- Open Access
Basic model for high energy cosmic ray interactions
Phys. Rev. D 113, 074001 – Published 1 April, 2026
DOI: https://doi.org/10.1103/nzzw-898t
Abstract
A Monte Carlo generator of high energy cosmic ray interactions, relying on a very basic and transparent theoretical formalism, in the framework of the Reggeon field theory, is presented. The main motivation for our work is to provide a new cosmic ray interaction model characterized by relatively transparent physics, sufficient parameter freedom, and a high computational efficiency, which can be easily managed by external users, including a retuning of the model parameters. Such a model can be used for studying potential modifications of the interaction treatment, necessary for describing particular sets of data on extensive air showers initiated by high energy cosmic rays, at a microscopic level, thereby keeping a consistency with general restrictions, like the unitarity, energy-momentum and charge conservation, and Lorentz and isospin invariance. Importantly, this should allow one to study a compatibility of such modifications with relevant accelerator data. The model results for particle production and for basic extensive air shower characteristics are presented and discussed.
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References (94)
- M. Nagano and A. A. Watson, Observations and implications of the ultrahigh-energy cosmic rays, Rev. Mod. Phys. 72, 689 (2000).
- J. Blümer, R. Engel, and J. R. Hörandel, Cosmic rays from the knee to the highest energies, Prog. Part. Nucl. Phys. 63, 293 (2009).
- R. Engel, D. Heck, and T. Pierog, Extensive air showers and hadronic interactions at high energy, Annu. Rev. Nucl. Part. Sci. 61, 467 (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).
- F. Riehn, R. Engel, A. Fedynitch, T. K. Gaisser, and T. Stanev, Hadronic interaction model Sibyll 2.3d and extensive air showers, Phys. Rev. D 102, 063002 (2020).
- S. Ostapchenko, QGSJET-III model of high energy hadronic interactions: The formalism, Phys. Rev. D 109, 034002 (2024).
- P. Abreu et al. (Pierre Auger Collaboration), Interpretation of the depths of maximum of extensive air showers measured by the Pierre Auger Observatory, J. Cosmol. Astropart. Phys. 02 (2013) 026.
- A. Aab et al. (Pierre Auger Collaboration), Muons in air showers at the Pierre Auger Observatory: Measurement of atmospheric production depth, Phys. Rev. D 90, 012012 (2014).
- A. Aab et al. (Pierre Auger Collaboration), Muons in air showers at the Pierre Auger Observatory: Mean number in highly inclined events, Phys. Rev. D 91, 032003 (2015).
- A. Aab et al. (Pierre Auger Collaboration), Testing hadronic interactions at ultrahigh energies with air showers measured by the Pierre Auger Observatory, Phys. Rev. Lett. 117, 192001 (2016).
- 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).
- J. Ebr, J. Blazˆek, J. Vícha, T. Pierog, E. Santos, P. Trávníĉek, N. Denner, and R. Ulrich, Impact of modified characteristics of hadronic interactions on cosmic-ray observables for proton and nuclear primaries, Proc. Sci., ICRC2023 (2023) 245.
- V. N. Gribov, A Reggeon diagram technique, Sov. Phys. JETP 26, 414 (1968).
- V. N. Gribov, Glauber corrections and the interaction between high-energy hadrons and nuclei, Sov. Phys. JETP 29, 483 (1969).
- A. B. Kaidalov and K. A. Ter-Martirosyan, Pomeron as quark-gluon strings and multiple hadron production at SPS collider energies, Phys. Lett. 117B, 247 (1982).
- A. Capella, U. Sukhatme, C.-I. Tan, and J. Tran Thanh Van, Dual parton model, Phys. Rep. 236, 225 (1994).
- A. Donnachie and P. V. Landshoff, Total cross-sections, Phys. Lett. B 296, 227 (1992).
- N. N. Kalmykov and S. S. Ostapchenko, The nucleus-nucleus interaction, nuclear fragmentation, and fluctuations of extensive air showers, Phys. At. Nucl. 56, 346 (1993).
- S. Ostapchenko, Monte Carlo treatment of hadronic interactions in enhanced Pomeron scheme: QGSJET-II model, Phys. Rev. D 83, 014018 (2011).
- S. Ostapchenko, QGSJET-III model of high energy hadronic interactions: II. Particle production and extensive air shower characteristics, Phys. Rev. D 109, 094019 (2024).
