Reinterpreting the development of extensive air showers initiated by nuclei and photons

Ultra-high energy cosmic rays (UHECRs) interacting with the atmosphere generate extensive air showers (EAS) of secondary particles. The depth corresponding to the maximum development of the shower, Xmax, is a well-known observable for determining the nature of the primary cosmic ray which initiated the cascade process. In this paper, we present an empirical model to describe the distribution of Xmax for EAS initiated by nuclei, in the energy range from 1017 eV up to 1021 eV, and by photons, in the energy range from 1017 eV up to 1019.6 eV. Our model adopts the generalized Gumbel distribution motivated by the relationship between the generalized Gumbel statistics and the distribution of the sum of non-identically distributed variables in dissipative stochastic systems. We provide an analytical expression for describing the Xmax distribution for photons and for nuclei, and for their first two statistical moments, namely 〈Xmax〉 and σ(Xmax). The impact of the hadronic interaction model is investigated in detail, even in the case of the most up-to-date models accounting for LHC observations. We also briefly discuss the differences with a more classical approach and an application to the experimental data based on information theory.