CERN Accelerating science

Granularity of the electromagnetic barrel LAr calorimeters~\cite{CERN-LHCC-96-040}. The accordion geometry is indicated with blue and orange lines on the side of the tower.

Granularity of the electromagnetic barrel LAr calorimeters~\cite{CERN-LHCC-96-040}. The accordion geometry is indicated with blue and orange lines on the side of the tower.

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: Example of the steps in the \PCAI transformation for 65~\GeV\ photons with $0.2 < |\eta| < 0.25$: (a) shows the distribution of energy fractions in EMB1, (b) the Gaussian distribution, and (c) is the leading principal component of the \PCAI with bin borders (dotted pink lines) showing five PCA bins. The steps of the \PCAII are identical to those of the \PCAI but performed in each PCA bin separately to generate uncorrelated Gaussian distributions using all principal components of the \PCAII. The errors bars indicate the size of the statistical uncertainty.

: : Example of the steps in the \PCAI transformation for 65~\GeV\ photons with $0.2 < |\eta| < 0.25$: (a) shows the distribution of energy fractions in EMB1, (b) the Gaussian distribution, and (c) is the leading principal component of the \PCAI with bin borders (dotted pink lines) showing five PCA bins. The steps of the \PCAII are identical to those of the \PCAI but performed in each PCA bin separately to generate uncorrelated Gaussian distributions using all principal components of the \PCAII. The errors bars indicate the size of the statistical uncertainty.

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: Correlations between the transformed energies deposited in several layers, before PCA rotation, showing (a) Presampler barrel vs EM barrel 1, (b) Presampler vs EM barrel 2 and (c) EM barrel 1 vs EM barrel 2. The energies were transformed into Gaussian distributions. The correlation factors obtained from these 2D histograms are displayed.

: : Correlations between the transformed energies deposited in several layers, before PCA rotation, showing (a) Presampler barrel vs EM barrel 1, (b) Presampler vs EM barrel 2 and (c) EM barrel 1 vs EM barrel 2. The energies were transformed into Gaussian distributions. The correlation factors obtained from these 2D histograms are displayed.

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: Correlations between the \PCAIh components after the PCA rotation. The individual components are approximately Gaussian distributed.

: : Correlations between the \PCAIh components after the PCA rotation. The individual components are approximately Gaussian distributed.

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: : Validation of the energy, $E$, parameterization is shown for 65~\GeV\ photons with $0.2 < |\eta| < 0.25$, comparing the input \GEANT sample (black triangles) with \FCaloSII (red dots). Good agreement is observed for all layers and the total energy. The errors bars indicate the size of the statistical uncertainty.

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: Model: equal hit energy

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: Comparison of mean and RMS of the models

: pion : The lateral shower shape parameterization for (a) photons and (b) pions with energies of 265~\GeV{} in the range $0.55 < |\eta| < 0.60$ and parameterized in the second layer of the EM barrel and Tile barrel respectively. To visualize the core of the shower, these plots have a cut-off at $\Delta R^\text{mm} \sim 100~\SI{}{\milli\meter}$.

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: \GEANT : Model: equal hit energy

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: Model: weighted hit energy : Comparison of mean and RMS of the models

Schematic representation of the architecture of the GANs used by \FGAN. The input to the generator is at the top left and the output from the discriminator is at the bottom left. The Rectified Linear Unit (ReLU) activation function is used in all layers of the discriminator with the exception of the last.

: Ratio $E_\text{voxel}/E_\text{hit}$ as function of $\Delta R^{\SI{}{\milli\meter}}$ for deposited energy from a 65~\GeV\ charged pion in EMB2 in the range $0.20<|\eta|<0.25$ in the first bin of the leading PCA (PCA=1). Entries with $E_\text{voxel}=0$ are shown in the underflow bin below $10^{-9}$. Lateral shower shape model (a) in \GEANT, (b) in a model using equal deposited energies, (c) in a model using weighted hit deposited energies. (d) Comparison of the mean (central value) and the RMS (error bars) for the equal hit, weighted hit and \GEANT models. The yellow band indicates the $1\sigma$ uncertainty for \GEANT. : Caption not extracted

The $\chi^2$ sum divided by the number of degree of freedom (NDF) calculated between the reference samples and the GAN as a function of the number of epochs. The lowest point (in red) represents the selected epoch.

