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\author{Mathieu Ouillon}
\date{\today}


\begin{document}

\title{{RG-D Common Analysis Note}\\{\normalsize{\itshape Electron and pion cuts for Run Group D}}}
\affiliation{
Mississippi State University\\
}
\emailAdd{ouillon@jlab.org}
\maketitle

\newpage
\pagestyle{fancynotes}


\section{Electron cuts}
\label{sec:electron-cuts}
In this section, we detail the particle identification and selection of electrons in the CLAS12 detector. Electrons are measured in the Forward Detector (FD); charged-pion identification is documented separately in \cref{sec:pion-cuts}. Particle identification and refined selection follow procedures developed by Run Group A.

Data has been taken from Pass 1 with a beam energy of 10.54 GeV and has been produced with CLAS12 software \href{https://github.com/JeffersonLab/coatjava/releases/tag/13.0.0}{COATJAVA 13.0.0}. The data are available at \texttt{/cache/clas12/rg-d/production/pass1/}.

\subsection{Particle identification}
\label{sec:electron-pid}

The electron is typically the first particle required to define an event for physics analysis. To accurately identify electron candidates, a series of selection criteria, or cuts, is applied to detector responses, specifically targeting negatively charged tracks. These cuts are designed to distinguish electrons from minimum ionizing particles (MIPs), such as negatively charged pions ($\pi^-$). Additionally, the Event Builder assigns a classification (\texttt{status}) based on the detector system in which the electron track is located.

\subsubsection{Event Builder requirements}
\label{sec:eb-requirements}

In the initial step, the Event Builder assigns electron or positron identification ($e^- \rightarrow 11,e^+ \rightarrow -11$) to tracks that exhibit appropriate responses in both the High Threshold Cherenkov Counter (HTCC) and the Electromagnetic Calorimeter (ECAL), following the criteria outlined in \cref{tab:electron-requierement}. Additionally, the EB requires an associated matched hit in FTOF, either in panel-1b or, if there is no panel-b hit, in panel-1a.


\begin{table}[hbt]
  \centering
\begin{tabularx}{\textwidth}{|l|l|X|}
\hline
\multicolumn{1}{|c|}{\textbf{Cut}} & \multicolumn{1}{c|}{\textbf{Requirement}} & \multicolumn{1}{c|}{\textbf{Purpose}} \\ \hline
Charge                             & $Q = -1$                                   & Select negative tracks                \\ \hline
HTCC Photoelectrons                & $N_{\text{phe}} > 2$                       & Reject charged hadrons                \\ \hline
PCAL Energy                        & $E_{\text{PCal}} > 60$ MeV                 & Reject MIPs                           \\ \hline
Sampling Fraction                  & $\pm5\sigma$ vs $E_{\text{Tot,dep}}$       & Electromagnetic shower identification \\ \hline
Track Matching                     & Geometric matching                         & Ensure correct track      \\ \hline
\end{tabularx}
 \caption{EB electron ($\texttt{pid} = 11$) assignment requirements. The EB sampling fraction is parameterized based as a function of the total energy deposited in the calorimeters. Electrons defining the event start time are prioritized to the first column of the \texttt{REC::Particle} data structure (bank) and have a negative status.}
  \label{tab:electron-requierement}
\end{table}

Since CLAS12 can detect particles across a broad kinematic range, electron detection is restricted to the Forward Detector. This restriction is implemented by selecting events with a negative status in the range (-4000, -2000]. A negative status indicates that the track contributed to the event trigger. The particle status reflects the detector topology and is determined by summing the numerical values assigned to individual detector hits:
\begin{equation*}
	\texttt{status} = 1000 \times \text{FT} + 2000 \times \text{FD} + 4000 \times \text{CD} + 100  \times N_{\text{scint}} + 10  \times N_{\text{cal}} + 1  \times N_{\text{cher}} 
\end{equation*}
With the FT subsystem being 1 if used, else 0, same for CD and FD for the central subsystem and forward subsystem, respectively.

The charge of a particle determines its curvature as it moves through the toroidal magnetic field. Depending on the field polarity, particles experience inbending or outbending deflections along the polar angle. Electrons, being negatively charged, bend inwardly during inbending configurations and outwardly during outbending configurations.

\subsubsection{HTCC photoelectron cut}
\label{sec:htcc-cut}

The High Threshold Cherenkov Counter (HTCC) is used to minimize negative pion contamination in electron samples for candidate tracks below 4.5 GeV momentum. Below this momentum threshold, selecting electrons based on the number of detected photo-electrons ($N_{\text{phe}}$) is sufficient. Electron candidate tracks produce more than 2 photo-electrons (see \cref{fig:nphe-OB} and \cref{fig:nphe-IB}).

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\dimexpr\textwidth+\marginparwidth+\marginparsep\relax]{images/electron/plots_event_builder/nphe_neg_all_targets_OB.pdf}
  \caption{HTCC photo-electron distribution for all negative tracks in the Forward Detector in outbending polarity for the torus, apply cut at $N_{\text{phe}} > 2$ for pion rejection.}
  \label{fig:nphe-OB}
\end{figure}


\subsubsection{PCAL energy cut}
\label{sec:pcal-cut}

Two calorimetric cuts discriminate between electrons and minimum ionizing particles (MIPs), primarily negative pions. The first cut utilizes energy deposition in the preshower calorimeter (PCAL) and electromagnetic calorimeter (ECAL), with the latter divided into inner and outer regions. Electrons develop extended electromagnetic showers and deposit significant energy, while pions deposit consistently lower energy as MIPs. This distinct energy signature enables electron selection through a 0.06 GeV threshold cut, rejecting pions below this value (see \cref{fig:EPCal-comparison}).


\begin{figure}[hbt]
  \centering
  \begin{subfigure}[b]{0.48\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/electron/plots_event_builder/EPCal_vs_ECal_S1_all_LD2_OB.pdf}
    \caption{All negative tracks}
    \label{fig:EPCal-before}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.48\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/electron/plots_event_builder/EPCal_vs_ECal_S1_nphe_sf_LD2_OB.pdf}
    \caption{After $N_{\text{phe}}$ and SF cuts}
    \label{fig:EPCal-after}
  \end{subfigure}
  \caption{PCAL minimum energy cut $E_{\text{PCal,dep}} > 60$ MeV (black line), for sector 1, to remove pions. (a) All negative tracks. (b) Negative tracks which pass the conditions on the number of photoelectrons in the HTCC and on the sampling fraction in the calorimeter.}
  \label{fig:EPCal-comparison}
\end{figure}

In the RG-A Common Analysis Note, a fixed threshold of 0.07 GeV was used for the PCAL energy cut. However, as shown in \cref{fig:EPCal-cut_fit}, this threshold was found to be too high for RG-D, leading to the exclusion of valid electron tracks. Indeed a cut a 0.06 GeV is already equivalent to a $7\sigma$ cut on the PCAL energy distribution for $\pi^-$. The $\sigma$ have been obtained by fitting with a Gaussian the top of the peak of the PCal energy distribution after the $N_{\text{phe}}$ and SF cuts.

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\dimexpr\textwidth+\marginparwidth+\marginparsep\relax]{images/electron/plots_event_builder/EPCal_S1_nphe_sf.pdf}
  \caption{PCAL minimum energy cut $E_{\text{PCal,dep}} > 70$ MeV (purple line) used in RG-A, for sector 1, to remove pions. The orange line is the $7\sigma$ cut on the PCAL energy distribution for $\pi^-$.}
  \label{fig:EPCal-cut_fit}
\end{figure}

\subsubsection{Sampling fraction vs deposited energy}
\label{sec:sf-energy-cut}

In addition to the minimum PCAL energy threshold cut, a sampling fraction (SF) cut is applied to further eliminate negative pion tracks. The sampling fraction is defined as the ratio of the total deposited energy
\begin{equation}
	\text{SF} = \frac{E_{\text{PCAL,dep}} + E_{\text{Inner,dep}} + E_{\text{Outer,dep}}}{p}
	\label{eq:SF-definition}
\end{equation}
to the reconstructed track momentum $p$. For electrons, this fraction remains nearly constant at $\sim 0.25$ across all momenta, indicating that the deposited energy scales proportionally with the electron's momentum. In contrast, negative pions, as minimum ionizing particles (MIPs), deposit a relatively constant amount of energy regardless of momentum.
To accurately select electrons, tracks are required to be within $\pm 5\sigma$ of the sampling fraction as a function of total deposited energy. This is achieved by binning the data in $E_{\text{Tot,dep}}$, fitting a Gaussian to the sampling fraction distribution in each bin to extract the mean ($\mu_b$) and standard deviation ($\sigma_b$), and then parameterizing these values using \cref{eq:mean-sigma-SF}. The resulting functions define the energy-dependent sampling fraction cut.
\begin{align}
  \mu_{\text{SF}} = a(b+c/E_{\text{Tot,dep}} + d/E_{\text{Tot,dep}}^2) \\
  \sigma_{\text{SF}} = e(b+f/E_{\text{Tot,dep}} + g/E_{\text{Tot,dep}}^2)
  \label{eq:mean-sigma-SF}	
\end{align}
The parameters $a,b,c,d,e,f$ and $g$  can be found in the \href{https://clasweb.jlab.org/cgi-bin/ccdb/versions?table=/calibration/eb/electron_sf}{CCDB for EB}.

The electron tracks are selected $\pm5\sigma$ from the sampling fraction as a function of the $E_{\text{Tot,dep}}$. as seen in \cref{fig:SF-neg}.