- L. Gribov, E. Levin, and M. Ryskin, Semihard processes in QCD, Phys. Rep. 100, 1 (1983).
- A. Donnachie and P. V. Landshoff, Small x: Two pomerons!, Phys. Lett. B 437, 408 (1998).
- S. Ostapchenko, H. J. Drescher, F. M. Liu, T. Pierog, and K. Werner, Consistent treatment of soft and hard processes in hadronic interactions, J. Phys. G 28, 2597 (2002).
- S. Ostapchenko, On the re-summation of enhanced Pomeron diagrams, Phys. Lett. B 636, 40 (2006).
- A. B. Kaidalov, L. A. Ponomarev, and K. A. Ter-Martirosyan, Total cross-sections and diffractive scattering in a theory of interacting Pomerons with , Sov. J. Nucl. Phys. 44, 468 (1986).
- P. D. B. Collins, An Introduction to Regge Theory and High Energy Physics (Cambridge University Press, Cambridge, England, 1977).
- L. Frankfurt, M. Strikman, D. Treleani, and C. Weiss, Evidence for color fluctuations in the nucleon in high-energy scattering, Phys. Rev. Lett. 101, 202003 (2008).
- M. L. Good and W. D. Walker, Diffraction disssociation of beam particles, Phys. Rev. 120, 1857 (1960).
- S. Ostapchenko, Total and diffractive cross sections in enhanced pomeron scheme, Phys. Rev. D 81, 114028 (2010).
- V. A. Abramovsky, V. N. Gribov, and O. V. Kancheli, Character of inclusive spectra and fluctuations produced in inelastic processes by multi-pomeron exchange, Sov. J. Nucl. Phys. 18, 308 (1974).
- A. B. Kaidalov, Quark and diquark fragmentation functions in the model of quark gluon strings, Sov. J. Nucl. Phys. 45, 902 (1987).
- J. Engel, T. K. Gaisser, T. Stanev, and P. Lipari, Nucleus-nucleus collisions and interpretation of cosmic ray cascades, Phys. Rev. D 46, 5013 (1992).
- V. R. Zoller, Diffractive scattering off nuclei in the multiple scattering theory with inelastic screening, Sov. J. Nucl. Phys. 48, 361 (1988).
- R. L. Workman et al. (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
- S. Ostapchenko and G. Sigl, Model uncertainties for the predicted maximum depth of extensive air showers, Phys. Rev. D 110, 063041 (2024).
- S. Ostapchenko, QGSJET-II: Physics, recent improvements, and results for air showers, EPJ Web Conf. 52, 02001 (2013).
- S. Ostapchenko, QGSJET-III model: Novel features, Phys. Atom. Nucl. 84, 1017 (2021).
- A. B. Kaidalov, V. A. Khoze, A. D. Martin, and M. G. Ryskin, Leading neutron spectra, Eur. Phys. J. C 47, 385 (2006).
- S. Ostapchenko and M. Bleicher, Taming the energy rise of the total proton-proton cross-section, Universe 5, 106 (2019).
- G. Antchev et al. (TOTEM Collaboration), First measurement of elastic, inelastic and total cross-section at by TOTEM and overview of cross-section data at LHC energies, Eur. Phys. J. C 79, 103 (2019).
- G. Aad et al. (ATLAS Collaboration), Measurement of the total cross section and -parameter from elastic scattering in pp collisions at with the ATLAS detector, Eur. Phys. J. C 83, 441 (2023).
- R. J. Glauber, High-energy collision theory, in Lectures in Theoretical Physics, edited by W. E. Brittin and L. G. Dunham (Interscience Publishers, New York, 1959), Vol. 1, pp. 315–414.
- G. Bellettini, G. Cocconi, A. N. Diddens, E. Lillethun, G. Matthiae, J. P. Scanlon, and A. M. Wetherell, Proton-nuclei cross sections at 20 GeV, Nucl. Phys. 79, 609 (1966).
- J. Engler et al., Neutron-nucleus total cross-sections between and , Phys. Lett. B 32, 716 (1970).
- L. W. Jones, M. J. Longo, T. P. McCorriston, E. F. Parker, S. T. Powell, and M. N. Kreisler, Neutron total cross-sections on protons and nuclei in the 10 to momentum range, Phys. Lett. 36B, 509 (1971).
