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Beyond standard model particles in the atmospheric flux: a long-lived stau example

T0 review · 1 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that long-lived staus from cosmic-ray air collisions would give IceCube-Gen2 at most a few tracks per decade, so LHC limits on stau pair production remain the stronger constraint in the gauge-mediated SUSY scenarios…

desk verdict Solid, honest phenomenology: the paper's main conclusion—that LHC constraints beat IceCube-Gen2 for these GMSB staus—is robust, despite rough edges around flux normalization and event-rate details. read the letter →

arxiv 2507.10745 v1 pith:HQVV2424 submitted 2025-07-14 hep-ph

Atri Bhattacharya , Mary Hall Reno , Ina Sarcevic This is my paper
classification hep-ph PACS 14.80.Ly95.55.Vj96.50.sd
keywords long-livedstausatmosphericfluxcosmic-rayairshowersZ-momentformalismgauge-mediatedsupersymmetrybreakingIceCube-Gen2promptparticleproductionbeyond-standard-model
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 asks whether ultra-high-energy cosmic rays hitting the atmosphere could make enough long-lived supersymmetric staus for a large neutrino telescope to detect, and it answers with concrete numbers: almost none. Using the same Z-moment cascade machinery that computes prompt atmospheric neutrino fluxes, the authors evaluate the stau flux produced through squark pair production and decay ($N\to\tilde{q}\to\tilde{\tau}$), including the stau energy distribution relative to the incident cosmic-ray nucleon. For a 290 GeV stau, IceCube-Gen2 would collect at most a few track events in ten years; for a 490 GeV stau, the mass that LHC searches already allow, the rate is $O(10^{-4})$ events. Within the gauge-mediated SUSY framework considered, current collider constraints on stau pair production are therefore stronger than a decade of IceCube-Gen2 data. The method is the broader point: the Z-moment approach is a template for computing prompt atmospheric fluxes of any quasi-stable beyond-standard-model particle produced through a short-lived intermediary.

What carries the argument

The load-bearing object is the Z-moment formalism for atmospheric cascade equations, in particular the reduced z-moment $z_{N\tilde{\tau}}(E) = \int dE_N\, [\phi_N(E_N)/\phi_N(E)]\, [\sigma_{N\to\tilde{\tau}}(E_N)/\sigma_{N\to\tilde{\tau}}(E)]\, (1/E_N)\, dn/dx$, which separates the stau production probability $P(E)=\sigma_{pp\to\tilde{\tau}}/\sigma_{pp}$ from the spectrum-weighted energy distribution of the produced staus. The central identity is eq. (3.8), $\phi_{\tilde{\tau}}(E) \simeq 2 Z_{N\tilde{\tau}}(E)/(1-Z_{NN}(E))\, \phi_N(E,0)$, which turns the squark-pair production cross section into the prompt atmospheric stau flux. On the detection side, stau propagation uses the energy-loss law $dE_{\tilde{\tau}}/d\ell = -b\, \rho(\ell)\, E_{\tilde{\tau}}$ with $b \propto (10^3\,\mathrm{GeV}/m_{\tilde{\tau}})^{1.25}$, and the track signal is calibrated by the muon-equivalent energy $E_{\tilde{\tau}}^{\mathrm{eff}} \simeq (b_{\tilde{\tau}}/b_\mu) E_{\tilde{\tau}} \simeq 5\times10^{-4} E_{\tilde{\tau}}$. The exponential slope $b=7.55$ in the stau energy distribution and the cross-section power $E_N^{0.58}$ are the numerical anchors that turn the formalism into numbers.

What would settle it

Evaluate the differential $N\to\tilde{q}\to\tilde{\tau}$ cross section with a parton-level Monte Carlo at several fixed-target nucleon energies from $10^8$ to $10^{11}$ GeV: if the scaled energy distribution $(d\sigma/dx)/\sigma$ varies with $E_N$, or if the production cross section departs from $\sigma \propto E_N^{0.58}$, then the single-exponential Z-moment flux in eq. (3.8) is wrong and the event-rate projections change. On the experimental side, a ten-year IceCube-Gen2 search for up-going tracks at $E_{\min}=100$ TeV in the band $68^\circ \lesssim \theta_{\mathrm{Zenith}} \lesssim 70^\circ$ would either produce the few tracks predicted for $m_{\tilde{\tau}}=290$ GeV or, finding none, exclude that benchmark.