\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_event_builder/Etot_vs_SF_S1_cut_LD2_OB.pdf}
  \caption{Sampling fraction, the ratio of the sum of energy deposited in the calorimeter layers to reconstructed track momentum, vs deposited energy for each pre-shower calorimeter sector. The electron candidate tracks shown are with the $N_{\text{phe}}$ and $E_{\text{PCal}}$ cuts applied. The red line is the result of the fit to the mean, and black lines are at $\pm5\sigma$. }
  \label{fig:SF-neg}
\end{figure}

\subsubsection{Sampling fraction vs momentum}
\label{sec:sf-momentum-cut}

To improve the electron identification, another cut on the sampling fraction as a function of the momentum has been developed. For that, the SF is binned in momentum, and the resulting SF distribution is fit with a Gaussian distribution to extract the mean and sigma for a given momentum range. The momentum or energy dependence of the mean and sigma can then be fit to \cref{eq:fit-SF-p}
\begin{align}
	\mu_{\text{SF}}=a_{\mu}+b_{\mu}p+c_{\mu}p^2+d_{\mu}p^3 \\
	\sigma_{\text{SF}}=a_{\sigma}+b_{\sigma}p+c_{\sigma}p^2+d_{\sigma}p^3
	\label{eq:fit-SF-p}
\end{align}
The cut is then defined as $\mu_{\text{SF}} + 3\sigma_{\text{SF}}$ as seen in \cref{fig:CxC:p-vs-SF} for the carbon target, in \cref{fig:LD2:p-vs-SF} for the LD2 target, and in \cref{fig:CuSn:p-vs-SF} for the copper and tin target (CuSn). The fits to the SF mean function are given in red line and the sigma function upper and lower limits as a black line.

\begin{figure}[hbt]
  \begin{minipage}{\dimexpr\textwidth+\marginparwidth+\marginparsep\relax}
    \centering
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_sampling_fraction/sector_1_LD2_OB/p_vs_SF_S1_with_fit.pdf}
      \caption{LD2}
      \label{fig:LD2:p-vs-SF}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_sampling_fraction/sector_1_CxC_OB/p_vs_SF_S1_with_fit.pdf}
      \caption{C}
      \label{fig:CxC:p-vs-SF}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_sampling_fraction/sector_1_CuSn_OB/p_vs_SF_S1_with_fit.pdf}
      \caption{CuSn}
      \label{fig:CuSn:p-vs-SF}
    \end{subfigure}
  \end{minipage}
  \caption{Sampling fraction vs momentum for electron candidate tracks in outbending polarity for different targets: (a) LD2, (b) carbon, and (c) copper and tin (CuSn). The red lines is the fit to the mean and the $\pm3.5\sigma$ for the sector 1. The blue lines are the same fit but for the sector average.}
  \label{fig:p-vs-SF-all-targets}
\end{figure}

This cut is obtained by slicing the SF distribution in 50 momentum bins from 0 to 11 GeV, fitting the peak from half of the height on the left to half of the height on the right with a Gaussian to each slice to extract the mean and sigma, and then fitting these values to \cref{eq:fit-SF-p}.

The plot in \cref{fig:SF-LD2-comparison} shows a comparison between the sampling fraction mean and standard deviation as a function of momentum for the LD2 target in outbending polarity for different sectors and the overall distribution. The same plots are available for the CxC and CuSn targets in the appendix. 

\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_sampling_fraction/OB/sectors_comparison_LD2.pdf}
  \caption{Comparison of the sampling fraction mean and sigma as a function of momentum for the LD2 target in outbending polarity for different sectors and the overall distribution.}
  \label{fig:SF-LD2-comparison}
\end{figure}

The \cref{fig:SF-all-targets-comparison} shows a comparison between the sampling fraction mean and standard deviation as a function of momentum for all the targets in outbending polarity for the average over all sectors.

\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_sampling_fraction/OB/targets_comparison_average.pdf}
  \caption{Comparison of the sampling fraction mean and sigma as a function of momentum for all the targets in outbending polarity for the average over all sectors.}
  \label{fig:SF-all-targets-comparison}
\end{figure}

The sampling fraction behavior also depends on the magnetic field polarity of the torus. A comparison between inbending (IB) and outbending (OB) polarities for the LD2 target is shown in \cref{fig:SF-polarity-comparison-LD2}. Both polarities exhibit similar sampling fraction mean values and sigma dependencies on momentum, indicating consistent electron identification performance across different field configurations. The slight differences observed are accounted for in the polarity-dependent parameterizations used in the analysis.

\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_sampling_fraction/polarity_comparison_LD2.pdf}
  \caption{Comparison of the sampling fraction mean and sigma as a function of momentum for the LD2 target between inbending and outbending polarities, averaged over all sectors.}
  \label{fig:SF-polarity-comparison-LD2}
\end{figure}

\subsubsection{Momentum cut}
\label{sec:electron-momentum}

A cut on the momentum of the electron is applied. The cut is defined as $p > 2.0$ GeV/$c$. It is applied to ensure a clean trigger-electron selection by rejecting low-momentum candidates whose reconstruction and trigger efficiency are less reliable.

\subsubsection{Triangular cut (not applied)}
\label{sec:triangular-cut}

The RG-A Common Analysis Note also employed a so-called "triangular" cut. This cut was designed to reduce pion contamination in the electron sample and was developed using the sampling fractions of the preshower calorimeter and the inner calorimeter. However, with our sampling fraction versus momentum cut, DC fiducial cuts, and calorimeter cuts, the pion contamination is already very low, and this cut would remove valid electron tracks. Therefore, this cut is not applied in this analysis. \Cref{fig:triangular-before} shows the distribution before any refinement cuts (only the \texttt{pid} and \texttt{status} cuts applied), and \cref{fig:triangular-after} shows the distribution after all refinement cuts have been applied.

\begin{figure}[hbt]
  \centering
  \begin{subfigure}[b]{0.48\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/electron/LD2_electron_SFIn_vs_SFPcal_5_6_raw.pdf}
    \caption{Before refinement cuts}
    \label{fig:triangular-before}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.48\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/electron/LD2_electron_SFIn_vs_SFPcal_5_6_cuts.pdf}
    \caption{After all refinement cuts}
    \label{fig:triangular-after}
  \end{subfigure}
  \caption{Sampling fraction of the preshower calorimeter vs sampling fraction of the inner calorimeter for electron candidate tracks in the LD2 target in outbending polarity. (a) Before any refinement cuts. (b) After all refinement cuts.}
  \label{fig:triangular-comparison}
\end{figure}

\subsubsection{Vertex cut}
\label{sec:electron-vz}

Vertex cuts are applied to ensure that selected electron tracks originate from the target region, suppressing background from interactions outside the target. The vertex position along the beamline, $V_z$, is used to define these cuts. For the solid-target variants (CuSn and CxC), the raw $V_z$ distribution is distorted by sector-, $\phi$-, momentum-, and $\theta$-dependent biases that smear the two cell-wall peaks. We therefore apply a per-event $V_z$ correction first and then fit the corrected distribution with a Voigt profile.

\paragraph{Sigmoid-blended dual-target $V_z$ correction}
\label{sec:vz-correction}

The raw $V_z$ distribution carries $(s, p, \phi, \theta)$-dependent biases --- from beam-line alignment, drift-chamber survey residuals, and the toroidal field map --- that broaden each cell-wall peak by 20--30\% and shift its centroid by up to $0.5$~cm depending on the kinematics. We remove these biases per event by mapping $V_z^{\text{raw}}$ onto a common reference frame in which the two cell-wall peaks (Sn / Cu for the CuSn target and C$_1$ / C$_2$ for CxC) sit at fixed positions.

For every $(s, p, \phi)$ cell, the raw peak positions $\mu_{\text{lo,hi}}(\theta)$ and widths $\sigma_{\text{lo,hi}}(\theta)$ are extracted from a double-Gaussian fit in each $\theta$ slice and parameterized as a third-order polynomial in $\theta$. References are taken from the inclusive sample at $\theta \geq 20^\circ$ and $p \geq 5$ GeV/$c$. Each event's $V_z$ is then standardized against both predicted peaks and remapped to the references; the two resulting branches are blended with the logistic posterior of a two-Gaussian mixture so that the upstream branch dominates near the upper peak, the downstream branch near the lower peak, and the two interpolate smoothly across the valley:
\begin{equation}
	V_z^{\text{corr}} = V_z^{\text{corr,hi}} + \sigma_{\text{sgm}}\!\left(\tfrac{1}{2}\left(z_{\text{hi}}^2 - z_{\text{lo}}^2\right) + \ln\!\left(\sigma_{\text{hi}}/\sigma_{\text{lo}}\right)\right) \cdot \left(V_z^{\text{corr,lo}} - V_z^{\text{corr,hi}}\right),
	\label{eq:vz-blend}
\end{equation}
with $z_{\text{lo,hi}} = \left(V_z^{\text{raw}} - \mu_{\text{lo,hi}}(\theta)\right)/\sigma_{\text{lo,hi}}(\theta)$ and $\sigma_{\text{sgm}}(x) = 1/(1 + e^{-x})$. \Cref{fig:vz-correction-method} illustrates the reference region and the per-cell fits for the CuSn outbending variant.

\begin{figure}[hbt]
  \centering
  \includegraphics[page=1,width=\dimexpr\textwidth+\marginparwidth+\marginparsep\relax]{images/electron/vz/vz_correction_methodology_cusn_ob.pdf}
  \caption{Reference-region density and per-cell fits for the CuSn outbending variant. The same pipeline is applied to every $(s, p, \phi)$ cell of every variant.}
  \label{fig:vz-correction-method}
\end{figure}

\Cref{fig:vz-migration} validates that the correction behaves as a smooth mapping rather than a re-binning artefact. The raw $V_z$ axis is split into 10 contiguous buckets per sector (top row); the same events, after correction, are restacked using the \emph{original} bucket colour (bottom row). Each coloured stack remains compact and contracts toward the reference peak positions, exactly the $(s, p, \phi, \theta)$-dependent behaviour the correction is designed to remove.

\begin{figure}[hbt]
  \centering
  \includegraphics[page=42,width=\dimexpr\textwidth+\marginparwidth+\marginparsep\relax]{images/electron/vz/vz_correction_methodology_cusn_ob.pdf}
  \caption{Per-sector $V_z$ migration for the CuSn outbending variant. \textbf{Top row:} raw $V_z$ split into 10 contiguous buckets per sector. \textbf{Bottom row:} the same events after the per-event correction of \cref{eq:vz-blend}, restacked using the original bucket colour.}
  \label{fig:vz-migration}
\end{figure}

