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

Lorenzo Cazon ¹, Ruben Conceição ^2,3, Miguel A. Martins ^1,*, and Felix Riehn ⁴

¹Instituto Galego de Física de Altas Enerxías (IGFAE), Rúa de Xoaquín Díaz de Rábago, s/n, Campus Vida, Universidade de Santiago de Compostela, 15705, Santiago de Compostela, Galicia, Spain
²Departamento de Física, Instituto Superior Técnico (IST), Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisbon, Portugal
³Laboratório de Instrumentação e Física Experimental de Partículas (LIP)-Lisbon, Avenida Prof. Gama Pinto, 2, 1649-003 Lisbon, Portugal
⁴Technische Universität Dortmund, August-Schmidt-Straße 4, 44221 Dortmund, Germany

^*Contact author: miguelalexandre.jesusdasilva@usc.es

Phys. Rev. D 112, 043016 – Published 14 August, 2025

DOI: https://doi.org/10.1103/1664-pht6

Abstract

The interaction of ultrahigh-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 $Δ X_{\max} \equiv X_{\max} - X_{1}$ . We introduce new multiparticle production variables, $α_{had}$ , $ζ_{had}$ , and $ζ_{EM}$ , built from the energy spectra of secondaries in the primary interaction. Their linear combination, $ξ$ , predicts over 50% of the fluctuations in $Δ X_{\max}$ . Moreover, we build a probabilistic mapping based on the causal connection between $ξ$ and $Δ X_{\max}$ that enables model-independent predictions of $X_{\max}$ moments with biases below $3 g c m^{- 2}$ . Therefore, measurements of the distribution of $X_{\max}$ allow the 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 ultrahigh-energy cosmic rays.

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References (42)

