Exploring the limits of high-energy proton-pion separation in granular calorimeters

Abstract: Highly granular calorimeters provide detailed information about hadronic-shower development that may enable particle identification beyond their conventional role in energy measurement. We investigate how well this information can distinguish protons from positively charged pions and how the achievable discrimination depends on detector segmentation and particle energy. The study uses Geant4 simulations of isolated particles with energies from 10 to 100 GeV in a homogeneous lead-tungstate calorimeter. A Deep Sets model operating directly on cell positions and detected energy and time outperforms a boosted decision tree based on reconstructed shower observables. With cells measuring $3 \times 3 \times 6$ mm$<sup>3$, Deep Sets achieves an accuracy of 93.8% at 10 GeV, decreasing to 67.2% at 100 GeV. Shower topology is independently informative, deposited energy provides the largest additional contribution, and timing supplies complementary information. Coarser segmentation reduces discrimination, with performance more sensitive to longitudinal than transverse granularity. These results provide an encouraging benchmark for calorimeter-based hadron identification and motivate its inclusion among the optimization targets for future highly granular calorimeters.

• The paper quantifies high-energy proton-pion separation limits and their dependency on detector segmentation and incident energy in PbWO4 granular calorimeters and achieves 93.8% accuracy at 10 GeV.
• Deposited energy and topology are a key indicators, though performance degrades to 67.2% at 100 GeV.
• Longitudinal granularity substantially affects discrimination, underscoring a potential for more focused segmentation strategies in future calorimeter designs.

Overview and motivation

This paper quantifies how much particle-identification (PID) information about an incident charged hadron can be extracted from the spatial, energetic, and temporal structure of its shower in a highly granular homogeneous calorimeter, and how that information depends on detector segmentation and incident energy. The study builds on a previous three-species classification study [DeVita2025] but focuses specifically on proton versus $\pi^+$ discrimination, motivated by the observation that proton- and pion-induced showers differ systematically in electromagnetic fraction, shower radius, longitudinal depth profile, and timing structure. The central question is practical: at what granularity, and up to what energy, can a calorimeter alone serve as a complementary PID system for future collider detectors.

The physical origin of the separation is well established in the paper. Baryon-number conservation forces a baryon into the final state of proton-induced showers, suppressing large transfers of energy to $\pi^0$ production, whereas pions can convert much of their energy through charge exchange ($\pi^+ + p \rightarrow \pi^0 + n$), whose subsequent decay feeds the electromagnetic component. Pion showers therefore deposit on average more visible energy and are more collimated, providing correlated discriminating observables beyond total deposited energy.

Simulation setup and data representations

The study uses Geant4 simulations with the FTFP_BERT physics list of isolated protons and $\pi^+$ at kinetic energies of 10, 25, 50, and 100 GeV, generated 3 m upstream and propagating parallel to the calorimeter axis without angular spread. The detector is a homogeneous PbWO$_4$ block segmented into $100\times100\times200$ cells of $3\times3\times6~\mathrm{mm}^3$, spanning $7.66$ Molière radii laterally and $5.92$ nuclear interaction lengths longitudinally. Only quantities accessible in realistic conditions — deposited energy, hit position, and global time — are used; electronic noise, signal shaping, saturation, and other readout effects are explicitly not modeled, which the authors acknowledge will likely reduce achievable performance relative to the reported numbers.

Two shower representations are constructed. The first compresses each shower into high-level features (HLFs) describing global properties, the reconstructed first nuclear interaction vertex, local energy distribution, and longitudinal/transverse development; these are classified with XGBoost. In this branch, hit times are smeared with a Gaussian resolution of $\sigma = 30$ ps. The second representation preserves full granularity as an enriched point cloud $\pi^0$0 over active cells, with per-cell energies summed over deposits and per-cell times computed as energy-weighted averages; these are processed by a Deep Sets architecture, exploiting permutation invariance. Coarser segmentations are obtained by merging cells post-simulation, so results across configurations share the same underlying events and are intrinsically correlated — small differences between neighboring points are genuine granularity effects rather than statistical fluctuations.

