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Comparative study and optimization of SDHCAL hadronic energy reconstruction methods

This paper evaluates various hadronic energy reconstruction methods for the SDHCAL within the ILD detector using the APRIL Particle Flow Algorithm, finding that split and polynomial regression techniques offer the best balance of linearity and resolution while highlighting the potential of integrating precise timing information to further reduce PFA confusion.

Original authors: Tanguy Pasquier, Gérald Grenier, Imad Laktineh

Published 2026-07-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tanguy Pasquier, Gérald Grenier, Imad Laktineh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic puzzle where the pieces are the tiniest building blocks of matter. To solve this puzzle, scientists build massive machines called particle colliders, which smash particles together at nearly the speed of light to see what happens. But when these particles crash, they don't just bounce off; they explode into showers of new particles, like a firework display made of invisible sparks. To understand the explosion, physicists need a way to catch every single spark and measure how much energy it carries. This is where "calorimeters" come in. Think of a calorimeter not as a thermometer, but as a giant, high-tech net made of layers of steel and sensors. When a particle hits the net, it leaves a trail of signals, kind of like a footprint in the snow. The trick is that these footprints can be messy. Sometimes, two particles land on the same spot, or the snow is too deep in some places and shallow in others, making it hard to count exactly how many sparks there were.

The goal of this research is to figure out the best way to translate those messy footprints into a precise energy number. The scientists are testing different "math recipes" to turn the raw data from their detector into a clear answer. They want to know: if a particle has a certain amount of energy, can our math tell us exactly what that energy is, no matter how messy the explosion gets? Getting this right is crucial for the next generation of particle colliders, which aim to discover new physics by measuring these energy explosions with incredible precision. If the math is off, we might miss a tiny clue that points to a whole new universe of secrets.

The Paper: A Detective Story for Particle Showers

This paper is a comparative study of different math recipes used to reconstruct the energy of hadronic showers (explosions of heavy particles) in a specific detector called the SDHCAL, which is part of a larger machine concept known as the ILD. The researchers, T. Pasquier, G. Grenier, and I. Laktineh, put four different reconstruction methods to the test using computer simulations. They wanted to see which recipe gave the most accurate and consistent results across different energy levels.

First, the team had to deal with a sneaky problem caused by the shape of the detector. The SDHCAL is built like a giant barrel made of stacked wheels. When particles fly straight into the ends of the barrel (the "endcap"), they hit the layers head-on, leaving a nice, clear trail of signals. But when particles fly into the side of the barrel (the "barrel"), they often hit the layers at a slant. Imagine walking through a forest: if you walk straight down a row of trees, you pass a certain number of trunks. But if you walk diagonally across the rows, you might pass fewer trees because you're cutting corners, or you might hit the same tree twice depending on the angle. In the detector, this slanting path meant the particles were leaving fewer "footprints" (signals) than expected, causing the computer to think the particle had less energy than it actually did. The authors found that this effect was significant enough to throw off their measurements. To fix this, they developed a "geometric correction," a mathematical adjustment that acts like a pair of glasses, correcting the view so that a slanted path is counted as if it were straight. Once this correction was applied, the energy measurements in the barrel became much more accurate.

Next, they tested four different ways to calculate the final energy:

Linear: A simple recipe where the energy is just a straight sum of the signals.
Quadratic: A slightly more complex recipe that accounts for the fact that as particles get more energetic, the signals start to "saturate" or overlap, making the relationship non-straight.
Split: A clever approach that uses two different recipes: one for low-energy events (small clusters of signals) and another for high-energy events (large clusters), switching between them at a specific threshold.
Polynomial Regression: A method inspired by machine learning that uses a complex polynomial equation to find the best fit for the data.

When they tested these recipes on single particles (specifically neutral kaons, or $K^0_L$ ), the results were interesting. The simple linear method was okay but tended to overestimate the energy, while the quadratic method tended to underestimate it, especially at lower energies. The Split method and the Polynomial regression emerged as the winners in this category. They offered the best balance, providing excellent resolution (precision) at low energies without losing accuracy at high energies. The Split method, in particular, was very robust, keeping the linearity (how well the measured energy matches the true energy) within a few percent across the board.