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Sum of the energy in all voxels for photons with $0.2 < |\eta| < 0.25$. The calorimeter response for \GEANT (solid black line) compared with \FGAN (dashed red line).

Schematic representation of the architecture of the GANs used by \FGAN. The input to the generator is at the top left and the output from the discriminator is at the bottom left. The Rectified Linear Unit (ReLU) activation function is used in all layers of the discriminator with the exception of the last.

Sum of the energy in all voxels for pions with $0.2 < |\eta| < 0.25$. The calorimeter response for \GEANT (solid black line) is compared with \FGAN (dashed red line).

The $\chi^2$ sum divided by the number of degree of freedom (NDF) calculated between the reference samples and the GAN as a function of the number of epochs. The lowest point (in red) represents the selected epoch.

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Sum of the energy in all voxels for photons with $0.2 < |\eta| < 0.25$. The calorimeter response for \GEANT (solid black line) compared with \FGAN (dashed red line).

: Sum and RMS of the energy in all voxels normalized to the true momentum for (a) photons, (b) electrons and (c) pions with 0.2 < $|\eta|$ < 0.25 as a function of the true momentum. The calorimeter response for \GEANT (solid black line) is compared with \FGAN (dashed red line), which is also abbreviated to FGAN. The uncertainty bars in the top panel indicate the RMS of the total energy distribution. The ratio of the means of the two energy distributions is shown in the middle panel, and the ratio of the RMS values is shown in the bottom panel. The error bars in the ratio indicate its statistical uncertainty. For most points, this uncertainty is smaller than the size of the markers.

Sum of the energy in all voxels for pions with $0.2 < |\eta| < 0.25$. The calorimeter response for \GEANT (solid black line) is compared with \FGAN (dashed red line).

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: Sum and RMS of the energy in all voxels as a function of $|\eta|$ for (a) photons, (b) electrons and (c) pions of momentum 65~\GeV{}. The calorimeter response for \GEANT (solid black line) is compared with \FGAN (dashed red line), which is also abbreviated to FGAN, while their ratio is shown in the ratio plots. The uncertainty bars in the top panel indicate the RMS of the total energy distribution. The ratio of the means of the two energy distributions is shown in the middle panel, and the ratio of the RMS is shown in the bottom panel. The error bars in the ratio indicate its statistical uncertainty. For most points, this uncertainty is smaller than the size of the markers.

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The probability of a single-pion event to produce at least one punch-through particle with an energy of at least 50~\MeV\ as a function of the $\eta$ and $p$ of the incoming pion determined from \GEANT.

: Sum and RMS of the energy in all voxels normalized to the true momentum for (a) photons, (b) electrons and (c) pions with $0.2 < |\eta| < 0.25$ as a function of the true momentum. The calorimeter response for \GEANT (solid black line) is compared with \FGAN (dashed red line), which is also abbreviated to FGAN. The uncertainty bars in the top panel indicate the RMS of the total energy distribution. The ratio of the means of the two energy distributions is shown in the middle panel, and the ratio of the RMS values is shown in the bottom panel. The error bars in the ratio indicate its statistical uncertainty. For most points, this uncertainty is smaller than the size of the markers. : Caption not extracted

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: Sum and RMS of the energy in all voxels as a function of $|\eta|$ for (a) photons, (b) electrons and (c) pions of momentum 65~\GeV{}. The calorimeter response for \GEANT (solid black line) is compared with \FGAN (dashed red line), which is also abbreviated to FGAN, while their ratio is shown in the ratio plots. The uncertainty bars in the top panel indicate the RMS of the total energy distribution. The ratio of the means of the two energy distributions is shown in the middle panel, and the ratio of the RMS is shown in the bottom panel. The error bars in the ratio indicate its statistical uncertainty. For most points, this uncertainty is smaller than the size of the markers. : Caption not extracted

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The probability of a single-pion event to produce at least one punch-through particle with an energy of at least 50~\MeV\ as a function of the $\eta$ and $p$ of the incoming pion determined from \GEANT.