Hyperparameters were tuned by exhaustive grid search with three-fold cross-validation for XGBoost and Bayesian optimization for Deep Sets (fixed architecture, optimized learning rate $\pi^0$1 and batch size). Uncertainties on accuracies use Clopper–Pearson intervals.

Baseline: high-level features at 100 GeV

The HLF/XGBoost baseline uses 80,000 training and 20,000 validation events at 100 GeV with the finest segmentation considered in this branch, $\pi^0$2. Feature importance based on gain identifies the total deposited energy as among the most discriminative observables, followed by the shower radius $\pi^0$3, its energy-weighted counterpart $\pi^0$4, and $\pi^0$5 — consistent with pions' larger electromagnetic fraction producing smaller, more compact showers. Vertex-related features rank lower, attributed by the authors to two compounding effects: these observables benefit from finer longitudinal segmentation than tested, and the vertex-finding algorithm used is not perfectly accurate. This is a concrete limitation on the interpretation of the baseline result rather than evidence against the physics of first-interaction localization.

Accuracy degrades monotonically with increasing cell volume, from roughly 58% toward the homogeneous (unsegmented) limit to higher values at fine segmentation. Because all configurations derive from shared events, even modest neighboring-point differences reflect real granularity effects. A threshold on the winning probability trades efficiency for purity, with pion candidates identified more confidently than protons.

Point-cloud analysis across energy

The Deep Sets analysis doubles the dataset to 100,000 events per species and refines the finest segmentation to $\pi^0$6. Before training, cells with $\pi^0$7 MeV are discarded. The authors show this threshold is not merely noise reduction: it improves accuracy non-negligibly at 25 GeV across all longitudinal segmentations by making the shower core topology visible. However, they concede that without it the model's sensitivity to low-energy deposits indicates a lack of inherent robustness that a larger training set might mitigate; all headline results implicitly assume the 5 MeV threshold.

Relative contribution of spatial, energy, and timing information

A restricted-feature ablation at 10 GeV (assuming perfect timing resolution) isolates the contribution of each observable class:

Feature set	Accuracy
Topology only	74.59%
Topology + time	80.45%
Topology + energy	91.46%
Topology + energy + time	93.80%

Topology alone carries significant discrimination; deposited energy provides the largest single increment; timing adds a complementary gain. The authors caution that this decomposition is only approximate because cell time is an energy-weighted average intrinsically correlated with energy, and the timing result assumes ideal resolution unattainable by any physical detector — so timing-related numbers should be read as an optimistic upper bound.

Granularity dependence and the dominance of the longitudinal dimension

Across all four energy points, accuracy decreases monotonically with increasing cell volume, but the dominant factor is longitudinal segmentation. For fixed $\pi^0$8-cell size, varying transverse segmentation produces only minor changes, whereas coarsening along $\pi^0$9 yields substantial losses. Configurations of equal cell volume but different aspect ratios yield clearly different accuracy, demonstrating that cell volume alone does not characterize retained PID information. The physical explanation is supported visually: reducing transverse granularity from $\pi^+ + p \rightarrow \pi^0 + n$0 to $\pi^+ + p \rightarrow \pi^0 + n$1 elementary cells leaves shower topology largely intact, since the most discriminative structure resides in the longitudinal depth profile. The practical implication stated by the authors is direct: for isolated normally incident particles, allocating channel budget preferentially along the shower direction yields a better information-to-complexity ratio than uniform refinement.

