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Angular distribution of gamma rays produced in proton-proton collisions

Spencer Griffith1,2,*, John F. Beacom1,2,3, Jung-Tsung Li (李融宗)1,2,3, and Annika H. G. Peter1,2,3

  • *Contact author: griffith.1037@osu.edu
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Phys. Rev. D 112, 043009 – Published 8 August, 2025

DOI: https://doi.org/10.1103/yyvx-tnjl

Abstract

Accurate modeling of how high-energy proton-proton collisions produce gamma rays through the decays of pions and other secondaries is needed to correctly interpret astrophysical observations with the Fermi-LAT telescope. In the existing literature on cosmic-ray collisions with gas, the focus is on the gamma-ray yield spectrum, dNγ/dE. However, in some situations, the joint energy and angular distribution can be observed, so one needs instead d2Nγ/dEdΩ. We provide calculations of this distribution over the energy range from the pion production threshold to 100 GeV, basing our results on fluka simulations. We provide the results in tabular form and provide a python tool on github to aid in utilization. We also provide an approximate analytic formula that illuminates the underlying physics. We discuss simplified examples where this angular dependence can be observed to illustrate the necessity of taking the joint distribution into account.

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References (87)

  1. F. Stecker, Cosmic Gamma Rays, NASA SP (Scientific and Technical Information Office, National Aeronautics and Space Administration, 1971).
  2. F. Aharonian, Very High Energy Cosmic Gamma Radiation: A Crucial Window on the Extreme Universe, EBSCO ebook academic collection (World Scientific, Singapore, 2004).
  3. C. Dermer and G. Menon, High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos, Princeton Series in Astrophysics (Princeton University Press, Princeton, NJ, 2009).
  4. F. Aharonian et al. (H. E. S. S. Collaboration), Astron. Astrophys. 449, 223 (2006).
  5. C.-Y. Huang, S.-E. Park, M. Pohl, and C. Daniels, Astropart. Phys. 27, 429 (2007).
  6. M. Ackermann et al. (Fermi-LAT Collaboration), Science 339, 807 (2013).
  7. H. Abdalla et al. (HESS Collaboration), Astron. Astrophys. 612, A6 (2018).
  8. A. U. Abeysekara et al., Nat. Astron. 5, 465 (2021); 5, 724(E) (2021).
  9. V. A. Acciari et al. (MAGIC Collaboration), Nat. Astron. 6, 689 (2022); 6, 760(E) (2022).
  10. T. Sudoh and J. F. Beacom, Phys. Rev. D 107, 043002 (2023).
  11. T. Sudoh and J. F. Beacom, Phys. Rev. D 108, 043016 (2023).
  12. B.-H. Song, T.-Q. Huang, and K. Wang, Astrophys. J. 961, 254 (2024).
  13. K. Fang and F. Halzen, J. High Energy Astrophys. 43, 140 (2024).
  14. M. Aguilar et al. (AMS Collaboration), Phys. Rev. Lett. 114, 171103 (2015).
  15. M. Aguilar et al. (AMS Collaboration), Phys. Rev. Lett. 115, 211101 (2015).
  16. D. Maurin, M. Ahlers, H. Dembinski, A. Haungs, P.-S. Mangeard, F. Melot, P. Mertsch, D. Wochele, and J. Wochele, Eur. Phys. J. C 83, 971 (2023).
  17. J.-T. Li, J. F. Beacom, and A. H. G. Peter, Astrophys. J. 937, 27 (2022).
  18. D. Seckel, T. Stanev, and T. K. Gaisser, Astrophys. J. 382, 652 (1991).
  19. E. Orlando and A. W. Strong, Astron. Astrophys. 480, 847 (2008).
  20. B. Zhou, K. C. Y. Ng, J. F. Beacom, and A. H. G. Peter, Phys. Rev. D 96, 023015 (2017).
  21. H. S. Hudson, A. MacKinnon, M. Szydlarski, and M. Carlsson, Mon. Not. R. Astron. Soc. 491, 4852 (2019).
  22. Z. Li, K. C. Y. Ng, S. Chen, Y. Nan, and H. He, Chin. Phys. C 48, 045101 (2024).
  23. M. Gutiérrez and M. Masip, Astropart. Phys. 119, 102440 (2020).
  24. M. Gutiérrez, M. Masip, and S. Muñoz, Astrophys. J. 941, 86 (2022).
  25. M. N. Mazziotta, P. De La Torre Luque, L. Di Venere, A. Fassò, A. Ferrari, F. Loparco, P. R. Sala, and D. Serini, Phys. Rev. D 101, 083011 (2020).
  26. J.-T. Li, J. F. Beacom, S. Griffith, and A. H. G. Peter, Astrophys. J. 961, 167 (2024).
  27. E. Puzzoni, F. Fraschetti, J. Kóta, and J. Giacalone, Astrophys. J. 973, 118 (2024).