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Extended reading notes

Core claim

The paper's central claim is that the atmospheric stau flux from the chain $N\to\tilde{q}\to\tilde{\tau}$ is small enough that collider limits, not neutrino telescopes, bound the model. The flux at Earth's surface is written in Z-moment form, $\phi_{\tilde{\tau}}(E) \simeq 2\, Z_{N\tilde{\tau}}(E)/(1-Z_{NN}(E))\, \phi_N(E,0)$, where the factor of two counts the two squarks per pair and the Z-moment $Z_{N\tilde{\tau}}$ encodes both the production probability and the cosmic-ray-spectrum-weighted stau energy distribution. With an exponential fit to the scaled stau energy distribution ($d\sigma/dx \simeq A e^{-7.55 x}$, $x = E_{\tilde{\tau}}/E_N$) and a production cross section growing as $E_N^{0.58}$, the ten-year track rates at IceCube-Gen2 for a 100 TeV muon-equivalent threshold are a few events at $m_{\tilde{\tau}}=290$ GeV with $m_{\tilde{q}}=415$ GeV, and $O(10^{-4})$ events at $m_{\tilde{\tau}}=490$ GeV with $m_{\tilde{q}}=1.3$ TeV. Because staus lose little energy in matter ($b_{\tilde{\tau}}/b_\mu \sim 5\times10^{-4}$) and live long, their tracks keep a flat zenith-angle distribution down to about $70^\circ$ below the horizon, but the rates are too low to compete with the LHC bounds of roughly 430-490 GeV for this GMSB scenario. The stated conclusion is that a detector like IceCube-Gen2 over ten years is less constraining than existing collider searches, and that finding atmospheric staus would need a substantially larger instrumented volume.

Load-bearing premise

The calculation assumes the shape of the stau energy distribution, scaled to the incoming cosmic-ray nucleon energy, is the same at every incident energy from $10^8$ up to $10^{11}$ GeV, so that one exponential fit with slope $b=7.55$ and one power-law cross section $\sigma \propto E_N^{0.58}$ describe stau production everywhere; if the shape or the power law changes with energy, the predicted flux and event rates would shift.

Editorial extensions

If this is right

  • A ten-year IceCube-Gen2 run cannot constrain GMSB stau masses beyond LHC limits: roughly $O(10^{-4})$ tracks for the collider-allowed $m_{\tilde{\tau}}=490$ GeV case and at most a few tracks for the already-disfavoured $m_{\tilde{\tau}}=290$ GeV case.
  • Stau tracks keep a nearly flat zenith-angle distribution down to about $70^\circ$ below the horizon, so the up-going band is the place to look, not the full-solid-angle rate.
  • Lowering the stau mass to 200 GeV raises event rates by a factor of 3-4 relative to the 290 GeV benchmark, but such masses are already excluded, so atmospheric discovery would happen in parameter space colliders have ruled out.
  • Pion re-interactions ($N\to\pi^\pm\to\tilde{q}\to\tilde{\tau}$) add at most about 10% to the stau flux, so the $N\to\tilde{q}\to\tilde{\tau}$ route alone controls the prediction.
  • The Z-moment treatment is a template for computing atmospheric fluxes of any quasi-stable beyond-standard-model particle produced via a short-lived intermediate, at the EeV energy scale where cosmic-ray collisions exceed LHC energies.

Reading between the lines

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

  • Applied to gluino-mediated chains ($\tilde{g}\to\tilde{q}\to\tilde{\tau}$), the same machinery could give larger fluxes, since gluino pair-production cross sections can exceed squark-pair cross sections in parts of GMSB parameter space; the paper notes the cross-section dependence on squark and gluino masses but does not compute this channel.
  • The projected rate drops by roughly five orders of magnitude between $m_{\tilde{\tau}}=290$ GeV and 490 GeV, so the atmospheric channel is effectively a probe of sub-TeV spectra; heavier spectra are invisible to cubic-kilometre neutrino telescopes.
  • Because the reduced z-moment falls steeply below about $10^8$ GeV, the flux is set by a narrow window of incident cosmic-ray energies; better measurements of the nucleon spectrum and composition near $10^8$-$10^9$ GeV would sharpen, or overturn, the few-events-per-decade projection.
  • Read as a template, the result suggests a general ranking: any long-lived charged particle produced through a short-lived intermediary in air showers will be collider-dominated unless its production cross section substantially exceeds squark-pair production, and the Z-moment ratio derived here makes that comparison quantitative for any model.
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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