The configuration of the different tools used for \AFIII, which depends on the particle type, the detector and the particle energy.

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Reconstructed photon energy for photons generated at the calorimeter surface with an energy of 65~\GeV{} and 0.2 < $|\eta|$ < 0.25 by \GEANT (solid black line), \FCaloSII{} (dashed blue line), and \FGAN (dashed red line). The statistical uncertainties are shown but are similar in size to the points or smaller.

: : The punch-through probability as a function of the punch-through pion (a) multiplicity and (b) energy. The error bars indicate the statistical uncertainty and the overflow is not included in the final bins.

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Distribution of the number of constituents in the jets in a $1.8 < \pT < 2.5$ TeV dijet sample in \GEANT (black triangles) and the combination of \FCaloSII and \FGAN with transitions in the range 4--8~\GeV\ (blue stars), 8--16~\GeV\ (red diamonds) and 16--32~\GeV\ (green crosses). Here `hybrid' refers to the combination of \FCaloSII and \FGAN. The statistical uncertainties are shown but may be smaller than the markers.

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: : The punch-through probability as a function of (a) deflection angle in $\theta$ and energy, (b) deflection angle in $\phi$ and energy, (c) relative momentum deflection in $\theta$ and energy, and (d) relative momentum deflection in $\phi$ and energy. Secondary pions with an energy of 524~\GeV\ in the region $|\eta| \leq 0.4$ from the \GEANT reference samples were used.

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The configuration of the different tools used for \AFIII, which depends on the particle type, the detector and the particle energy.

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Reconstructed photon energy for photons generated at the calorimeter surface with an energy of 65~\GeV{} and $0.2 < |\eta| < 0.25$ by \GEANT (solid black line), \FCaloSII{} (dashed blue line), and \FGAN (dashed red line). The statistical uncertainties are shown but are similar in size to the points or smaller.

The simulated total energy before (blue stars) and after (red diamonds) probabilistic reweighting for a photon of energy 262~\GeV{} in the range $0.4 < |\eta| < 0.45$ compared with \GEANT (black triangles). The RMS of each distribution is indicated in the legend. The statistical uncertainties are shown but may be smaller than the markers.

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: : Ratio of the average energy response to the generated energy for $\pi^{\pm}$ for (a) $0.20 < |\eta| < 0.25$ and (b) as a function of $|\eta|$ and \Ekin. The error bars indicate the statistical uncertainty of the mean. For most points, this uncertainty is smaller than the size of the markers.

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Distribution of the number of constituents in the jets in a $1.8 < \pT < 2.5$ TeV dijet sample in \GEANT (black triangles) and the combination of \FCaloSII and \FGAN with transitions in the range 4--8~\GeV\ (blue stars), 8--16~\GeV\ (red diamonds) and 16--32~\GeV\ (green crosses). Here `hybrid' refers to the combination of \FCaloSII and \FGAN. The statistical uncertainties are shown but may be smaller than the markers.

Energy response correction factors as a function of the true kinetic energy for protons, neutrons and kaons (left) in the barrel and their antiparticles (right). The kinetic energy for antiparticles includes their mass. The coloured bands indicate the size of the statistical uncertainty in the correction.

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Energy response correction factors as a function of the true kinetic energy for protons, neutrons and kaons (left) in the barrel and their antiparticles (right). The kinetic energy for antiparticles includes their mass. The coloured bands indicate the size of the statistical uncertainty in the correction.

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Residual energy response correction factors as a function of the true kinetic energy for photons, electrons and pions in the endcap. The coloured bands indicate the size of the statistical uncertainty in the correction.

: : Number of cells in the leading cluster for pions in the barrel at different energies in \GEANT (black triangles), \FCaloSII (red diamonds) and \FGAN (blue stars). The statistical uncertainties are shown but may be smaller than the markers.

