospi0}
\end{figure}
% --------------------------------------------------------------------------
\subsection{Fiducial Volume Cuts}
% --------------------------------------------------------------------------
Geometric fiducial volume cuts have been applied according to the {\it nominal} scenario outlined in Section~5.3
of the analysis note~\cite{CLAS-NOTE_2017-002}. These volumes were regions of the detector that were not
well modeled and needed to be removed from the analysis. For example, the magnetic field varied rapidly close
to the torus coils making these regions difficult to simulate. Thus, any particle whose trajectory was near a torus
coil was identified and suvsequently, the event was excluded from our analysis. The effect of this particular cut
was most dramatic in the forward region, where the coils occupied a larger amount of the solid angle.\par\smallskip
In a brief summary~(taken directly from Ref.~\cite{CLAS-NOTE_2017-002}), such regions for all the g12 data,
where the detector acceptance was well behaved and reliably reproduced in simulations, were expressed as an
upper and lower limit of the difference in azimuthal angle between the center of a given sector and a particle
track. Because of the hyperbolic geometry of CLAS and the presence of the toroidal magnetic field, the fiducial
boundaries on $\phi$ were functions of momentum $p$, charge, and polar angle $\theta$ of each track. The
boundaries were evaluated separately in each sector, nominally defined as the $\phi$~values in which
occupancy drops below 50\,\% of that in the respective sector's flat region. The flat regions were defined as
$−10^\circ < \phi < 10^\circ$. The nominal upper and lower $\phi$~limits depended strongly on particle
charge, $p$ and $\theta$, hence the need for functional characterization and extrapolation.\par\smallskip
In order to determine the fiducial limits for charged hadrons, a sample of exclusive $p\,\pi^+\pi^-$~events
were sliced into $5\times 15\times 6$~bins in $p$, $\theta$, and sector, respectively. The $\phi$~distributions
for $\pi^+$ and $\pi^-$ were then plotted separately in each bin. The upper and lower $\phi$~limits of these
first-generation plots were found according to the {\it nominal} fiducial definition of 50\,\% occupancy. The
results from the first-generation fits were represented in second-generation plots of $\phi_{\rm min}$ and
$\phi_{\rm max}$ vs. $\theta$ and fit with hyperbolas, chosen since they replicate the projection of the detector.
In a last step, the second-generation fitting parameters were plotted vs. $p$ in third-generation plots. These
third-generation plots were fit to power functions and the fit results defined the sought-after functional form
$\phi_{\rm min} (\theta, p)$ and $\phi_{\rm max} (\theta, p)$ for each sector. The sector-integrated results for
positive and negative hadron tracks compose the nominal fiducial region.
% --------------------------------------------------------------------------
\subsection{Event Statistics after Applying all Cuts and Corrections}
% --------------------------------------------------------------------------
The process of developing and applying energy, momentum and other necessary corrections during the
course of this analysis served the purpose of correcting for the effects of the experimental setup, therefore
resulting in a data set that was as nature intended it. Additionally, determining and enforcing cuts used
in the analysis served not only to remove the remaining instrumental effects of the experimental setup
but also to remove the contributions from physics events not of interest to the analysis (the hadronic or
electromagnetic background). Through the application of the proper vertex position, photon and particle
identification variables, this background could be reduced considerably.
Table~\ref{Table:statistics_after_cuts} shows how many events survived after applying various cuts. %The
%number quoted within parentheses shows the percentage of surviving events.
\begin{table}[!ht]
\addtolength{\extrarowheight}{4pt}
\begin{center}
\begin{tabular}{ | c || c | c | c | c |}
\hline
Cuts & \multicolumn{4}{c|}{$\#$ of Events}\\
\hline
\hline
%No of initial events after all corrections & \multicolumn{4}{c|}{}\\
%\hline
Inclusive three tracks ($p, \pi^+, \pi^-$) &
\multicolumn{4}{c|}{not available}\\
\hline
& Topology 4 & \multicolumn{3}{c|}{Topology 5}\\\cline{2-5}
\rb{Vertex $\&$ $\Delta\beta$ cuts + Topology (CL) Cut} & 6.8\,M & \multicolumn{3}{c|}{23.8\,M}\\\hline
& $\gamma p\to p\,\pi^+\pi^-$ & $\gamma p\to p\,\omega$ & $\gamma p\to p\,\eta$ &
$\gamma p\to K^0\,\Sigma$\\\cline{2-5}
\rb{Final $\#$ of events} & 6.8\,M & 4.2\,M &138271 & 22890\\\hline
\end{tabular}
\caption{\it The table shows the remaining statistics after various cuts. Note that Topology~4 implies
kinematic fitting imposing no missing particle as well as energy and momentum conservation. The
exclusive two-pion final state is then considered background-free. Loose confidence-level (CL) cuts
of $p > 0.001$ were applied in both topologies and remaining background subtracted using $Q$~values.}
\label{Table:statistics_after_cuts}
\end{center}
\end{table}
\clearpage
% ---------------------------------------------------------------------------------------
\subsection{Beam and Target Polarization}
% ---------------------------------------------------------------------------------------
\subsubsection{Circularly-Polarized Photon Beam - Degree of Polarization}
% ---------------------------------------------------------------------------------------
Circularly-polarized photons were produced via bremsstrahlung of longitudinally-polarized electrons from
an amorphous radiator. The degree of circular polarization of these bremsstrahlung photons, $\delta_{\,\odot}$,
could be calculated from the longitudinal polarization of the electron beam, $\delta_{\,{\rm e}^-}$, multiplied
by a numerical factor. Using $x = E_{\gamma}/E_{\rm e^-}$, the degree of polarization was given by the
Maximon-Olson formula~\cite{Olsen:1959zz}:
\begin{equation}
\delta_{\,\odot}(x) \,=\, \delta_{\, {\rm e}^-} \,\cdot\, \frac{4x - x^2}{4 - 4x + 3x^2}\,.
\label{Equation:cir_beam_pol}
\end{equation}
\begin{figure}[!b]
\begin{center}
\includegraphics[width=0.49\textwidth,height=0.37\textheight]{g12_figures/Circular_Pol.pdf}\hfill
\includegraphics[width=0.49\textwidth,height=0.37\textheight]{g12_figures/polarization.pdf}
\caption{\it Left: Degree of circular polarization in units of $[\delta_\gamma/\delta_{\rm e^-}]$ as a
function of the fraction of the photon/electron energy. Right: Degree of circular-photon polarization
as a function of incident-photon energy for the g12 CEBAF-energy of 5.715~GeV; the electron-beam
polarization was 67.17\,\%.}
\label{Figure:cir_beam_pol}
\end{center}
\end{figure}
Figure~\ref{Figure:cir_beam_pol} shows that the degree of circular polarization is roughly proportional to the
photon-beam energy. In this figure, the incident-photon energy, $E_\gamma$, is given as a fraction of the
electron-beam energy, $E_{{\rm e}^-}$ (left) and for the actual g12 incident-photon energy range (right). In the
g12~experiment, the electron beam (CEBAF) energy was 5.715~GeV for all the runs that we used in this analysis.