\paragraph{Voigt-profile fit on corrected $V_z$}
\label{sec:vz-voigt}

After correction, the cell-wall peaks are well separated and approximately symmetric, but residual non-Gaussian tails from multiple scattering and binning persist. We model the corrected 1D $V_z$ distribution as the sum of two area-normalized Voigt profiles plus a flat pedestal:
\begin{equation}
	f(V_z) = A_{\text{lo}}\, V\!\left(V_z - \mu_{\text{lo}}; \sigma_{\text{lo}}^G, \gamma_{\text{lo}}^L\right) + A_{\text{hi}}\, V\!\left(V_z - \mu_{\text{hi}}; \sigma_{\text{hi}}^G, \gamma_{\text{hi}}^L\right) + C
	\label{eq:vz-voigt}
\end{equation}
where the Voigt profile $V(x; \sigma^G, \gamma^L)$ is the convolution of a Gaussian core (width $\sigma^G$) with a Lorentzian tail (HWHM $\gamma^L$). The Olivero--Longbothum approximation is used to convert each fit to its full width at half maximum,
\begin{equation}
	\text{FWHM} \approx 0.5346 \cdot 2\gamma^L + \sqrt{0.2166 \cdot \left(2\gamma^L\right)^2 + \left(2\sqrt{2\ln 2}\,\sigma^G\right)^2},
	\label{eq:voigt-fwhm}
\end{equation}
from which we define a Gaussian-equivalent width $\sigma^{\text{eff}} = \text{FWHM}/(2\sqrt{2\ln 2})$. The selected vertex window for each peak is then $\mu \pm 3\sigma^{\text{eff}}$.

The Voigt fit is preferred over a plain Gaussian (which inflates $\sigma$ to absorb the tails and undershoots the peak) and over a generalised-normal or core+tail double-Gaussian alternative (which require more parameters for comparable $\chi^2$). \Cref{fig:vz-voigt-grid} shows the Voigt fit for every (variant $\times$ species) combination.

\begin{figure}[hbt]
  \centering
  \includegraphics[page=9,width=\dimexpr\textwidth+\marginparwidth+\marginparsep\relax]{images/electron/vz/vz_fit_voigt_summary.pdf}
  \caption{Voigt-profile fit on the corrected $V_z$ distribution for every variant (CuSn / CxC $\times$ OB / IB) and species ($e^-$, $\pi^-$, $\pi^+$). Each panel shows the data (black), the fit (green), the fitted parameters $\mu_{\text{lo,hi}}$, $\sigma^G_{\text{lo,hi}}$, $\gamma^L_{\text{lo,hi}}$, the Olivero--Longbothum FWHM, and the $\mu \pm 3\sigma^{\text{eff}}$ cut edges (dashed lines). The selected window is centred on each cell-wall peak and excludes the inter-peak valley.}
  \label{fig:vz-voigt-grid}
\end{figure}

The corresponding electron cut windows are summarized in \cref{tab:Vz-cuts}. Cu and Sn (or C$_1$ and C$_2$) windows correspond to the upstream and downstream cell walls, respectively, in the corrected vertex frame.

\begin{table}[htb]
\centering
\small
\caption{Electron Voigt-fit parameters and $\mu \pm 3\sigma^{\text{eff}}$ vertex windows for the four solid-target variants, in the corrected $V_z$ frame. The first window selects the upstream cell wall (Sn for CuSn; C$_1$ for CxC); the second selects the downstream cell wall (Cu / C$_2$). For the LD2 target a fixed cut $V_z \in [-15, 5]$~cm is retained to encompass the full target length without requiring the per-event correction.}
\label{tab:Vz-cuts}
\makebox[\textwidth][c]{%
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\textbf{Variant} & \textbf{$\mu_{\text{lo}}$ [cm]} & \textbf{$\sigma^{\text{eff}}_{\text{lo}}$ [cm]} & \textbf{$\mu_{\text{hi}}$ [cm]} & \textbf{$\sigma^{\text{eff}}_{\text{hi}}$ [cm]} & \textbf{Cut windows [cm]} \\
\hline
CuSn OB & $-2.916$ & $0.370$ & $-7.861$ & $0.415$ & $[-4.03,\, -1.81]$; $[-9.11,\, -6.62]$ \\
\hline
CuSn IB & $-2.812$ & $0.315$ & $-7.765$ & $0.321$ & $[-3.76,\, -1.87]$; $[-8.73,\, -6.80]$ \\
\hline
CxC OB  & $-2.906$ & $0.373$ & $-7.887$ & $0.395$ & $[-4.02,\, -1.79]$; $[-9.07,\, -6.70]$ \\
\hline
CxC IB  & $-2.713$ & $0.307$ & $-7.670$ & $0.313$ & $[-3.63,\, -1.79]$; $[-8.61,\, -6.73]$ \\
\hline
\end{tabular}%
}
\end{table}

\FloatBarrier
\subsection{Fiducial cuts}
\label{sec:electron-fiducial}

To ensure uniform detector acceptance and reduce edge effects, fiducial cuts are applied to the electron candidates. These cuts define a geometric region within the detector where the response is well understood and consistent. The fiducial cuts are for the drift chambers (DC) and the electromagnetic calorimeter (ECAL).

\subsubsection{Calorimeters fiducial cuts}
\label{sec:cal-fiducial}
Fiducial cuts for the calorimeters are applied in the local coordinate system of each sector ($u, v$ and $w$). The RG-A Common Analysis Note provides a detailed description of the calorimeter fiducial cuts, which are implemented here without modification. The cuts are designed to exclude regions near the edges of the calorimeter sectors where the response may be non-uniform or poorly understood. For electron candidates, the fiducial cuts are $v > 9$ cm and $w > 9$ cm. It is done because the PCal bar are 4.5 cm wide, and to have a good clusters formation and reconstruction, the hit should have at least two bars. The profile of the distributions of sampling fraction as a function of $v$ and $w$ for electron candidate tracks in the LD2 target in outbending polarity are shown in \cref{fig:ECAL-fiducial-cut}. Similar plots for the CxC and CuSn targets are available in the appendix. The SF start to drop and fluctuate below 9 cm for both $v$ and $w$ coordinates.

\begin{figure}[hbt]
  \centering
  \begin{subfigure}[b]{0.48\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/electron/plots_calorimeter_fiducials/lv_vs_sf_LD2_all_sectors.pdf}
    \caption{SF vs $v$}
    \label{fig:ECAL-fiducial-cut-lv}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.48\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/electron/plots_calorimeter_fiducials/lw_vs_sf_LD2_all_sectors.pdf}
    \caption{SF vs $w$}
    \label{fig:ECAL-fiducial-cut-lw}
  \end{subfigure}
  \caption{Profile of the 2D distribution SF vs $v$ (a) and $w$ (b) for electron candidate tracks in the LD2 target in outbending polarity for each sector and the overall distribution.}
  \label{fig:ECAL-fiducial-cut}
\end{figure}

The comparison of the profile of the 2D distribution SF vs $v$ or $w$ for electron candidate tracks in outbending polarity for the three targets: LD2, CxC and CuSn is shown in \cref{fig:ECAL-fiducial-cut-comp-target}. The three targets present similar distributions and justify the use of the same fiducial cuts. 

\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.5\textwidth]{images/electron/plots_calorimeter_fiducials/target_comparison/lv_vs_sf_S1_target_comparison.pdf}
  \caption{Comparison of the profile of the 2D distribution SF vs $v$ or $w$ for electron candidate tracks in outbending polarity for the three targets: LD2, CxC and CuSn.}
  \label{fig:ECAL-fiducial-cut-comp-target}
\end{figure}

\FloatBarrier
\subsubsection{DC fiducial cuts}
\label{sec:dc-fiducial}
Fiducial cuts for the drift chambers are applied using the local variable \texttt{edge} defined in the \texttt{REC::Traj} bank. This variable represents the distance between the virtual edge of the DC (defined in the simulation) and the hit position. To define the cut value, the profile of the 2D distribution of the track $\chi^2/\text{ndf}$ versus \texttt{edge} for electron candidate tracks with outbending polarity is used for the three targets.

The methodology for determining the fiducial cut position is illustrated in \cref{fig:DC-variance-method} and employs a moving variance method. This approach identifies optimal fiducial cut positions by analyzing local stability in the $\chi^2/\text{ndf}$ distribution as a function of edge distance. The method applies a sliding window (typically 5 points) across the $\chi^2/\text{ndf}$ profile and calculates the variance within each window. In regions far from the detector edges, where tracking performance is optimal and stable, the $\chi^2/\text{ndf}$ values exhibit low variance. However, as tracks approach the physical boundaries of the drift chambers, detector inefficiencies and edge effects cause the $\chi^2/\text{ndf}$ to rise unpredictably, resulting in significantly increased local variance. By scanning from the outer edge inward and identifying the first position where variance exceeds a predefined threshold (typically 0.05), the method systematically determines the boundary between the stable detector region and the degraded edge region. This approach is particularly robust because it captures the transition point where tracking quality begins to deteriorate, rather than relying on arbitrary $\chi^2/\text{ndf}$ thresholds that may vary across different detector regions or experimental conditions.