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: Photons

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: Pions : Energy response, defined as the ratio of the reconstructed energy in the calorimeter cells to the kinetic energy of the particle, for (a) photons in $1.05 < |\eta| < 1.10 $ and (b) pions in $0.20 < |\eta| < 0.25$. The red dotted points represent the response derived at discrete energies, using \GEANT simulated single particles. The black line is a spline fit used to interpolate between discrete energy points. The statistical uncertainties are shown but are similar in size to the points or smaller.

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The simulated total energy before (blue stars) and after (red diamonds) probabilistic reweighting for a photon of energy 262~\GeV{} in the range $0.4 < |\eta| < 0.45$ compared with \GEANT (black triangles). The RMS of each distribution is indicated in the legend. The statistical uncertainties are shown but may be smaller than the markers.

: The transverse momentum distribution of the leading jets (a) and the pseudorapidity distribution of the sub-leading jets (b) in a \ttbar sample with \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The jets are EMPFlow jets with $R=0.4$. The statistical uncertainties are shown but may be smaller than the size of the markers.

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: (a) The total energy response exhibits a dependence on the impact position in $\phi$ of the particle in the calorimeter cell ($|\phi_\mathrm{mod}|$), shown for 65~\GeV\ photons with $0.2 < |\eta| < 0.25$ (\GEANT). The ratio has been shifted such that mean ratio of the energy from \GEANT to the true energy is unity. (b) The impact of the correction on \GEANT simulation (gray triangles are without correction, black are with corrections) and the result of the stand-alone simulation for 131~\GeV\ photons with $1.65 < |\eta| < 1.7$ to which the correction has been applied as well as the reweighting described in Section~\ref{sec:energy:res_corr}. The statistical uncertainties are shown in the error bars. : Caption not extracted

: Distribution of the number of constituents in the leading jets for EMPFlow jets with $R=0.4$ (a) and UFO jets with $R=1.0$ (b) in the $Z^\prime$ sample in \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

Energy response correction factors as a function of the true kinetic energy for protons, neutrons and kaons (left) in the barrel and their antiparticles (right). The kinetic energy for antiparticles includes their mass. The coloured bands indicate the size of the statistical uncertainty in the correction.

The $D_2$ variable for the leading jets in a $W^\prime$ sample reconstructed using the UFO algorithm with radius parameter $R=1.0$ with \GEANT (black triangles), \AFII (blue stars), and \AFIII (red dimaonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

Energy response correction factors as a function of the true kinetic energy for protons, neutrons and kaons (left) in the barrel and their antiparticles (right). The kinetic energy for antiparticles includes their mass. The coloured bands indicate the size of the statistical uncertainty in the correction.

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Residual energy response correction factors as a function of the true kinetic energy for photons, electrons and pions in the endcap. The coloured bands indicate the size of the statistical uncertainty in the correction.

: The $\tau_{32}$ variable for the leading jets in a $Z^\prime$ sample reconstructed using the UFO algorithm with radius parameter $R=1.0$ (a) and the LCTopo algorithm (b) with \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers, and the dark blue arrows indicate that a point is beyond the $y$-axis range.

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: after correction : The ratio of the energies assigned to each cuboid of the second electromagnetic barrel layer in \AFIII and \GEANT for a photon of 65~\GeV{} in the range $0.20 < |\eta| < 0.25$ using a simplified cuboid geometry and after applying the correction for the simplified geometry.

: Hadronic $\tau$-lepton decay modes for reconstructed $\tau$-leptons matched to true $\tau$-leptons (a) and reconstructed $\tau$-leptons not matched to true $\tau$-leptons (b) in a $Z^\star/\gamma^\star \rightarrow \tau\tau$ Drell--Yan sample filtered for an off-shell mass of 2.0--2.25~\TeV{}. The decays with one or three charged-particle tracks are denoted by 1p and 3p respectively. X$(=1,2,3)$ denotes the number of neutral particles. The statistical uncertainties are shown but may be smaller than the size of the markers.

