Atmospheric muons offer a sharper test of the cosmic-ray puzzle

Four experiments, one reference baseline

G. I. Rubtsov and S. V. Troitsky bring together published measurements from BESS, L3+C, KM3NeT and Jinping. Their question is whether the muon discrepancy also appears in observations that preferentially sample energetic particles produced early in atmospheric cascades.

These experiments do not measure identical quantities. BESS and L3+C measured differential surface fluxes: how many muons arrive within particular momentum ranges. KM3NeT measured reconstructed event rates in deep-sea detectors. Jinping reconstructed surface fluxes above energy thresholds from underground observations, accounting for muons lost while crossing rock.

The authors divide each measurement by its corresponding published prediction. A ratio of one means agreement with the central prediction; a larger ratio means the measurement exceeds it. This puts the discrepancies on a common relative scale without treating the underlying measurements as interchangeable.

The reference calculations share two ingredients: SIBYLL 2.3d, a model of particle collisions involving the strong nuclear force, and GSF19, a description of the incoming cosmic-ray flux and composition. The authors chose this combination for the availability of published predictions, not because they established it as the best model. Treatments of low-energy collisions, muon propagation and detector response remain experiment-specific.

A recurring excess, not a single combined discovery

All four experiments show measurements predominantly above their central reference predictions. L3+C ratios are mostly around 1.2–1.3, corresponding to roughly 20–30% more muons than predicted. BESS measurements also lie mainly above one.

For the KM3NeT detector configurations examined, the event-rate ratios range from 1.36 to 1.39 for ORCA6 and from 1.51 to 1.74 for ARCA6. Jinping's reconstructed integral fluxes show an excess too, although uncertainties grow at the highest thresholds.

Those numbers are not a unified measurement of how the excess changes with energy. A surface momentum bin, the range of muon energies contributing to a deep-detector event rate, and an underground-derived energy threshold describe different things.

Nor do the authors calculate a combined statistical significance. Shared uncertainties in cosmic-ray inputs and collision models, together with incomplete information about how errors are correlated, make separate comparisons more informative. An excess over a central prediction is not automatically a statistically established disagreement.

An apparent energy trend does not hold up

Combining BESS and L3+C initially suggests that the measured-to-predicted ratio rises with momentum. But the experiments have different absolute flux levels and cover different momentum ranges. That combination can create an apparent trend even without a rise within either data set.

At a momentum of 50 GeV/c, their independently fitted ratio levels differ by about 9%. Allowing each experiment its own constant level reduces the fitted momentum slope to a value consistent with zero. Including model uncertainties also weakens the apparent trend. The analysis therefore does not establish that the muon spectrum falls more slowly with energy than predicted.

The direction test has a similar qualification. L3+C shows a lower ratio in its most nearly vertical interval. An apparent broader angular slope disappears when that interval is removed. Its statistical interpretation also depends strongly on assumptions about correlated uncertainties, so the authors treat the localized feature—not a general directional trend—as the main observation.

Why energetic muons matter

Many muons counted in an extensive air shower are relatively low-energy particles produced after several generations of collisions. Energetic atmospheric muons preferentially trace an earlier part of the cascade, through the decay of energetic parent particles called mesons, particularly pions and kaons.

The reason is that increasingly energetic cosmic rays become much rarer. Producing an energetic muon after substantial energy has already been lost generally requires a rarer, higher-energy primary particle. Mesons that carry a large share of the initial collision energy consequently matter especially strongly.

This distinction limits explanations intended to solve both discrepancies. Simply producing more low-energy particles late in a shower could increase its total muon count while barely changing the energetic atmospheric-muon flux. A shared explanation must also increase the effective supply of energetic meson parents.

What remains unresolved

The excess alone cannot distinguish altered particle production from a higher incoming cosmic-ray nucleon flux. Detector effects matter too: the KM3NeT ratios, particularly ARCA6's, are sensitive to assumptions about light absorption in seawater. Some numerical inputs were reconstructed from published figures rather than supplied as original tables.

The preprint thus offers a test of proposed explanations, not a solution. More consistent predictions, better uncertainty correlations and complementary measurements—such as muon charge ratios and atmospheric-neutrino spectra—are needed to determine which explanation survives.