1 major / 5 minor

Summary. This paper calculates the flux of long-lived staus produced in cosmic-ray air showers through the two-step process N -> squark -> stau in Gauge-Mediated Supersymmetry Breaking models, using the Z-moment formalism that is standard for prompt atmospheric neutrino fluxes. The authors then estimate the track event rates at IceCube and IceCube-Gen2 for two benchmark stau masses (290 GeV and 490 GeV), compare them with the background from atmospheric and astrophysical muon neutrinos, and conclude that existing LHC limits on stau pair production are more constraining than a 10-year run of IceCube-Gen2 for the parameter space considered.

Significance. The paper is a useful and well-structured template for computing atmospheric fluxes of quasi-stable BSM particles that originate from short-lived intermediaries, extending the well-established Z-moment technique from Standard Model charm production to supersymmetric production. The central comparison with LHC constraints is robust: the gap between the predicted event rates (O(10^-4) per decade for m_tau = 490 GeV) and any conceivable detector sensitivity is several orders of magnitude, so even large systematic uncertainties in the flux derivation do not affect the main conclusion. The calculation is a forward prediction from published cross sections and cosmic-ray flux models, with no parameter fitted to the target observable, and the paper explicitly identifies the effect of the stau energy distribution on the flux. These features make the result a credible and reproducible baseline for future searches.

major comments (1)
  1. [Section 3.2, Eq. (3.11) and Fig. 5] The quantitative event rates in Fig. 5 rest on the assumption, stated in Eqs. (2.1) and (2.2), that the scaled stau energy distribution (dσ/dx)/σ is independent of the incident cosmic-ray energy and of the mass benchmark, so that a single exponential with b = 7.55 and a cross section proportional to E_N^0.58 describe production over the full energy range up to 10^11 GeV. The authors should quantify the uncertainty introduced by this approximation, for example, by recomputing the flux using the full energy-dependent distributions shown in Fig. 1 (rather than the fit), and by varying the power-law exponent of the cross section. Without such an estimate, the absolute event rates (a few per 10 years for m_tau = 290 GeV and about 10^-4 per 10 years for m_tau = 490 GeV) lack error bars. This does not affect the central comparison with LHC limits, which is robust to factor-of-few changes, but the quantitative predictions should be accompanied by an uncertainty band.
minor comments (5)
  1. [Section 4.2, Figure 5] The text refers to 'the left and right plots in figure 5', but Figure 5 shows a single panel; please correct the wording to match the actual figure layout.
  2. [Abstract and Title Page] There are missing spaces in 'IceCube-Gen2to' and 'IceCube-Gen2over' on the title page; these should be fixed for readability.
  3. [Section 4.1, below Eq. (4.4)] The word 'determiend' should be 'determined' in the sentence 'The effective stau energy that would be determiend by electromagnetic energy loss measurements'.
  4. [Section 3.2, first paragraph] The sentence beginning 'Since the decay length of staus is long...' is grammatically convoluted; consider splitting it into two sentences for clarity.
  5. [Section 2, Eq. (2.2)] When introducing the dashed line as the analytic approximation, state explicitly that it corresponds to the fit with b = 7.55, so the reader does not have to infer it from the figure.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stau flux is a forward Z-moment calculation with external LHC benchmarks; no fitted parameter is renamed as a prediction.

full rationale

The central derivation is a forward calculation. Squark production cross sections and the squark-to-stau decay are computed from GMSB benchmark spectra using standard pQCD, and the atmospheric stau flux is obtained from the same Z-moment cascade formalism used for prompt atmospheric neutrinos (eqs. 3.8-3.11). No parameter in the calculation is fitted to the stau flux or to the IceCube-Gen2 event-rate target. The analytic approximations in eqs. (2.1)-(2.2), with b = 7.55 and sigma ~ E_N^0.58, are internal parametrizations of the computed cross-section shapes and are explicitly described as a cross-check of the numerical results, not as inputs calibrated to the final observable. The stau propagation and energy-loss inputs are taken from published, independent calculations. The self-citations that appear, such as the value Z_NN = 0.23 from ref. [9] and stau energy-loss results from refs. [22] and [23], are not used to force the central conclusion; they are auxiliary numerical inputs that could be replaced by other published values. The final claim is benchmarked against external LHC limits [30]-[32], so the comparison has independent content. The assumption that the scaled stau energy distribution is independent of the incident cosmic-ray energy is a modeling approximation that could shift the absolute flux and event rates by a factor of a few, but it is not a circular input: it is a stated numerical uncertainty, not a parameter fitted to the quantity being predicted.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new particles or entities are introduced; the stau and squark are standard GMSB SUSY states from prior literature. The free parameters are shape fits, benchmark choices, and thresholds, none of which are fitted to the target observable.

