The mean mass of the primary cosmic rays that produce the 'knee' of the all-particle energy spectrum near 3-4 PeV is not settled. At roughly 2-10 PeV, published determinations of the mean mass number or mean logarithmic mass span helium to silicon, and reconstructions of the light/heavy balance above the knee change qualitatively with the assumed high-energy hadronic-interaction model. The open problem is to determine the actual mean mass of the primary cosmic rays at approximately 2-10 PeV - which anchors whether the knee region is dominated by light elements or a mixture and how the composition evolves through the knee - and to reconcile the conflicting experimental determinations: the precise but non-monotonic LHAASO-KM2A measurement of the mean logarithmic mass (light, near helium at the knee) against the monotonic-heavier trend of earlier air-shower experiments, small-sample estimates such as PRISMA-YBJ and emulsion-chamber analyses, and muon-bundle inferences that remain in tension with post-LHC hadronic models.

Background

Cosmic rays are atomic nuclei (plus some electrons) accelerated by astrophysical sources. The flux of all particles combined falls as a power law in energy but steepens near 3-4 PeV (1 PeV = 101510^{15} eV); this break is called the 'knee'. The leading explanation is a rigidity-dependent cutoff of Galactic accelerators - rigidity being momentum per unit charge, so that each nucleus with charge ZZ cuts off at E∼Z×E \sim Z \times (few PeV) and the composition would grow heavier through the knee - but propagation effects and more exotic mechanisms are also debated. The deciding evidence is the mean mass of the primaries at knee energies.

Because the flux at PeV energies is only about one particle per square meter per year, the composition above about 1 PeV cannot be measured directly and must be inferred from extensive air showers (EAS), the cascades of secondary particles that primaries initiate in the atmosphere. EAS observables sensitive to mass are the electron shower size NeN_e, the muon number NμN_\mu, the depth of shower maximum Xmax⁡X_{\max} (the atmospheric depth in g/cm2^2 where the cascade is largest, measured by fluorescence or Cherenkov light), the lateral distribution of Cherenkov light, and the hadronic or thermal-neutron content near the shower core. Converting any of these into a primary mass requires Monte Carlo simulations built on high-energy hadronic-interaction models (the QGSJET, SIBYLL and EPOS families), whose forward-region predictions - production of particles at very small angles, which dominates EAS development - are not directly constrained by central collider data. The composition is conventionally summarized by the mean logarithmic mass ⟨ln⁡A⟩\langle\ln A\rangle, where AA is the primary mass number; reference values are ln⁡A=0\ln A = 0 for protons, 1.39 for helium, about 2.6 for the CNO group (carbon, nitrogen, oxygen), and 4.03 for iron.

The experimental situation around 2-10 PeV is conflicting. Puchkov and Pyatovsky (2020) state that 'the absolute value of the average mass at the crucial energy of 10 PeV is needed to anchor the composition', with previous estimates placing it 'anywhere between helium and silicon'; from Tien-Shan muon data and the MC0-FANSY model (a phenomenological hadronic-interaction model) they themselves obtain ⟨A⟩≃8−9\langle A\rangle \simeq 8{-}9 at 10 PeV. Chernov et al. (2015) document that the rise of the heavy component between 2 and 20 PeV reported in KASCADE-Grande and Tunka-133 analyses is a consequence of analyzing the data with the QGSJET-II-02 or SIBYLL 2.1 models, while EPOS 1.99 yields a qualitatively different, several-times-lighter composition at the same energies. Shchegolev et al. (2019), using the ratio of electron to thermal-neutron signals recorded by the PRISMA-YBJ prototype array at Yangbajing, find ⟨ln⁡A⟩=1.6±0.1\langle\ln A\rangle = 1.6 \pm 0.1 at an average energy of about 2 PeV - a value not directly comparable with experiments reporting at 101610^{16} eV. Below the knee, the hybrid measurement combining ARGO-YBJ with a wide-field Cherenkov-telescope array of LHAASO (WFCTA; Bartoli et al. 2015) located the proton-helium knee below 1 PeV.

The landscape was then transformed by the Large High Altitude Air Shower Observatory (LHAASO), whose square-kilometer array KM2A (Cao et al. 2024), using an energy reconstruction that is nearly independent of the assumed composition and a muon-content calibration performed with post-LHC models (QGSJET-II-04, EPOS-LHC and SIBYLL-2.3d - model versions retuned after Large Hadron Collider measurements), measured ⟨ln⁡A⟩\langle\ln A\rangle from 0.3 to 30 PeV with a combined systematic uncertainty of about 9% (including a model spread below 6%) - the most precise determination in this energy region to date, compared with roughly 25% systematics for KASCADE and IceTop. The result is non-monotonic: ⟨ln⁡A⟩\langle\ln A\rangle falls from about 1.7 at 0.3 PeV to about 1.3 near the knee at 3.67±0.05±0.153.67 \pm 0.05 \pm 0.15 PeV, then turns heavier with a break at 5.36±0.08±0.465.36 \pm 0.08 \pm 0.46 PeV, and the knee is attributed to the cutoff of light components. The same paper emphasizes that earlier experiments (KASCADE, CASA-MIA, IceTop, Tunka-133) instead showed a gradual increase of ⟨ln⁡A⟩\langle\ln A\rangle through this energy range - the cross-experiment shape conflict is sharpened, not resolved, by the new measurement. Interpretation work built on these data (Lv et al. 2025; He et al. 2024) finds that the knee is the rigidity-dependent cutoff of light Galactic components and that most standard composition models (Gaisser, Horandel, GST) predict a heavier ⟨ln⁡A⟩\langle\ln A\rangle than LHAASO measures around 10 PeV, with only the Global Spline Fit (GSF) composition parametrization matching; LHAASO's own helium-spectrum measurement (2026) finds a helium knee near 7 PeV. Meanwhile Bogdanov et al. (2025) report that NEVOD-DECOR muon-bundle intensities at 101510^{15}-101610^{16} eV are in unresolved tension with the EPOS LHC-R and QGSJET-III-01 predictions, with the composition inference sensitive to the assumed all-particle spectrum - the 'muon puzzle', which entangles possible hadronic-model deficiencies with composition inferences exactly in the 2-10 PeV window.

Problem Statement

Determine the absolute mean mass - the mean mass number ⟨A⟩\langle A\rangle or the mean logarithmic mass ⟨ln⁡A⟩\langle\ln A\rangle, which are different summaries of the same mass distribution - of the primary cosmic rays at approximately 2-10 PeV, i.e. across the knee of the all-particle energy spectrum. This mean mass anchors whether the knee region is dominated by a single light element or by a mixture, and how the composition evolves through the knee. Establish the actual mean mass and its energy dependence in a way that holds up against the choice of high-energy hadronic-interaction model (the QGSJET, SIBYLL and EPOS families) and composition-model priors, and reconcile the conflicting experimental determinations: the precise but non-monotonic LHAASO-KM2A ⟨ln⁡A⟩(E)\langle\ln A\rangle(E) (light, near helium at the knee, turning heavier beyond about 5.4 PeV) with the monotonic-heavier reconstructions of KASCADE/KASCADE-Grande, Tunka-133, CASA-MIA and IceTop, with the small-sample PRISMA-YBJ and emulsion-chamber estimates (including the energy-gap issue that PRISMA-YBJ reports near 2 PeV while others report at 101610^{16} eV), and with muon-bundle inferences that remain in tension with post-LHC hadronic models - identifying for each discrepancy its origin, for example hadronic-model deficiencies, energy-scale differences between experiments, detector and analysis systematics, or other assumed inputs such as the all-particle spectrum.

References

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