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<  ILC e$^+$e$^-$ collider is presented using the \rm{$B$ = 1 T} test beam results for the measured electron PID resolution.
---
>  ILC e$^+$e$^-$ collider is presented using the \rm{$B$= 1 T} test beam results for the measured electron PID resolution.
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<  A GridPix has a very fine granularity of 256$\times$256 pixels of {55$\times$55}{ $\rm \mu m^2$} and a high efficiency of about 85\%  to detect single ionisation electrons. Besides the Time-of-Arrival (ToA) of the signals on a pixel, the Timepix3 also measures the Time-over-Threshold (ToT), related to the deposited charge of the signal.
---
>  The GridPixes have a very fine granularity of 256$\times$256 pixels of {55$\times$55}{ $\rm \mu m^2$} and a high efficiency of about 85\%  to detect single ionisation electrons. Besides the Time-of-Arrival (ToA) of the signals on a pixel, the Timepix3 also measures the Time-over-Threshold (ToT), related to the deposited charge of the signal.
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<  The data analysed were taken at electron beam momenta of 5 and 6 GeV/c and at magnetic fields of 0 and 1 T. The TPC was operated using a so-called T2K  gas mixture of 95/3/2 \% of {Ar}/{CF$_4$}/{iC$_4$H$_{10}$  (by volume) with a small amount of oxygen and water vapour of < 7000 ppm.
---
>  The data analysed were taken at electron beam momenta of 5 and 6 GeV/c and at magnetic fields of 0 and 1 T. The TPC was operated using a so-called T2K  gas mixture of 95/3/2 \% of {Ar}/{CF$_4$}/{iC$_4$H$_{10}$  (by volume) with a small amount of oxygen and water vapour.
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<  Particles can be identified by their characteristic energy loss per unit of track length, $dE/dx$ and/or the number of primary clusters, $dN/dx$ produced along the track. 
< In a classical TPC with pad read out, the charge is measured and used to estimate $dE/dx$ using a truncation method that reduces the Landau tail. In a pixel TPC based on a GridPix detector, the number of ionisation electrons is measured with high granularity as a function of the distance along the track.
< The number of ionisation electrons is proportional to the energy loss of the particle in the gas and has a Landau-like distribution. Due to the fine granularity a Gridpix is sensitive to the primary clusters that are described by a Poissonian distribution. The best PID performance is reached by counting the clusters (and separating the from the electrons by that cluster) and measuring $dN/dx$.
< A Pixel TPC is sensitive to $dE/dx$ and $dN/dx$ depending on the analysis method that is applied.
<          
< %In a GridPix detector, one can measure both the number of hits produced along the track and their relative distance.
---
>  Particles can be identified by their characteristic energy loss per unit of track length, $dE/dx$ and/or the number of primary clusters, $dN/dx$ produced along the track. In a GridPix detector one can measure both the number of hits produced along the track and their relative distance.
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< To measure the track performance of $dE/dx$ or $dN/dx$, a track selection was applied selecting tracks crossing the central chips - defined in \cite{Beuzekom:2024part1}. 
---
> In order to measure the track performance of $dE/dx$ or $dN/dx$, a track selection was applied selecting tracks crossing the central chips - defined in \cite{Beuzekom:2024part1}. 
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< %The number of hits of different GridPixes were weighted and calibrated to give the same mean number of hits per GridPix. 
< The number of hits of different GridPixes were weighted to give the same mean number of hits per GridPix. 
---
> The number of hits of different GridPixes were weighted and calibrated to give the same mean number of hits per GridPix. 
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< Using a large number of tracks, it is possible to determine from data the shape of the minimum distance distribution. At distances above approximately 10 pixels, the distribution follows an exponential distribution. At lower distance, weights for the \rm{$B$= 0 T} and  1 T data are determined and applied to ensure an exponential distribution over the whole range. The values of the weights at low distances depend on the transverse diffusion coefficient and the drift length.  
---
> Using a large number of tracks, it is possible to determine from data the shape of the minimum distance distribution. At distances above approximately 10 pixels the distribution follows an exponential distribution. At lower distance, weights for the \rm{$B$= 0 T} and  1 T data are determined and applied to ensure an exponential distribution over the whole range. The values of the weights at low distances depend on the transverse diffusion coefficient and the drift length.  
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<          Method & $B$ = 0 T  & $B$ = 1 T \\
---
>          Method & $B$= 0 T  & $B$= 1 T \\
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<  To estimate the performance for different particles it is important to quantify the linearity of the methods.  
---
>  In order to estimate the performance for different particles it is important to quantify the linearity of the methods.  
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< E.g. the ALICE experiment published a $dE/dx$ resolution of 5\% for a track length of 1.65 m \cite{YU201355}. 
---
> 
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< \section{Single-electron detection efficiency at high hit rates}
---
> \section{Single-electron efficiency at high hit rates}
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< The single-electron detection efficiency of the GridPix detector can be reduced in a high background environment due to the charge up of the resistive layers and the effective reduction of the amplification field. It is therefore important to measure the single-electron efficiency in different background conditions.