\par\smallskip
The polarization of the electron beam was measured regularly using the M\o ller polarimeter, which makes use
of the helicity-dependent nature of M\o ller scattering. Table~\ref{Table:moller_mea} summarizes the M\o ller
measurements of the electron-beam polarization, $\delta_{\,{\rm e}^-}$. Note that only the measurement for the
second run range (56476 - 56643) was used here. During the g12~experiment, Hall~B did not have priority and
as a result, the polarization of the beam was delivered as a byproduct (based on the requirements of the other
halls). Although the polarization fluctuated, the majority of the g12 runs had a beam polarization close to 70\,\%
with a total uncertainty estimated to be 5\,\%.\par\smallskip
The degree of circular polarization was not a continuous function of the center-of-mass energy. Therefore,
we used the following equation to determine the polarization for center-of-mass bins:
\begin{equation}
\bar{\delta}_{\, \odot} \,=\, \frac{1}{N^{+} + N^{-}} \sum_{i\,\in\,\Delta\tau} \delta_{\,\odot}\,(W)\,,
\label{Equation:ave_cir_beam_pol}
\end{equation}
where $N^{\pm}$ was the total number of $\gamma p \to p\,\pi^{+}\pi^{-}$ events (used for the observable $I^\odot$)
for the two helicity states and $W$ was the center-of-mass energy; $\delta_{\,\odot}\,(W)$ was calculated from
Equation~\ref{Equation:cir_beam_pol}. Average values were derived for each center-of-mass bin and are shown in
Table~\ref{Table:cir_beam_pol}. %Figure~\ref{Figure:beam_pol} shows the degree of circular polarization and their
%averages for the g12 electron beam energy of 5.715~GeV.
\begin{table}[t!]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ c | c }
\hline
Run Range & Electron-Beam Polarization $\delta_{{\rm e}^-}$ (M\o ller Readout)\\
\hline\hline
56355 - 56475 & $(81.221\pm 1.48)$\,\% \\
\bl{56476 - 56643} & \bl{$(67.166\pm 1.21)$\,\%} \\
56644 - 56732 & $(59.294\pm 1.47)$\,\% \\
56733 - 56743 & $(62.071\pm 1.46)$\,\% \\
56744 - 56849 & $(62.780\pm 1.25)$\,\% \\
56850 - 56929 & $(46.490\pm 1.47)$\,\% \\
56930 - 57028 & $(45.450\pm 1.45)$\,\% \\
57029 - 57177 & $(68.741\pm 1.38)$\,\% \\
57178 - 57249 & $(70.504\pm 1.46)$\,\% \\
57250 - 57282 & $(75.691\pm 1.46)$\,\% \\
57283 - 57316 & $(68.535\pm 1.44)$\,\%
\end{tabular}
\caption{\it M\o ller measurements of the electron-beam polarization. Only the measurement for the
run range 56476 - 56643 (highlighted in \bl{blue}) was used in our analysis (see also
Table~\ref{Table:TriggerConfigurations}).}
\label{Table:moller_mea}
\end{center}
\end{table}
\begin{table}[h!]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ c | c }
\hline
& Average Degree of Circular Polarization, $\bar\delta_{\,\odot}$\\
\rb{Center-of-Mass Energy [GeV]} & $E_{{\rm e}^-} = 5.715$ GeV\\
\hline\hline
1.70 - 1.75 & 0.150\\
\hline
1.75 - 1.80 & 0.164\\
\hline
1.80 - 1.85 & 0.179\\
\hline
1.85 - 1.90 & 0.194\\
\hline
1.90 - 1.95 & 0.210\\
\hline
1.95 - 2.00 & 0.226\\
\hline
2.00 - 2.05 & 0.243\\
\hline
2.05 - 2.10 & 0.261\\
\hline
2.10 - 2.15 & 0.279\\
\hline
2.15 - 2.20 & 0.298\\
\hline
2.20 - 2.25 & 0.317\\
\hline
2.25 - 2.30 & 0.336\\
\hline
2.30 - 2.35 & 0.356\\
\hline
2.35 - 2.40 & 0.377\\
\hline
2.40 - 2.45 & 0.397\\
\hline
2.45 - 2.50 & 0.418\\
\hline
2.50 - 2.55 & 0.438\\
\hline
2.55 - 2.60 & 0.459\\
\hline
2.60 - 2.65 & 0.479\\
\hline
2.65 - 2.70 & 0.500\\
\hline
2.70 - 2.75 & 0.519\\
\hline
2.75 - 2.80 & 0.538\\
\hline
2.80 - 2.85 & 0.556\\
\hline
2.85 - 2.90 & 0.574\\
\hline
2.90 - 2.95 & 0.590\\
\hline
2.95 - 3.00 & 0.605\\
\hline
3.00 - 3.05 & 0.619\\
\hline
3.05 - 3.10 & 0.631\\
\hline
3.10 - 3.15 & 0.642\\
\hline
3.15 - 3.20 & 0.651\\
\hline
3.20 - 3.25 & 0.658\\
\hline
%3.25 - 3.30 & 0.664\\
%\hline
\end{tabular}
\caption{\it The average degree of circular (incident-photon) polarization for g12 $W$-bins.}
\label{Table:cir_beam_pol}
\end{center}
\end{table}
\subsubsection{Circularly-Polarized Photon Beam - Orientation of the Helicity States}
The direction of the beam polarization depended on the condition of the half-wave plate (HWP) which was either
IN or~OUT. In CLAS-g12, the longitudinal polarization of the electron beam was flipped pseudo-randomly at a
high rate with many sequences of helicity ($+$\,,\,$-$) or ($-$\,,\,$+$) signal per second. Occasionally, the HWP
was inserted in the circularly-polarized laser beam of the electron gun to reverse helicities and thus, the beam
polarization phase was changed by $180^\circ$. The HWP was inserted and removed at semi-regular intervals
throughout the experimental run to ensure that no polarity-dependent bias was manifested in the measured
asymmetries.\par\smallskip
For most of the g12 runs, we had direct reporting of the electron-beam helicity and the information could be
retrieved from the ``level1-trigger latch word'' of the TGBI bank. Bit~16 in this word described the photon
helicity-state corresponding to the sign of the electron-beam polarization as shown in
Table~\ref{Table:Helicity_Signal}.\\[20ex]
\noindent
Alternatively, the g12 run-group provided the following method:
\begin{center}
\begin{minipage}{15cm}
\begin{verbatim}
int GetHelicity(clasHEVT_t *HEVT)
{
int helicity = 0;
int readout = HEVT->hevt[0].trgprs;
if(readout > 0) helicity = 1;
if(readout < 0) helicity = -1;
return helicity;
}
\end{verbatim}
\end{minipage}\\[4ex]
\end{center}
\noindent
When the HWP was OUT, a bit-16 value of ``one'' meant that the beam polarization was parallel to the beam
direction and a value of ``zero'' that the beam polarization was antiparallel to the beam. When the HWP was IN,
the directions of the beam polarization were switched.
%In g12, the HWP setting had to be taken into account by the user performing the analysis.