Karl-Heinz Kampert and Michael Unger, Measurements of the cosmic ray composition with air shower experiments, Astropart. Phys. 35, 660 (2012).
A. Abdul Halim et al. (Pierre Auger Collaboration), Testing hadronic-model predictions of depth of maximum of air-shower profiles and ground-particle signals using hybrid data of the Pierre Auger Observatory, Phys. Rev. D 109, 102001 (2024).
Johannes Albrecht et al., The Muon Puzzle in cosmic-ray induced air showers and its connection to the Large Hadron Collider, Astrophys. Space Sci. 367, 27 (2022).
A. Aab et al. (Pierre Auger Collaboration), Depth of maximum of air-shower profiles at the Pierre Auger Observatory. I. Measurements at energies above $10^{17.8} eV$ , Phys. Rev. D 90, 122005 (2014).
P. Abreu et al. (Pierre Auger Collaboration), Measurement of the proton-air cross section at $\sqrt{s} = 57 TeV$ with the Pierre Auger Observatory, Phys. Rev. Lett. 109, 062002 (2012).
Isabel Astrid Goos, Xavier Bertou, and Tanguy Pierog, Determination of high-energy hadronic interaction properties from observables of proton initiated extensive air showers, arXiv:2304.08007.
Lorenzo Cazon, Ruben Conceição, Miguel Alexandre Martins, and Felix Riehn, Proton-air interactions at ultra-high energies in muon-depleted air showers with different depths, Phys. Lett. B 859, 139115 (2024).
W. Heitler, The Quantum Theory of Radiation, International Series of Monographs on Physics Vol. 5 (Oxford University Press, Oxford, 1936).
Thomas K. Gaisser, Ralph Engel, and Elisa Resconi, Cosmic Rays and Particle Physics: 2nd Edition (Cambridge University Press, Cambridge, England, 2016).
J. Matthews, A Heitler model of extensive air showers, Astropart. Phys. 22, 387 (2005).
Lorenzo Cazon, Ruben Conceição, and Felix Riehn, Probing the energy spectrum of hadrons in proton air interactions at ultrahigh energies through the fluctuations of the muon content of extensive air showers, Phys. Lett. B 784, 68 (2018).
Isabel Astrid Goos, Determination of physical properties of high-energy hadronic interactions from the $X_{\max} - N_{μ}$ anticorrelation, Ph.D. thesis, Karlsruher Institut für Technologie (KIT), KIT-Bibliothek, Karlsruhe, KIT, Karlsruhe, Dept. Phys., 2023.
Lorenzo Cazon, Probing high-energy hadronic interactions with extensive air showers, Proc. Sci. ICRC2019 (2020) 005 [arXiv:1909.02962].
Bruno Rossi, High Energy Particles (Prentice-Hall, New York, 1952).
K. Greisen, Cosmic ray showers, Annu. Rev. Nucl. Part. Sci. 10, 63 (1960).
This dependence was verified by fitting the energy evolution of $N_{\max}$ to $y = p_{1} {(E_{0} / ξ_{c}^{e})}^{γ} + p_{2}$ , yielding $γ = 1.004 \pm 0.001$ , in agreement with [1,10]. The values of $N_{\max}$ were obtained with conex simulations, as described in Section 3, for primary energies $E_{0} \in [10^{12}, 1 0^{19}] eV$ .
Jaime Alvarez-Muñiz, Ralph Engel, T. K. Gaisser, Jeferson A. Ortiz, and Todor Stanev, Hybrid simulations of extensive air showers, Phys. Rev. D 66, 033011 (2002).
This functional form is motivated by the fact that for equipartition of energy $ζ_{EM} / α_{EM} = \ln m_{tot}$ and the average $m_{tot}$ increases approximately as a power law in energy [17].
T. Pierog et al., First results of fast one-dimensional hybrid simulation of EAS using conex, Nucl. Phys. B, Proc. Suppl. 151, 159 (2004).
T. Bergmann, R. Engel, D. Heck, N. N. Kalmykov, S. Ostapchenko, T. Pierog, T. Thouw, and K. Werner, One-dimensional hybrid approach to extensive air shower simulation, Astropart. Phys. 26, 420 (2007).
Tanguy Pierog and Klaus Werner, epos LHC-R: Up-to-date hadronic model for EAS simulations, Proc. Sci. ICRC2023 (2023) 230.
Sergey Ostapchenko, qgsjet-III model of high energy hadronic interactions: The formalism, Phys. Rev. D 109, 034002 (2024).
Felix Riehn, Ralph Engel, Anatoli Fedynitch, Thomas K. Gaisser, and Todor Stanev, Hadronic interaction model SIBYLL 2.3d and extensive air showers, Phys. Rev. D 102, 063002 (2020).
Alexander Aab et al. (Pierre Auger Collaboration), The Pierre Auger Cosmic Ray Observatory, Nucl. Instrum. Methods Phys. Res., Sect. A 798, 172 (2015).
Ralf Ulrich, Ralph Engel, and Michael Unger, Hadronic multiparticle production at ultra-high energies and extensive air showers, Phys. Rev. D 83, 054026 (2011).
T. Pierog, Iu. Karpenko, J. M. Katzy, E. Yatsenko, and K. Werner, epos LHC: Test of collective hadronization with data measured at the CERN Large Hadron Collider, Phys. Rev. C 92, 034906 (2015).
Sergey Ostapchenko, Monte Carlo treatment of hadronic interactions in enhanced Pomeron scheme: I. qgsjet-II model, Phys. Rev. D 83, 014018 (2011).
Lorenzo Cazon, Ruben Conceição, Miguel Alexandre Martins, and Felix Riehn, Constraining the energy spectrum of neutral pions in ultra-high-energy proton-air interactions, Phys. Rev. D 103, 022001 (2021).
Ralph Engel, Dieter Heck, and Tanguy Pierog, Extensive air showers and hadronic interactions at high energy, Annu. Rev. Nucl. Part. Sci. 61, 467 (2011).
L. Cazon, R. Conceição, and F. Riehn, Universality of the muon component of extensive air showers, J. Cosmol. Astropart. Phys. 03 (2023) 022.
Z. Citron et al., Report from Working Group 5: Future physics opportunities for high-density QCD at the LHC with heavy-ion and proton beams, CERN Yellow Rep. Monogr. 7, 1159 (2019).
A. Abada et al. (FCC Collaboration), FCC Physics Opportunities: Future circular collider conceptual design report volume 1, Eur. Phys. J. C 79, 474 (2019).
S. Chatrchyan et al. (CMS Collaboration), The CMS experiment at the CERN LHC, J. Instrum. 3, S08004 (2008).
A. Augusto Alves, Jr. et al. (LHCb Collaboration), The LHCb detector at the LHC, J. Instrum. 3, S08005 (2008).
Jonathan L. Feng et al., The forward physics facility at the high-luminosity LHC, J. Phys. G 50, 030501 (2023).
Felix Kling and Laurence J. Nevay, Forward neutrino fluxes at the LHC, Phys. Rev. D 104, 113008 (2021).
O. Adriani et al. (LHCf Collaboration), The LHCf detector at the CERN Large Hadron Collider, J. Instrum. 3, S08006 (2008).
O. Adriani et al. (LHCf Collaboration), Measurements of longitudinal and transverse momentum distributions for neutral pions in the forward-rapidity region with the LHCf detector, Phys. Rev. D 94, 032007 (2016).
O. Adriani et al. (LHCf Collaboration), Measurement of energy flow, cross section and average inelasticity of forward neutrons produced in $\sqrt{s} = 13 TeV$ proton-proton collisions with the LHCf Arm2 detector, J. High Energy Phys. 07 (2020) 016.
Olena Tkachenko for the Pierre Auger Collaboration, Studies of the mass composition of cosmic rays and proton-proton interaction cross sections at ultra-high energies with the Pierre Auger Observatory, Proc. Sci. ICRC2023 (2023) 438.
Alexander Aab et al. (Pierre Auger Collaboration), Evidence for a mixed mass composition at the ankle’ in the cosmic-ray spectrum, Phys. Lett. B 762, 288 (2016).
https://doi.org/10.54499/2024.06879.CERN

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

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