Probabilistic mapping between multiparticle production variables and the depth of maximum in proton-induced extensive air showers

TL;DR

This work formulates a data-driven, model-independent link between the primary interaction in proton-induced air showers and the depth of shower maximum, by introducing new multiparticle production variables $\alpha_{\text{had}}$, $\zeta_{\text{had}}$, and $\zeta_{\text{EM}}$, and their linear combination $\xi$. The authors show that $\xi$ captures the majority of event-by-event fluctuations in $\Delta X_{\max}$ and that $X_{\max}$ distributions can be predicted from the distribution of $\xi$ via a universal, probabilistic framework with biases below $3$ g cm$^{-2}$, independent of the hadronic interaction model. They decompose the shower response into a universal kernel $p(R_X|\langle\xi\rangle)$ and demonstrate that accelerator-forward measurements of the new variables can constrain hadron spectra in the kinematic regimes relevant for extensive air showers. The approach provides a principled, data-driven path to constrain high-energy hadronic interactions beyond collider reach, with implications for improving mass composition inferences and hadronic-model discrimination in UHECR physics.

Abstract

The interaction of ultra-high-energy cosmic rays with air nuclei triggers extensive air showers that reach their maximal energy deposition at the atmospheric depth $X_{\max}$. The distribution of this shower observable encodes information about the proton-air cross-section via fluctuations of the primary interaction point, $X_1$, and hadron production through $ΔX_{\max} \equiv X_{\max} - X_1$. We introduce new multiparticle production variables, $α_{\textrm{had}}$, $ζ_{\textrm{had}}$, and $ζ_{\mathrm{EM}}$, built from the energy spectra of secondaries in the primary interaction. Their linear combination, $ξ$, predicts over $50 \%$ of the fluctuations in $ΔX_{\max}$. Moreover, we build a probabilistic mapping based on the causal connection between $ξ$ and $ΔX_{\max}$ that enables model-independent predictions of $X_{\max}$ moments with biases below $3\,\mathrm{g\,cm^{-2}}$. Therefore, measurements of the distribution of $X_{\max}$ allow a data-driven probing of secondary hadron spectra from the cosmic-ray-air interaction, in proton-induced showers. The distributions of the new multiparticle production variables can be measured in rapidity regions accessible to current accelerators and are strongly dependent on the hadronic interaction model in the kinematic regions exclusive to ultra-high-energy cosmic rays.

Figures (18)

• Figure 1: Scheme of the primary interaction, guiding the deduction of the model for fluctuations of $\Delta X_{\max}$. The variables $x$ represent the fraction of the primary energy carried by each secondary, in the laboratory frame. The number of hadronically interacting particles is given by $m_{\text{had}}$, and that of neutral pions is $m_{\text{EM}}$. Each hadronically interacting particle further interacts, after a depth $\lambda_i$, producing a subsequent EM cascade characterized by the multiparticle production variables $\alpha_{\text{EM}}$ and $\zeta_{\text{EM}}$, defined in the main text.
• Figure 2: Correlation between the predictor of $\Delta X_{\max}$ from the first $p$-air interaction, $\xi$, and the event-by-event values of $\Delta X_{\max}$. The contours containing 68 % and 95 % of the events are represented by the dashed and solid black lines, respectively. The 1:1 line is represented in solid grey and the linear regression curve as a dotted black line. This figure was produced using the library of proton-induced Conex simulations described in Section \ref{['sec:xi_model_performance']}, with the high-energy hadronic interaction model QGSjet -III.01.
• Figure 3: Upper panel: distribution of the residuals $\Delta X_{\max} - \xi$, along with the bias and resolution in the determination of $\Delta X_{\max}$. Lower panel: distributions of $\Delta X_{\max}$ (dotted lines) and its predictor from the first interaction $\xi$ (solid lines). The hadronic interaction models Epos LHC-R, QGSjet -III.01 and Sibyll2.3e are represented in blue, orange and purple, respectively. This figure was produced with the library of proton-induced Conex described in Section \ref{['sec:xi_model_performance']}.
• Figure 4: Correlation plot between $\xi$ and $\Delta X_{\max}|_{1^{\text{st}}}$, for stochastic first interactions and a deterministic prediction of the rest of the shower. The distribution is obtained with Conex proton-induced simulations as described in the main text, but with the energy threshold between Monte-Carlo and Cascade Equations set to $E_{\text{th}} = 0.999 \times E_0$, using the high-energy hadronic interaction model QGSjet -III.01.
• Figure 5: Top panel: Average values of $\xi$ and $\Delta X_{\max}$ as a function of the primary energy. Bottom panel: Bias $\expval{\Delta X_{\max} - \xi}$ as a function of the primary energy, for the hadronic interaction models Epos LHC-R, QGSjet -III.01 and Sibyll2.3e.