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Exploring ultra-high energy neutrino experiments through the lens of the transport equation

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A first-principles line-of-sight transport calculation shows the KM3NeT 220 PeV muon is in 3.1 sigma tension with IceCube under a diffuse power-law neutrino flux, with only a transient source capable of lowering the tension to 1.6 sigma.

desk verdict Solid transport formalism, but the headline 3.1σ tension is inflated by a Bayes-factor-to-significance conversion error; corrected calibration gives about 2σ. read the letter →

arxiv 2507.10665 v3 pith:5T3GJPAW submitted 2025-07-14 hep-ph astro-ph.HEhep-ex

Stefano Palmisano , Diego Redigolo , Michele Tammaro , Andrea Tesi This is my paper · ORCID
classification hep-phastro-ph.HEhep-ex
keywords ultra-high-energyneutrinosneutrinotelescopestransportequationline-of-sightapproximationKM3-230213AIceCubeKM3NeTBayesfactors
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 derives the expected muon rate at a neutrino telescope from the transport equation in the line-of-sight approximation, so that the only free parameters are the normalization, spectral index, and directionality of the neutrino flux at Earth's surface. It applies this machinery to the KM3-230213A event, a 120+110-60 PeV muon track seen by KM3NeT at elevation 0.54 degrees, and compares the predicted rates with IceCube's non-observation of similar events. The central quantitative result is a 3.1 sigma tension between the two experiments under a diffuse power-law flux, corresponding to a Bayes factor of about 21 and an expected event ratio of about 70 in the 10 to 400 PeV bin. A steady point source sharpens the tension to 3.8 sigma, while a transient point source would relax it to 1.6 sigma. The formalism also clarifies how detector volume, Earth density, and lepton energy loss enter the event rate, and it provides a roadmap for including beyond-Standard-Model muon sources.

What carries the argument

The load-bearing object is the muon phase-space distribution $f_\mu$ computed by the line-of-sight solution of the Boltzmann transport equation: one integrates along the straight neutrino trajectory from the crust to the detector, using the Earth's density profile, the deep-inelastic-scattering neutrino cross section, and the charged-lepton energy-loss coefficients as inputs. The master outputs are Eq. (4.7) for muon-neutrino events and Eq. (4.13) for tau-neutrino events, each an integral over the line of sight of a transport function times the surface flux. These formulas expose the effective detector volume as $A_{\rm disk}(R_{\rm det} + 1/[b_\mu(2+\gamma-\lambda)])$, so the inverse radiative stopping length $1/b_\mu$, not the geometric volume, sets the collecting power for through-going muons.

What would settle it

Recompute the expected IceCube count in the 10 to 400 PeV bin using the detector's simulated effective area and the full 12-year livetime rather than a geometric disk; if the predicted ratio to KM3NeT drops from about 70 toward about 10, the 3.1 sigma tension falls below 3 sigma. More directly, a single IceCube muon above 10 PeV from the KM3-230213A sky region would falsify the paper's central claim.

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Extended reading notes

Core claim

The central claim is that the differential muon event rate at a neutrino telescope can be written, without Monte Carlo event generation, as an integral along the observer's line of sight through Earth of transport functions acting on the surface neutrino flux, with muons produced either by neutrino up-scattering near the detector or by tau decays at a distance. For a diffuse power-law flux the resulting master formulas, Eq. (4.7) for $\nu_\mu$ and Eq. (4.13) for $\nu_\tau$, imply an IceCube-to-KM3NeT expected-event ratio near 70 in the ultra-high-energy bin, an effective-volume enhancement $1/b_\mu$ beyond the geometric detector volume, and a 3.1 $\sigma$ tension between the KM3NeT event and the IceCube null. The same calculation gives a stronger tension for steady point sources and for energy-localized diffuse sources, and a milder 1.6 $\sigma$ tension if the source is transient.

Load-bearing premise

The result rests on modeling IceCube's ultra-high-energy exposure as a single Poisson bin from 10 to 400 PeV with zero observed events, and on treating both detectors as unit-efficiency spheres with geometric transverse area.

