The lateral distribution function of cosmic-ray induced air showers studied with the HAWC observatory

J. A. Morales-Soto; J. C. Arteaga-Vel\'azquez; J. D. \'Alvarez (for the HAWC Collaboration)

REVIEW 3 major objections 6 minor 15 references

The lateral distribution function of cosmic-ray induced air showers studied with the HAWC observatory

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read HAWC's modified NKG lateral distribution function best describes cosmic-ray air showers from $10^{3.5}$ to $10^{5.5}$ GeV, and its fitted lateral age separates proton from iron primaries above $10^{3.8}$ GeV.

desk verdict Useful first HAWC cosmic-ray LDF comparison, but the ranking and composition claim rest on an uncalibrated error model and an internal consistency issue. read the letter →

arxiv 1908.07930 v1 pith:DYPPB7HV submitted 2019-08-21 astro-ph.HE

J. A. Morales-Soto , J. C. Arteaga-Velázquez , J. D. Álvarez (for the HAWC Collaboration) This is my paper · ORCID

classification astro-ph.HE

keywords lateraldistributionfunctionextensiveairshowersHAWCobservatoryageparametercosmic-raymasscompositionNKGsensitivity

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

This paper sets out to find which parameterization best describes the lateral distribution of particles in cosmic-ray air showers recorded by HAWC at 4100 m altitude. Using 1.3 days of 2016 data with energies between $10^{3.5}$ and $10^{5.5}$ GeV, it compares four lateral distribution functions by fitting the average deposited charge in 2 m radial bins and ranking the fits by $\chi^2$ per degree of freedom. It concludes that HAWC's own modified NKG function, with a fixed Moliere radius $r_M = 124.21$ m and exponents $(s-3, s-4.5)$, gives the best overall description of cosmic-ray showers in that energy range. It then shows that the lateral age parameter $s$ recovered from that fit separates simulated proton and iron showers with a figure of merit above 1 for energies from $10^{3.8}$ to $10^{5.5}$ GeV. If correct, this makes a single LDF fit a usable mass-composition observable for HAWC's data.

What carries the argument

The load-bearing object is the modified NKG lateral distribution function $f(r)=A(r/r_M)^{s-3}(1+r/r_M)^{s-4.5}$, with $r_M=124.21$ m fixed at HAWC's altitude, where $A$ is a normalization and $s$ is the lateral age. The argument runs through three pieces: (i) a charge-error model $\log_{10} Q_{\mathrm{error}} = 0.3 - 0.06667\log_{10}Q_{\mathrm{eff}}$ for $\log_{10}Q_{\mathrm{eff}} \le 3$ and $0.1$ above, which sets the weights in every fit; (ii) a $\chi^2$ per degree of freedom comparison of four LDFs on binned mean effective charge; and (iii) the figure of merit $\mathrm{FOM}=|s_{\mathrm{Fe}}-s_p|/\sqrt{\sigma_p^2+\sigma_{\mathrm{Fe}}^2}$ from simulated proton and iron showers, which converts the fitted age into a composition-separation metric.

What would settle it

Recompute the four LDF fits using charge uncertainties taken from PMT charge-resolution calibrations or from shower-to-shower fluctuations in place of Eq. (3.1), and check whether HAWC's LDF still has the lowest $\chi^2/\mathrm{NDOF}$; as a second check, evaluate the lateral-age proton-iron separation on independent simulated showers generated with a different hadronic interaction model.

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Extended reading notes

Core claim

The central claim is that the NKG-style function previously used by HAWC for gamma-ray showers, $f(r) = A (r/r_M)^{s-3}(1+r/r_M)^{s-4.5}$ with $r_M = 124.21$ m, also gives a good description of cosmic-ray-induced showers, and in a $\chi^2/\mathrm{NDOF}$ comparison over $10^{3.5}$ to $10^{5.5}$ GeV it performs at least as well as three literature LDFs across the full energy range. The fitted lateral age parameter $s$ varies with primary mass: iron showers are older (larger $s$) and protons are younger, and the separation, quantified by the figure of merit $|s_{\mathrm{Fe}}-s_p|/\sqrt{\sigma_p^2+\sigma_{\mathrm{Fe}}^2}$, exceeds 1 for $E > 10^{3.8}$ GeV, peaks near $E = 10^{4.8}$ GeV, and remains about 1.75 at $E = 10^{5.5}$ GeV. The paper also reports that the average measured lateral distributions lie within the band predicted by the high-energy hadronic interaction model used in the simulations.