< The time-over-threshold (ToT) is related to the deposited charge on the pixel and the single-electron detection efficiency of the detector.
< The relative change in the single-electron detection efficiency  $\delta \epsilon / \epsilon$  
---
> The single-electron efficiency of the GridPix detector can be reduced in a high background environment due to the charge up of the resistive layers and the effective reduction of the amplification field. It is therefore important to measure the single-electron efficiency in different background conditions.
> The time-over-threshold ToT is related to the deposited charge on the pixel and the single-electron efficiency of the detector.
> The relative change in the single-electron efficiency  $\delta \epsilon / \epsilon$  
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< By measuring the mean ToT in high (low) rate environment at $B$ fields of 0 and 1 T, the relative change in single-electron detection efficiency is extracted. 
---
> By measuring the mean ToT in high (low) rate environment runs at $B$ fields of 0 and 1 T, the relative change in single-electron efficiency is extracted. 
362,364c356,358
<  To obtain a precise result for the mean ToT, hits associated to TPC tracks were used. The track selection is the same as in section 2.
<  The analysed runs with a trigger rate above 90 Hz were taken at a beam momentum (E) of 5 GeV/c. 
<  The runs that were taken at a beam momentum of 6 GeV/c have a trigger hit rate below 3 Hz. 
---
>  In order to obtain a precise result for the mean ToT, hits associated to TPC tracks were used. The track selection is the same as in section 2.
>  The analysed runs runs 6934 and 6935 and 6983 have a high hit rate and were taken at a beam momentum of 5 GeV/c. 
>  Runs 6916 and 6969 were taken at a beam momentum of 6 GeV/c and have a low hit rate. 
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<     \scalebox{0.8}{
< 	\begin{tabular}{c c c c c c c c c c c c} \toprule
< 	      run & E & B   & ToT1 & ToT2 & triggers & run time  & Hits1 & Hits2 & trig rate & Rate1 & Rate2 \\
< 	          &GeV & [T] & [$\rm \mu$s]  & $[\rm \mu$s] & $10^{3}$ &  [$10^{3}$s]      &  $10^{6}$  & $10^{6}$ &  [Hz]  & [$10^{3}$hits/s] & [$10^{3}$hits/s]  \\ \midrule
< 	     6916 & 6 & 0 & 0.628 & 0.653 & 16.8 & 5.81 & 6.25    & 13.1     & 2.9 & 1.08 & 2.26  \\
< 	     6934 & 5 & 0 & -     & 0.651 & 73.4 & 0.60 & -       & 20.5     & 121.7 & -    & 33.92 \\
< 	     6935 & 5& 0 & 0.620 &   -    & 73.9 & 0.60 & 6.95    & -        & 122.5 & 11.51 & - \\
< 	     6969 & 6& 1 & 0.650 & 0.666 & 7.94 & 3.45 & 1.93    & 2.16     & 2.3 & 0.56 & 0.62 \\
< 	     6983 & 5 & 1 & 0.657 & 0.678 & 67.9 & 0.70 & 11.6    & 14.1     & 96.2 & 16.44 & 19.94 \\
---
>      \begin{tabular}{c c c c c c c c c c c} \toprule
> 	      run & B   & ToT1 & ToT2 & triggers & run time  & Hits1 & Hits2 & trig rate & Rate1 & Rate2 \\
> 	          & [T] & [$\rm \mu$s]  & $[\rm \mu$s] & $10^{3}$ &  [$10^{3}$s]      &  $10^{6}$  & $10^{6}$ &  [Hz]  & [$10^{3}$hits/s] & [$10^{3}$hits/s]  \\ \midrule
> 	     6916 & 0 & 0.628 & 0.653 & 16.8 & 5.81 & 6.25    & 13.1     & 2.9 & 1.08 & 2.26  \\
> 	     6934 & 0 & -     & 0.651 & 73.4 & 0.60 & -       & 20.5     & 121.7 & -    & 33.92 \\
> 	     6935 & 0 & 0.620 &   -    & 73.9 & 0.60 & 6.95    & -        & 122.5 & 11.51 & - \\
> 	     6969 & 1 & 0.650 & 0.666 & 7.94 & 3.45 & 1.93    & 2.16     & 2.3 & 0.56 & 0.62 \\
> 	     6983 & 1 & 0.657 & 0.678 & 67.9 & 0.70 & 11.6    & 14.1     & 96.2 & 16.44 & 19.94 \\
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<     } 
<    \label{tab:hitrates}
---
>     \label{tab:hitrates}
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<   The relative change in the mean ToT for the \rm{$B$ = 1 T} data is +1.1\% (upper half) and +1.8\% (lower half). 
