Comparative study and optimization of SDHCAL hadronic energy reconstruction methods

G\'erald Grenier; Imad Laktineh; Tanguy Pasquier

REVIEW 3 major objections 6 minor 26 references

Comparative study and optimization of SDHCAL hadronic energy reconstruction methods

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Split and polynomial-regression formulas give the best compromise between linearity and resolution for hadronic showers, except at high jet energies where particle-flow confusion favors the simple linear formula.

desk verdict Useful SDHCAL reconstruction study with a real but fixable flaw: the single-hadron ranking is partly in-sample, while the dijet and PerfectPFA results hold up independently. read the letter →

arxiv 2607.15023 v1 pith:KZE6VMQ2 submitted 2026-07-16 physics.ins-det hep-ex

Tanguy Pasquier , Gérald Grenier , Imad Laktineh This is my paper · ORCID

classification physics.ins-dethep-ex

keywords semi-digitalhadroniccalorimeterenergyreconstructionparticleflowalgorithmjetresolutionpolynomialregressionangularcorrectionMonteCarlosimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper compares four ways to turn hit counts in a semi-digital hadronic calorimeter into a hadronic energy measurement: a linear formula, a quadratic formula, a split formula that uses two quadratics depending on cluster size, and a degree-2 polynomial regression. Using simulated single neutral kaons and dijets, it asks which formula gives the most linear response and best energy resolution, and how the particle-flow reconstruction's confusion degrades jet resolution at high energies. The central finding is that the split and polynomial formulas offer the best overall compromise: they improve resolution at low energies without hurting linearity at high energies. The paper also shows that a geometry-based angular correction is necessary in the barrel region to remove a systematic underestimation from non-perpendicular incidence. At jet energies above about 100 GeV, however, confusion in the particle-flow reconstruction dominates the resolution, so no re-parametrization of the formula can help and the simplest linear formula becomes competitive.

What carries the argument

The central object is the reconstruction formula E = αN1 + βN2 + γN3, where N1, N2, N3 are the numbers of calorimeter pads firing above each of three charge thresholds; the linear version fixes α, β, γ as constants, while the quadratic, split, and polynomial variants make the coefficients or add terms depending on total hit count N_hit, introducing saturation behavior. The coefficients are tuned by a chi-squared-like minimization on a simulated sample of contained single-hadron events. The other load-bearing mechanism is the angular correction, which multiplies hit counts by geometric factors 1/sinθ and 1/cosφ to compensate for the path-length variation of inclined tracks in the tilted barre

What would settle it

Take the same four formulas and a fixed detector geometry, then apply them to real test-beam data for 5–80 GeV single hadrons using the same thresholds and angular corrections; if the split method's low-energy resolution improvement over the linear formula does not appear, the simulation-based ranking does not transfer to the physical detector.

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Extended reading notes

Core claim

The paper's central claim, stated in its conclusion, is that the split method and polynomial regression provide the best compromise across different energy regimes, combining stability with good resolution. For single neutral hadrons, linearity better than 5% is obtained above 10 GeV, with resolution from about 25% at 5 GeV to 7–8% at 80 GeV. For dijets, the optimal choice depends on the reconstruction quality: under realistic particle-flow confusion at high energies, the linear formula is favored because it is less sensitive to hit-count fluctuations, whereas with perfect truth-based reconstruction the nonlinear formulas are superior across the full energy range. The jet energy resolution r

Load-bearing premise

The entire comparison lives in a Monte Carlo simulation whose digital thresholds and hit multiplicities are tuned to reproduce test-beam data, so the formula ranking is only as reliable as that tuning, particularly at low energies where the paper reports an unresolved residual bias at 5 GeV.