\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/variance_method_explanation.pdf}
  \caption{Illustration of the moving variance method used to determine DC fiducial cut positions. A sliding window (typically 5 points) calculates local variance across the $\chi^2/\text{ndf}$ profile. The method scans from the outer edge inward, identifying the first position where the local variance exceeds the threshold of 0.05, marking the transition from the stable detector region to the degraded edge region.}
  \label{fig:DC-variance-method}
\end{figure}

The resulting fiducial cut for sector 1 and region 1 for the LD2 target is shown in \cref{fig:DC-fiducial-cut-comp-target}. The same procedure is applied for each sector and each target. The resulting profiles for each sector for regions 1, 2, and 3 are shown in \cref{fig:DC-fiducial-cut-comp-sector-region1}, \cref{fig:DC-fiducial-cut-comp-sector-region2}, and \cref{fig:DC-fiducial-cut-comp-sector-region3}, respectively, for the LD2 target.

\begin{figure}[hbt]
  \centering
  \makebox[\textwidth][c]{%
    \hspace{2cm}
    \begin{minipage}{\dimexpr\textwidth+\marginparwidth+\marginparsep\relax}
      \centering
      % Top row: Sector 1 Region 1 detail and Region 1 all sectors
      \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/fiducial_cut_Tracks_ElectronTracks_edge_vs_chi2ndf_r1_S1_1.pdf}
        \caption{Sector 1, Region 1 detail}
        \label{fig:DC-fiducial-cut-comp-target}
      \end{subfigure}
      \hfill
      \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/sector_comparison/sector_comparison_r1.pdf}
        \caption{Region 1 - all sectors}
        \label{fig:DC-fiducial-cut-comp-sector-region1}
      \end{subfigure}

      \vspace{0.5cm}

      % Bottom row: Region 2 and Region 3 all sectors
      \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/sector_comparison/sector_comparison_r2.pdf}
        \caption{Region 2 - all sectors}
        \label{fig:DC-fiducial-cut-comp-sector-region2}
      \end{subfigure}
      \hfill
      \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/sector_comparison/sector_comparison_r3.pdf}
        \caption{Region 3 - all sectors}
        \label{fig:DC-fiducial-cut-comp-sector-region3}
      \end{subfigure}
    \end{minipage}%
  }
  \caption{DC fiducial cuts for electron candidate tracks in outbending polarity for the LD2 target. (a) Application of the moving variance method for sector 1, region 1, showing the $\chi^2/\text{ndf}$ profile with the local variance calculation and the resulting fiducial cut position where variance exceeds the 0.05 threshold. (b-d) Profiles of $\chi^2/\text{ndf}$ vs \texttt{edge} for all sectors in regions 1, 2, and 3, respectively, with determined fiducial cut positions indicated.}
  \label{fig:DC-fiducial-cuts-combined}
\end{figure}

To validate the robustness of the DC fiducial cuts, the dependence of the $\chi^2/\text{ndf}$ profiles on electron momentum and scattering angle is investigated. The $\chi^2/\text{ndf}$ versus \texttt{edge} profiles are examined in different momentum bins (0.5-2 GeV, 2-4 GeV, 4-6 GeV, and $>$6 GeV) for each region, as shown in \cref{fig:DC-momentum-dep-r1,fig:DC-momentum-dep-r2,fig:DC-momentum-dep-r3}. The profiles remain consistent across momentum ranges, with the fiducial cut positions showing minimal momentum dependence. This indicates that the single fiducial cut value determined from the integrated momentum distribution is appropriate for all electron momenta.

\begin{figure}[hbt]
  \begin{minipage}{\dimexpr\textwidth+\marginparwidth+\marginparsep\relax}
    \centering
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/momentum_bins/momentum_comparison_r1.pdf}
      \caption{Region 1}
      \label{fig:DC-momentum-dep-r1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/momentum_bins/momentum_comparison_r2.pdf}
      \caption{Region 2}
      \label{fig:DC-momentum-dep-r2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/momentum_bins/momentum_comparison_r3.pdf}
      \caption{Region 3}
      \label{fig:DC-momentum-dep-r3}
    \end{subfigure}
  \end{minipage}
  \caption{Momentum dependence of DC fiducial cuts for the LD2 target in outbending polarity. The $\chi^2/\text{ndf}$ vs \texttt{edge} profiles are shown for different momentum bins in regions 1, 2, and 3, demonstrating consistent behavior across the full momentum range.}
  \label{fig:DC-momentum-dependence}
\end{figure}

Similarly, the angular dependence is studied by binning the data in polar angle $\theta$ ranges corresponding to different detector geometries. As shown in \cref{fig:DC-theta-dep-r1,fig:DC-theta-dep-r2,fig:DC-theta-dep-r3}, the $\chi^2/\text{ndf}$ profiles exhibit consistent behavior across different $\theta$ bins within each region. The slight variations observed are within the statistical uncertainties and do not require $\theta$-dependent fiducial cuts.

\begin{figure}[hbt]
  \begin{minipage}{\dimexpr\textwidth+\marginparwidth+\marginparsep\relax}
    \centering
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/theta_bins/theta_comparison_r1.pdf}
      \caption{Region 1}
      \label{fig:DC-theta-dep-r1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/theta_bins/theta_comparison_r2.pdf}
      \caption{Region 2}
      \label{fig:DC-theta-dep-r2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/LD2_OB/theta_bins/theta_comparison_r3.pdf}
      \caption{Region 3}
      \label{fig:DC-theta-dep-r3}
    \end{subfigure}
  \end{minipage}
  \caption{Angular dependence of DC fiducial cuts for the LD2 target in outbending polarity. The $\chi^2/\text{ndf}$ vs \texttt{edge} profiles are shown for different $\theta$ bins in regions 1, 2, and 3, showing consistent edge effects independent of scattering angle.}
  \label{fig:DC-theta-dependence}
\end{figure}

The DC fiducial cuts are also compared across the three different targets (LD2, CxC, and CuSn) for sector 1 in each region, as shown in \cref{fig:DC-target-comp-r1,fig:DC-target-comp-r2,fig:DC-target-comp-r3}. The profiles are nearly identical for all three targets, with fiducial cut positions differing by less than 0.1 cm on average (see \cref{tab:fiducial_cuts}). This similarity justifies using the same DC fiducial cut methodology across all targets.

\begin{figure}[hbt]
  \begin{minipage}{\dimexpr\textwidth+\marginparwidth+\marginparsep\relax}
    \centering
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/target_comparison/OB/target_comparison_r1_S1.pdf}
      \caption{Region 1}
      \label{fig:DC-target-comp-r1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/target_comparison/OB/target_comparison_r2_S1.pdf}
      \caption{Region 2}
      \label{fig:DC-target-comp-r2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.32\linewidth}
      \centering
      \includegraphics[width=\textwidth]{images/electron/plots_chi2ndf_edge/target_comparison/OB/target_comparison_r3_S1.pdf}
      \caption{Region 3}
      \label{fig:DC-target-comp-r3}
    \end{subfigure}
  \end{minipage}
  \caption{Target comparison of DC fiducial cuts for sector 1 in outbending polarity. The $\chi^2/\text{ndf}$ vs \texttt{edge} profiles for LD2, CxC, and CuSn targets are nearly identical across all three regions, demonstrating target-independent edge effects.}
  \label{fig:DC-target-comparison}
\end{figure}

The average fiducial cut positions for each region for the outbending polarity are summarized in \cref{tab:fiducial_cuts}.

\begin{table}[htbp]
\centering
\caption{Average Fiducial Cut Positions by Region for the outbending polarity.}
\label{tab:fiducial_cuts}
\begin{tabular}{|c|c|c|}
\hline
\textbf{Region 1} (cm) & \textbf{Region 2} (cm) & \textbf{Region 3} (cm) \\
\hline
 $1.88\pm0.16$ & $2.08 \pm 0.10$ & $8.62 \pm 0.55$  \\
\hline
\end{tabular}
\end{table}

\FloatBarrier
\subsection{Summary}
\label{sec:summary}

This subsection recapitulates all the cuts applied for electron identification and selection in RG-D analysis.
\begin{table}[htb]
\centering
\begin{tabular}{|l|l|l|}
\hline
\textbf{Cut Type} & \textbf{Cut Description} & \textbf{Value/Requirement} \\
\hline
Particle ID & Event Builder assignment & $\texttt{pid} = 11$ (electron) \\
\hline
Status & Forward Detector trigger & $-4000 < \texttt{status} \leq -2000$ \\
\hline
Momentum & Trigger-electron selection & $p > 2.0$ GeV/$c$ \\
\hline
SF vs Momentum & Momentum-dependent SF & $\mu_{\text{SF}} \pm 3.5\sigma$ vs $p$ \\
\hline
Calorimeter $v,w$ coordinate & Local coordinate cut & $v,w > 9$ cm \\
\hline
Drift Chamber Edge & Distance from DC edge & \texttt{edge} $> $ region dependent \\
\hline
Vertex $V_z$ & Target vertex position & Target dependent (see \cref{tab:Vz-cuts}) \\
\hline
\end{tabular}
\caption{Summary of all cuts applied for electron identification and selection in RG-D analysis.}
\label{tab:electron-cuts-summary}
\end{table}

The survival rates after each cut for the different targets are summarized in \cref{tab:cut_efficiencies}.
\begin{table}[htb]
\centering
\label{tab:cut_efficiencies}
\begin{tabular}{|l|l|c|c|c|}
\hline
\textbf{Target} & \textbf{Cut} & \textbf{Cumulative} & \textbf{Step} \\
\hline
\multirow{5}{*}{LD2} 
    & Momentum cut              & 1.0000 & 1.0000 \\
    & Sampling fraction cut     & 0.9714 & 0.9715 \\
    & PCAL cut                  & 0.9714 & 1.0000 \\
    & DC cut                    & 0.9522 & 0.9802 \\
    & Vz cut                    & 0.8995 & 0.9447 \\
\hline
\multirow{5}{*}{CxC} 
    & Momentum cut              & 1.0000 & 1.0000 \\
    & Sampling fraction cut     & 0.9740 & 0.9740 \\
    & PCAL cut                  & 0.9740 & 1.0000 \\
    & DC cut                    & 0.9550 & 0.9805 \\
    & Vz cut                    & 0.7779 & 0.8146 \\
\hline
\multirow{4}{*}{CuSn} 
    & Momentum cut              & 1.0000 & 1.0000 \\
    & Sampling fraction cut     & 0.9824 & 0.9824 \\
    & PCAL cut                  & 0.9823 & 0.9999 \\
    & DC cut                    & 0.9642 & 0.9816 \\
\hline
\end{tabular}
\caption{Electron survival rates for different targets}
\end{table}


\section{Pion cuts}
\label{sec:pion-cuts}
Pion identification and selection are necessary for the color transparency study and the nuclear TMDs analysis in RG-D. The pion selection is done by applying cuts on the reconstructed mass of the particle, calculated using the time-of-flight information from the time-of-flight detectors and the momentum from the drift chambers.

