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Description
The distribution of the number of muons over an ensemble of proton-induced showers at ultra-high energies is shaped by the energy spectrum of secondary neutral pions in cosmic ray-air interactions.
This work finds that the steepness of the muon number distribution in muon-poor showers, quantified by the parameter $\Lambda_\mu$, decreases with the depth of the shower maximum, $X_{\max}$, with increasing dependence on the hadronic interaction model. Additionally, we show that $X_{\max}$ can probe the hadronic activity of the first interaction, so that the evolution of $\Lambda_\mu$ with $X_{\max}$ can be used to constrain the hardness of the energy spectrum of neutral pions in different regions of the kinematic phase-space of the cosmic ray-air interaction.
We verify that an unbiased measurement of $\Lambda_\mu$ is feasible for realistic mass composition expectations. Finally, we check that the statistical precision in $\Lambda_\mu$ required to distinguish between hadronic interaction models can be achieved with current extensive air shower experiments, given their resolution and exposure. Thus, the proposed measurement has the potential to constrain hadronic interactions at center-of-mass energies beyond the reach of human-made colliders.