Table~\ref{Table:Con_Half_wave} shows the HWP settings in the g12 data sets. The information shown in this table
was experimentally confirmed by studying the beam asymmetries $I^\odot$ in the two-pion channel.\par\smallskip
\begin{table}[t!]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{c | c | c}
\hline
\hline
TGBI latch1 & \multicolumn{2}{c}{Beam Helicity} \\
\hline
Bit 16 & $\lambda$/2 (OUT) & $\lambda$/2 (IN) \\
\hline
1 & $+$ & $-$ \\
0 & $-$ & $+$ \\
\hline
\hline
\end{tabular}
\caption{\it Helicity signal from the TGBI bank for the two half-wave-plate positions. In the table, the
sign $+$\,($-$) denotes the beam polarization was parallel (anti-parallel) to the beam direction. We
believe that this is the correct assignment. However, this information is not crucial for our analysis
since we also double-checked the polarization in different ways.}
\label{Table:Helicity_Signal}
\end{center}
\end{table}
\begin{table}[h!]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ c | c | c }
\hline
\hline
Period & Run Range & HWP Condition\\
\hline
1 & 56519 and earlier & \\
\hline
\bl{2} & \bl{56520 - 56594, 56608 - 56646} & \\
\hline
3 & 56601 - 56604, 56648 - 56660 & \\
\hline
4 & 56665 - 56667 & \\
\hline
5 & 56605, 56607, 56647 & \\
\hline
6 & 56668 - 56670 & \\
\hline
7 & 56897 and later & \\
\hline
8 & 57094 and later & \\
\hline
\hline
\end{tabular}
\caption{\it The half-wave plate (HWP) condition in the g12 data sets. In our analysis, only Period~2
(highlighted in \bl{blue}) was used (see also Table~\ref{Table:TriggerConfigurations}).}
\label{Table:Con_Half_wave}
\end{center}
\end{table}
% ----------------------------------------------------------------------------------
\subsubsection{Beam-Charge Asymmetry in Data Sets with Circularly-Polarized Photons}
\label{Subsubsection:BCA}
% ----------------------------------------------------------------------------------
The electron-beam polarization was toggled between the helicity-plus ($h^+$) and the helicity-minus
($h^-$) state at a rate of about 30~Hz. At this large rate, the photon-beam flux for both helicity states
should be the same, on average. However, small beam-charge asymmetries of the electron beam could
cause instrumental asymmetries in the observed {\it hadronic} asymmetries and had to be considered.
The beam-charge asymmetry could be calculated from the luminosities of $h^+$ and $h^-$~events:
\begin{equation}
\Gamma^{\pm} \,=\, \alpha^{\pm}\,\Gamma \,=\, \frac{1}{2}\,( 1\,\pm\, \bar{a}_{c} )\,\Gamma\,,
\label{Equation:BCA}
\end{equation}
where $\Gamma$ was the total luminosity and the parameters $\alpha^{\pm}$ denoted the fraction of $h^+$
and $h^-$~events. The parameters $\alpha^{\pm}$ depended on the mean value of the electron-beam
charge asymmetry, $\bar{a}_c$, which was studied in other CLAS experiments and typically less than
$0.2\,\%$, e.g. Ref.~\cite{Strauch:2005cs,g9b:circ_beam_pol}. Since the beam-charge asymmetry was
very small, it could be considered negligible.\par\smallskip
%----------------------------------------------------------------------------------
\subsection{\texorpdfstring{Signal-Background Separation: $Q$-Factor Method}
{Signal-Background Separation: Q-Factor Method}}
\label{Subsection:Q-factor_method}
%----------------------------------------------------------------------------------
The remaining step in preparing a clean event sample of the reaction in question is the removal of background
underneath the signal peak. Figure~\ref{Figure:DoublePionBackgroundFree} shows an example of the
missing-mass distribution for the exclusive $p\,\pi^+\pi^-$ final state where the proton was artificially
removed from the data sample and then the missing mass was calculated. The figure shows an almost
background-free distribution and thus, no further background-subtraction method was applied.
\begin{figure}[!b]
\begin{center}
\includegraphics[width=0.7\textwidth,height=0.34\textheight]{g12_figures/mass_proton_unfit.pdf}\hfill
%\includegraphics[width=0.49\textwidth,height=0.3\textheight]{g12_figures/polarization.pdf}
\caption{\it Example of a (background-free) missing-mass distribution for the exclusive
$\gamma p\to p\,\pi^+\pi^-$ reaction. Though detected, the proton was removed from the event
sample and the missing-mass was calculated.}
\label{Figure:DoublePionBackgroundFree}
\end{center}
\end{figure}
The (event-based) $Q$-factor method used for the background separation in the $p\,\phi\to p\,K^+K^-$ and
$p\,\pi^+\pi^-\,\pi^0$ final states (including $\gamma p\to p\,\omega\to p\,\pi^+\pi^-\pi^0$, $\gamma p\to
p\,\eta\to p\,\pi^+\pi^-\pi^0$, as well as $\gamma p\to K^0\,\Sigma^+\to p\,\pi^+\pi^-\pi^0$) is described in the
following sections.
\clearpage
%----------------------------------------------------------------
\subsubsection{General Description of the Method}
\label{Subsubsection:Qfactor_equations}
%----------------------------------------------------------------
In this event-based method, the set of coordinates that described the multi-dimensional phase space of the
reaction was categorized into two types: {\it reference} and {\it non-reference} coordinates. The signal and
background shapes had to be known {\it a priori} in the reference coordinate but this knowledge was not required
in the non-reference coordinates. Mass was typically chosen as the reference coordinate. For each event, we then
set out to find the $N_{c}$ nearest neighbors in the phase space of the non-reference coordinates. This was similar
to binning the data using a dynamical bin width in the non-reference coordinates and making sure that we had
$N_{c}$~events per fit.\par\smallskip
The mass distribution of the $N_{c}$ events (including the candidate event) in the reference coordinate was then
fitted with a total function defined as:
\begin{equation}
f(x) \,=\, N\,\cdot [ f_{s}\,\cdot\, S(x) \,+\, ( 1 \,-\, f_{s} )\,\cdot\, B(x) ]\,,
\label{Equation:total_function}
\end{equation}
where $S(x)$ denoted the signal and $B(x)$ the background probability density function. $N$ was a normalization
constant and $f_{s}$ was the signal fraction with a value between 0 and 1. The RooFit package of the CERN ROOT
software~\cite{RooFit_Users_Manual} was used for the fit procedure. Since $N_{c}$ was usually a small number (of
the order of a few hundred events), an unbinned maximum likelihood method was used for the fitting. The $Q$~factor
itself was given by:
\begin{equation}
Q \,=\, \frac{s(x)}{s(x) \,+\, b(x)}\,,
\label{Equation:Q_factor}
\end{equation}
where $x$ was the value of the reference coordinate for the candidate event, $s(x) = f_{s} \cdot S(x)$ and $b(x) = (1-f_{s})
\cdot B(x)$. The $Q$~factor could then be used as an event weight to determine the signal contribution to any
physical distribution.
%----------------------------------------------------------------------------
\subsubsection{\texorpdfstring{The $Q$-Factor Method for the Reaction
$\gamma p\to p\,\omega\to p\,\pi^+\pi^-\pi^0$}{The Q-Factor Method for the omega Reaction}}
%----------------------------------------------------------------------------
The kinematic variables that described the reaction $\gamma p \to p\,\omega$ were chosen to be the
incident-photon energy, $E_\gamma$, and the center-of-mass angle of the outgoing~$\omega$,
cos\,$\theta^{\,\omega}_{\rm c.\,m.}$. Since we reconstructed the $\omega$ from its decay into $\pi^+\pi^-\,(\pi^0)$,
we also considered the relevant kinematic variables which described the five-dimensional phase space of the
$3\pi$~system. The $\omega$~decay was thus entirely defined by five independent kinematic variables (including
the invariant $\pi^+\pi^-\pi^0$~mass we used as {\it reference} variable). In total, we chose six {\it non-reference}
variables:
\begin{itemize}
\item The incident photon energy $E_\gamma$ (or alternatively, the total center-of-mass energy $W$),
\item The two angles of the $\omega$~meson in the helicity frame, cos\,$\theta_{\rm \,HEL}$ and $\phi_{\rm \,HEL}$,
\item The center-of-mass azimuthal and polar angles of the $\omega$, and
\item The decay parameter $\lambda \propto |\,\vec{p}_{\pi^+}\,\times\,\vec{p}_{\pi^-}\,|^2$
\cite{Williams:2007thesis}\,.