\subsection{$\Delta t$ vs $p$}
\label{sec:pion-deltat}
Pion identification relies on time-of-flight measurements to distinguish pions from other charged hadrons such as kaons and protons. The quantity $\Delta t$ is defined as the difference between the measured time-of-flight and the expected time-of-flight computed under the pion mass hypothesis:
\begin{equation}
	\Delta t = t_{\text{measured}} - t_{\text{start time}} - t_{\text{expected}}(\pi)
\end{equation}
where the expected time is calculated from the reconstructed momentum $p$ and the path length using the pion mass. For correctly identified pions, $\Delta t$ is centered near zero, while other particle species produce shifted distributions.

To develop a momentum-dependent $\Delta t$ cut, the two-dimensional distribution of $\Delta t$ versus momentum is analyzed separately for $\pi^+$ and $\pi^-$ candidates. The distribution is sliced into 30 momentum bins, and in each bin the $\Delta t$ projection is fitted with a Gaussian to extract the mean $\mu_{\Delta t}$ and standard deviation $\sigma_{\Delta t}$. The momentum dependence of the mean is parameterized with a second-order polynomial:
\begin{equation}
	\mu_{\Delta t}(p) = a_\mu p^2 + b_\mu p + c_\mu
	\label{eq:deltat-mean}
\end{equation}
The standard deviation is parameterized with a physically motivated timing-resolution model,
\begin{equation}
	\sigma_{\Delta t}(p) = \sqrt{\sigma_0^2 + \left(\frac{k}{p}\right)^2} + c_\sigma \, p
	\label{eq:deltat-sigma}
\end{equation}
with three free parameters: $\sigma_0$, $k$, and $c_\sigma$.

The pion selection cut is then defined as $\mu_{\Delta t}(p) \pm 3\sigma_{\Delta t}(p)$, keeping tracks within three standard deviations of the expected pion timing. The fitted parameters of \cref{eq:deltat-mean,eq:deltat-sigma} are listed in \cref{tab:deltat-coefficients} for outbending polarity and in \cref{tab:deltat-coefficients-ib} for inbending polarity.

\begin{table}[htb]
\centering
\small
\caption{Fitted parameters of the $\Delta t$ cut for each target and pion species in outbending polarity. Mean coefficients $(a_\mu, b_\mu, c_\mu)$ correspond to the second-order polynomial of \cref{eq:deltat-mean}; width parameters $(\sigma_0, k, c_\sigma)$ correspond to the resolution model of \cref{eq:deltat-sigma}. Cut edges at any momentum $p$ are obtained as $\mu_{\Delta t}(p) \pm n_\sigma \sigma_{\Delta t}(p)$, with the default $n_\sigma = 3$. The intrinsic resolution $\sigma_0 \approx 75$~ps is consistent across targets; $k$ is roughly twice as large for $\pi^-$ as for $\pi^+$, reflecting the larger multiple-scattering and nuclear-interaction contribution at low momentum for negative pions.}
\label{tab:deltat-coefficients}
\makebox[\textwidth][c]{%
\begin{tabular}{|l|c|r|r|r|r|r|r|}
\hline
\textbf{Target} & \textbf{Species} & \textbf{$a_\mu$} & \textbf{$b_\mu$} & \textbf{$c_\mu$} & \textbf{$\sigma_0$} & \textbf{$k$} & \textbf{$c_\sigma$} \\
\hline
\multirow{2}{*}{LD$_2$}
    & $\pi^+$ & $+0.000944$ & $-0.010445$ & $+0.046100$ & $+0.076738$ & $+0.054354$ & $-0.000929$ \\
    & $\pi^-$ & $-0.001640$ & $+0.009816$ & $-0.025958$ & $+0.080510$ & $+0.118046$ & $-0.001205$ \\
\hline
\multirow{2}{*}{CxC}
    & $\pi^+$ & $+0.001103$ & $-0.011081$ & $+0.046219$ & $+0.075224$ & $+0.054430$ & $-0.000447$ \\
    & $\pi^-$ & $-0.001866$ & $+0.010890$ & $-0.026872$ & $+0.078231$ & $+0.118102$ & $-0.000447$ \\
\hline
\multirow{2}{*}{Cu}
    & $\pi^+$ & $+0.001381$ & $-0.013593$ & $+0.049945$ & $+0.072581$ & $+0.059173$ & $+0.000113$ \\
    & $\pi^-$ & $-0.001433$ & $+0.008117$ & $-0.022846$ & $+0.076203$ & $+0.117692$ & $+0.000187$ \\
\hline
\multirow{2}{*}{Sn}
    & $\pi^+$ & $+0.001150$ & $-0.010677$ & $+0.041478$ & $+0.076579$ & $+0.050554$ & $-0.000595$ \\
    & $\pi^-$ & $-0.001777$ & $+0.010759$ & $-0.029889$ & $+0.075510$ & $+0.120755$ & $+0.000220$ \\
\hline
\end{tabular}%
}
\end{table}

\begin{table}[htb]
\centering
\small
\caption{Fitted parameters of the $\Delta t$ cut for all targets for inbending polarity.}
\label{tab:deltat-coefficients-ib}
\makebox[\textwidth][c]{%
\begin{tabular}{|l|c|r|r|r|r|r|r|}
\hline
\textbf{Target} & \textbf{Species} & \textbf{$a_\mu$} & \textbf{$b_\mu$} & \textbf{$c_\mu$} & \textbf{$\sigma_0$} & \textbf{$k$} & \textbf{$c_\sigma$} \\
\hline
\multirow{2}{*}{LD2}
    & $\pi^+$ & $-0.001731$ & $+0.011267$ & $-0.054537$ & $+0.085218$ & $+0.120109$ & $-0.003085$ \\
    & $\pi^-$ & $+0.000405$ & $-0.006009$ & $+0.004971$ & $+0.076147$ & $+0.054397$ & $-0.001330$ \\
\hline
\multirow{2}{*}{C}
    & $\pi^+$ & $-0.001542$ & $+0.009979$ & $-0.054565$ & $+0.088016$ & $+0.119062$ & $-0.003361$ \\
    & $\pi^-$ & $+0.000399$ & $-0.006449$ & $+0.004927$ & $+0.074921$ & $+0.056532$ & $-0.000445$ \\
\hline
\multirow{2}{*}{Cu}
    & $\pi^+$ & $-0.001792$ & $+0.011443$ & $-0.052059$ & $+0.087525$ & $+0.118368$ & $-0.003076$ \\
    & $\pi^-$ & $+0.000546$ & $-0.007970$ & $+0.012212$ & $+0.078201$ & $+0.054772$ & $-0.001775$ \\
\hline
\multirow{2}{*}{Sn}
    & $\pi^+$ & $-0.001860$ & $+0.012422$ & $-0.059442$ & $+0.089254$ & $+0.123236$ & $-0.003298$ \\
    & $\pi^-$ & $+0.001120$ & $-0.010184$ & $+0.008613$ & $+0.080358$ & $+0.050856$ & $-0.001380$ \\
\hline
\end{tabular}%
}
\end{table}

An example of the Gaussian fits to the $\Delta t$ projections in each momentum bin is shown in \cref{fig:pip-gaussian-fits-LD2} for $\pi^+$ candidates in the LD2 target. The extracted mean and standard deviation from each bin, together with the parameterized fits of \cref{eq:deltat-mean,eq:deltat-sigma}, are shown in \cref{fig:deltat-fit-results-LD2}.

The resulting cuts overlaid on the two-dimensional $\Delta t$ versus momentum distributions are shown in \cref{fig:deltat-vs-p-LD2} for the LD2 target, and in \cref{fig:deltat-vs-p-CxC,fig:deltat-vs-p-Cu,fig:deltat-vs-p-Sn} for the C, Cu, and Sn targets, respectively.

The comparison of the extracted mean and sigma parameters across all targets is shown in \cref{fig:deltat-comparison-mean,fig:deltat-comparison-sigma}, demonstrating consistent timing behavior across the different nuclear targets. The resulting $\mu_{\Delta t}(p) \pm 3\sigma_{\Delta t}(p)$ cut boundaries for all targets are compared in \cref{fig:deltat-comparison-fit}, confirming that the cuts are nearly identical across targets.

The sector dependence of the $\Delta t$ parameters is examined in \cref{fig:deltat-sector-comparison-LD2}, showing the extracted mean and sigma for each of the six CLAS12 sectors separately for the LD2 target. The sector-by-sector $\pm3\sigma$ cut boundaries are compared in \cref{fig:deltat-sector-fit-LD2}. While small sector-to-sector variations are observed, the overall behavior is consistent, justifying the use of a single sector-averaged parameterization.