Editorial extensions

If this is right

Adopting the split or polynomial formula for single-hadron reconstruction improves resolution below 20 GeV without worsening linearity above 10 GeV.
For jet energies above roughly 100 GeV, the resolution ceiling is set by particle-flow confusion, not by the calorimeter formula; formula choice should be made with the reconstruction algorithm in mind.
The angular correction removes a multi-percent systematic underestimation in the barrel and should be part of any calibration of a modular hadronic calorimeter.
If particle-flow confusion were removed (via better clustering or timing), nonlinear formulas would give better jet resolution at all energies, making the linear formula's high-energy advantage a temporary artifact of the current reconstruction.
A jet energy resolution of 3.4–4% above 100 GeV is achievable with the semi-digital technology, within a point of the 3% benchmark of the alternative scintillator-based design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same fitting machinery could take timing information (from a planned timing-capable version of the calorimeter) as an additional input to the polynomial regression, attacking confusion at its source; the paper lists this as a future direction, but the natural extension is to quantify how much confusion the polynomial's cross terms already absorb.
The reversal of formula ranking at high jet energies is essentially a statement about the bias-variance tradeoff under noise: the linear formula is the highest-bias, lowest-variance estimator, so it wins when hit-count noise is amplified by clustering mistakes. This suggests the same qualitative ranking would appear for any high-granularity calorimeter with a noisy clustering step, not just this o
A testable extension is to rerun the dijet comparison at the specific jet energies of planned lepton colliders (for example around 100 GeV for ZH events); the paper covers 30–280 GeV, so the formula of choice for the actual physics runs can be pinned down.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is a solid, useful detector-reconstruction study. It documents a new angular correction for the Videau-geometry barrel and gives the first APRIL-based comparison of four SDHCAL energy formulas in ILD_l2_v02, including a scikit-learn polynomial regression. The PerfectPFA check is the best part: it cleanly separates formula performance from PFA confusion and confirms that at high dijet energies confusion dominates. That is a genuinely useful result for the ILD/APRIL program.

Second, the headline claim that the split method and polynomial regression are the best compromise rests more heavily than it should on in-sample numbers. Section 3.1 explicitly says the single-K0L methods were tested on the same samples used to fit the formulas, and Section 2.4.3 describes scanning the split threshold from 200 to 600 hits on that same data. So the split method's apparent low-energy advantage in Figs. 8–9 could be an artifact of tuning. The polynomial regression has an internal train/test split, but that doesn't protect the cross-method ranking. This is a real weakness, but it is fixable: report k-fold cross-validation or a proper held-out test set, and disclose the threshold scan as a fitted hyperparameter.

Other soft spots are minor. The paper reports no uncertainties on the ΔJER differences, so we don't know if the split method's gain is significant even in-sample. The digitization thresholds are tuned to reproduce test-beam data, but there is no direct validation of the digitization here; that is a domain assumption. The 5 GeV residual after angular correction is acknowledged and left for future work.

What is good: the analysis is internally consistent, the dijet results are genuinely out-of-sample (coefficients fitted on single K_L then applied to jets), and the PerfectPFA and PandoraPFA cross-checks independently support the confusion claim. The paper is honest about its own limitations, which is more than many detector papers do.

Who is this for: the ILD/CALICE community, especially people working on APRIL and SDHCAL calibration. A serious referee should see it—the core results are useful, and the in-sample issue is a methodological blemish rather than a fatal flaw. I would engage with this work and would want the authors to fix the cross-validation before final acceptance.

Referee Report

3 major / 6 minor

Summary. This paper compares four methods for reconstructing hadronic energy in the SDHCAL within the ILD_l2_v02 simulation: linear, quadratic, split, and polynomial regression. It introduces a geometrical angular correction for the barrel, calibrates coefficients on single K_L samples, and evaluates linearity and resolution on single K_L and dijet events using APRIL, with PerfectPFA and PandoraPFA cross-checks. The central claims are: (i) the angular correction is essential; (ii) all methods achieve few-percent linearity; (iii) for single neutral hadrons the split method and polynomial regression give the best compromise; and (iv) for dijets above roughly 100 GeV, PFA confusion dominates and favors the linear parametrization, while non-linear methods are superior when confusion is removed. Full parameter tables are provided.

Significance. The dijet confusion conclusion is well supported by the independent PerfectPFA and PandoraPFA checks, which is a genuine strength. The angular correction is physically motivated and addresses a real geometry effect. If the single-hadron ranking were validated on independent data, the paper would provide useful calibration guidance for the SDHCAL option in ILD. However, the single-hadron 'best compromise' claim currently rests on an in-sample evaluation; the split threshold is tuned on the same K_L sample used for Fig. 8, and no uncertainties are given for the resolution differences. The paper is therefore not yet convincing on its headline conclusion, although the methodology and the dijet analysis are sound.

major comments (3)

