A bi-effect model of muon deflections in air showers

Chao Zhang; Ruo-Yu Liu; Si-Zhe Wu; Xi-Shui Tian; Zhuo Li

REVIEW 3 major objections 5 minor 22 references

A bi-effect model of muon deflections in air showers

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A two-term vector formula, split between geomagnetic bending and atmospheric absorption, predicts where muons land in inclined air showers.

desk verdict A useful two-parameter muon deflection model with a convincing azimuthal shape test, but the unresolved B-coupling question makes it a conditional tool rather than a proven predictive one. read the letter →

arxiv 2508.17475 v2 pith:WZZHZ2UM submitted 2025-08-24 astro-ph.HE hep-ph

Si-Zhe Wu , Chao Zhang , Ruo-Yu Liu , Xi-Shui Tian , Zhuo Li This is my paper · ORCID

classification astro-ph.HEhep-ph

keywords airshowersmuonsgeomagneticdeflectionatmosphericabsorptionlateraldistributiondetectorlayoutzenithangledependencecosmicraycomposition

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 tries to establish that the sideways spreading of muons reaching the ground from inclined air showers reduces to a two-piece vector formula. One piece comes from the Lorentz force of Earth's magnetic field bending the muon's path; the other comes from atmospheric absorption, which removes muons preferentially on one side of the shower axis. The two effects are assumed to act independently and add linearly. Compared against Monte Carlo air-shower simulations at three sites with different geomagnetic fields, the formula reproduces the simulated average deflections of muon number and energy with 68% relative-error intervals of 11% and 8%. If correct, this gives a simple predictive tool for placing detector stations and for reconstructing the direction and composition of the primary cosmic ray.

What carries the argument

The central object is the vector decomposition in Eq. (3): every muon's ground-plane deflection is split into a magnetic term along the ground-plane north-south axis and an atmospheric term along the projection of the shower direction. The magnetic linearity hypothesis approximates Larmor motion as projectile motion, making the magnetic deflection proportional to $\vec{e}_p \times \vec{B}$ with a single scalar $c_{\mathrm{mag}}$; the independence hypothesis makes atmospheric absorption contribute a displacement along $-\vec{a}$ with scalar $c_{\mathrm{atm}}$. The pair $(c_{\mathrm{mag}}, c_{\mathrm{atm}})$ is constant over azimuth, so once they are known the whole deflection pattern for a site is a geometric projection, and the empirical formulas for $c_{\mathrm{mag}}$ and $c_{\mathrm{atm}}$ connect those parameters to the primary's zenith angle, first-interaction height, and atmospheric depth.

What would settle it

Run air-shower simulations at a site with a substantially stronger magnetic field, or at zenith angles beyond 80°, and compare the simulated ground-plane muon deflection with Eq. (3) using parameters fitted at current sites; if the residual grows with the size of the magnetic deflection, the linear superposition fails. A second check is to add a cross term proportional to $c_{\mathrm{mag}} c_{\mathrm{atm}}$ to Eq. (3) and see whether it improves the fit by more than the reported 8-11% uncertainties.

Watch

Extended reading notes

Core claim

The central claim is Eq. (3): the ground-plane deflection vector $\Delta \vec{r}$ equals $c_{\mathrm{mag}} [\vec{b}\cdot(\vec{e}_p \times \vec{B})]\vec{b} + [c_{\mathrm{atm}} - c_{\mathrm{mag}} \vec{c}\cdot(\vec{e}_p \times \vec{B})/(\vec{e}_p\cdot\vec{n})]\vec{a}$, where $\vec{b},\vec{a},\vec{n}$ are the ground-plane and shower-plane axes, $\vec{e}_p$ is the shower direction, $\vec{B}$ is the geomagnetic field, and $c_{\mathrm{mag}}$, $c_{\mathrm{atm}}$ are two scalar parameters independent of the azimuth angle of the primary. The paper reports that one fixed pair $(c_{\mathrm{mag}}, c_{\mathrm{atm}})$ reproduces the simulated deflection pattern for all azimuths, including its two-lobed 'double-circle' structure, and that the empirical functions $c_{\mathrm{mag}} = A\tau^{B}(\cos\theta)^{C}$ and $c_{\mathrm{atm}} = A\tau^{B}\tan\theta \exp(C\tau/\tan\theta)$ capture the dependence on zenith angle, first-interaction height, and site. The stated accuracy is a 68% relative-error interval of 11% for the number-weighted deflection and 8% for the energy-weighted deflection.