\end{itemize}
The six non-reference coordinates and their maximum ranges used in the $Q$-factor method are summarized in
Table~\ref{Table:non_ref_cor_omega}.
\begin{table}[t!]
\addtolength{\extrarowheight}{6pt}
\begin{center}
\begin{tabular}{ c | l | l }
\hline
$\Gamma_{i}$ & Non-Reference Coordinate & Maximum Range $\Delta_{i}$ \\
\hline
\hline
\rbbb{$\Gamma_{0}$} & \rbbb{cos\,$\Theta_{\rm c.\,m.}^{\,\omega}$} & \rbbb{2}\\
\hline
\rbbb{$\Gamma_{1}$ \& $\Gamma_{2}$} & \rbbb{cos\,$\theta_{\rm \,HEL}$} and \rbbb{$\phi_{\rm \,HEL}$}
& \rbbb{2 \& $2\pi$ [radians]}\\
\hline
\rbbb{$\Gamma_{3}$} & \rbbb{$\phi^{\,\omega}_{\rm \,lab}$} & \rbbb{2$\pi$ [radians]} \\
\hline
\rbbb{$\Gamma_{4}$} & \rbbb{$\lambda$} & \rbbb{1}\\
\hline
\rbbb{$\Gamma_{5}$} & \rbbb{incident photon energy $E_\gamma$ (or $W$)}
& \rbbb{20~MeV (10~MeV below $W=2.1$~GeV)}\\
\hline
\end{tabular}
\caption{\it The non-reference coordinates $\Gamma_i$ and their ranges $\Delta_i$.}
\label{Table:non_ref_cor_omega}
\end{center}
\end{table}
For the signal-background separation in the $\omega\to\pi^+\pi^-\pi^0$~analysis, we initially applied a small
$CL > 0.001$~cut (from kinematic fitting) on the $\gamma p \to p\,\pi^+\pi^-\,(\pi^0)$ final state. This loose
$CL$~cut significantly reduced the background, in particular from $\gamma p \to p\,\pi^+\pi^-$~events. We then
used the event-based technique to select $\omega$~events.\par\smallskip
The data were divided into data subsets based on the photon energy (20-MeV wide bins). We chose the number
of 1000~nearest-neighbor events for each candidate event in the phase space spanned by the non-reference
coordinates. The $\pi^{+}\pi^{-}\pi^{0}$~invariant mass distribution of these 1000~events was then fitted over
the mass range 650\,-\,900~MeV using the unbinned maximum-likelihood technique. Since the natural width
of the $\omega$~meson is 8.49~MeV and thus, at the level of the detector resolution, we chose a Voigtian function
for the signal pdf. The Voigtian function is a convolution of a Gaussian, which was used to describe the resolution,
and a Breit-Wigner, which described the natural line shape of the resonance. The background shape was modeled
with a second-order Chebychev polynomial for incident photon energies above~1400~MeV. Close to the reaction
threshold of $E_\gamma\approx 1109$~MeV, the $\omega$~signal peak is located very close to the upper
$3\pi$~phase space boundary. For this reason, we chose an Argus function instead of a Chebychev polynomial
to describe the background shape.\par\smallskip
Table~\ref{Table:pdf_constraints_omega} shows the parameters of the signal and background pdfs and the constraints
imposed on them. The two pdfs were used to construct a total pdf (see Equation~\ref{Equation:total_function}) and the
$Q$~factor of the candidate event was extracted using Equation~\ref{Equation:Q_factor}. %Figure~\ref{Figure:Q_omega}
%shows examples of fits to invariant $\pi^{+} \pi^{-}\pi^{0}$ mass distributions of 1000~events. The figure shows
%a comparison of the data with appropriately normalized signal and background pdfs after the fit parameters were
%determined in the unbinned-maximum likelihood fit.
\par\smallskip
\begin{table}[!b]
\addtolength{\extrarowheight}{4pt}
\begin{center}
\begin{tabular}{ c | c | c | c }
\hline
Probability Density Function & Parameters & Initial Value & Fit Range\\
\hline
\hline
\multirow{3}{*}{Voigtian} & mean, $\mu$ & 782.65~MeV~\cite{Olive:2016xmw} & fixed\\
\cline{2-4}
& width, $\sigma$ & $8.0$~MeV & 0\,-\,30~MeV\\
\cline{2-4}
& natural width, $\Gamma$ & 8.49~MeV~\cite{Olive:2016xmw} & fixed\\
\hline
\multirow{2}{*}{Chebychev ($E_\gamma > 1.4$~GeV)} & $c_{0}$ & 0.5 & 0.0\,-\,1.8\\
\cline{2-4}
& $c_{1}$ & 0.1 & $-1.2$\,-\,1.2\\
\hline
\multirow{2}{*}{Argus ($E_\gamma < 1.4$~GeV)} & endpoint, $m_{0}$ & 820~MeV & 790.0\,-\,950.0~MeV\\
\cline{2-4}
& slope, $c$ & $-1.0$ & $-10.0$\,-\,0.2\\
\hline
\end{tabular}
\caption{\it Parameters of the signal and background probability-density functions. A Voigtian was used to
describe the $\omega$~signal and a second-order Chebychev polynomial (an Argus function for
$E_\gamma < 1.4$~GeV) was used to describe the background over the $\pi^+\pi^-\pi^0$~mass range
650\,-\,900~MeV.}
\label{Table:pdf_constraints_omega}
\end{center}
\end{table}
\clearpage
\subsubsection*{Quality Checks\label{Subsubsection:QualityChecks}}
\begin{enumerate}
\item Once the fit parameters were determined in an individual likelihood fit, we performed a least-square ``fit''
of the same mass distribution from the 1000~events. Among other things, this allowed us to plot the distribution
of reduced-$\chi^2$ values as a goodness-of-fit measure. The left column of Figure~\ref{Figure:QualityChecks}
shows several such reduced-$\chi^2$ distributions for a few randomly-selected example $E_\gamma$~bins: (top
to bottom row) $E_\gamma\in [1.64,\,1.66]$~GeV, $E_\gamma\in [2.10,\,2.12]$~GeV, $E_\gamma\in [4.00,\,4.02]$~GeV,
$E_\gamma\in [5.00,\,5.02]$~GeV. These reduced-$\chi^{2}$ distributions peak fairly close to the ideal value of
{\it one}. Given the fairly small number of events in these distributions, we also concluded that the fitter picks
up statistical fluctuations. This resulted in overconstrained fits and slightly smaller reduced-$\chi^2$ values,
about 0.7\,-\,0.8 on average.
\item Defined in terms of the pion momenta in the rest frame of the $\omega$~meson, the quantity
$\lambda = |\,\vec{p}_{\pi^+}\,\times\,\vec{p}_{\pi^-}|^2 \,/\,\lambda_{\rm \,max}$ is proportional to the
$\omega\to\pi^+\pi^-\pi^0$~decay amplitude as a consequence of isospin conservation~\cite{Williams:2009ab}
with $\lambda_{\rm \,max}$ defined as~\cite{Weidenauer:1993mv}:
\begin{equation}
\lambda_{\rm \,max} \,=\, T^2\,\bigg(\frac{T^2}{108}\,+\,\frac{mT}{9}\,+\,\frac{m^2}{3} \bigg)
\end{equation}
for a totally symmetric decay, where $T = T_1 + T_2 + T_3$ is the sum of the $\pi^{\pm,\,0}$~kinetic energies and
$m$ is the $\pi^\pm$~mass. The parameter~$\lambda$ varies between 0 and 1 and shows a linearly-increasing
distribution as expected for a vector meson.\par\smallskip
Figure~\ref{Figure:QualityChecks} (center column) shows the $\lambda$~distributions for the same energy bins
as for the corresponding reduced-$\chi^2$ distributions in the left column. The (red) signal was generated by
weighting event-by-event the (black) full distribution with the $Q$~values; the (blue) background distribution
was generated by weighting the full distribution with $1-Q$. The linear behavior of the $\omega$~signal events
is clearly visible.
