Forward hadron production in proton-air collisions above LHC energies through the fluctuations of extensive air showers

Abstract

Primary proton-air interactions at ultra-high energies leave a physically interpretable imprint on the correlated fluctuations of the depth of shower maximum and the muon content in extensive air showers. This imprint reflects the stochasticity in the partition of the primary energy among secondary particles in the first interaction. We show that these fluctuations can be accessed through a probabilistic description that isolates sensitivity to hadronic physics in the initial collision, while treating the subsequent shower development as effectively universal. The uncertainties resulting from this universality are smaller than the spread among current hadronic interaction models and comparable to current experimental uncertainties. Consequently, the joint observable space defined by these two quantities provides a new probe of hadron production in kinematic regimes far beyond the reach of human-made accelerators.

Figures (5)

• Figure 1: Mean values of several multiparticle--production variables of primary proton--air interactions across the joint distribution of $n_\mu$ and $X_{\max}$. The top row shows classical variables, and the bottom row shows variables encoding information about the shape of the secondary--particle energy spectra. Produced with the ensemble of Conex simulations described in the Supplemental Material using Epos LHC-R.
• Figure 2: Left: joint distribution of $\alpha_1$ and $\xi$. Middle and right: examples of joint distributions of $n_\mu$ and $X_{\max}$ corresponding to the particular values of $(\alpha_1,\xi)$ in the bins highlighted in the left panel. Star markers indicate the centroids; black contours show the $1\sigma$ boundary of each response; grey contours show the $1\sigma$ (dotted) and $2\sigma$ (solid) boundaries of the full $f(n_\mu,X_{\max})$.
• Figure 3: Central panel: ratio between the predicted and true $f(n_\mu, X_{\max})$. The prediction is obtained by applying Equation \ref{['eq:pred_nmu_xmax_univ_kernel']} to a prior $f(\alpha_1, \xi)$ provided by Epos LHC-R. Peripheral panels: True and predicted distributions of $n_\mu$ (top) and $X_{\max}$ (right). The figure was produced using an ensemble of Conex simulations of proton-induced air showers with $E_0 = 10^{19}$ eV and $\theta = 60^\circ$, employing the high-energy hadronic interaction model Epos LHC-R.
• Figure 4: Primary energy evolution of the uncertainty on the main moments of $f(X_{\max})$: $\delta\left[\expval{X_{\max}} \right]$ (left), $\delta \left[ \sigma_{\rm{left}}(X_{\max})\right]$ (middle) and $\delta\left[\Lambda_\eta\right]$ (right). This uncertainty is represented in blue and it is half the spread of the biases induced on these moments by predicting $f(n_\mu, X_{\max})$ using Equation \ref{['eq:pred_nmu_xmax_univ_kernel']} on $f(\alpha_1, \xi)$ priors provided by: Epos LHC-R, Epos LHC, QGSjet -III.01, QGSjet -II.04 and Sibyll2.3e. All uncertainties are fitted to linear functions by minimizing $\chi^2$. The spread of model predictions for each moment isrepresented by the grey dashed line. The experimental systematic uncertainties are taken from Ref. 2018_Bellido_HeCoXmax ("Syst. HeCo (ICRC17)") and Refs. 2014_Auger_xmax2023_Thomas_XmaxFDrec ("Syst. FD (PRD14)"), and are represented in black.
• Figure 5: Primary energy evolution of the uncertainty on the main moments of $f(n_\mu)$: $\delta \left[ \sigma_{\rm{right}}(n_\mu)\right]$ (left) and $\delta\left[\Lambda_\mu\right]$ (right). This uncertainty is represented in blue and it is half the spread of the biases induced on these moments by predicting $f(n_\mu, X_{\max})$ using Equation \ref{['eq:pred_nmu_xmax_univ_kernel']} on $f(\alpha_1, \xi)$ priors provided by: Epos LHC-R, Epos LHC, QGSjet -III.01, QGSjet -II.04 and Sibyll2.3e. All uncertainties are fitted to linear functions by minimizing $\chi^2$. The spread of model predictions for each moment isrepresented by the grey dashed line. The experimental systematic uncertainty is taken from Ref. 2021_Auger_muonfluctuations ("Syst. (PRL21)"), and it is represented in black.