Load-bearing premise

The comparison that picks HAWC's LDF rests on the empirical charge-error function of Eq. (3.1), whose coefficients are asserted without calibration; if those errors do not represent the real PMT charge uncertainties, the $\chi^2$ values and the LDF ranking could change.

Editorial extensions

If this is right

The lateral age parameter from a single HAWC LDF fit can serve as a mass-composition observable between $10^{3.8}$ and $10^{5.5}$ GeV, with proton-iron separation above one standard deviation.
HAWC's existing gamma-ray LDF can be reused for cosmic-ray analyses without introducing a new parameterization.
Average lateral distributions measured by HAWC are consistent with the predictions of the QGSJET-II-03 hadronic interaction model across $10^{3.5}$ to $10^{5.5}$ GeV.
The proton-iron figure of merit peaks near $E = 10^{4.8}$ GeV and is still about 1.75 at $E = 10^{5.5}$ GeV.

Reading between the lines

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

Beyond the paper: replacing the empirical charge-error function of Eq. (3.1) with PMT-level calibrated uncertainties could change the $\chi^2$ ranking of the four LDFs, although the age-based composition result depends mainly on the HAWC LDF itself.
Beyond the paper: combining lateral age with other composition-sensitive observables, such as muon content or shower-front curvature, may extend the useful range beyond the energy where the FOM begins to decrease.
Beyond the paper: repeating the four-function comparison on showers simulated with alternative hadronic interaction models would test whether the LDF ranking is model-dependent.

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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

What you should know: this is a short ICRC proceedings paper that does the first systematic comparison of four lateral distribution functions on HAWC cosmic-ray data, and then uses the lateral age parameter from the HAWC LDF to estimate proton-iron separation with a figure of merit. That is a useful and honest incremental step, not a breakthrough.

The paper earns credit for using real HAWC data with explicit quality cuts, fitting standard LDFs from KASCADE, ARGO-YBJ, and AGASA with a clear chi-square procedure, and for reporting the FOM rather than overclaiming composition resolution. It also openly states in the Discussion that none of the LDFs describes the data satisfactorily across all radial ranges and energies, which is a sign of care. Citations look appropriate, mostly to the original LDF papers and HAWC references.

The soft spots are real but addressable. The biggest one is that every chi2/NDOF value in Fig. 2 (left) is computed with the empirical PMT charge error model of Eq. (3.1), whose coefficients (0.3, -0.06667, 0.1) are simply asserted. No calibration against single-PMT charge resolution, photo-electron statistics, or bin-to-bin scatter is shown. Since the weights enter every fit, an unvalidated error model can change the ranking of the four LDFs, and therefore the claim that the HAWC LDF is the best description. That claim is further weakened by an internal tension: Section 6 says different LDFs do better at different energies, while the Conclusion first says none describes the data satisfactorily and then says the HAWC LDF gives a good description. Without a pre-specified acceptance threshold, 'good' is doing too much work. The modified scaling formalism parameters (beta, phi, delta) are taken from MC without showing the derivation, and only one hadronic interaction model (QGSJET-II-03) is tested. Also, the energy reconstruction and the MC predictions share the same HAWC simulation framework, so the agreement in Fig. 2 (right) is not an independent check of the interaction model.

That said, the central idea is plausible and the flaws are not fatal. This is a conference-length paper that should not carry the full weight of a composition analysis without further validation, but it is a legitimate first look. Who it is for: cosmic-ray and air-shower colleagues, especially those working on lateral distributions and composition with water-Cherenkov arrays. It deserves a serious referee: the data are real, the question is relevant, and the requested revisions (calibrate the error model, define a fit-acceptance criterion, show the MC parameter derivation) are reasonable. I would not desk-reject it; I would send it to review with moderate confidence that it can be strengthened.