\end{enumerate}
\begin{figure}[!t]
\begin{tabular}{ccc}
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/chi2_164367_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/lambdahisto_NewEBin28_Per20_164367_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/MMhisto_NewEBin028_Per20_164367_events.pdf}\\
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/chi2_109936_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/lambdahisto_NewEBin51_Per20_109936_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/MMhisto_NewEBin051_Per20_109936_events.pdf}\\
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/chi2_4606_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/lambdahisto_NewEBin146_Per20_4606_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/MMhisto_NewEBin146_Per20_4606_events.pdf}\\
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/chi2_1139_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/lambdahisto_NewEBin196_Per20_1139_events.pdf} &
\includegraphics[width=0.32\textwidth,height=0.2\textheight]{g12_figures/MMhisto_NewEBin196_Per20_1139_events.pdf}\\
\end{tabular}
\caption{\it Quality checks - shown are randomly selected $E_\gamma$~bins across a wide range in the incident photon energy:
(top to bottom row) $E_\gamma\in [1.64,\,1.66]$~GeV, $E_\gamma\in [2.10,\,2.12]$~GeV, $E_\gamma\in [4.00,\,4.02]$~GeV,
$E_\gamma\in [5.00,\,5.02]$~GeV. (Left column) Examples of reduced-$\chi^{2}$ \text{distributions.} (Center) Examples of
$\lambda$~distributions. (Right) The full mass distribution for the energy bin. The black line denotes the full distribution,
the red line the signal, and the blue solid line the background distribution.}
\label{Figure:QualityChecks}
\end{figure}
Finally, $\omega\to\pi^+\pi^-\pi^0$-mass distributions showing the full statistics in a given energy bin are
presented in Figure~\ref{Figure:QualityChecks} (right column) for the selected $E_\gamma$~bins discussed above
and in 20-MeV-wide bin for the entire CLAS-g12 energy range in
Figures~\ref{Figure:MM_Q_factor_omega_I}\,-\,\ref{Figure:MM_Q_factor_omega_IV}.
Since we analyzed a total of 215 energy bins, we show the mass distribution for every sixth energy bin in these figures.
\par\smallskip
\subsubsection*{The Photon-Energy Range below 2~GeV}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
%\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin052_Per20.pdf} &
%\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin058_Per20.pdf}\\
%\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin064_Per20.pdf} &
&\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin022_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin028_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin034_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin040_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin046_Per20.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ mass distributions for the reaction $\gamma p\to p\,\omega$. Shown is every sixth
20-MeV-wide $E_\gamma$~bin starting at $E_\gamma\in [1200,\,1220]$~MeV (top left), $E_\gamma\in [1220,\,1240]$~MeV
(top right), etc.}
\label{Figure:MM_Q_factor_omega_I}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin052_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin058_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin064_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin070_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin076_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin082_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin088_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin094_Per20.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ mass distributions for the reaction $\gamma p\to p\,\omega$. Shown is every sixth
20-MeV-wide $E_\gamma$~bin starting at $E_\gamma\in [2120,\,2140]$~MeV (top left), $E_\gamma\in [2140,\,2160]$~MeV
(top right), etc.}
\label{Figure:MM_Q_factor_omega_II}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
&\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin106_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin112_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin118_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin124_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin130_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin136_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin144_Per20.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ mass distributions for the reaction $\gamma p\to p\,\omega$. Shown is every sixth
20-MeV-wide $E_\gamma$~bin starting at $E_\gamma\in [3200,\,3220]$~MeV (top right). The $[3080,\,3100]$~MeV bin is
missing owing to tagger inefficiencies.}
\label{Figure:MM_Q_factor_omega_III}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin150_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin156_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin162_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin168_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin174_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin180_Per20.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin186_Per20.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/OMEGA-Masses/MMhisto_NewEBin192_Per20.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ mass distributions for the reaction $\gamma p\to p\,\omega$. Shown is every sixth
20-MeV-wide $E_\gamma$~bin starting at $E_\gamma\in [4080,\,4100]$~MeV (top left), $E_\gamma\in [4200,\,4220]$~MeV
(top right), etc.}
\label{Figure:MM_Q_factor_omega_IV}
\end{figure}
\clearpage
\subsubsection{\texorpdfstring{The Reaction $\gamma p\to p\,\eta\to p\,\pi^+\pi^-\pi^0$}
{The eta Reaction}}
\label{Subsubsection:Q-Factor-eta}
The reconstruction of the $\eta$~meson is based on its $\pi^+\pi^-\pi^0$~decay mode and therefore, the invariant
$\pi^+\pi^-\pi^0$~mass was used as the reference coordinate. The non-reference coordinates are the same as those
used for the $\omega$~meson and are summarized in Table~\ref{Table:non_ref_cor_omega}. The quantity $\lambda$
may not have a direct physical meaning in the $\eta$~decay but in defining the phase space for the nearest-neighbors
search, it still serves as an independent kinematic variable.\par\smallskip
The $\eta$~meson has a natural width of $\Gamma = 1.31\pm 0.05$~keV~\cite{Olive:2016xmw}. For this reason, the
observed width of the $\eta$~signal is dominated by the experimental resolution and the lineshape should be describable
by a Gaussian. However, we noticed that a simple Gaussian could not describe very well the very low- and high-mass tails
of the signal peak, which caused some enhancements in the background description. A double-Gaussian in combination
with a second-order Chebychev polynomial for the background solved this issue. Table~\ref{Table:pdf_constraints_eta}
summarizes the parameters of the signal and background pdfs and the constraints imposed on them.\par\smallskip
\noindent
To compare with published CLAS results~\cite{Williams:2009yj}, we have used the following binning scheme
in~$W$:\\[-3.5ex]
\begin{enumerate}
\item $W\in [1720,\,2100]$~MeV in 10-MeV-wide $W$~bins $(E_\gamma\in[1707,\,1881]$~MeV),\\[-3.7ex]
\item $W\in [2100,\,2360]$~MeV in 20-MeV-wide $W$~bins $(E_\gamma\in[1881,\,2499]$~MeV),\\[-3.7ex]
\item $W\in [2360,\,3320]$~MeV in 40-MeV-wide $W$~bins $(E_\gamma\in[2499,\,5405]$~MeV).\\[-3.5ex]
\end{enumerate}
Figures~\ref{Figure:MM_Q_factor_eta_I}\,-\,\ref{Figure:MM_Q_factor_eta_IV} show the invariant $\pi^+\pi^-\pi^0$~mass
distributions for the $W\in [1720,\,2070]$~MeV range, which corresponds to about $E_\gamma \in [1107,\,1814]$~MeV.