Load-bearing premise

The load-bearing premise is that magnetic bending and atmospheric absorption act independently and add as vectors, even though a magnetically bent muon travels through a different amount of atmosphere than an unbent one.

Editorial extensions

If this is right

Detector upgrade footprints can be computed directly from the model: the required extra area in each direction falls out of the deflection curve, without running new simulations.
Because $c_{\mathrm{mag}}$ and $c_{\mathrm{atm}}$ are azimuth-independent, the deflection pattern for any arrival direction at a given site is predicted by the same two parameters.
The fitted parameters differ between proton and iron primaries, so measured deflection patterns could be used to infer the mass composition of the primary cosmic ray.
The model explicitly covers zenith angles from 50° to 80°, a regime beyond the 60° limit of earlier universality descriptions of particle distributions.

Reading between the lines

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

Read as an extension: if the linear superposition holds, the same two-parameter vector form should describe muon deflections at any site, so the model's core physics content is really the scaling of $c_{\mathrm{mag}}$ and $c_{\mathrm{atm}}$ with atmospheric depth.
A natural stress test beyond the paper: push the model into the regime where muon decay is non-negligible, which the paper notes is where the energy-weighted fit degrades; separating decay losses from magnetic-atmospheric coupling would clarify whether the linear superposition or the muon physics is responsible.
An untested consequence of the geometry: the double-circle deflection pattern is essentially the projection of a fixed shower-plane vector onto the ground plane, which implies the same pattern should appear in electromagnetic particles if their absorption can be modelled, a topic the paper leaves for separate work.

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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

The headline is that this is a plausible empirical parameterization, and the azimuthal double-circle test is the real gem: a single pair of parameters (cmag, catm) reproduces the deflection pattern across 72 azimuths. That is a genuine geometric check, not a trivial fit. The novelty is moderate — combining Lorentz force and slant-depth absorption in one formula is new, but the ingredients are standard and the output is an engineering tool, not new physics.

What the paper does well: the projection in Eqs. (2)–(3) is clean, the boundary conditions for the fitting formulas are sensible, and Table II gives coefficients for three sites, proton/iron primaries, and a decent energy span. The reported 8–11% relative errors are honest, though they are in-sample.

The soft spot is exactly what the stress test flagged: the Independence Hypothesis in Sec. II is load-bearing and never directly tested. The authors validate only at each site's real magnetic field, so a two-parameter fit can absorb B-dependent coupling between magnetic displacement and atmospheric absorption without revealing it. A CORSIKA run with B set to zero, and another with, say, twice the field, would settle whether cmag is truly B-independent and whether the superposition holds. Without that, the predictive uses for detector layout and mass composition are conditional. The lack of released code or data makes it harder to check; and there is no baseline comparison against a simpler one-parameter model. The energy range stops at 1e17 eV, which is fine for the stated scope but not for UHECR regimes.

Minor quibbles: the muon-decay discrepancy at high zenith and high FIXHEI is waved off, and the empirical formulas are matched to boundary conditions, which is acceptable but limits extrapolation.

Verdict: worth a serious referee. The geometric form is likely right and the model can inform array design at LHAASO, SKA, and Auger. I would send it to peer review with an explicit request for a B-switch test and a data release; I would not use it predictively before that check, but I would cite it as a useful parameterization.

Referee Report

3 major / 5 minor

Summary. This paper proposes a 'bi-effect model' for the ground-plane lateral deflection of muons in extensive air showers. The model combines two effects linearly: the geomagnetic Lorentz-force deflection and an atmospheric-absorption-induced displacement along the shower asymmetry direction, leading to Eq. (3) for the mean deflection vector as a function of the primary direction, the local magnetic field, and two scalar parameters cmag and catm. The parameters are fitted to CORSIKA simulations for proton and iron primaries, energies between 10^14 and 1.024×10^17 eV, zenith angles 50°–80°, 72 azimuth angles, and the magnetic-field geometries of LHAASO, SKA, and the Pierre Auger Observatory. The authors propose empirical analytic forms cmag=A τ^B (cosθ)^C and catm=A τ^B tanθ exp(Cτ/tanθ), where τ is the atmospheric depth, and tabulate fitted coefficients. They report that the model reproduces the simulated azimuthal double-circle structure and quote 68% relative-error intervals of 11% for the number deflection and 8% for the energy deflection, and discuss applications to detector layout and mass-composition reconstruction. The central claim is that this simple formula can describe the overall deflection of muons and accurately fit the deflection in simulated air showers, thereby validating the two underlying hypotheses.