- S. P. Denisov, S. V. Donskov, Yu. P. Gorin, R. N. Krasnokutsky, A. I. Petrukhin, Yu. D. Prokoshkin, and D. A. Stoyanova, Absorption cross-sections for pions, kaons, protons and anti-protons on complex nuclei in the 6 to momentum range, Nucl. Phys. B61, 62 (1973).
- A. Babaev et al., Total cross-section measurement of neutrons on protons and nuclei over the energy range 28–54 GeV, Phys. Lett. 51B, 501 (1974).
- A. S. Clough et al., Pion-nucleus total cross-sections from 88 to 860 MeV, Nucl. Phys. B76, 15 (1974).
- P. V. R. Murthy, C. A. Ayre, H. R. Gustafson, L. W. Jones, and M. J. Longo, Neutron total cross sections on nuclei at Fermilab energies, Nucl. Phys. B92, 269 (1975).
- A. S. Carroll et al., Total cross-Sections of , , and on protons and deuterons between and , Phys. Lett. 80B, 319 (1979).
- U. Dersch et al. (SELEX Collaboration), Total cross-section measurements with , and protons on nuclei around , Nucl. Phys. B579, 277 (2000).
- A. Aduszkiewicz et al. (NA61/SHINE Collaboration), Measurements of total production cross sections for , , , and at and and at , Phys. Rev. D 98, 052001 (2018).
- A. Aduszkiewicz et al. (NA61/SHINE Collaboration), Measurements of production and inelastic cross sections for , , and at and and at , Phys. Rev. D 100, 112001 (2019).
- A. Acharya et al. (NA61/SHINE Collaboration), Measurement of the production cross section of protons on carbon via beam attenuation in a 90-cm-long target, Phys. Rev. D 103, 012006 (2021).
- C. Alt et al. (NA49 Collaboration), Inclusive production of charged pions in collisions at beam momentum, Eur. Phys. J. C 45, 343 (2006).
- T. Anticic et al. (NA49 Collaboration), Inclusive production of protons, anti-protons and neutrons in collisions at beam momentum, Eur. Phys. J. C 65, 9 (2010).
- T. Anticic et al. (NA49 Collaboration), Inclusive production of charged kaons in collisions at beam momentum and a new evaluation of the energy dependence of kaon production up to collider energies, Eur. Phys. J. C 68, 1 (2010).
- C. Alt et al. (NA49 Collaboration), Inclusive production of charged pions in collisions at beam momentum, Eur. Phys. J. C 49, 897 (2007).
- B. Baatar et al. (NA49 Collaboration), Inclusive production of protons, anti-protons, neutrons, deuterons and tritons in collisions at beam momentum, Eur. Phys. J. C 73, 2364 (2013).
- H. Adhikary et al. (NA61/SHINE Collaboration), Evidence of isospin-symmetry violation in high-energy collisions of atomic nuclei, Nat. Commun. 16, 2849 (2025).
- S. Ostapchenko and G. Sigl, On the model uncertainties for the predicted muon content of extensive air showers, Astropart. Phys. 163, 103004 (2024).
- H. Adhikary et al. (NA61/SHINE Collaboration), Measurement of hadron production in -C interactions at 158 and with NA61/SHINE at the CERN SPS, Phys. Rev. D 107, 062004 (2023).
- A. Aduszkiewicz et al. (NA61/SHINE Collaboration), Measurement of meson resonance production in interactions at SPS energies, Eur. Phys. J. C 77, 626 (2017).
- J. Whitmore, S. J. Barish, D. C. Colley, and P. F. Schultz, Invariant cross-section for the inclusive reaction at , Phys. Rev. D 11, 3124 (1975).
- I. V. Ajinenko et al. (EHS/NA22 Collaboration), Strange and nonstrange baryon production in and interactions at , Z. Phys. C 44, 573 (1989).
- F. Oljemark, Single diffraction in proton-proton scattering with TOTEM at the Large Hadron Collider., Ph.D. thesis, University of Helsinki, 2020.
- A. Aduszkiewicz et al. (NA61/SHINE Collaboration), Measurements of , , and spectra in proton-proton interactions at 20, 31, 40, 80 and with the NA61/SHINE spectrometer at the CERN SPS, Eur. Phys. J. C 77, 671 (2017).
- A. Aduszkiewicz et al. (NA61/SHINE Collaboration), Measurements of hadron production in and interactions at , Phys. Rev. D 100, 112004 (2019).