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Number of clusters in hadronic $\tau$-decay candidates reconstructed with one charged track (1p) and either matched (a) or not matched (b) to a true $\tau$-lepton in an $Z^\star/\gamma^\star \rightarrow \tau\tau$ Drell--Yan sample filtered for an off-shell mass of 2.0--2.25~\TeV{}. The statistical uncertainties are shown but may be smaller than the size of the markers.

: `Tight' identification efficiencies for single electrons with true energy greater than 20\,\GeV (a) and photons from $H \rightarrow \gamma\gamma$ decays (b) inclusive in $|\eta| < 2.5$ as a function of their reconstructed \pT for \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers. : Caption not extracted

Number of clusters in hadronic $\tau$-decay candidates reconstructed with one charged track (1p) and either matched (a) or not matched (b) to a true $\tau$-lepton in an $Z^\star/\gamma^\star \rightarrow \tau\tau$ Drell--Yan sample filtered for an off-shell mass of 2.0--2.25~\TeV{}. The statistical uncertainties are shown but may be smaller than the size of the markers.

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: The transverse momentum distribution of the leading jets (a) and the pseudorapidity distribution of the sub-leading jets (b) in a \ttbar sample with \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The jets are EMPFlow jets with $R=0.4$. The statistical uncertainties are shown but may be smaller than the size of the markers. : Caption not extracted

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Comparison of muon punch-through simulation in \AFIII and \GEANT as a function of the \pT of misidentified muons from 500~\GeV\ single-pion events. The statistical uncertainties are shown but may be smaller than the size of the markers.

: Distribution of the number of constituents in the leading jets for EMPFlow jets with $R=0.4$ (a) and UFO jets with $R=1.0$ (b) in the $Z^\prime$ sample in \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers. : Caption not extracted

Comparison of muon segments in jets reconstructed with a radius parameter of 0.4 using the EMPFlow algorithm in a $Z^\prime\rightarrow\ttbar$ sample with a $Z^\prime$ mass of 4~\TeV\ in \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

The $D_2$ variable for the leading jets in a $W^\prime$ sample reconstructed using the UFO algorithm with radius parameter $R=1.0$ with \GEANT (black triangles), \AFII (blue stars), and \AFIII (red dimaonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

The difference between the true \MET{} and the reconstructed \MET{} in the $x$ (a) and $y$ (b) directions for a \ttbar{} sample for \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

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The reconstructed diphoton invariant mass distribution from a selection targeting events with Higgs boson decays into two photons. Events are selected by requiring two photons with $\pT > 0.35 m_{\gamma\gamma}$ and $\pT > 0.25 m_{\gamma\gamma}$, and with $|\eta| < 1.37$ or $1.52 < |\eta| < 2.47$. The statistical uncertainties are shown but may be smaller than the size of the markers.

: The $\tau_{32}$ variable for the leading jets in a $Z^\prime$ sample reconstructed using the UFO algorithm with radius parameter $R=1.0$ (a) and the LCTopo algorithm (b) with \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers, and the dark blue arrows indicate that a point is beyond the $y$-axis range. : Caption not extracted

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: Invariant mass distribution from a selection targeting events with a $Z$ boson decaying into (a) two muons or (b) two electrons with $\pT > 25$~\GeV\ and $|\eta| < 1.37$ or $1.52 < |\eta| < 2.47$, and (c) the visible part of the invariant mass of two hadronically decaying $\tau$-leptons in Drell--Yan $Z^\star/\gamma^\star \rightarrow \tau\tau$ events filtered for an off-shell mass of 2.0--2.25~\TeV{}. The statistical uncertainties are shown but may be smaller than the size of the markers.