The black solid line denotes the full mass distribution, the red solid is the signal, and the blue solid line represents the
background distribution. Note that $E_\gamma = 1100$~MeV is at the very low end of the tagging range. For this reason,
the first truly available $W$~bin is 1750\,-\,1760~MeV; the statistics is still low, though. Full statistics is then available
in CLAS-g12 from $W = 1760$~MeV (Fig.~\ref{Figure:MM_Q_factor_eta_I}, top row). The background exhibits an almost
linear behavior but we chose a second-order polynomial and a slightly broader fit range of 455\,-\,650~MeV to avoid
ambiguities between the background pdf and the second (broader) signal Gaussian. The broader fit range also required
us to use 500~events (up from initially 300~events) in the search for nearest neighbors to accumulate sufficient signal
statistics in the individual mass distributions. Finally, Fig.~\ref{Figure:MM_Q_factor_eta_V} shows the
$\pi^+\pi^-\pi^0$~distributions for the $W\in [2360,\,2680]$~MeV range.
\begin{table}[!b]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ c | c | c | c }
\hline
Probability Density Function & Parameters & Initial Value & Fit Range\\
\hline
\hline
& Mean, $\mu$ & 547.86~\cite{Olive:2016xmw}~MeV & fixed\\\cline{2-4}
\rb{Gaussian I} & Width, $\sigma$ & 5.0~MeV & 1.5\,-\,9.5~MeV\\\hline
& Mean, $\mu$ & 548.40~MeV & 540.0\,-\,560.0~MeV\\\cline{2-4}
\rb{Gaussian II} & Width, $\sigma$ & 10.0~MeV & 6.0\,-\,28.0~MeV\\\hline
\multirow{2}{*}{Chebychev} & $c_{0}$ & 0.8 & 0.0\,-\,1.6\\
\cline{2-4}
& $c_{1}$ & 0.19 & $-0.6$\,-\,1.5\\
\hline
\end{tabular}
\caption{\it Parameters of the signal and background probability-density functions. A double-Gaussian was used to
describe the $\eta$~signal and a second-order Chebychev polynomial was used to describe the background over the
$\pi^+\pi^-\pi^0$~mass range 455-650~MeV.}
\label{Table:pdf_constraints_eta}
\end{center}
\end{table}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin04_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin05_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin06_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin07_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin08_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin09_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin10_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin11_Per11.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ distributions for the reaction $\gamma p\to p\eta$. Shown are 10-MeV-wide
$W$~bins starting at $W\in [1750,\,1760]$~MeV (top left), $W\in [1760,\,1770]$~MeV (top right), etc.}
\label{Figure:MM_Q_factor_eta_I}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin12_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin13_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin14_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin15_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin16_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin17_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin18_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin19_Per11.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ distributions for the reaction $\gamma p\to p\eta$. Shown are 10-MeV-wide
$W$~bins starting at $W\in [1830,\,1840]$~MeV (top left), $W\in [1840,\,1850]$~MeV (top right), etc.}
\label{Figure:MM_Q_factor_eta_II}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin20_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin21_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin22_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin23_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin24_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin25_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin26_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin27_Per11.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ distributions for the reaction $\gamma p\to p\eta$. Shown are 10-MeV-wide
$W$~bins starting at $W\in [1910,\,1920]$~MeV (top left), $W\in [1920,\,1930]$~MeV (top right), etc.}
\label{Figure:MM_Q_factor_eta_III}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin28_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin29_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin30_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin31_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin32_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin33_Per11.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin34_Per11.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin35_Per11.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ distributions for the reaction $\gamma p\to p\eta$. Shown are 10-MeV-wide
$W$~bins starting at $W\in [1990,\,2000]$~MeV (top left), $W\in [2000,\,2010]$~MeV (top right), etc.}
\label{Figure:MM_Q_factor_eta_IV}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin01_Per13.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin02_Per13.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin03_Per13.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin04_Per13.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin05_Per13.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin06_Per13.pdf}\\
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin07_Per13.pdf} &
\includegraphics[width=0.49\textwidth,height=0.29\textwidth]{g12_figures/ETA-Masses/MMhisto_NewEBin08_Per13.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-\pi^0$ distributions for the reaction $\gamma p\to p\eta$. Shown are 40-MeV-wide
$W$~bins starting at $W\in [2360,\,2400]$~MeV (top left), $W\in [2400,\,2440]$~MeV (top right), etc.}
\label{Figure:MM_Q_factor_eta_V}
\end{figure}
\clearpage
\subsubsection{\texorpdfstring{The Reaction $\gamma p\to K_S^0\,\Sigma^+$}
{The KS Reaction}}
The reconstruction of the $K_S^0\,\Sigma^+$~final state differs from the $\omega$ and the $\eta$. While the
latter two are based directly on the $\pi^+\pi^-\pi^0$~system, the strange~$K_S$ is reconstructed from the
$\pi^+\pi^-$~system and the remaining $\pi^0$ originates from the baryon decay. Since the $K_S\to\pi^+\pi^-$
and the $\Sigma^+\to p\pi^0$ are highly correlated (associated strangeness production), the reference quantity
can either be the invariant $\pi^+\pi^-$~mass or the invariant $p\pi^0$~mass. We determined $Q$~values
independently applying both approaches, which serves as a cross check when comparing the cross sections.
Table~\ref{Table:non_ref_cor_KS} shows the non-reference variables used for the background subtraction in
this reaction. The quantities in parentheses are the non-reference coordinates used for the $Q$~values based
on the invariant $p\pi^0$~mass.\par\smallskip
\begin{table}[b!]
\addtolength{\extrarowheight}{7pt}
\begin{center}
\begin{tabular}{ c | l | c}
\hline
\hline
\rbbb{$\Gamma_{i}$} & \rbbb{Non-Reference Coordinate} & \rbbb{Maximum Range $\Delta_{i}$}\\
\hline
\hline
\rbbb{$\Gamma_{0}$} & \rbbb{incident-photon energy $E_\gamma$} & \rbbb{50 MeV}\\
\hline
\rbbb{$\Gamma_{1}~\&~\Gamma_{2}$} & \rbbb{cos$\,\theta_{\pi^+}$~(cos$\,\theta_{p}$)
\& $\phi_{\,\pi^+}$~($\phi_{\,\pi^0}$) in the $\pi^+\pi^-$~($p\pi^0$) rest frame} & \rbbb{2 \& $2\pi$}\\
\hline
\rbbb{$\Gamma_{3}$} & \rbbb{cos\,$\Theta_{\rm \,c.\,m.}^{\,K_S}$ in the center-of-mass frame} & \rbbb{2}\\
\hline
\rbbb{$\Gamma_{4}$} & \rbbb{$\phi_{\rm \,lab}^{\,K_S}$} & \rbbb{$2\pi$}\\
\hline
\rbbb{$\Gamma_{5}$} & \rbbb{cos\,(opening angle $\angle\,(p,\,\pi^0)$)} & \rbbb{2}\\
\hline
\hline
\end{tabular}
\caption{\it The non-reference coordinates $\Gamma_i$ and their ranges $\Delta_i$. Note that we used
100-MeV-wide incident-photon bins for the induced polarization.}
\label{Table:non_ref_cor_KS}
\end{center}
\end{table}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.49\textwidth,height=0.29\textheight]{g12_figures/KS-Masses/MassDistribution-Total.pdf}
\includegraphics[width=0.49\textwidth,height=0.29\textheight]{g12_figures/KS-Masses/MassDistribution-SigmaCut.pdf}
\includegraphics[width=0.49\textwidth,height=0.29\textheight]{g12_figures/KS-Masses/MassDistribution-KaonOmega.pdf}
\includegraphics[width=0.49\textwidth,height=0.29\textheight]{g12_figures/KS-Masses/MassDistribution-SigmaOmegaCut.pdf}
\caption{\it Top row: Invariant $\pi^+\pi^-$ mass distribution of all g12 $\pi^+\pi^-\pi^0$~events in Period~2
(left) and the same invariant $\pi^+\pi^-$ mass distribution after the $\Sigma^+$~cut (right). Bottom row: Invariant
$\pi^+\pi^-\pi^0$ mass vs. the corresponding $\pi^+\pi^-$ mass of all g12 $\pi^+\pi^-\pi^0$ events in Period~2
(left) and the same invariant $\pi^+\pi^-$ mass distribution shown in the top row after the $\omega$~and the
$\Sigma^+$~cuts (right).}
\label{Figure:Masses-KS-Cuts}
\end{center}
\end{figure}
Since the cross section for the reaction $\gamma p\to K^0\,\Sigma^+$ is relatively small, the observed statistics
is low and the invariant $\pi^+\pi^-$ mass is dominated by background in the mass region of the $K_S$ (see
Fig.~\ref{Figure:Masses-KS-Cuts}, top left). Therefore, we considered two mass cuts before we applied the
$Q$-factor method:
\begin{enumerate}
\item Strangeness is conserved in electromagnetic and strong interactions. For this reason, the $K_S$~meson is
produced together with a $\Sigma^+$~baryon (in our analysis). The life time of the $\Sigma^+$
($\tau = (0.8018\pm 0.0026)\times 10^{-10}$~s) is fairly long since it can only decay weakly. We thus applied
a narrow cut of 20~MeV around the $\Sigma^+$~mass of 1189.37~MeV~\cite{Olive:2016xmw}. The effect can be
seen in Figure~\ref{Figure:Masses-KS-Cuts} (top row). The left side shows the {\it raw} $\pi^+\pi^-$~distribution
of all g12 $\pi^+\pi^-\pi^0$~events in Period~2 (see Table~\ref{Table:TriggerConfigurations}), whereas the
right side shows the same distribution after the $\Sigma^+$~cut. The background is significantly reduced and
the $K_S$~peak is clearly visible; the $K_S\,\Sigma^+$~statistics is only marginally affected.