free parameters (5)
  • b (exponential slope of stau energy distribution) = 7.55
    Fit to the numerically computed dsigma/dx for m_q=415 GeV, m_tau=290 GeV; assumed energy-independent in eq. (2.2).
  • sigma power-law exponent = 0.58
    Extracted from the high-energy behavior of sigma_N->tau(E_N) in Fig. 1; used in the analytic cross-check of the reduced z-moment.
  • stau proper decay length c tau = >= 10 m
    Chosen so staus are quasi-stable for neutrino-telescope searches; event rates depend on survival probability.
  • track energy threshold E_min = 100 TeV
    Chosen to suppress atmospheric neutrino background; event rates in Fig. 5 depend on this choice.
  • benchmark masses (m_tau, m_q) = (290,415) GeV and (490,1300) GeV
    GMSB-inspired benchmark points; 290 GeV is in tension with LHC limits, 490 GeV is allowed by current searches.
assumptions (6)
  • domain assumption The Z-moment cascade equation solution for prompt atmospheric neutrinos applies to stau production via N -> q -> tau (eq. 3.8).
    Borrowed from prompt neutrino flux calculations (refs. [5-12]); validity at EeV energies and for long-lived BSM particles is assumed.
  • ad hoc to paper Squarks decay to staus with 100% branching fraction.
    Assumed in Fig. 1 and Sec. 2; maximizes the flux and may overestimate event rates.
  • ad hoc to paper The stau energy distribution shape is independent of primary energy and is fit by A exp(-b x) with b = 7.55.
    Stated in Sec. 2 and used in eqs. (2.1)-(2.2) and (3.11) for all E_N.
  • domain assumption Z_NN = 0.23 is energy-independent.
    Taken from ref. [9] and assumed for all energies in eq. (3.8).
  • domain assumption Pion re-interactions contribute less than 10% to the stau flux and are neglected.
    Estimated in Sec. 3.2 using Z-moments from ref. [38]; allows the use of eq. (3.8) as the main flux.
  • domain assumption Cosmic ray nucleon flux is described by the H3p model (most optimistic) as default.
    Choice of input spectrum; H3a gives ~25% lower flux at relevant energies (Sec. 5).

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Cite this review

Pith. "Pith review of Beyond standard model particles in the atmospheric flux: a long-lived stau example." pith.science (2026). https://pith.science/paper/HQVV2424

@misc{pith2026250710745,
  author       = {Pith},
  title        = {Pith review of: Beyond standard model particles in the atmospheric flux: a long-lived stau example},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HQVV2424}},
  note         = {Machine review of arXiv:2507.10745}
}
read the original abstract

We discuss fluxes of long-lived supersymmetric (SUSY) particles produced in the atmosphere from ultra-high energy cosmic-ray interactions with air nuclei. We consider long-lived particle production which proceeds first via the on-shell formation of heavier particles in the SUSY spectrum and then their decays. Specifically, we focus on the production of stau pairs from the decays of squarks produced in the initial cosmic ray-air collision under the purview of gauge-mediated symmetry breaking models that have the stau as the next-to-lightest SUSY particle. The calculation of the resulting stau flux schematically mirrors that of prompt atmospheric neutrino production, however, with the energy scale of initial hadronic collision set to EeV energy scales of the incident cosmic rays rather than the PeV energies where the prompt atmospheric neutrino fluxes dominate over neutrino fluxes from pions and kaons, and with the appropriate stau production cross section. We show the effect of accounting for the energy distribution of the staus relative to the squark energy distributions. We discuss the potential for future ultra-high energy detectors similar in scope to IceCube-Gen2 to detect atmospheric staus via tracks going through the detector volume. Current collider constraints on stau pair production within the SUSY framework considered here are more constraining than those from a detector like IceCube-Gen2 over ten years.

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