Referee Report

3 major / 6 minor

Summary. The paper studies the lateral distribution function (LDF) of cosmic-ray induced air showers recorded by the HAWC observatory, using 1.3 days of 2016 data with reconstructed energies between 10^3.5 and 10^5.5 GeV and zenith angles below about 16.7 degrees. Four LDF parameterizations are fitted to the mean effective charge in radial bins via chi-square minimization: the HAWC/NKG-like function of Eq. (3.2), the KASCADE function of Eq. (3.3), the ARGO function of Eq. (3.5), and a modified scaling formalism of Eq. (3.7). The authors compare the resulting chi-square per degree of freedom as a function of energy, conclude that HAWC's LDF provides the best description of the data among the tested forms, and use the lateral age parameter from that fit to assess proton/iron separation through a figure of merit. The paper also compares the measured average lateral distributions with CORSIKA/QGSJET-II-03 predictions for proton and iron primaries.

Significance. If the central claims were fully supported, the paper would establish a useful result: the lateral age parameter derived from a single LDF fit could serve as a mass-composition-sensitive observable at HAWC over a multi-TeV energy range, and HAWC's standard gamma-ray LDF would be validated for cosmic-ray showers. The work has clear strengths: it compares four LDFs on the same data set, presents the energy dependence of fit quality, evaluates composition sensitivity with a quantitative FOM, and checks the data against Monte Carlo predictions. However, the main conclusions rest on chi-square comparisons that use an unvalidated empirical charge-error model, and the text contains an internal tension between the statements that no LDF describes the data satisfactorily and that HAWC's LDF gives a good description. These issues are directly load-bearing for the paper's central claims and need to be resolved before the results can be accepted.

major comments (3)

[Section 3, Eq. (3.1)] The empirical effective-charge error function log10(Qerror) = 0.3 - 0.06667*log10(Qeff) for log10(Qeff) <= 3 and 0.1 otherwise is introduced without any calibration or validation. This function provides the uncertainties that enter every chi-square fit, so it directly controls the LDF ranking in Fig. 2 (left) and the conclusion that HAWC's LDF is the best description. The authors need to demonstrate that this error model is a faithful representation of the PMT charge uncertainties, for example by comparing it with single-PMT charge resolution measurements, repeated-event studies, or the bin-by-bin scatter of the radial distributions, and to show that the ranking is robust to plausible variations of the error model. As it stands, the central comparison is not yet established.
[Sections 6 and 7] There is an internal inconsistency in the goodness-of-fit claim. Section 6 concludes that for E < 10^4.5 GeV Eqs. (3.3) and (3.2) give a better description and for E > 10^4.5 GeV Eqs. (3.5) and (3.2) fit better, while Section 7 states that none of the selected LDFs describes the measured data satisfactorily for all radial ranges and energies but that HAWC's LDF gives a good description in 10^3.5-10^5.5 GeV in comparison with the other LDFs. Since all example chi2/NDOF values in Fig. 1 exceed 1, 'best among the tested functions' and 'good description in an absolute sense' are different statements. The authors need to specify a pre-defined acceptance criterion for chi2/NDOF (or equivalent), discuss the statistical and systematic errors affecting the chi-square values, and reconcile the two statements with the same numbers.
[Section 5 and Fig. 2 (right)] The comparison of the average lateral distributions with Monte Carlo predictions and the subsequent composition-sensitivity analysis are performed using only one high-energy hadronic interaction model, QGSJET-II-03, and only proton and iron primaries. The reconstructed energies also rely on the HAWC MC-based method described in [10], which introduces an additional model dependence into the comparison. Since the modified-scaling parameters beta, phi, and delta in Eq. (3.7) are stated to be derived from MC simulations, the comparison involving this LDF is partly circular. At minimum, the authors should state these caveats and quantify the effect of the interaction-model choice on the FOM values and on the energy dependence of the lateral age parameter.

minor comments (6)