\item The dominant reaction contributing to the $p\,\pi^+\pi^-\pi^0$~final state is $\omega$~production. The
bottom row of Figure~\ref{Figure:Masses-KS-Cuts} shows the invariant $\pi^+\pi^-\pi^0$ mass vs. the corresponding
$\pi^+\pi^-$~mass (left side). The vertical band for the $\omega$ is clearly visible and moreover, it exhibits a
maximum intensity in the vicinity of the $K_S$ in the projection onto the $\pi^+\pi^-$~axis. Therefore, we applied
a mass cut to remove contributions from $\omega$~production: $m_{\pi^+\pi^-\pi^0} <$ 752~MeV and
$m_{\pi^+\pi^-\pi^0} >$ 812~MeV. The resulting (final) $\pi^+\pi^-$~mass distribution showing the $K_S$~peak is
given on the right side. A comparison of Figure~\ref{Figure:Masses-KS-Cuts} (top right) and
Figure~\ref{Figure:Masses-KS-Cuts} (bottom right) indicates that only little $K_S\,\Sigma^+$ statistics is lost due
to the $\omega$~cut.\\[2ex]
\end{enumerate}
The two-dimensional distribution also explains the two structures which can be observed in the projection onto the
$\pi^+\pi^-$~axis (right side of Figure~\ref{Figure:Masses-KS-Cuts}): (1) The peak around 400~MeV is the reflection
of the $\eta\to\pi^+\pi^-\pi^0$ which is cut off at the phase-space boundary, and (2) the enhancement around
550~MeV is most likely based on the $\eta$ decaying into $\pi^+\pi^-\gamma$.\par\smallskip
To subtract the background for the $K_S\,\Sigma^+$~final state, the selected g12 data were divided into 50-MeV-wide
incident-photon energy bins for the cross-section measurement and 100-MeV-wide energy bins for the measurement
of the induced polarization. We then chose a 1000~nearest-neighbor events for each signal candidate in the phase
space spanned by the non-reference coordinates. The invariant $\pi^+\pi^-$~mass distribution of these 1000~events
was fitted over the mass range 473\,-\,523~MeV for the $K_S$ and independently, the $p\pi^0$ distribution was
fitted over the mass range 1149\,-\,1229~MeV for the $\Sigma^+$ using the unbinned maximum-likelihood technique.
Since the $K_S$~decays weakly into $\pi^+\pi^-$ with a mean life $\tau$ of about \mbox{$(8.954\pm 0.004)\times
10^{-11}$~s}~\cite{Olive:2016xmw} (and has thus a narrow natural width), we chose a Gaussian function for the signal
pdf and a second-order Chebychev polynomial for the background. Table~\ref{Table:pdf_constraints_KS} shows the
parameters of the signal and background pdfs and the constraints imposed on them.\par\smallskip
\begin{table}[!t]
\addtolength{\extrarowheight}{3pt}
\begin{center}
\begin{tabular}{ | c | c || c | c || c | c | }
\cline{3-6}
\multicolumn{2}{c|}{} & \multicolumn{2}{c||}{Ref. Coordinate: $\pi^+\pi^-$ Mass}
& \multicolumn{2}{c|}{Ref. Coordinate: $p\pi^0$ Mass}\\
\cline{3-6}
\multicolumn{2}{r|}{} & Initial Value & Fit Range & Initial Value & Fit Range\\
\hline
\hline
& Mean, $\mu$ & 497.61~MeV~\cite{Olive:2016xmw} & fixed & 1189.37~MeV~\cite{Olive:2016xmw} & fixed\\
\cline{2-6}
\rb{Gaussian pdf} & Width, $\sigma$ & 4.5~MeV & 2.0\,-\,8.0~MeV & 4.5~MeV & 0.0\,-\,9.0~MeV\\\hline
\multirow{2}{*}{Chebychev pdf} & $c_{0}$ & 0.1 & $-1.5$\,-\,1.5 & 0.1 & $-1.5$\,-\,1.5\\
\cline{2-6}
& $c_{1}$ & 0.1 & $-1.5$\,-\,1.5 & 0.1 & $-1.5$\,-\,1.5\\
\hline
\end{tabular}
\caption{\it Parameters of the signal $\&$ background probability-density functions. A Gaussian was used to
describe the signal and a second-order Chebychev polynomial to describe the background.}
\label{Table:pdf_constraints_KS}
\end{center}
\end{table}
\begin{figure}[!b]
\begin{center}
\includegraphics[width=0.49\textwidth,height=0.4\textheight]{g12_figures/KS-Masses/MassDistribution_EBin_04_ABin_03.pdf}
\includegraphics[width=0.49\textwidth,height=0.4\textheight]{g12_figures/KS-Masses/MassDistribution_EBin_04_ABin_06.pdf}
\includegraphics[width=0.49\textwidth,height=0.4\textheight]{g12_figures/KS-Masses/MassDistribution_EBin_06_ABin_03.pdf}
\includegraphics[width=0.49\textwidth,height=0.4\textheight]{g12_figures/KS-Masses/MassDistribution_EBin_06_ABin_06.pdf}
\caption{\it Examples of $\pi^+\pi^-$ distributions for $\gamma p\to K_S\,\Sigma^+$. Top row: $E_\gamma\in
[1400,\,1500]$~MeV. Bottom row: $E_\gamma\in [1600,\,1700]$~MeV. The left side is for $-0.6 < {\rm cos}\,
\theta_{\rm \,c.m.}^{\,K_S} < -0.4$, the right side is for $0.0 < {\rm cos}\,\theta_{\rm \,c.m.}^{\,K_S} < 0.2$ (according
to Table~\ref{Table:KS-Statistics}).}
\label{Figure:MM_Q_factor_KS_examples}
\end{center}
\end{figure}
Figures~\ref{Figure:MM_Q_factor_KS_I}\,-\,\ref{Figure:MM_Q_factor_KS_V} show the complete set of invariant
$\pi^+\pi^-$ mass distributions (left) and the corresponding $p\pi^0$ mass distributions (right) for 100-MeV-wide
incident-photon energy bins in the range $E_\gamma \in [\,1100,\,3000\,]$~MeV (full statistics used in this analysis).