---
>   The relative change in the mean ToT for the \rm{$B$ = 1 T} data is +1.1\% (upper half) and +1.8\% (lower half) 
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< Using Eq.~\ref{eq:tot}, this means that the relative change in the single-electron detection efficiency  $\delta \epsilon / \epsilon$ is stable at the level of +0.9\% (${B \rm = 1~T}$) and -0.6\% (${B \rm = 0~T}$) for hit  rates up to 3.3 (5.7) kHz per GridPix.
---
> Using Eq.~\ref{eq:tot}, this means that the relative change in the single-electron efficiency  $\delta \epsilon / \epsilon$ is stable at the level of +0.9\% (${B \rm = 1~T}$) and -0.6\% (${B \rm = 0~T}$) for hit  rates up to 3.3 (5.7) kHz per GridPix.
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< The single-electron resolution in the $xy$ precision plane as a function of the local track angle $\phi$, defined as the azimuthal angle in the $xy$ plane, has been measured in the run 6969 of the \rm{$B$ = 1 T} data set taken at a beam momentum of 6 GeV/c. For a pad based readout system the resolution has a strong dependence on the local track 
---
> The single-electron resolution in the $xy$ precision plane as a function of the local track angle $\phi$ has been measured in run 6969 of the \rm{$B$ = 1 T} data set taken at a beam momentum of 6 GeV/c. For a pad based readout system the resolution has a strong dependence on the local track 
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< To test experimentally this hypothesis, reconstructed circular tracks were selected. Examples of circular tracks can be observed in the event display shown in Fig.~\ref{fig:eventDisplay}. For circular tracks, the local track angle $\phi$ depends on the position of the individual hits on the circle
---
> In order to test experimentally this hypothesis, reconstructed circular tracks were selected. Examples of circular tracks can be observed in the event display shown in Fig.~\ref{fig:eventDisplay}. For circular tracks, the local track angle $\phi$ depends on the position of the individual hits on the circle
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< In the circle fit, the coarse uncertainty in $xy$ was estimated to be about 4 pixels and in $z$ equal to 1 mm.
---
> In the circle fit, the coarse uncertainty in $xy$ was estimated to be about 4 pixels and in $z$  1 mm.
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< The selected data set has 973 circular tracks, with a mean radius of 155 pixels ((8.5 mm) and a mean number of hits of 194.
---
> The selected data set has 973 circular tracks, with a mean radius of 155 pixels and a mean number of hits of 194.
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< Finally, the resolution in $xy$ was extracted using a Gaussian fit to the track residuals in the range of $\pm 2 \sigma$ around the centre.
---
> Finally, the resolution in $xy$ was extracted - using a Gaussian fit to the track residuals in the range of $\pm 2 \sigma$ around the centre.
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<  The electron PID resolution in the ILD TPC is expected to be 2.4\% (template fit) and 3\% (truncation method) at polar angles of $\theta =\pi/2$ ($\cos\theta=0$) and a track length ($t^{0}_{\rm length}$) of 1441 mm. 
---
>  The electron PID resolution in the ILD TPC is expected to be 2.4\% (template fit) and 3\% (truncation method) at polar angles of $\theta =\pi/2$ ($\cos\theta=0$) and a track length ($\rm tlength_0$) of 1441 mm. 
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< 	 \sigma_{i} =  \sigma_{e} \sqrt{{t^0_{\rm length}} ~\cdot ~E_{e}}/\sqrt{{t_{\rm length}} ~\cdot ~E_{i}},
---
> 	 \sigma_{i} =  \sigma_{e} \sqrt{{\rm tlength}_0 ~\cdot ~E_{e}}/\sqrt{{\rm tlength} ~\cdot ~E_{i}},
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< where $t_{\rm length}$ is the track length and $E_i$ is the expected energy loss for particle $i$ (electron = $e$, muon = $\rm \mu$ , pion = $\pi$, kaon = $K$, proton = $p$).    
---
> where $\rm tlength$ is the track length and $E_i$ is the expected energy loss for particle $i$ (electron = $e$, muon = $\rm \mu$ , pion = $\pi$, kaon = $K$, proton = $p$).    
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< It is clear from the above that a GridPix Pixel TPC in ILD will provide powerful particle identification. Studies have shown that for a pad readout in ILD, the PID resolution is expected to be 5\% \cite{einhaus2019}. 
---
> It is clear from the above that a GridPix Pixel TPC in ILD will provide powerful particle identification. 
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> 
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<   The single-electron detection efficiency at high hit rates was studied. 
---
>   The single-electron efficiency at high hit rates was studied. 
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< \bibitem{YU201355}
<  Weilin Yu, {Particle identification of the ALICE TPC via dE/dx},
< Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 706,
< (2013), 55--58.
< \newblock \href {https://doi.org/10.1016/j.nima.2012.05.022}
< {\path{doi.org/10.1016/j.nima.2012.05.022}}.
< 
< 
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< \bibitem{einhaus2019}
< 	U. Einhaus, U. Krämer and P. Malek, {Studies on Particle Identification with dE/dx for the ILD TPC},\newblock \href {https://arxiv.org/abs/1902.05519} {\path{arXiv:1902.05519}}.
< 