[Abstract and Section 2] The zenith-angle cuts are given as '< 16.7 degrees' in the abstract, 'θ < 16.71°' in Section 2, and 'θ < 16°' in the introduction; these should be made consistent.
[Figure 1 caption] The caption states that the error bars represent both 'the error on the mean and the sigma error' without explaining which quantity is shown; please define both and indicate which is plotted.
[Section 4] The fitting procedure does not specify the number of radial bins, the number of degrees of freedom, the treatment of bins with low occupancy, or how the mean effective charge and its variance are computed in each bin; these details are needed for the chi-square comparison to be reproducible.
[Throughout] The name of the hadronic interaction model appears both as 'QGSJet-II-03' and 'QGSJET-II-03'; please use a single spelling consistently.
[Introduction] The text contains a typo ('V olcano') and would benefit from a brief explanation of why the NKG lateral age parameter, originally derived for electromagnetic showers, is applicable to hadron-induced air showers in the TeV-PeV range.
[Section 4, Eq. (4.1)] The FOM definition does not include systematic uncertainties from energy reconstruction, shower-core location, or PMT saturation; given that these potentially dominate at large radii and high energies, a sentence on the impact of these systematics on the FOM would be valuable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LDF comparison is an empirical fit to independent data, and the QGSJET-II-03 and FOM analyses are direct simulation-based tests rather than reductions to inputs.

full rationale

The paper's central derivation chain is data-driven and self-contained. The four LDFs are fitted directly to HAWC effective-charge distributions, and the ranking is based on the resulting chi-square per degree of freedom; the HAWC LDF is not assumed to be optimal but is tested against three literature forms. Equation (3.1) is an uncalibrated empirical error model that affects the fit-quality comparison, but this is an input systematic, not a circular step: the conclusion is not guaranteed by the error model, and the ranking could in principle change with a different model. The energy reconstruction is cited from the collaboration's own spectrum analysis [10], but that calibration does not contain or presuppose the paper's LDF result or its composition-sensitivity claim. The QGSJET-II-03 comparison in Fig. 2 (right) is a direct data-versus-simulation comparison; even though the same hadronic interaction model was used in the simulation and reconstruction calibration, the measured lateral distributions are not derived from the model's predictions. The FOM analysis is a simulation-based sensitivity estimate, not a claim that QGSJET-II-03 is uniquely correct, and the lateral age is obtained from fits rather than from the model as an input. No self-citation is load-bearing in a way that makes a prediction equivalent to its inputs. The internal tension in Section 7 between 'none of the selected LDFs describes satisfactorily' and 'HAWC's LDF gives a good description' is a possible logical or statistical inconsistency, but it is not circularity under the specified definitions.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or physical entities are introduced. The 'modified scaling formalism' is a re-parameterized fit function, not a new entity. The free parameters are the fitted LDF parameters and the unvalidated error-function coefficients that drive the chi-square comparison.

free parameters (4)

Lateral age parameter s = not quoted; obtained per energy bin
Free parameter in the LDF fits (Eq. 3.2 and others); its mass-composition sensitivity is the main result, so the result is a measurement of a fitted quantity, not a prediction.
Normalization factor A (or N, C) = not quoted
Free normalization for each LDF fit; standard and not physically significant.
Modified scaling parameters beta, phi, delta = beta=3.59, phi=0.19, delta=3.61
Derived from MC simulations (Sec. 3, Eq. 3.7) and then fixed in fits to data; no derivation or uncertainties are shown in the paper.
Empirical charge error coefficients = 0.3, -0.06667, 0.1
Coefficients of Eq. (3.1) used to assign uncertainties to Qeff for chi-square fits; chosen empirically without validation, yet central to the LDF comparison.

assumptions (5)

domain assumption The NKG function and the lateral age parameter, originally derived for electromagnetic cascades, can be applied to hadronic air showers.
Section 1 extends the electromagnetic age concept to hadronic showers without deriving a new formalism.
domain assumption The lateral age parameter is sensitive to the mass composition of primary cosmic rays.
Motivated by prior literature (e.g., [2,13]) and used to interpret the FOM; not independently demonstrated in the paper aside from MC.
domain assumption QGSJET-II-03 and FLUKA describe hadronic interactions at HAWC energies, and the GEANT4-based detector simulation models HAWC response.
Section 2 uses CORSIKA with these models for MC; the model comparison in Sec. 5 is only as good as these models.
domain assumption The HAWC event reconstruction and energy reconstruction from [5,10] are unbiased.
Section 2 applies the reconstruction procedure of [5] and the energy method of [10]; these are from the same collaboration and not independently validated here.
ad hoc to paper The empirical error function (Eq. 3.1) gives correct uncertainties for Qeff.
The coefficients are stated without derivation or validation, yet they set the chi-square values that rank the LDFs.