Editorial extensions

If this is right

  • For a diffuse power-law flux, the expected ratio of ultra-high-energy events between IceCube and KM3NeT is about 70 and only mildly dependent on the spectral index; with IceCube's fit as reference this translates into a Bayes factor of 21 and a 3.1 sigma tension.
  • A steady point source at the KM3-230213A position increases the ratio to about 140 and the tension to 3.8 sigma, so point sources do not relieve the discrepancy.
  • An energy-localized diffuse source behaves like a diffuse power law, with ratio about 67 and tension 2.4 sigma, while a transient point source lowers the ratio to about 11 and the tension to 1.6 sigma, the only considered Standard Model scenario that substantially reduces it.
  • Tau-neutrino point sources do not favor KM3NeT once tau energy loss is accounted for; even a pure $\nu_\tau$ source keeps the event ratio above about 70 at the relevant energies.
  • Because the effective volume grows as $1/b_\mu$, differences in muon energy loss in ice versus water materially change the expected-event ratio; ignoring this enhancement would push the ratio closer to 100.

Reading between the lines

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

  • Editorially, the same line-of-sight integrals can be applied to other large neutrino telescopes and to future high-energy upgrades; the framework's practical payoff is that nuisance parameters such as the cross-section slope, Earth density, and energy-loss coefficients enter as calculable theory inputs rather than as Monte Carlo systematics.
  • The 3.1 sigma number is tied to treating IceCube's exposure as a single zero-event bin; a revised fit using a longer livetime or a different high-energy binning could move the significance, so the exact sigma should be read as a model-dependent estimate.
  • A discriminating test the paper leaves implicit is that if more ultra-high-energy events accumulate, the energy distribution distinguishes transient from steady sources: a transient source populates a narrow energy window, whereas a power-law diffuse source predicts a particular falling spectrum.
  • If the tension persists, the paper's BSM discussion points to a concrete signature: a long-lived particle with decay length between the two detectors' chord lengths would preferentially produce muons at KM3NeT, visible as a high-energy track without a strong gamma-ray counterpart.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a first-principles transport-equation formalism, in the line-of-sight approximation, to compute the muon event rate at neutrino telescopes from the neutrino flux at the Earth's surface. It includes both muon production from νμ charged-current scattering and from ντ→τ→μ decay chains, and it derives master formulas, Eqs. (4.7) and (4.13), that depend on the surface neutrino flux and Standard Model/environmental inputs. The formalism is applied to the KM3-230213A event in comparison with the IceCube non-observation under several flux hypotheses: diffuse power-law, energy-localized diffuse, point source, transient point source, τ-neutrino point source, and a BSM two-state scenario. The headline result is a claimed 3.1σ tension between KM3NeT and IceCube for a diffuse power-law flux, with a Bayes factor of about 21, stronger tension for point sources, and reduced tension for transient sources.

Significance. If the quantitative claims survive scrutiny, the paper provides a transparent and physically motivated alternative to full detector Monte Carlo for UHE neutrino telescopes. The explicit treatment of tau propagation, Earth density profiles, and the derivation of effective-area-like quantities from first principles are genuine strengths, and the master formulas are stated in a form that can be reused and checked. The paper is also honest about its main approximations, notably the unit-efficiency spherical detector and the neglect of tau energy loss. However, the central quantitative significance is weakened by a statistical conversion error and by an exposure inconsistency between the fitted and predicted IceCube data, so the headline conclusion cannot be accepted as it stands.