Significance. If the central claim holds, the paper offers a compact analytic description of a nontrivial geometric effect, with practical value for detector design and shower reconstruction at current and future observatories. The work has notable strengths: it uses a large and systematic simulation library (about 11,000 showers, three sites, 72 azimuthal bins), tests azimuthal independence over a full circle, checks the energy dependence over three decades, and provides tabulated coefficients that allow immediate use of the model. The geometric part of the model, namely the projection of the Lorentz deflection onto the ground plane with a single azimuth-independent parameter, is physically well motivated and is supported by the observed double-circle pattern. However, the empirical validation is currently in-sample for the fitted parameters, and the linear superposition of magnetic and atmospheric effects is assumed rather than directly tested. The significance of the paper therefore hinges on the additional holdout and field-switch tests requested below.

major comments (3)

[Sec. III, Model uncertainties; Fig. 6] The quoted 68% errors of 11% and 8% are computed with ε = |Δr_model − Δr_simulation| / |Δr_simulation| using simulations and fitted parameters from the same analysis chain. As written, the text does not state whether the 'several hundreds showers for testing' mentioned in Sec. II are actually held out from the fits that determine cmag and catm and from the empirical coefficient fits of Eqs. (5) and (7). If they are not, the quoted errors are in-sample residuals and cannot support the predictive claim in the Abstract. Please specify the train/test split explicitly and report the relative-error distribution separately for the fitted and held-out samples; if no true holdout was used, rerun the validation on an independent set, for example a random subset of azimuth angles or a different set of FIXHEI values.
[Sec. II, Model hypotheses; Eq. (3)] The Independence Hypothesis states Δr = Δr_mag + Δr_atm. Because a magnetically deflected muon traverses a different atmospheric depth, the number- and energy-weighted means of the detected muon population are not guaranteed to be additive in this way. The validation included in the paper uses only the real geomagnetic fields of three sites, so the possible B-dependence of cmag and catm through coupling is never probed. The central predictive claim, that the model is generic across sites and zenith angles, therefore requires a direct test: rerun a representative set of showers with the magnetic field switched off, compute Δr(B=0), and compare Δr(B) − Δr(B=0) with the magnetic term in Eq. (3); also vary the field strength (for example B=0, B, and 2B) to verify that cmag remains independent of B and that catm is unchanged. If the superposition fails at large zenith angles or high-field sites, cmag and catm are effective parameters that vary with the magnetic field, and the application in Fig. 7 overstates the predictive reach.
[Sec. III, Fitting formula for deflection parameters; Eqs. (5), (7), (4), (6)] The functional forms for cmag and catm are chosen to satisfy the boundary conditions in Eqs. (4) and (6), but boundary condition (iv), 'FIXHEI→112 km, |Δr|>0', is not checked against simulations, and the model's own text notes that the energy deflection of highly inclined showers initiated at very high altitudes deviates from the fitted form because of muon decay, with the discrepancy dismissed because the magnetic effect dominates. This breakdown regime is exactly the regime where the independence hypothesis is least safe, so it cannot be dismissed without a quantitative estimate. Please state the ranges of zenith angle and FIXHEI over which Eqs. (5) and (7) reproduce the fitted parameters to better than, say, 10%, and quantify the muon-decay effect, for example by comparing against a simulation run with muon decay switched off, rather than ignoring it.

minor comments (5)

[Throughout] The figure cross-references are inconsistent: 'FIG. III' in Sec. III should be Fig. 3, and the relative-error distribution is described in the text as Fig. 5 but the caption identifies it as Fig. 6.
[Table II] The units in the Table II header are unclear: the column header '([m/μT])' is attached to all three coefficients A, B, and C, but these coefficients have different dimensions in Eqs. (5) and (7). Please state the units of each coefficient explicitly.
[Sec. III, Fitting formula for deflection parameters] The sentence 'Since the definition of τ contains cos(θ)−1, when B > C, and B > 0, the boundary conditions are satisfied' is too terse; please spell out how each boundary condition in Eq. (4) follows from the parameter conditions.
[References] References [13] and [21] are identical (Bae and Chatzidakis, PTEP 2022, 043F01); please remove the duplicate.
[Sec. III, Model validation] The notation for the magnetic deflection parameter is inconsistent: the text uses both 'cmag' and 'Cmag' in the validation paragraph; please unify to a single symbol.