[§3.1, §2.4.3] The single-K_L performance comparison is in-sample. Section 3.1 states that all methods were 'tested on the same samples of single K_L events used to fit the formulas'; Section 2.4.3 describes the split transition threshold as scanned from 200 to 600 hits on the same data, with 'optimal results' chosen. Thus the ranking in Figs. 8 and 9 is not an independent test: the split method's improvement at 5–20 GeV could be an overfit artifact. Only the polynomial regression has a within-method train/test split, which does not protect the cross-method ranking. Please re-evaluate at least the split method on a disjoint holdout sample or with k-fold cross-validation, and quote statistical uncertainties on the ΔJER values in Fig. 9. The dijet analysis is less affected because it applies pre-trained coefficients to an independent topology, but the single-hadron claim in §4 is load-bearing.
[§2.4, Tables 1–4] Parameter uncertainties are not reported. The text says the coefficients are intended for direct use and their individual uncertainties are 'not relevant,' but the paper's central claim is a comparison of methods. Without uncertainties on the fitted coefficients or on the reconstructed-energy differences, the reader cannot judge whether the small ΔJER differences in Figs. 9 and 11 are statistically significant, even in-sample. Please provide covariance matrices or bootstrap error bars for the key energy points.
[§2.1, §2.2] The entire study is simulation-based. The digitization thresholds are fixed to reproduce test-beam multiplicity and efficiency (Ref. [21]), but no direct validation of the simulated response against test-beam data is shown in this paper, and the 5 GeV residual in §2.2 is left unresolved. The ranking of the methods, especially at low energies, could change if the simulated multiplicity/threshold response differs from the real detector. Please state this limitation explicitly and, if possible, include a comparison with test-beam data or an alternative digitization for at least one energy point.

minor comments (6)

[Eq. (2.1)] Define N_i^barrel and N_i^barrel,module; the current notation is not self-explanatory.
[§2.2] The text mentions the 'standard PandoraPFA calibration procedure' but Fig. 7 uses the quadratic formula; clarify which reconstruction is used in the angular-correction study.
[§2.4.3] The scan of transition thresholds from 200 to 600 hits is mentioned but not shown; include the scan curve or a table of the tested thresholds and corresponding resolutions.
[Figs. 8, 9, 11] No error bars are shown; even if statistical uncertainties are small, state the procedure used to compute errors or state that they are negligible.
[General] Typos/typesetting: 'dĳet' appears as a ligature; 'K^0_L' and '𝜙_folded' are inconsistently formatted; some equations lack punctuation.
[§3.2] The PandoraPFA confirmation is described in the text but no plot is shown; consider adding a reference to an existing figure or a supplementary plot.

Circularity Check

2 steps flagged · score 6.0 of 10

Single-hadron method ranking is in-sample: the calibration coefficients and the split threshold are fitted and evaluated on the same K_L samples; dijet and PerfectPFA parts are independent.

fitted input called prediction [§2.3, §2.4.1-2.4.4, §3.1, Figs. 8-9]
"All the methods described in section 2.4 were first tested on the same samples of single K_L events used to fit the formulas. However, for the performance evaluation presented in this section, the containment and pure-HCAL selection criteria applied to define the calibration dataset are not imposed."

Section 2.3 constructs the calibration set by selecting contained, pure-HCAL single-K_L events and fits all formulas on it; Section 3.1 then evaluates the formulas on the same samples, only dropping the selection cuts and running APRIL. The linearity/resolution in Figs. 8-9 is therefore in-sample. The claim that the split method and polynomial regression 'offer the best compromise' for single hadrons is a description of the fitted sample, not an independent prediction; no cross-validation protects the cross-method ranking.
fitted input called prediction [§2.4.3, §3.1, Fig. 9]
"Various transition thresholds, ranging from 200 to 600 hits, were tested. Ultimately, the optimal results were achieved by applying the function fitted on events with N_hit≤600 to clusters with N_hit≤400, and the function fitted on events with N_hit>400 to clusters with N_hit>400."

The 400-hit split threshold is a hyperparameter chosen by scanning 200-600 hits and keeping the 'optimal results' on the same single-K_L data used for the Fig. 8-9 evaluation. The subsequent conclusion that 'the split method yields improved performance between 5 GeV and 20 GeV' is therefore partly a restatement of that optimization, not an out-of-sample validation. This makes the headline low-energy advantage partially circular by construction.