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\textwidth]{images/pions/delta_t_cut/pip_gaussian_fits_LD2.pdf}
  \caption{Gaussian fits to the $\Delta t$ projections in 30 momentum bins for $\pi^+$ candidates in the LD2 target. Each panel shows the $\Delta t$ distribution (black points) and the Gaussian fit (red curve) for a given momentum range, with the extracted mean $\mu$, standard deviation $\sigma$, and reduced $\chi^2$ indicated.}
  \label{fig:pip-gaussian-fits-LD2}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\textwidth]{images/pions/delta_t_cut/pion_delta_t_fit_results_LD2.pdf}
  \caption{Momentum dependence of the extracted $\Delta t$ mean (top) and standard deviation (bottom) for $\pi^+$ (left) and $\pi^-$ (right) candidates in the LD2 target. The red curves show the parameterized fits of \cref{eq:deltat-mean,eq:deltat-sigma}.}
  \label{fig:deltat-fit-results-LD2}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=400px]{images/pions/delta_t_cut/delta_t_vs_pion_momentum_LD2.pdf}
  \caption{Two-dimensional distribution of $\Delta t$ versus momentum for $\pi^+$ (left) and $\pi^-$ (right) candidates in the LD2 target with outbending polarity. The red solid line shows the fitted mean $\mu_{\Delta t}(p)$ and the red dashed lines indicate the $\pm3\sigma_{\Delta t}(p)$ cut boundaries.}
  \label{fig:deltat-vs-p-LD2}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=400px]{images/pions/delta_t_cut/delta_t_vs_pion_momentum_CxC.pdf}
  \caption{Two-dimensional distribution of $\Delta t$ versus momentum for $\pi^+$ (left) and $\pi^-$ (right) candidates in the C target with outbending polarity.}
  \label{fig:deltat-vs-p-CxC}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=400px]{images/pions/delta_t_cut/delta_t_vs_pion_momentum_Cu.pdf}
  \caption{Two-dimensional distribution of $\Delta t$ versus momentum for $\pi^+$ (left) and $\pi^-$ (right) candidates in the Cu target with outbending polarity.}
  \label{fig:deltat-vs-p-Cu}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=400px]{images/pions/delta_t_cut/delta_t_vs_pion_momentum_Sn.pdf}
  \caption{Two-dimensional distribution of $\Delta t$ versus momentum for $\pi^+$ (left) and $\pi^-$ (right) candidates in the Sn target with outbending polarity.}
  \label{fig:deltat-vs-p-Sn}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\textwidth]{images/pions/delta_t_cut/delta_t_comparison_mean.pdf}
  \caption{Comparison of the fitted $\Delta t$ mean $\mu_{\Delta t}(p)$ across all targets for $\pi^+$ (a) and $\pi^-$ (b). The solid curves show the second-order polynomial parameterization of \cref{eq:deltat-mean}.}
  \label{fig:deltat-comparison-mean}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\textwidth]{images/pions/delta_t_cut/delta_t_comparison_sigma.pdf}
  \caption{Comparison of the fitted $\Delta t$ standard deviation $\sigma_{\Delta t}(p)$ across all targets for $\pi^+$ (a) and $\pi^-$ (b). The solid curves show the timing resolution parameterization of \cref{eq:deltat-sigma}.}
  \label{fig:deltat-comparison-sigma}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\textwidth]{images/pions/delta_t_cut/delta_t_comparison_fit.pdf}
  \caption{Comparison of the $\mu_{\Delta t}(p) \pm 3\sigma_{\Delta t}(p)$ cut boundaries across all targets for $\pi^+$ (a) and $\pi^-$ (b). The solid lines show the mean and the dashed lines show the $\pm3\sigma$ boundaries.}
  \label{fig:deltat-comparison-fit}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\textwidth]{images/pions/delta_t_cut/delta_t_sector_comparison_LD2.pdf}
  \caption{Sector-by-sector comparison of the extracted $\Delta t$ mean (top) and standard deviation (bottom) for $\pi^+$ (left) and $\pi^-$ (right) in the LD2 target.}
  \label{fig:deltat-sector-comparison-LD2}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\textwidth]{images/pions/delta_t_cut/delta_t_sector_fit_LD2.pdf}
  \caption{Comparison of the sector-by-sector $\mu_{\Delta t}(p) \pm 3\sigma_{\Delta t}(p)$ cut boundaries for $\pi^+$ (a) and $\pi^-$ (b) in the LD2 target. The solid lines show the mean and the dashed lines show the $\pm3\sigma$ boundaries for each sector.}
  \label{fig:deltat-sector-fit-LD2}
\end{figure}

\subsection{$V_z$ cuts}
\label{sec:pion-vz}
Vertex cuts along the beamline ($V_z$) are applied to pion candidates to ensure they originate from the target region and to suppress background from secondary interactions in the detector material. Pions follow the same two-step procedure introduced for electrons in \cref{sec:electron-vz}: the per-event $V_z$ correction described in \cref{sec:vz-correction} is re-applied with pion-specific polynomial parameters, and the corrected 1D $V_z$ distribution is fit with the two-Voigt + pedestal model of \cref{eq:vz-voigt}. Pion-specific correction parameters are necessary because multiple-scattering and energy-loss contributions broaden the per-cell peaks more than for electrons; using the electron parameters directly would systematically misalign the pion peaks by a few hundred microns. The cut convention is unchanged: $\mu \pm 3\sigma^{\text{eff}}$ around each cell-wall peak, with $\sigma^{\text{eff}} = \text{FWHM}/(2\sqrt{2\ln 2})$ from \cref{eq:voigt-fwhm}.

The Voigt fits for $\pi^-$ and $\pi^+$ across all four solid-target variants are shown in columns 2 and 3 of \cref{fig:vz-voigt-grid}. The extracted parameters and the $\mu \pm 3\sigma^{\text{eff}}$ windows are summarized in \cref{tab:Vz-pion-cuts}. Compared with the electron fits in \cref{tab:Vz-cuts}, pion peaks are systematically broader by 10--15\% (larger $\gamma^L$), as expected from the heavier-particle multiple-scattering contribution. Peak positions differ from electrons by $\lesssim 0.15$~cm and the cut windows by $\lesssim 0.3$~cm, well within the inter-peak gap of $\sim 5$~cm.

\begin{table}[htb]
\centering
\small
\caption{Pion Voigt-fit parameters and $\mu \pm 3\sigma^{\text{eff}}$ vertex windows for each solid-target variant, in the corrected $V_z$ frame. The two windows per row select the upstream (Sn / C$_1$) and downstream (Cu / C$_2$) cell walls, respectively. For the LD2 target a fixed window $V_z \in [-15, 5]$~cm is retained.}
\label{tab:Vz-pion-cuts}
\makebox[\textwidth][c]{%
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\textbf{Variant} & \textbf{Species} & \textbf{$\mu_{\text{lo}}$ [cm]} & \textbf{$\sigma^{\text{eff}}_{\text{lo}}$ [cm]} & \textbf{$\mu_{\text{hi}}$ [cm]} & \textbf{$\sigma^{\text{eff}}_{\text{hi}}$ [cm]} & \textbf{Cut windows [cm]} \\
\hline
\multirow{2}{*}{CuSn OB}
    & $\pi^-$ & $-2.823$ & $0.382$ & $-7.777$ & $0.449$ & $[-3.97,\, -1.68]$; $[-9.12,\, -6.43]$ \\
    & $\pi^+$ & $-2.821$ & $0.393$ & $-7.782$ & $0.403$ & $[-4.00,\, -1.64]$; $[-8.99,\, -6.57]$ \\
\hline
\multirow{2}{*}{CuSn IB}
    & $\pi^-$ & $-2.921$ & $0.425$ & $-7.830$ & $0.463$ & $[-4.20,\, -1.65]$; $[-9.22,\, -6.44]$ \\
    & $\pi^+$ & $-2.951$ & $0.401$ & $-7.840$ & $0.426$ & $[-4.15,\, -1.75]$; $[-9.12,\, -6.56]$ \\
\hline
\multirow{2}{*}{CxC OB}
    & $\pi^-$ & $-2.805$ & $0.389$ & $-7.815$ & $0.412$ & $[-3.97,\, -1.64]$; $[-9.05,\, -6.58]$ \\
    & $\pi^+$ & $-2.802$ & $0.409$ & $-7.783$ & $0.364$ & $[-4.03,\, -1.58]$; $[-8.88,\, -6.69]$ \\
\hline
\multirow{2}{*}{CxC IB}
    & $\pi^-$ & $-2.782$ & $0.437$ & $-7.744$ & $0.463$ & $[-4.09,\, -1.47]$; $[-9.13,\, -6.36]$ \\
    & $\pi^+$ & $-2.849$ & $0.410$ & $-7.799$ & $0.421$ & $[-4.08,\, -1.62]$; $[-9.06,\, -6.54]$ \\
\hline
\end{tabular}%
}
\end{table}

A side-by-side overlay of $e^-$, $\pi^-$, and $\pi^+$ corrected $V_z$ distributions for the CuSn OB variant is shown in \cref{fig:vz-species-compare}. The three species share their corrected peak positions to within 0.15~cm; pion distributions exhibit slightly wider Lorentzian wings, which is well captured by the Voigt model.

\begin{figure}[hbt]
  \centering
  \includegraphics[page=3,width=\dimexpr\textwidth+\marginparwidth+\marginparsep\relax]{images/electron/vz/vz_compare_species_cusn_ob.pdf}
  \caption{Corrected $V_z$ distributions for $e^-$, $\pi^-$, and $\pi^+$ in the CuSn outbending variant. All three species are corrected with their own pion- or electron-specific parameters and are shown overlaid on shared axes; peak positions agree within $\lesssim 0.15$~cm.}
  \label{fig:vz-species-compare}
\end{figure}

\subsection{$\Delta V_z$ cuts}
\label{sec:pion-delta-vz}
An additional cut is applied on the difference in vertex $z$-position between the electron and the pion, $\Delta V_z = V_z(e^-) - V_z(\pi^\pm)$. This cut rejects pion candidates that do not originate from the same interaction vertex as the scattered electron, suppressing contributions from secondary interactions and accidental coincidences.