major comments (3)
  1. [§5.3 and Table 3 footnote] The conversion of the Bayes factor into a Gaussian significance is incorrect. The quoted formula Δσ = √2 erfc⁻¹(Q(#dof/2; 2 ln B)) with #dof = 2 evaluates Q(1; 2 ln B) = exp(−2 ln B) = B⁻². The survival probability of a chi-square distribution with two degrees of freedom at χ² = 2 ln B is instead Q(1; ln B) = exp(−ln B) = B⁻¹. For the reported B ≈ 21, the correct two-sided significance is about 2.0σ, not 3.1σ. Because the abstract and Table 1 present the 3.1σ value as the central result, the significance must be recalibrated, and the related statements ('stark contrast', and the σ values for point and transient sources) revised accordingly.
  2. [§5.1, §5.3, Table 2] The IceCube likelihood in Section 5.3 is fit to the 9.5-year dataset of Ref. [8], using 'the last 31 bins in the right panel of Fig. 1 of [8]', while Section 5.1 and Table 2 state a 12-year IceCube exposure at the time of KM3-230213A, and Eq. (5.5) uses the combination [T A]_IC. If the flux normalization fitted to 9.5 years is used together with the 12-year exposure to predict the UHE-bin counts without an explicit rescaling, the expected numbers entering the Bayes factor are biased by a factor of 12/9.5 ≈ 1.26. The paper should specify which exposure is used for each step and make the rescaling explicit and consistent.
  3. [§5.2] The single high-energy bin [10 PeV, 400 PeV] is introduced specifically to contain the KM3-230213A event after that event was observed. The reported tension is therefore conditional on a bin choice made with knowledge of the data. If the bin has a physics motivation independent of the event, that motivation should be stated; otherwise the significance should be accompanied by a trials-factor correction or be described as exploratory. This issue compounds the calibration problem identified above.
minor comments (5)
  1. [Fig. 13 caption] The contour labels 'egg' and 'vegan bacon' are informal and unexplained; they should be replaced with standard posterior-density labels or a legend description.
  2. [Table 3 caption] The caption defines Δσ but not the meanings of B and ̃B in a self-contained way; the definitions in the text should be repeated briefly in the caption.
  3. [§4.1] The statement that more detailed efficiency choices 'do not change the result of Section 5' would be easier to trust if a quantitative robustness bound or a short scan were shown.
  4. [§4.4] The approximation bτ = 0 is stated, but Section 5.5 later explains that tau energy loss can affect the conclusions; a quantitative estimate of the size of the bτ correction on Eq. (4.13) would be useful.
  5. [§5 heading] There is a typo in the heading 'statystical treatment', which should read 'statistical treatment'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transport-equation event rates are a self-contained forward model, and the IceCube-to-KM3NeT tension is a genuine Bayes-factor extrapolation.

full rationale

The derivation chain is linear and self-contained. Eq. (2.14) expresses the muon phase-space density as a convolution of the surface neutrino flux with a transport function built from external inputs (PREM density, DIS cross sections from MadGraph/PDF fits, and charged-lepton energy losses); Eqs. (4.7) and (4.13) are the corresponding master formulas, with the flux parameters left free and later fitted to IceCube data. The tension analysis in Section 5.3 evaluates the KM3NeT likelihood under a flat prior and under the IceCube-informed prior through Eq. (5.4); no KM3NeT observable is used as an input to predict itself, and no parameter is fitted to KM3NeT and then presented as a prediction. The point-source and BSM sections similarly use the same forward model. There are no load-bearing self-citations and no ansatz is imported solely from the authors' prior work. The only notable caveat is statistical rather than circular: the conversion of the reported Bayes factor B = 21 into 3.1 sigma using Delta-sigma = sqrt(2) erfc^-1(Q(#dof/2; 2 ln B)) appears to miss the conventional factor of 1/2 in the chi-square survival-function argument (a p = 1/B calibration would give roughly 2 sigma). This affects the headline significance and should be corrected or verified, but it does not make the derivation equal to its inputs.

Assumptions & free parameters 3 free parameters · 8 assumptions · 2 invented entities

The central tension result rests on the muon-neutrino channel, whose ingredients are standard transport theory, externally fitted PDFs and cross sections, a spherical unit-efficiency detector model, and a Poisson binning of the ultra-high-energy data. The tau-channel and beyond-Standard-Model extensions rely on additional approximations (b_tau = 0, sterile mixing) that the paper itself flags. No parameters are fitted to the KM3NeT event itself; the flux parameters are fitted to IceCube, which is the design of the tension test.