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Cite this review

Pith. "Pith review of The lateral distribution function of cosmic-ray induced air showers studied with the HAWC observatory." pith.science (2026). https://pith.science/paper/DYPPB7HV

@misc{pith2026190807930,
  author       = {Pith},
  title        = {Pith review of: The lateral distribution function of cosmic-ray induced air showers studied with the HAWC observatory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYPPB7HV}},
  note         = {Machine review of arXiv:1908.07930}
}

abstract

The particle lateral distribution function (LDF) of air showers at a given altitude is sensitive to the mass composition and primary energy of cosmic rays. Studies of the LDF are difficult to perform due to experimental effects such as sampling bias, as well as shower-to-shower fluctuations in particle density. The High Altitude Water Cherenkov (HAWC) observatory, a dense air shower array located in central Mexico at 4100 m a.s.l., is well-suited to perform detailed event-by-event studies of the LDF of multi-TeV cosmic-ray showers. The detector is instrumented with 1,200 photomultipliers (PMTs) in close-packed water Cherenkov tanks containing a total of 60 ML of water. We present a study of the LDF of cosmic-ray air showers recorded by HAWC in 2016 with energies between 3 TeV and 300 TeV and zenith angle < 16.7$^{\circ}$. The data are used to determine the optimal parameterization of the LDF at the HAWC site. From here the lateral shower age is obtained and its sensitivity to the cosmic ray mass composition is analyzed.

Figures

Figures reproduced from arXiv: 1908.07930 by the authors.

**Figure 1.** Left panel: lateral distribution of an air shower from the experimental data set of an energy of E = 105 GeV and a zenith angle of 1.3 ◦ . The gray markers correspond to the effective charge points per PMT and their corresponding errors were calculated using Eq (3.1), the solid black circles correspond to the mean Qe f f per radial bin and there the error bars represent the error on the mean and the sigma error. Rig… view at source ↗

**Figure 2.** Left panel: comparison χ 2/NDOF of the resulting fits of each of the selected LDFs for the experimental data. Right panel: Average lateral distribution of an EAS for different energy intervals corresponding to the MC (blue bands: proton; red bands: iron) and experimental (solid markers) data. In both plots, the errors on the mean are smaller than the marker size. 5. Comparison of LDF data with model predictions To … view at source ↗

**Figure 3.** Left panel: lateral age parameter estimated by the fit performed from the HAWC LDF (3.2) as a function of the energy. The error bands of each component correspond to a 1σ containment error. Right panel: FOM distribution for the proton and iron calculated with MC data. of the data, but in the high energy regime, E > 104.5 GeV, Eqs. (3.5) and (3.2) fit better to the data. That means that the HAWC LDF gives also a good… view at source ↗

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

HAWC Collaboration, All-particle cosmic ray energy spectrum measured by the HAWC experiment from 10 to 500 TeV, Physical Review D, 96 (2017) [arXiv:1710.00890]

arXiv 2017

[2] [1]

P. K. F. Grieder, Extensive Air Showers, Springer-Verlag, Berlin Heidelberg, 2010

work page 2010

[3] [2]

Kamata and J

K. Kamata and J. Nishimura, Prog. of Theor. Phys. Supp. 6 (1958) 93

1958

[4] [3]

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M. Kachelrieß, Lecture Notes on High Energy Cosmic Rays, 17.th Jyväsklyä Summer School (2017) [arXiv:0801.4376]

work page Pith review arXiv 2017

[5] [4]

KASCADE Collaboration, Electron, Muon, and Hadron Lateral Distributions Measured in Air-Showers by the KASCADE Experiment, Astroparticle Physics 14 (2008) 1 [arXiv:astro-ph/0004233]

work page Pith review arXiv 2008

[6] [5]

HAWC Collaboration, Observation of the Crab Nebula with the HAWC Gamma-Ray Observatory, The Astrophysical Journal, 843 (2017) 39 [arXiv:1701.01778]

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work page 2005