Circularity Check

2 steps flagged · score 5.0 of 10

Partial circularity: cmag and catm are fitted to the same CORSIKA data used to report the 11%/8% errors, and the Independence Hypothesis is embedded in the fitted form; the azimuthal double-circle test provides independent content.

fitted input called prediction [Sec. III, 'Fitting formula for deflection parameters' and 'Model uncertainties' (Eq. (3))]
"By fitting the particle deflection data using these parameters, cmag and catm are expected to be expressed as functions of the parameters of the primary particles. ... Using the parameters of the Pierre Auger observatory site, by comparing theoretical and simulated deflections, the distribution of the relative error for ∆⃗ rN ∆⃗ rE are presented by Fig. 5, with 68% confidence intervals for these errors are 11% and 8%, respectively — confirming the robustness of this model."

The two parameters defining the model, cmag and catm, are obtained by fitting the CORSIKA mean deflections, and the same simulated deflections are then used to compute the reported relative error ε = |∆⃗ r_model − ∆⃗ r_simulation| / |∆⃗ r_simulation|. The 68% intervals (11% and 8%) are therefore in-sample residuals of a two-parameter fit, not out-of-sample predictions. Calling these residuals 'confirming the robustness of this model' reduces the headline accuracy claim to a goodness-of-fit statement. The azimuthal double-circle test, which uses one (cmag, catm) pair across 72 azimuth angles, is an independent check of the model's angular form, so the circularity is partial rather than total.
self definitional [Sec. II 'Model hypotheses' and Sec. III 'Model validation']
"We assume that the magnetic deflection (∆⃗ rmag) caused by the Lorentz force and the atmospheric absorption-induced displacement (∆⃗ ratm) are independent. This relationship is expressed as: ∆⃗ r= ∆⃗ rmag + ∆⃗ ratm. ... using a single pair of cmag, catm, the model effectively reproduce the double-circle structure of the magnetic deflection, further validating the hypotheses and the azimuth angle independence."

The model's total deflection in Eq. (3) is constructed from exactly this Independence Hypothesis, with the atmospheric contribution represented by a fitted vector catm a and the magnetic contribution by a fitted scalar cmag. Fitting this two-parameter additive form to simulated mean deflections cannot validate additivity: the simulations always use the full real local magnetic field and never switch off B to isolate ∆⃗ ratm or vary B to isolate ∆⃗ rmag. A two-parameter additive form can absorb some coupling between the two effects while still matching the smooth azimuthal curve. Thus the claim that the data are 'further validating the hypotheses' is the independence assumption re-encountered through the fitted model, not an independent test of that assumption.

full rationale

The paper's central formula is a geometric projection of two assumed effects, and its validation rests on fits to the same CORSIKA showers used to evaluate the model's errors. This makes the reported 11% (number) and 8% (energy) relative-error intervals in-sample residuals rather than predictive tests. The azimuthal double-circle structure is nonetheless a genuine check of the functional form: one parameter pair reproduces the angle dependence across 72 azimuths, which a poorly chosen model form would not do. Similarly, the empirical formulas for cmag and catm in Eqs. (5) and (7) are openly presented as fitting functions satisfying boundary conditions; they are not disguised predictions. The self-citation [16] is motivational and not load-bearing for the derivation. Overall, the model form has independent geometric content, but the quantitative validation and the claimed confirmation of the Independence Hypothesis are partly circular because the fitted parameters and the fitted model embody the hypotheses being 'validated'. This warrants a moderate score of 5, not a higher score, because the azimuthal test and the explicit fit-based empirical parameterizations provide independent substance.

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

The model rests on two physical approximations (independence and magnetic linearity), on the adopted atmospheric density profile, and on empirical functional forms chosen to match boundary conditions. The parameters cmag and catm, plus the A, B, C coefficients in Table II, are all fitted to CORSIKA-simulated data, so the predictive content of the model is limited to interpolation within the simulated parameter space.

free parameters (4)

cmag = Site- and species-dependent; e.g., AUGER proton Delta r_N Amag = 3.463 ± 0.010 m/uT
Magnetic deflection parameter fitted to each CORSIKA shower set; central to Eq. (3).
catm = Site- and species-dependent; e.g., AUGER proton Delta r_N Aatm = -130.4 ± 1.0 m
Atmospheric deflection parameter fitted to each CORSIKA shower set; central to Eq. (3).
Empirical coefficients A, B, C for cmag = Table II, e.g., Amag = 3.463, Bmag = 3.024, Cmag = 1.381 for AUGER proton Delta r_N
Coefficients in the empirical formula cmag = A tau^B (cos theta)^C, fitted to the simulation-derived cmag values.
Empirical coefficients A, B, C for catm = Table II, e.g., Aatm = -130.4, Batm = 1.768, Catm = -1.013 for AUGER proton Delta r_N
Coefficients in the empirical formula catm = A tau^B tan theta exp(C tau / tan theta), fitted to the simulation-derived catm values.