free parameters (3)
  • Diffuse flux normalization phi0 = 0.10 +/- 0.01 in units of 1e-18 GeV^-1 cm^-2 s^-1 sr^-1 at E* = 100 TeV
    Fitted to IceCube bin data via MCMC (Eq. 5.6); used to predict KM3NeT event counts.
  • Spectral index gamma = 2.46 +/- 0.05
    Fitted jointly with phi0 to IceCube data; controls energy distribution of events and the effective-volume correction (2 + gamma - lambda).
  • Low-x PDF power lambda = 0.4 (scanned over 0.3 to 0.5)
    Sets the high-energy scaling of the neutrino-nucleon cross section (Eq. 3.13); taken from external PDF fits (Refs [26,27]), fixed to 0.4 in the main analysis.
assumptions (8)
  • domain assumption Muon direction is constant along the line of sight (n_hat dot product equal to zero).
    Used to solve the transport equation by characteristics (Eq. 2.12). Valid at ultra-high energies where multiple scattering is small; stated in Section 2.2.
  • domain assumption Muon energy loss is continuous with dE/dx = -a - bE, with b constant in the ultra-high-energy limit; ionization neglected.
    Sections 2.1 and 3.3; underlies the exponential momentum scaling in Eq. (2.12) and the effective-volume factor 1/b_mu.
  • domain assumption Neutrino flux factorizes into energy and direction, and for diffuse sources is isotropic; flavor composition is equal.
    Eq. (3.4) and Section 2; not valid for all source classes, e.g., Galactic plane sources would need direction-dependent spectra.
  • domain assumption Detector is a spherical fiducial volume with unit efficiency (epsilon = 1).
    Eqs. (4.2) to (4.4); the paper checks that more detailed choices do not change the tension, but the exposure and effective area are otherwise uncalibrated.
  • ad hoc to paper Tau energy loss is neglected (b_tau = 0) in the tau-neutrino event formula.
    Eq. (4.13); the paper explicitly says this is the biggest underlying assumption in the tau channel and later (Section 5.5) notes that including b_tau removes the naive KM3NeT advantage.
  • domain assumption Neutrino regeneration from tau decays is neglected.
    Section 2.3; stated to conservatively reduce the muon yield.
  • domain assumption Inelasticity distribution P(y) is taken from MadGraph Monte Carlo, not derived analytically.
    Section 3.2, Eq. (3.14); inputs to the master formulas.
  • standard math Small-x parton distributions follow a power law xq ~ x^-lambda with lambda independent of Q^2.
    Eqs. (3.9) to (3.13); used to obtain analytic cross-section scaling; justified by PDF fits in Refs [26,27].
invented entities (2)
  • Heavy neutral lepton X (sterile neutrino model)
    purpose: Decays to muons to enhance KM3NeT relative to IceCube via decay length d_X between L_IC and L_KM (Eq. 5.14).
    Section 5.6; constrained by CHARM bounds; the paper concludes the yield is too small in realistic flux compositions, so no independent handle is provided.
  • Light sterile neutrino nu_X (two-state scenario)
    purpose: Carries an incoming flux that up-scatters to produce the unstable heavy neutral lepton X.
    Section 5.6; the paper notes the beyond-Standard-Model rate is negligible unless the nu_X flux dominates, which is not expected for democratic flavor compositions.

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Pith. "Pith review of Exploring ultra-high energy neutrino experiments through the lens of the transport equation." pith.science (2026). https://pith.science/paper/5T3GJPAW

@misc{pith2026250710665,
  author       = {Pith},
  title        = {Pith review of: Exploring ultra-high energy neutrino experiments through the lens of the transport equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5T3GJPAW}},
  note         = {Machine review of arXiv:2507.10665}
}
abstract

We develop a first-principles formalism, based on the transport equation in the line-of-sight approximation, to link the expected number of muons at neutrino telescopes to the flux of neutrinos at the Earth's surface. We compute the distribution of muons inside Earth, arising from the up-scattering of neutrinos close to the detector, as well as from the decay of taus produced farther away. This framework allows one to account for systematic uncertainties, as well as to clarify the assumptions behind definitions commonly used in the literature, such as the effective area. We apply this formalism to analyze the high-energy muon event recorded by KM3NeT, with a reconstructed energy of $ 120^{+110}_{-60} \, \mathrm{PeV}$ and an elevation angle of $\left(0.54\pm 2.4\right)^\circ$, in comparison with the non-observation of similar events by IceCube. We find a $3.1\,\sigma$ tension between the two experiments, assuming a diffuse neutrino source with a power-law energy dependence. Combining both datasets leads to a preference for a very low number of expected events at KM3NeT, in stark contrast to the observed data. The tension increases both in the case of a diffuse source peaking at the KM3NeT energy and of a steady point source, whereas a transient source may reduce the tension down to $1.6\,\sigma$. The formalism allows one to treat potential beyond-the-Standard-Model sources of muons, and we speculate on this possibility to explain the tension.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Four, One, and None: Quantifying the Ultra-High-Energy Neutrino Anomaly Across ANITA-IV, KM3NeT, and IceCube

    astro-ph.HE 2026-07 conditional novelty 6.0 of 10

    A joint three-detector analysis finds that ANITA-IV's four events and KM3NeT's 220 PeV event cannot be reconciled with IceCube's silence under any Standard Model diffuse or transient flux, leaving a 5.9-7.9σ anomaly.

  2. The soft volume of ultra-high energy neutrinos experiments

    hep-ph 2026-07 conditional novelty 6.0 of 10

    A drift-diffusion approximation of muon energy loss maps ultra-high-energy neutrino fluxes to through-going track rates, with a soft volume several times the instrumented volume.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.