full rationale

The single-K_L comparison is partially circular: the coefficients are fitted on the calibration subset of these events, and the paper explicitly evaluates the methods on 'the same samples ... used to fit the formulas.' The split method adds a scanned hyperparameter (200-600 hits, 400 chosen as 'optimal') on the same data, so its low-energy resolution advantage is statistically forced to look good on those curves. This does not invalidate the whole paper: the angular correction is an independent geometric correction; the dijet analysis applies pre-trained coefficients to a separate topology and is therefore a genuine transfer test; and the PerfectPFA comparison independently supports the confusion-dominance interpretation. There is no load-bearing self-citation: APRIL, CALICE test-beam data, and ILD benchmarks are external reference points. The 5 GeV residual and lack of uncertainties on ΔJER are limitations, but not circularity. Score 6 reflects a partial circularity confined to the central single-hadron ranking claim.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The reconstruction formulas are pure fits: 3 + 9 + 18 + 9 coefficients plus a scanned split threshold and an arbitrary chi-squared weight. The angular correction adds no fitted constants but rests on an unverified geometric model. No invented physical entities; the T-SDHCAL timing is mentioned only as future work.

free parameters (7)

Linear energy factors alpha, beta, gamma = 0.03670, 0.07453, 0.36304 GeV (Table 1)
Fit to the calibration K0L sample via chi-squared (Eq. 2.3); the central calibration of the linear method.
Quadratic coefficients alpha1-alpha3, beta1-beta3, gamma1-gamma3 = Table 2 (9 values; gamma1 bounded at 1e-10 GeV)
Fit to the same calibration sample; introduces N_hit dependence into the weights.
Split-method coefficients (18 values) = Table 3 (two sets of 9)
Two quadratic functions fit to low/high N_hit subsets of the calibration sample.
Split transition thresholds = N_hit = 400 (application), 600 (fit range)
Chosen by testing thresholds from 200 to 600 hits and selecting the best-performing; a tuned hyperparameter, not derived.
Polynomial regression coefficients (9 values) = Table 4
Degree-2 polynomial fitted with scikit-learn on the same calibration dataset.
Chi-squared event weight sigma_i = sqrt(E_mc)
Modeling choice in Eq. 2.3, inherited from [9]; different weights would change all fitted coefficients.
Resolution-extraction window = [mu1 +/- 1.5 sigma1] from first Gaussian fit
Hand-chosen analysis definition for extracting mean and width in §3.1; affects all reported resolutions.

assumptions (5)

domain assumption GEANT4 with QGSP_BERT physics list models hadronic showers in the SDHCAL steel/GRPC structure adequately
Used for all samples; FTFP_BERT 'yielded similar results' per §2.1, but no validation against test-beam data is presented in this paper.
domain assumption Digitization thresholds (114 fC, 6.12 pC, 16.83 pC) reproduce prototype multiplicity and efficiency
§2.1, citing [21]; the N1, N2, N3 counts that feed every formula depend on this.
domain assumption Geometric angular correction Eq. (2.1): hit deficit scales as 1/sin(theta) and 1/cos(phi) per module
§2.2; purely geometric, neglects shower-shape effects; residual biases (2-3% above 20 GeV, under-correction at 5 GeV) show the model is incomplete.
domain assumption Cluster-level application of event-level-calibrated formulas is valid
Calibration at event level, reconstruction at cluster level (§2.4.3); the paper assumes fragmentation does not break the transfer, with only a brief caveat.
domain assumption Chi-squared weighting with sigma = sqrt(E_mc) is appropriate
Eq. 2.3; standard in CALICE analyses but not justified from first principles in this paper.

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Cite this review

Pith. "Pith review of Comparative study and optimization of SDHCAL hadronic energy reconstruction methods." pith.science (2026). https://pith.science/paper/KZE6VMQ2

@misc{pith2026260715023,
  author       = {Pith},
  title        = {Pith review of: Comparative study and optimization of SDHCAL hadronic energy reconstruction methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZE6VMQ2}},
  note         = {Machine review of arXiv:2607.15023}
}

abstract

We present a detailed study of hadronic shower energy reconstruction methods for the Semi-Digital Hadronic Calorimeter (SDHCAL) within the ILD detector concept, using the Particle Flow Algorithm (PFA) APRIL. Using samples of single $K^0_L$ and dijet ($u,d,s$) events, we compare linear, quadratic, split, and polynomial regression-based reconstruction formulas, focusing on their impact on linearity and resolution. The study also addresses angular corrections required in the barrel region due to non-perpendicular particle incidence. Results show that while all methods achieve good overall performance, the split method and the polynomial regression provide the best compromise across different energy regimes, offering improved resolution at low energies without compromising linearity at higher energies. For dijets, sensitivity to PFA confusion dominates the resolution at high energies. These findings highlight the potential of future improvements, notably the integration of precise timing information from the T-SDHCAL into APRIL, to further reduce confusion and enhance hadronic energy reconstruction for next-generation lepton colliders.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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