: Hadronic $\tau$-lepton decay modes for reconstructed $\tau$-leptons matched to true $\tau$-leptons (a) and reconstructed $\tau$-leptons not matched to true $\tau$-leptons (b) in a $Z^\star/\gamma^\star \rightarrow \tau\tau$ Drell--Yan sample filtered for an off-shell mass of 2.0--2.25~\TeV{}. The decays with one or three charged-particle tracks are denoted by 1p and 3p respectively. X$(=1,2,3)$ denotes the number of neutral particles. The statistical uncertainties are shown but may be smaller than the size of the markers. : Caption not extracted

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Number of clusters in hadronic $\tau$-decay candidates reconstructed with one charged track (1p) and either matched (a) or not matched (b) to a true $\tau$-lepton in an $Z^\star/\gamma^\star \rightarrow \tau\tau$ Drell--Yan sample filtered for an off-shell mass of 2.0--2.25~\TeV{}. The statistical uncertainties are shown but may be smaller than the size of the markers.

Comparison of the CPU performance of \AFIII with \GEANT and \AFII. The average CPU time to simulate an event is estimated using 10\,000 single photons at $0.20 < |\eta| < 0.25$ for three different energies: 8~\GeV{}, 65~\GeV{} and 256~\GeV{}. These photons are generated on the calorimeter surface and provide a comparison for calorimeter-only simulation time.

Number of clusters in hadronic $\tau$-decay candidates reconstructed with one charged track (1p) and either matched (a) or not matched (b) to a true $\tau$-lepton in an $Z^\star/\gamma^\star \rightarrow \tau\tau$ Drell--Yan sample filtered for an off-shell mass of 2.0--2.25~\TeV{}. The statistical uncertainties are shown but may be smaller than the size of the markers.

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: The (a) reconstructed muon transverse momentum distribution and (b) identification efficiency for different muon working points for a $Z \rightarrow \mu\mu$ sample generated with $\pT(Z) = 0$ for \GEANT, \AFII, and \AFIII. The statistical uncertainties are shown but may be smaller than the size of the markers. : Caption not extracted

Comparison of muon punch-through simulation in \AFIII and \GEANT as a function of the \pT of misidentified muons from 500~\GeV\ single-pion events. The statistical uncertainties are shown but may be smaller than the size of the markers.

Comparison of muon segments in jets reconstructed with a radius parameter of 0.4 using the EMPFlow algorithm in a $Z^\prime\rightarrow\ttbar$ sample with a $Z^\prime$ mass of 4~\TeV\ in \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

The difference between the true \MET{} and the reconstructed \MET{} in the $x$ (a) and $y$ (b) directions for a \ttbar{} sample for \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

The reconstructed diphoton invariant mass distribution from a selection targeting events with Higgs boson decays into two photons. Events are selected by requiring two photons with $\pT > 0.35 m_{\gamma\gamma}$ and $\pT > 0.25 m_{\gamma\gamma}$, and with $|\eta| < 1.37$ or $1.52 < |\eta| < 2.47$. The statistical uncertainties are shown but may be smaller than the size of the markers.

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: Invariant mass distribution from a selection targeting events with a $Z$ boson decaying into (a) two muons or (b) two electrons with $\pT > 25$~\GeV\ and $|\eta| < 1.37$ or $1.52 < |\eta| < 2.47$, and (c) the visible part of the invariant mass of two hadronically decaying $\tau$-leptons in Drell--Yan $Z^\star/\gamma^\star \rightarrow \tau\tau$ events filtered for an off-shell mass of 2.0--2.25~\TeV{}. The statistical uncertainties are shown but may be smaller than the size of the markers. : Caption not extracted

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: : Distribution of the (left) number of constituents in the leading $R=0.4$ EMPFlow jets in the $W^\prime$ sample and (right) the mass of trimmed $R=1.0$ UFO jets in the $Z^\prime$ sample in \GEANT (black triangles), \AFII (blue stars), and \AFIII (red diamonds). The statistical uncertainties are shown but may be smaller than the size of the markers.

Comparison of the CPU performance of \AFIII with \GEANT and \AFII. The average CPU time to simulate an event is estimated using 10\,000 single photons at $0.20 < |\eta| < 0.25$ for three different energies: 8~\GeV{}, 65~\GeV{}, and 256~\GeV{}. These photons are generated on the calorimeter surface and provide a comparison for calorimeter-only simulation time.