Moreover, Table~\ref{Table:KS-Statistics} shows the total number of events (as a sum over all $Q$ values) for all
100-MeV-wide energy bins and for two selected angle bins. Finally, Fig.~\ref{Figure:MM_Q_factor_KS_examples}
presents example distributions of $E_\gamma$~Bins~4 $\&$ 5 (shown in Table~\ref{Table:KS-Statistics}).\par\smallskip
Note that a full set of Q~values for all events is not necessarily unique. If the $Q$~values are determined for the
$K_S$, then the weighted $\pi^+\pi^-$~mass distribution will show a clear separation of $K_S$~signal and background.
However, the $p\pi^0$~mass distribution weighted with the same $Q$~values will still exhibit some background
under the $\Sigma^+$~signal. The same is true if the $Q$~values are determined for the $\Sigma^+$, in which case
some background under the $K_S$ will be observed. For a counting experiment like a cross-section measurement,
either approach can be used. For an analysis however which requires the full event information, a more sophisticated
method would be needed, e.g. a simultaneous fit of both mass distributions.\par\smallskip
The measurement of the $\Sigma^+$~recoil polarization was based on the asymmetry between the proton count
rate above and below the reaction plane, taken in the $\Sigma^+$~rest frame (for more details, see
Section~\ref{Subsection:HyperonPolarization}). For this reason, we used the invariant $p\pi^0$~mass as the reference
coordinate in the determination of $Q$~values for both the final cross sections and the polarization observable.
The (kinematic) decay information -- the crucial opening angle between the proton and the $\pi^0$ -- was added
to the distance metric (Table~\ref{Table:non_ref_cor_KS}). The $K_S$-peak-beased $Q$~values were used to
cross-check our final cross-section results.
\begin{table}[t!]
\addtolength{\extrarowheight}{2pt}
\begin{center}
\begin{tabular}{c|r|r|r}
\hline
& & &\\
\rb{Energy Bin} & \rb{$\#$ of Events in $E_\gamma$~Bin}
& \rb{$-0.6 < {\rm cos}\,\theta_{\rm c.m.}^{K_S} < -0.4$} & \rb{$0.0 < {\rm cos}\,\theta_{\rm c.m.}^{K_S} < 0.2$}\\\hline\hline
0 & 45918.5 & 992.7 & 7014.6\\\hline
1 & 248.0 & 17.5 & 44.1\\
2 & 3253.5 & 121.2 & 411.1\\
3 & 4024.9 & 148.0 & 559.6\\
4 & 5624.9 & 172.1 & 866.0\\
5 & 5684.8 & 75.8 & 842.9\\
6 & 5483.8 & 73.7 & 795.9\\
7 & 3989.4 & 44.0 & 630.8\\
8 & 3304.3 & 61.3 & 603.1\\
9 & 2249.3 & 34.8 & 378.8\\
10 & 2305.7 & 35.2 & 428.2\\
11 & 2078.9 & 41.8 & 354.6\\
12 & 2022.2 & 37.5 & 342.6\\
13 & 1463.9 & 32.8 & 241.0\\
14 & 1086.5 & 18.9 & 145.5\\
15 & 925.2 & 14.5 & 111.3\\
16 & 604.4 & 18.1 & 86.1\\
17 & 828.9 & 25.5 & 91.7\\
18 & 431.0 & 9.5 & 47.9\\
19 & 309.0 & 10.7 & 33.4\\\hline
\end{tabular}
\caption{\it Total number of $\gamma p\to K_S\,\Sigma^+$ events in
100-MeV-wide energy bins (full statistics of Period~2), where Bin~0 denotes the full energy range $1.1 < E_\gamma < 3.0$~GeV,
and Bin~1 corresponds to $1.1 < E_\gamma < 1.2$~GeV, etc. The statistics is also given for two randomly-chosen angle bins.}
\label{Table:KS-Statistics}
\end{center}
\end{table}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MassDistribution_EBin_00_ABin_00.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MassDistributionSigma_EBin_00_ABin_00.pdf}\\
\hline
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin01_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin01_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin02_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin02_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin03_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin03_Per02.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-$ distributions (left column) and the corresponding $p\pi^0$ distributions (right column)
for the reaction $\gamma p\to K_S\,\Sigma^+$. Shown are the full statistics (top row) and 100-MeV-wide energy bins starting
at $E_\gamma\in [1.1,\,1.2]$~GeV (second row), $E_\gamma\in [1.2,\,1.3]$~GeV (third row), etc.}
\label{Figure:MM_Q_factor_KS_I}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin04_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin04_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin05_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin05_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin06_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin06_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin07_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin07_Per02.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-$ distributions (left column) and the corresponding $p\pi^0$ distributions (right column)
for the reaction $\gamma p\to K_S\,\Sigma^+$. Shown are 100-MeV-wide energy bins starting at $E_\gamma\in
[1400,\,1500]$~MeV (top row), $E_\gamma\in [1500,\,1600]$~MeV (second row), etc.}
\label{Figure:MM_Q_factor_KS_II}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin08_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin08_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin09_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin09_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin10_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin10_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin11_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin11_Per02.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-$ distributions (left column) and the corresponding $p\pi^0$ distributions (right column)
for the reaction $\gamma p\to K_S\,\Sigma^+$. Shown are 100-MeV-wide energy bins starting at $E_\gamma\in
[1800,\,1900]$~MeV (top row), $E_\gamma\in [1900,\,2000]$~MeV (second row), etc.}
\label{Figure:MM_Q_factor_KS_III}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin12_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin12_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin13_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin13_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin14_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin14_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin15_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin15_Per02.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-$ distributions (left column) and the corresponding $p\pi^0$ distributions (right column)
for the reaction $\gamma p\to K_S\,\Sigma^+$. Shown are 100-MeV-wide energy bins starting at $E_\gamma\in
[2200,\,2300]$~MeV (top row), $E_\gamma\in [2300,\,2400]$~MeV (second row), etc.}
\label{Figure:MM_Q_factor_KS_IV}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin16_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin16_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin17_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin17_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin18_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin18_Per02.pdf}\\
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin19_Per01.pdf} &
\includegraphics[width=0.47\textwidth,height=0.29\textwidth]{g12_figures/KS-Masses/MMhisto_NewEBin19_Per02.pdf}\\
\end{tabular}
\end{center}
\caption{\it Invariant $\pi^+\pi^-$ distributions (left column) and the corresponding $p\pi^0$ distributions (right column)
for the reaction $\gamma p\to K_S\,\Sigma^+$. Shown are 100-MeV-wide energy bins starting at $E_\gamma\in
[2600,\,2700]$~MeV (top row), $E_\gamma\in [2700,\,2800]$~MeV (second row), etc.}
\label{Figure:MM_Q_factor_KS_V}
\end{figure}
\clearpage
\subsubsection{\texorpdfstring{The Reaction $\gamma p\to p\,\phi$}
{The phi Reaction}}
\label{Subsubsection:Q-Factor-phi}
The reconstruction of the $\phi$~meson was based on the $K^+ K^-$~decay mode and therefore, the invariant
$K^+ K^-$~mass was used as the reference coordinate.\par\smallskip
\noindent
\bl{Under construction ... Material will be added in the second round of this review.}
\clearpage