The $\Delta V_z$ distribution is fitted with a Gaussian function for each target and pion charge separately. The fit is performed on the top 50\% of the peak to minimize sensitivity to the non-Gaussian tails of the distribution. The cut is then defined as $\mu \pm 3\sigma$, where $\mu$ and $\sigma$ are the mean and width of the fitted Gaussian. The resulting cut values are summarized in \cref{tab:delta-vz-cuts}. The fitted parameters are consistent across all targets, with $\pi^-$ distributions centered near zero and $\pi^+$ distributions exhibiting a systematic offset of approximately $-0.8$~cm. The widths are similar across targets, with $\sigma \approx 1.3$~cm for $\pi^-$ and $\sigma \approx 1.0$~cm for $\pi^+$.

\begin{table}[htb]
\centering
\caption{$\Delta V_z$ cut values ($\mu \pm 3\sigma$) for pion candidates in outbending polarity.}
\label{tab:delta-vz-cuts}
\begin{tabular}{|l|c|c|c|c|}
\hline
\textbf{Target} & \textbf{Particle} & \textbf{$\mu$ [cm]} & \textbf{$\sigma$ [cm]} & \textbf{Cut range [cm]} \\
\hline
LD2 & $\pi^-$ & $0.18$  & $1.34$ & $[-3.85,\; 4.21]$ \\
    & $\pi^+$ & $-0.85$ & $1.05$ & $[-3.99,\; 2.29]$ \\
\hline
C   & $\pi^-$ & $0.19$  & $1.33$ & $[-3.81,\; 4.19]$ \\
    & $\pi^+$ & $-0.84$ & $1.05$ & $[-3.98,\; 2.30]$ \\
\hline
Cu  & $\pi^-$ & $0.20$  & $1.34$ & $[-3.82,\; 4.21]$ \\
    & $\pi^+$ & $-0.85$ & $1.03$ & $[-3.95,\; 2.26]$ \\
\hline
Sn  & $\pi^-$ & $0.17$  & $1.26$ & $[-3.60,\; 3.94]$ \\
    & $\pi^+$ & $-0.81$ & $1.04$ & $[-3.93,\; 2.30]$ \\
\hline
\end{tabular}
\end{table}

\Cref{fig:delta-vz-LD2,fig:delta-vz-CxC,fig:delta-vz-Cu,fig:delta-vz-Sn} show the $\Delta V_z$ distributions with the Gaussian fits and the resulting $\pm 3\sigma$ cut boundaries for each target.

\begin{figure}[hbt]
  \centering
  \begin{subfigure}[b]{\textwidth}
    \centering
    \includegraphics[width=200pt]{images/pions/delta_vz_analysis_LD2.pdf}
    \caption{LD$_2$}
    \label{fig:delta-vz-LD2}
  \end{subfigure}
  \vfill
  \begin{subfigure}[b]{\textwidth}
    \centering
    \includegraphics[width=200pt]{images/pions/delta_vz_analysis_CxC.pdf}
    \caption{C}
    \label{fig:delta-vz-CxC}
  \end{subfigure}
  \vfill
  \begin{subfigure}[b]{\textwidth}
    \centering
    \includegraphics[width=200pt]{images/pions/delta_vz_analysis_Cu.pdf}
    \caption{Cu}
    \label{fig:delta-vz-Cu}
  \end{subfigure}
  \vfill
  \begin{subfigure}[b]{\textwidth}
    \centering
    \includegraphics[width=200pt]{images/pions/delta_vz_analysis_Sn.pdf}
    \caption{Sn}
    \label{fig:delta-vz-Sn}
  \end{subfigure}
  \caption{$\Delta V_z = V_z(e^-) - V_z(\pi^\pm)$ distributions for each target in outbending polarity, shown separately for $\pi^-$ (left) and $\pi^+$ (right). The Gaussian fits to the core of the distributions are overlaid in red, with the $\pm 3\sigma$ cut boundaries indicated by the vertical dashed lines.}
  \label{fig:delta-vz-all}
\end{figure}

\subsection{DC fiducial cuts}
\label{sec:pion-dc-fiducial}
The same methodology as for the electron DC fiducial cuts is applied for the pion candidate tracks. The resulting profiles for each sector for regions 1, 2, and 3 are shown for the sector average in \cref{fig:DC-fiducial-cut-pion-LD2}, \cref{fig:DC-fiducial-cut-pion-CxC}, and \cref{fig:DC-fiducial-cut-pion-CuSn}, respectively.

For $\pi^-$, the optimal cut positions are 2.125~cm, 3.625~cm, and 9.625~cm for regions 1, 2, and 3, respectively. For $\pi^+$, the cut positions are 3.875~cm, 4.375~cm, and 2.625~cm for regions 1, 2, and 3, respectively.


\begin{figure}[hbt]
  \centering
  \begin{subfigure}[b]{\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/pions/LD2_OB/pion_charge_comparison.pdf}
    \caption{LD2}
    \label{fig:DC-fiducial-cut-pion-LD2}
  \end{subfigure}
  \vfill
  \begin{subfigure}[b]{\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/pions/CxC_OB/pion_charge_comparison.pdf}
    \caption{CxC}
    \label{fig:DC-fiducial-cut-pion-CxC}
  \end{subfigure}
  \vfill
  \begin{subfigure}[b]{\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/pions/CuSn_OB/pion_charge_comparison.pdf}
    \caption{CuSn}
    \label{fig:DC-fiducial-cut-pion-CuSn}
  \end{subfigure}
  \caption{DC fiducial cuts for pion candidate tracks in outbending polarity for the three targets (sector average).}
  \label{fig:DC-fiducial-cut-pion}
\end{figure}

\FloatBarrier

\clearpage

\appendix

\section{Sector Parameters}
\label{sec:sector-params}

\subsection*{Outbending (OB)}
\begin{table}[h]
\tiny
\centering
\caption{Sampling fraction polynomial parameters for outbending polarity.} \label{tab:sf-params-ob}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{Target} & \textbf{Sector} & \textbf{$a_\mu$} & \textbf{$b_\mu$} & \textbf{$c_\mu$} & \textbf{$d_\mu$} & \textbf{$a_\sigma$} & \textbf{$b_\sigma$} & \textbf{$c_\sigma$} & \textbf{$d_\sigma$} \\ \hline
\multirow{6}{*}{LD2} & S1 & 2.284817E-01 & 5.741607E-03 & -7.956694E-04 & 2.165831E-05 & 2.278400E-02 & -3.478097E-03 & 3.644345E-04 & -1.564506E-05 \\ \cline{2-10}
& S2 & 2.272353E-01 & 6.035907E-03 & -5.379217E-04 & -1.432191E-06 & 2.214450E-02 & -2.992410E-03 & 2.810789E-04 & -1.021893E-05 \\ \cline{2-10}
& S3 & 2.248222E-01 & 9.544562E-03 & -1.331928E-03 & 4.146476E-05 & 2.455364E-02 & -4.547263E-03 & 5.091026E-04 & -2.082045E-05 \\ \cline{2-10}
& S4 & 2.262726E-01 & 6.742876E-03 & -6.707265E-04 & 6.027435E-06 & 2.270259E-02 & -3.090211E-03 & 2.849498E-04 & -1.108599E-05 \\ \cline{2-10}
& S5 & 2.294593E-01 & 3.253914E-03 & -6.716294E-05 & -2.106351E-05 & 2.353437E-02 & -4.471507E-03 & 5.546300E-04 & -2.564773E-05 \\ \cline{2-10}
& S6 & 2.248109E-01 & 7.709680E-03 & -9.387105E-04 & 1.955346E-05 & 2.320134E-02 & -3.707398E-03 & 4.086606E-04 & -1.827203E-05 \\ \hline
\multirow{6}{*}{CxC} & S1 & 2.297164E-01 & 4.850482E-03 & -6.473612E-04 & 1.386706E-05 & 2.160027E-02 & -2.804498E-03 & 2.481187E-04 & -9.444168E-06 \\ \cline{2-10}
& S2 & 2.284924E-01 & 5.444945E-03 & -4.057949E-04 & -9.745369E-06 & 2.097248E-02 & -2.301250E-03 & 1.681611E-04 & -4.820942E-06 \\ \cline{2-10}
& S3 & 2.257831E-01 & 8.799251E-03 & -1.210729E-03 & 3.520214E-05 & 2.364466E-02 & -3.974250E-03 & 4.048261E-04 & -1.517698E-05 \\ \cline{2-10}
& S4 & 2.290273E-01 & 4.774422E-03 & -3.194710E-04 & -1.291518E-05 & 2.131817E-02 & -2.278156E-03 & 1.421861E-04 & -3.384835E-06 \\ \cline{2-10}
& S5 & 2.312938E-01 & 1.779729E-03 & 1.881011E-04 & -3.479554E-05 & 2.238608E-02 & -3.777670E-03 & 4.241412E-04 & -1.815833E-05 \\ \cline{2-10}
& S6 & 2.271200E-01 & 6.051471E-03 & -6.286571E-04 & 1.997614E-06 & 2.170337E-02 & -2.868470E-03 & 2.639296E-04 & -1.041341E-05 \\ \hline
\multirow{6}{*}{CuSn} & S1 & 2.320707E-01 & 3.994131E-03 & -4.894629E-04 & 1.898325E-06 & 2.241820E-02 & -3.277630E-03 & 3.136628E-04 & -1.234151E-05 \\ \cline{2-10}
& S2 & 2.302078E-01 & 3.476050E-03 & -1.567172E-04 & -2.212723E-05 & 2.141103E-02 & -2.354403E-03 & 1.075100E-04 & 1.855098E-06 \\ \cline{2-10}
& S3 & 2.268107E-01 & 8.877708E-03 & -1.179578E-03 & 3.071273E-05 & 2.378785E-02 & -4.104492E-03 & 4.372052E-04 & -1.750814E-05 \\ \cline{2-10}
& S4 & 2.290050E-01 & 6.702587E-03 & -7.847463E-04 & 1.459749E-05 & 2.250505E-02 & -2.997400E-03 & 2.528880E-04 & -8.680370E-06 \\ \cline{2-10}
& S5 & 2.298645E-01 & 4.027068E-03 & -2.501656E-04 & -1.062495E-05 & 2.240025E-02 & -3.576763E-03 & 3.577542E-04 & -1.355748E-05 \\ \cline{2-10}
& S6 & 2.268440E-01 & 6.788151E-03 & -8.409054E-04 & 1.710382E-05 & 2.268680E-02 & -3.427790E-03 & 3.396480E-04 & -1.371903E-05 \\ \hline
\multirow{6}{*}{Average} & S1 & 2.300896E-01 & 4.862074E-03 & -6.441645E-04 & 1.247456E-05 & 2.226749E-02 & -3.186742E-03 & 3.087387E-04 & -1.247691E-05 \\ \cline{2-10}
& S2 & 2.286452E-01 & 4.985634E-03 & -3.668113E-04 & -1.110160E-05 & 2.150934E-02 & -2.549354E-03 & 1.855833E-04 & -4.394923E-06 \\ \cline{2-10}
& S3 & 2.258053E-01 & 9.073840E-03 & -1.240745E-03 & 3.579321E-05 & 2.399538E-02 & -4.208668E-03 & 4.503779E-04 & -1.783519E-05 \\ \cline{2-10}
& S4 & 2.281016E-01 & 6.073295E-03 & -5.916479E-04 & 2.569914E-06 & 2.217527E-02 & -2.788589E-03 & 2.266746E-04 & -7.717066E-06 \\ \cline{2-10}
& S5 & 2.302059E-01 & 3.020237E-03 & -4.307581E-05 & -2.216133E-05 & 2.277357E-02 & -3.941980E-03 & 4.455085E-04 & -1.912118E-05 \\ \cline{2-10}
& S6 & 2.262583E-01 & 6.849767E-03 & -8.027577E-04 & 1.288497E-05 & 2.253050E-02 & -3.334553E-03 & 3.374127E-04 & -1.413483E-05 \\ \hline
\end{tabular}
\end{table}