- M. Apollonio et al. (HARP Collaboration), Forward production of charged pions with incident protons on nuclear targets at the CERN PS, Phys. Rev. C 80, 035208 (2009).
- M. Apollonio et al. (HARP Collaboration), Forward production of charged pions with incident on nuclear targets measured at the CERN PS, Nucl. Phys. A821, 118 (2009).
- G. Aad et al. (ATLAS Collaboration), Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC, New J. Phys. 13, 053033 (2011).
- G. Aad et al. (ATLAS Collaboration), Charged-particle distributions in interactions measured with the ATLAS detector at the LHC, Phys. Lett. B 758, 67 (2016).
- 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 inclusive forward neutron production cross section in proton-proton collisions at with the LHCf Arm2 detector, J. High Energy Phys. 11 (2018) 073.
- 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).
- T. Pierog and K. Werner, EPOS LHC-R: A global approach to solve the muon puzzle, Proc. Sci., ICRC2025 (2025) 358.
- H. Fesefeldt, The simulation of hadronic showers: Physics and applications, Report No. PITHA-85/02, Rheinisch-Westfälische Technische Hochschule Aachen, 1985.
- T. Pierog and K. Werner, Muon production in extended air shower simulations, Phys. Rev. Lett. 101, 171101 (2008).
- G. Battistoni, F. Cerutti, A. Fassò, A. Ferrari, S. Muraro, J. Ranft, S. Roesler, and P. R. Sala, The FLUKA code: Description and benchmarking, AIP Conf. Proc. 896, 31 (2007).
- M. Bleicher et al., Relativistic hadron hadron collisions in the ultrarelativistic quantum molecular dynamics model, J. Phys. G 25, 1859 (1999).
- S. S. Ostapchenko, Contemporary models of high-energy interactions: Present status and perspectives, J. Phys. G 29, 831 (2003).
- R. D. Parsons, C. Bleve, S. S. Ostapchenko, and J. Knapp, Systematic uncertainties in air shower measurements from high-energy hadronic interaction models, Astropart. Phys. 34, 832 (2011).
- S. Ostapchenko, M. Bleicher, T. Pierog, and K. Werner, Constraining high energy interaction mechanisms by studying forward hadron production at the LHC, Phys. Rev. D 94, 114026 (2016).
- S. Ostapchenko, LHC data on inelastic diffraction and uncertainties in the predictions for longitudinal extensive air shower development, Phys. Rev. D 89, 074009 (2014).
- S. Ostapchenko and M. Bleicher, Constraining pion interactions at very high energies by cosmic ray data, Phys. Rev. D 93, 051501(R) (2016).
- H. J. Drescher, A. Dumitru, and M. Strikman, High-density QCD and cosmic ray air showers, Phys. Rev. Lett. 94, 231801 (2005).
- J. Albrecht et al., Global tuning of hadronic interaction models with accelerator-based and astroparticle data, Nat. Rev. Phys. 8, 98 (2026).
- S. R. Kelner, F. A. Aharonian, and V. V. Bugayov, Energy spectra of gamma-rays, electrons and neutrinos produced at proton-proton interactions in the very high energy regime, Phys. Rev. D 74, 034018 (2006).
- T. Kamae, N. Karlsson, T. Mizuno, T. Abe, and T. Koi, Parameterization of , and neutrino spectra produced by interaction in astronomical environment, Astrophys. J. 647, 692 (2006).
- M. Korsmeier, F. Donato, and M. Di Mauro, Production cross sections of cosmic antiprotons in the light of new data from the NA61 and LHCb experiments, Phys. Rev. D 97, 103019 (2018).
- M. Kachelrieß, I. V. Moskalenko, and S. Ostapchenko, AAfrag: Interpolation routines for Monte Carlo results on secondary production in proton-proton, proton-nucleus and nucleus-nucleus interactions, Comput. Phys. Commun. 245, 106846 (2019).
- L. Orusa, M. Di Mauro, F. Donato, and M. Korsmeier, New determination of the production cross section for secondary positrons and electrons in the Galaxy, Phys. Rev. D 105, 123021 (2022).
- L. Orusa, M. Di Mauro, F. Donato, and M. Korsmeier, New determination of the production cross section for rays in the Galaxy, Phys. Rev. D 107, 083031 (2023).
- L. Orusa, M. Di Mauro, and F. Donato, New determination of the neutrino hadronic production cross sections from GeV to beyond PeV energies, Phys. Rev. D 113, 2 (2026).