At 100 GeV the transverse segmentation is fixed at $\pi^+ + p \rightarrow \pi^0 + n$2 and only the longitudinal scan is performed. Deep Sets outperforms the HLF baseline at every granularity, attributed both to doubled training statistics and to topological information lost when showers are compressed into aggregate features. Notably, the Deep Sets "no-segmentation" accuracy falls below the BDT value — explained by the absence of primary-particle time-of-flight information in the point-cloud representation, a candid asymmetry between the two pipelines. The authors also stress that the numerical gap between the two classifiers should not be read as a controlled architecture comparison, since their training samples and timing treatments differ.

Degradation with energy

At maximum granularity, accuracy decreases monotonically from 93.8% at 10 GeV to 67.2% at 100 GeV. The authors attribute this not to changing interaction probability (the nuclear interaction length is approximately energy-independent here) but to the growing complexity of high-energy cascades: more numerous and energetic secondary particles generate additional interaction vertices and sub-showers, so observable shower properties become averages over multiple contributions, washing out species-dependent signatures. Recovering sensitivity at high energy would require explicit resolution of secondary vertices and local substructure — itself dependent on fine granularity — leading the authors to conclude that high-energy calorimeter-based PID would not be feasible without sufficient segmentation.

Discussion and implications for detector design

Three findings carry design consequences. First, the strong asymmetry between longitudinal and transverse granularity suggests multi-objective optimization of future calorimeters should weight the longitudinal dimension more heavily for PID purposes — though the authors immediately qualify that this conclusion holds only for isolated, normally incident particles; in realistic environments transverse segmentation remains essential for pileup rejection, track-cluster association, and particle-flow reconstruction, so the relative weighting must be reassessed in multi-particle events. Second, the demonstration that PID performance varies substantially at fixed cell volume argues for including PID sensitivity as an explicit objective alongside energy resolution, reconstruction efficiency, and cost when comparing detector concepts. Third, prospective applications — particle-dependent calibration within particle flow, strange-quark tagging via kaon identification, boosted-object reconstruction — remain speculative pending dedicated studies in realistic conditions.

The limitations section is unusually explicit. Reported accuracies are optimistic benchmarks: readout effects (noise, thresholds, saturation, finite resolution) are absent; simulations assume perfect time resolution for point clouds; and Geant4 shower modeling, though community-standard, may imperfectly reproduce some of the very shower properties the classifiers exploit. Conversely, the numbers are not strict upper bounds, since alternative architectures could recover correlations the chosen models miss.

Limitations and open questions

Several questions remain open and are identified by the authors themselves. Whether the observed performance survives realistic readout simulation — including the interplay between the 5 MeV threshold and actual electronics thresholds and noise — is untested. The role of timing under finite resolution is unresolved: the restricted-feature study assumes perfect resolution, and the authors state that a scan over realistic timing resolutions is required before any conclusion about ultimate detector performance. The extension to multi-particle events with pileup and overlapping showers, where transverse granularity regains importance, has not been attempted. Finally, whether targeted substructure methods can recover the discrimination lost above tens of GeV — where accuracy drops to 67.2% — is left as the key open problem for extending calorimeter-based PID to the high-energy regime relevant at future colliders.

Conclusion

This work establishes quantitative benchmarks for proton–pion separation using calorimeter information alone in a highly granular PbWO$\pi^+ + p \rightarrow \pi^0 + n$3 geometry. A Deep Sets model on enriched point clouds reaches 93.8% accuracy at 10 GeV with $\pi^+ + p \rightarrow \pi^0 + n$4 cells, degrading monotonically to 67.2% at 100 GeV, and consistently outperforms a feature-based BDT at segmented configurations while losing at zero segmentation due to missing time-of-flight information. Shower topology is independently informative; deposited energy contributes the largest increment; timing adds complementary power under idealized assumptions. Performance is far more sensitive to longitudinal than transverse cell dimensions, so cell volume is insufficient to characterize retained PID information. The results support incorporating PID sensitivity as an objective in the co-design of future highly granular calorimeters, while the reported values should be treated as optimistic benchmarks pending realistic readout, timing-resolution, and multi-particle-event studies.