\subsection*{Inbending (IB)}
\begin{table}[h]
\tiny
\centering
\caption{Sampling fraction polynomial parameters for inbending polarity.} \label{tab:sf-params-ib}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{Target} & \textbf{Sector} & \textbf{$a_\mu$} & \textbf{$b_\mu$} & \textbf{$c_\mu$} & \textbf{$d_\mu$} & \textbf{$a_\sigma$} & \textbf{$b_\sigma$} & \textbf{$c_\sigma$} & \textbf{$d_\sigma$} \\ \hline
\multirow{6}{*}{LD2} & S1 & 2.311090E-01 & 4.309850E-03 & -5.451468E-04 & 4.964926E-06 & 2.235234E-02 & -3.117219E-03 & 2.887163E-04 & -1.130302E-05 \\ \cline{2-10}
& S2 & 2.304337E-01 & 3.585701E-03 & -1.672183E-04 & -2.195166E-05 & 2.170119E-02 & -2.428043E-03 & 1.302137E-04 & -2.341453E-09 \\ \cline{2-10}
& S3 & 2.269121E-01 & 8.797654E-03 & -1.171428E-03 & 3.051411E-05 & 2.423452E-02 & -4.301838E-03 & 4.742717E-04 & -1.982933E-05 \\ \cline{2-10}
& S4 & 2.288870E-01 & 6.786554E-03 & -8.022669E-04 & 1.557290E-05 & 2.313618E-02 & -3.260224E-03 & 3.018553E-04 & -1.156727E-05 \\ \cline{2-10}
& S5 & 2.288645E-01 & 4.637371E-03 & -3.662910E-04 & -4.233388E-06 & 2.310886E-02 & -3.892809E-03 & 4.115044E-04 & -1.656696E-05 \\ \cline{2-10}
& S6 & 2.272092E-01 & 6.581994E-03 & -8.197215E-04 & 1.632205E-05 & 2.321750E-02 & -3.665701E-03 & 3.804620E-04 & -1.587632E-05 \\ \hline
\multirow{6}{*}{CxC} & S1 & 2.311089E-01 & 4.309932E-03 & -5.451631E-04 & 4.965863E-06 & 2.235235E-02 & -3.117219E-03 & 2.887137E-04 & -1.130271E-05 \\ \cline{2-10}
& S2 & 2.304337E-01 & 3.585739E-03 & -1.672262E-04 & -2.195117E-05 & 2.170108E-02 & -2.427968E-03 & 1.302009E-04 & -1.715883E-09 \\ \cline{2-10}
& S3 & 2.269124E-01 & 8.797523E-03 & -1.171404E-03 & 3.051277E-05 & 2.423476E-02 & -4.301977E-03 & 4.742961E-04 & -1.983062E-05 \\ \cline{2-10}
& S4 & 2.288871E-01 & 6.786452E-03 & -8.022473E-04 & 1.557178E-05 & 2.313635E-02 & -3.260315E-03 & 3.018706E-04 & -1.156805E-05 \\ \cline{2-10}
& S5 & 2.288645E-01 & 4.637396E-03 & -3.662963E-04 & -4.233103E-06 & 2.310898E-02 & -3.892887E-03 & 4.115187E-04 & -1.656775E-05 \\ \cline{2-10}
& S6 & 2.272093E-01 & 6.581935E-03 & -8.197067E-04 & 1.632106E-05 & 2.321749E-02 & -3.665704E-03 & 3.804646E-04 & -1.587659E-05 \\ \hline
\multirow{6}{*}{CuSn} & S1 & 2.311087E-01 & 4.310017E-03 & -5.451781E-04 & 4.966680E-06 & 2.235247E-02 & -3.117310E-03 & 2.887315E-04 & -1.130379E-05 \\ \cline{2-10}
& S2 & 2.304337E-01 & 3.585708E-03 & -1.672224E-04 & 2.195131E-05 & 2.170126E-02 & -2.428101E-03 & 1.302294E-04 & -3.452408E-09 \\ \cline{2-10}
& S3 & 2.269123E-01 & 8.797570E-03 & -1.171414E-03 & 3.051337E-05 & 2.423472E-02 & -4.301969E-03 & 4.742963E-04 & -1.983071E-05 \\ \cline{2-10}
& S4 & 2.288870E-01 & 6.786555E-03 & -8.022665E-04 & 1.557285E-05 & 2.313625E-02 & -3.260272E-03 & 3.018643E-04 & -1.156777E-05 \\ \cline{2-10}
& S5 & 2.288646E-01 & 4.637320E-03 & -3.662803E-04 & -4.234038E-06 & 2.310900E-02 & -3.892931E-03 & 4.115288E-04 & -1.656837E-05 \\ \cline{2-10}
& S6 & 2.272092E-01 & 6.582013E-03 & -8.197249E-04 & 1.632221E-05 & 2.321744E-02 & -3.665649E-03 & 3.804500E-04 & -1.587559E-05 \\ \hline
\multirow{6}{*}{Average} & S1 & 2.311088E-01 & 4.309933E-03 & -5.451627E-04 & 4.965823E-06 & 2.235239E-02 & -3.117249E-03 & 2.887205E-04 & -1.130317E-05 \\ \cline{2-10}
& S2 & 2.304337E-01 & 3.585716E-03 & -1.672223E-04 & -7.317175E-06 & 2.170118E-02 & -2.428037E-03 & 1.302147E-04 & -2.503248E-09 \\ \cline{2-10}
& S3 & 2.269123E-01 & 8.797582E-03 & -1.171416E-03 & 3.051342E-05 & 2.423467E-02 & -4.301928E-03 & 4.742880E-04 & -1.983022E-05 \\ \cline{2-10}
& S4 & 2.288870E-01 & 6.786520E-03 & -8.022602E-04 & 1.557251E-05 & 2.313626E-02 & -3.260271E-03 & 3.018634E-04 & -1.156770E-05 \\ \cline{2-10}
& S5 & 2.288645E-01 & 4.637362E-03 & -3.662892E-04 & -4.233509E-06 & 2.310895E-02 & -3.892876E-03 & 4.115173E-04 & -1.656769E-05 \\ \cline{2-10}
& S6 & 2.272093E-01 & 6.581981E-03 & -8.197177E-04 & 1.632178E-05 & 2.321748E-02 & -3.665685E-03 & 3.804588E-04 & -1.587616E-05 \\ \hline
\end{tabular}
\end{table}



\section{Number of photo-electrons in HTCC}
\label{sec:appendix-nphe}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=\dimexpr\textwidth+\marginparwidth+\marginparsep\relax]{images/electron/plots_event_builder/nphe_neg_all_targets_IB.pdf}
  \caption{HTCC photo-electron distribution for all negative tracks in the Forward Detector in inbending polarity for the torus, apply cut at $N_{\text{phe}} > 2$ for pion rejection.}
  \label{fig:nphe-IB}
\end{figure}

\section{Sampling fraction versus momentum}
\label{sec:appendix-sf}
\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_sampling_fraction/OB/sectors_comparison_CxC.pdf}
  \caption{Comparison of the sampling fraction mean and sigma as a function of momentum for the CxC target in outbending polarity for different sectors and the overall distribution.}
  \label{fig:SF-CxC-comparison}
\end{figure}

\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_sampling_fraction/OB/sectors_comparison_CuSn.pdf}
  \caption{Comparison of the sampling fraction mean and sigma as a function of momentum for the CuSn target in outbending polarity for different sectors and the overall distribution.}
  \label{fig:SF-CuSn-comparison}
\end{figure}


\section{Calorimeter fiducial cuts}
\label{sec:appendix-cal-fiducial}
\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_calorimeter_fiducials/lv_vs_sf_CxC_all_sectors.pdf}
  \caption{Profile of the 2D distribution SF vs $v$ or $w$ for electron candidate tracks in the CxC target in outbending polarity for each sectors and the overall distribution.}
  \label{fig:ECAL-CxC-fiducial-cut}
\end{figure}


\begin{figure}[hbt]
  \centering
  \includegraphics[width=0.7\textwidth]{images/electron/plots_calorimeter_fiducials/lv_vs_sf_CuSn_all_sectors.pdf}
  \caption{Profile of the 2D distribution SF vs $v$ or $w$ for electron candidate tracks in the CuSn target in outbending polarity for each sectors and the overall distribution.}
  \label{fig:ECAL-CuSn-fiducial-cut}
\end{figure}



\end{document}