assumptions (4)

domain assumption Independence Hypothesis: Delta r = Delta rmag + Delta ratm; magnetic deflection and atmospheric absorption displacement are independent and add linearly.
Used to construct Eq. (3). Not rigorously derived; breaks for large deflections where the deflected path changes the traversed atmospheric depth.
domain assumption Magnetic Linearity Hypothesis: Delta l_mag = cmag (ep cross B), treating Larmor motion as projectile motion with constant Lorentz-force direction.
Used to derive the magnetic term in Eq. (2). This approximation is valid only when the change in particle direction is minimal.
domain assumption Linsley's standard atmosphere model provides the density profile for computing atmospheric depth tau via GDASTOOL.
The results depend on this adopted atmospheric model, which may differ from real atmospheric conditions at each site.
ad hoc to paper The functional forms cmag = A tau^B (cos theta)^C and catm = A tau^B tan theta exp(C tau / tan theta) are chosen to satisfy boundary conditions and to fit simulations.
These are not derived from first principles; they are empirical forms selected to match the stated boundary conditions in Eqs. (4) and (6).

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Cite this review

Pith. "Pith review of A bi-effect model of muon deflections in air showers." pith.science (2026). https://pith.science/paper/WZZHZ2UM

@misc{pith2026250817475,
  author       = {Pith},
  title        = {Pith review of: A bi-effect model of muon deflections in air showers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZZHZ2UM}},
  note         = {Machine review of arXiv:2508.17475}
}

abstract

Recent progress has shown that the geomagnetic field exerts a more significant impact than expected on the behavior of charged secondary particles in inclined air showers. In this study, we for the first time combine it with atmospheric effects to construct a bi-effect model, aiming to investigate the lateral distribution of particles on the ground plane. Despite the complex physical interactions during the development of air showers, a simple formula can describe the overall deflection of $\mu^{\pm}$ and accurately fit the deflection in simulated air showers, thereby validating the hypotheses about these effects in this study. Furthermore, we have obtained the relationship between model parameters and primary particle information for different experimental sites. This new model is highly successful and is promising to provide new insights for improving detector layout design and air shower reconstruction.

Figures

Figures reproduced from arXiv: 2508.17475 by the authors.

**Figure 3.** FIG. 3. Deflection model V.s simulations with the LHAASO parameters, and the species, zenith, altitude of first interaction [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4. Deflection parameters V.s fitting function for altitude of different first interaction points (FIXHEI) and zenith angles. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 7.** Figure 7: FIG. 7. The magnitude relationship between [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

Review history (2 revisions) →

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Reference graph

Works this paper leans on

22 extracted references · 20 canonical work pages

[1]

This relationship is expressed as: ∆⃗ r= ∆⃗ rmag + ∆⃗ ratm

Independence Hypothesis: We assume that the magnetic deflection (∆⃗ rmag) caused by the Lorentz force and the atmospheric absorption-induced displacement (∆⃗ ratm) are independent. This relationship is expressed as: ∆⃗ r= ∆⃗ rmag + ∆⃗ ratm

work page
[2]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

This relationship is expressed as: ∆⃗ r= ∆⃗ rmag + ∆⃗ ratm

Independence Hypothesis: We assume that the magnetic deflection (∆⃗ rmag) caused by the Lorentz force and the atmospheric absorption-induced displacement (∆⃗ ratm) are independent. This relationship is expressed as: ∆⃗ r= ∆⃗ rmag + ∆⃗ ratm

work page

[2] [2]

Magnetic Linearity Hypothesis: Charged particles undergo Larmor motion in the geomagnetic field. When the change in the direction of particle motion is mini- mal, we approximate the direction of the Lorentz force as constant; under this condition, Larmor motion is approx- imated as projectile motion. The deflection of muons in the shower plane ∆⃗l is prop...

work page

[3] [3]

Bhattacharjee and G

P. Bhattacharjee and G. Sigl, Phys. Rept. 327, 109 (2000)

work page 2000

[4] [4]

Alves Batista et al

R. Alves Batista et al. , Front. Astron. Space Sci. 6, 23 (2019). 7

work page 2019

[5] [5]

Aab et al

A. Aab et al. (Pierre Auger), Science 357, 1266 (2017)

work page 2017

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A. Addazi et al. (LHAASO), Chin. Phys. C 46, 035001 (2022)

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T. Abu-Zayyad et al. (Telescope Array), Astrophys. J. Lett. 768, L1 (2013)

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A. Aab et al. (Pierre Auger), Phys. Lett. B 762, 288 (2016)

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