However, the story gets more complicated when they moved from single particles to "dijets"—events where two jets of particles are created, mimicking the messy, high-energy collisions scientists actually care about. In these scenarios, the biggest enemy isn't the math recipe itself, but "confusion." Imagine a crowded party where everyone is shouting; it's hard to tell who said what. In a high-energy jet, thousands of particles are created, and the reconstruction algorithm (called APRIL) sometimes gets confused about which signal belongs to which particle.

The authors found that at high energies (above 100 GeV), this confusion becomes the dominant factor in the error. Surprisingly, when confusion is high, the simple Linear method actually performed quite well, sometimes better than the complex non-linear methods. Why? Because the complex methods are so sensitive to the tiny fluctuations caused by confusion that they amplify the errors. The linear method, being simpler, is less easily thrown off by the noise. However, the authors note that if they used a "PerfectPFA" (a hypothetical scenario where the computer knows exactly which particle created every signal, removing all confusion), the non-linear methods (like the Split and Polynomial ones) would once again be superior across all energy levels. This suggests that the current limitation isn't the math formulas themselves, but the ability of the reconstruction software to untangle the messy particle showers.

In conclusion, the paper suggests that while all methods work reasonably well, the Split method and Polynomial regression offer the best overall compromise for the current setup. They provide improved resolution at low energies and maintain good linearity at high energies. However, the authors emphasize that the future of this technology lies in reducing that "confusion." They suggest that integrating precise timing information from a new version of the detector (the T-SDHCAL) and improving the clustering algorithms could help separate the overlapping particles. If they can do that, the more sophisticated math recipes will likely unlock even better performance, bringing us closer to the ultra-precise measurements needed for the next generation of particle physics discoveries.

Technical Summary: Comparative Study and Optimization of SDHCAL Hadronic Energy Reconstruction Methods

Problem Statement
The Semi-Digital Hadronic Calorimeter (SDHCAL) is a proposed high-granularity detector for the International Large Detector (ILD) at future lepton colliders (Higgs factories). While high granularity is essential for Particle Flow Algorithms (PFA) to separate overlapping particle showers, the SDHCAL's semi-digital readout (providing 2-bit information across three charge thresholds) presents challenges for precise energy reconstruction. Specifically, the study identifies two primary issues:

Angular Dependence in the Barrel: Due to the "Videau" geometry of the calorimeter modules, particles incident at non-perpendicular angles in the barrel region traverse fewer sensitive layers per unit path length compared to the endcap. This leads to a systematic underestimation of energy in the barrel.
Reconstruction Formula Limitations: Standard linear reconstruction formulas, often used for analog calorimeters, may not adequately capture the non-linear saturation effects of hadronic showers at higher energies or the varying particle densities within a shower. The study seeks to determine which reconstruction strategy (linear, quadratic, split, or machine-learning-based) offers the best compromise between linearity and energy resolution across different energy regimes and physics cases (single neutral hadrons vs. dijets).

Methodology
The study utilizes the ILD_l2_v02 detector configuration within the k4geo package, simulated using GEANT4 (QGSP_BERT physics list) and reconstructed with the APRIL PFA framework. The analysis proceeds through the following steps:

Angular Corrections: A preliminary study revealed a ~5% energy underestimation in the barrel compared to the endcap for 20 GeV $K^0_L$ events. To address this, a purely geometrical correction was implemented. This applies angular weights to the number of hits ( $N_1, N_2, N_3$ ) based on the cluster's polar angle ( $\theta$ ) and a local azimuthal angle ( $\phi$ ) relative to the module orientation, effectively correcting for the reduced path length through sensitive layers.
Calibration Dataset: Calibration parameters were derived using samples of single $K^0_L$ events (5–90 GeV) fully contained within the HCAL, ensuring no leakage or upstream interactions.
Reconstruction Methods: Four distinct energy reconstruction formulas were compared:
1. Linear: A weighted sum of hits ( $E = \alpha N_1 + \beta N_2 + \gamma N_3$ ) with constant coefficients.
2. Quadratic: The coefficients ( $\alpha, \beta, \gamma$ ) are parametrized as quadratic functions of the total number of hits ( $N_{hit}$ ) to account for saturation.
3. Split Method: Two separate quadratic functions are fitted: one for low-energy showers ( $N_{hit} \le 400$ ) and one for high-energy showers ( $N_{hit} > 400$ ), with a smooth transition between them.
4. Polynomial Regression: A machine-learning-inspired approach using scikit-learn to fit a degree-2 polynomial combination of $N_1, N_2, N_3$ and their cross-terms.
Performance Metrics: Methods were evaluated on:
- Single $K^0_L$ samples: To assess intrinsic calorimeter performance (linearity and resolution) without PFA confusion.
- Dijet ( $u, d, s$ ) samples: To assess performance in a realistic PFA environment where "confusion" (mis-association of hits to particles) dominates resolution at high energies.
- PerfectPFA: A control study using Monte Carlo truth information to isolate detector resolution from reconstruction confusion.

Key Contributions and Results

Angular Correction: The implementation of the geometrical angular correction successfully mitigated the systematic underestimation in the barrel. While a slight residual overestimation (2–3%) was observed above 20 GeV, the correction significantly reduced the dependence on the incidence angle, restoring linearity.
Single $K^0_L$ Performance:
- All methods achieved linearity better than 5% above 10 GeV.
- The Split Method and Polynomial Regression demonstrated the best overall performance. The Split Method provided improved resolution at low energies (5–20 GeV) compared to linear and standard quadratic methods, while maintaining linearity at higher energies.
- The Linear method consistently showed the poorest resolution across all energy points.
Dijet Performance and PFA Confusion:
- In realistic APRIL reconstruction, the performance hierarchy shifted at high energies. For jet energies $E_j < 60$ GeV, the Quadratic and Split methods outperformed the Linear method.
- However, for $E_j \ge 90$ GeV, the trend reversed: the Linear method provided competitive or superior resolution compared to non-linear methods.
- Significance of Confusion: The study attributes this reversal to PFA confusion. Non-linear formulas are more sensitive to fluctuations in hit multiplicity caused by hit mis-assignment. When "PerfectPFA" (truth-based reconstruction) was used, the non-linear methods (Quadratic, Split, Polynomial) regained superiority across the entire energy range, confirming that the degradation of non-linear methods in realistic scenarios is driven by reconstruction confusion rather than the formulas themselves.
- The Polynomial Regression consistently provided the best linearity for dijets.
Endcap vs. Barrel: Results in the endcap region were consistent with the barrel, showing comparable linearity and resolution, confirming the robustness of the methods across detector geometries.

Significance and Claims
The paper concludes that while all tested methods achieve acceptable performance, the Split Method and Polynomial Regression offer the best compromise across different energy regimes and physics cases. The Split Method is particularly effective at improving low-energy resolution without sacrificing high-energy linearity.

The study emphasizes that the optimal choice of reconstruction formula is contingent upon the reconstruction quality and the level of PFA-induced confusion. At high jet energies, where confusion dominates, the simplicity of the Linear method reduces sensitivity to fluctuations, making it competitive. Conversely, in ideal conditions (PerfectPFA) or at lower energies, non-linear methods are superior.

The authors state that future improvements in hadronic energy reconstruction for next-generation lepton colliders will likely depend less on the choice of formula alone and more on:

Integrating precise timing information from the T-SDHCAL (Time-resolved SDHCAL) to disentangle overlapping showers.
Implementing advanced reclustering or cluster-splitting techniques within the PFA to mitigate confusion.
Addressing specific detector non-uniformities, such as the gaps between barrel wheels in the Videau geometry.

The jet energy resolution achieved in this study (3.4–4% above 100 GeV) is noted as competitive, though slightly below the ILD benchmark using scintillator HCAL and PandoraPFA (~3%), a difference primarily attributed to the current lack of optimized reclustering in the APRIL framework rather than the calorimeter technology itself.

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