  28. A. A. Abdo et al. (Fermi-LAT Collaboration), Astrophys. J. 734, 116 (2011).
  29. T. Linden, B. Zhou, J. F. Beacom, A. H. G. Peter, K. C. Y. Ng, and Q.-W. Tang, Phys. Rev. Lett. 121, 131103 (2018).
  30. Q.-W. Tang, K. C. Y. Ng, T. Linden, B. Zhou, J. F. Beacom, and A. H. G. Peter, Phys. Rev. D 98, 063019 (2018).
  31. T. Linden, J. F. Beacom, A. H. G. Peter, B. J. Buckman, B. Zhou, and G. Zhu, Phys. Rev. D 105, 063013 (2022).
  32. B. Arsioli and E. Orlando, Astrophys. J. 962, 52 (2024).
  33. T. Linden, J.-T. Li, B. Zhou, I. John, M. Crnogorčević, A. H. G. Peter, and J. F. Beacom, arXiv:2505.04625.
  34. A. Acharyya et al., arXiv:2505.06348.
  35. G. D. Badhwar, S. A. Stephens, and R. L. Golden, Phys. Rev. D 15, 820 (1977).
  36. K. Hanssgen and J. Ranft, Nucl. Sci. Eng. 88, 537 (1984).
  37. J. Ranft and S. Ritter, Z. Phys. C 20, 347 (1983).
  38. K. Hanssgen, H. J. Mohring, and J. Ranft, Nucl. Sci. Eng. 88, 551 (1984).
  39. H. J. Mohring and J. Ranft, Nucl. Sci. Eng. 89, 247 (1985).
  40. C. D. Dermer, Astron. Astrophys. 157, 223 (1986).
  41. K. Hanssgen, Nucl. Sci. Eng. 95, 135 (1987).
  42. A. Capella, U. Sukhatme, C.-I. Tan, and J. Tran Thanh Van, Phys. Rep. 236, 225 (1994).
  43. A. B. Kaidalov, Phys. At. Nucl. 66, 1994 (2003).
  44. T. Kamae, N. Karlsson, T. Mizuno, T. Abe, and T. Koi, Astrophys. J. 647, 692 (2006); 662, 779(E) (2007).
  45. N. Karlsson and T. Kamae, Astrophys. J. 674, 278 (2008).
  46. S. R. Kelner, F. A. Aharonian, and V. V. Bugayov, Phys. Rev. D 74, 034018 (2006); 79, 039901(E) (2009).
  47. E. Kafexhiu, F. Aharonian, A. M. Taylor, and G. S. Vila, Phys. Rev. D 90, 123014 (2014).
  48. L. Orusa, M. Di Mauro, F. Donato, and M. Korsmeier, Phys. Rev. D 107, 083031 (2023).
  49. H. B. J. Koers, A. Pe’er, and R. A. M. J. Wijers, arXiv:hep-ph/0611219.
  50. I. V. Moskalenko and T. A. Porter, Astrophys. J. 670, 1467 (2007).
  51. A. A. Abdo et al., Astrophys. J. 758, 140 (2012).
  52. M. Ackermann et al. (Fermi-LAT Collaboration), Phys. Rev. D 93, 082001 (2016).
  53. A. Ferrari, P. R. Sala, A. Fasso, and J. Ranft (2005), 10.2172/877507.
  54. T. T. Böhlen, F. Cerutti, M. P. W. Chin, A. Fassò, A. Ferrari, P. G. Ortega, A. Mairani, P. R. Sala, G. Smirnov, and V. Vlachoudis, Nucl. Data Sheets 120, 211 (2014).
  55. G. Battistoni et al., Ann. Nucl. Energy 82, 10 (2015).
  56. C. Ahdida et al., Front. Phys. 9, 788253 (2022).
  57. M. Kachelriess, I. V. Moskalenko, and S. S. Ostapchenko, Astrophys. J. 789, 136 (2014).
  58. M. N. Mazziotta, F. Cerutti, A. Ferrari, D. Gaggero, F. Loparco, and P. R. Sala, Astropart. Phys. 81, 21 (2016).
  59. F. Ballarini et al., EPJ Nucl. Sci. Technol. 10, 63 (2024).
  60. W. B. Atwood et al. (Fermi-LAT Collaboration), Astrophys. J. 697, 1071 (2009).
  61. C. D. Dermer, Astrophys. J. 307, 47 (1986).
  62. R. L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  63. W. Lock and D. Measday, Intermediate Energy Nuclear Physics (Methuen, Malton, North Yorkshire, 1970).
  64. J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
  65. T. K. Gaisser, R. Engel, and E. Resconi, Cosmic Rays and Particle Physics: 2nd Edition (Cambridge University Press, Cambridge, England, 2016).
  66. S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024).
  67. F. A. Aharonian and A. M. Atoyan, Astron. Astrophys. 309, 917 (1996).
  68. M. Basile et al., Nuovo Cimento A 66, 129 (1981).
  69. C. Bierlich et al., SciPost Phys. Codebases 2022, 8 (2022).
  70. E.-J. Ahn, R. Engel, T. K. Gaisser, P. Lipari, and T. Stanev, Phys. Rev. D 80, 094003 (2009).
  71. S. Ostapchenko, Nucl. Phys. B, Proc. Suppl. 151, 143 (2006).
  72. S. Agostinelli et al., Nucl. Instrum. Methods Phys. Res., Sect. A 506, 250 (2003).
  73. H. S. Fesefeldt, The simulation of hadronic showers: Physics and applications, Technical Report No. PITHA-85-02, Aachen TU 3, Inst. Phys., Aachen, 1985.
  74. M. Bleicher et al., J. Phys. G 25, 1859 (1999).
  75. D. Heck, Nucl. Phys. B, Proc. Suppl. 151, 127 (2006).
  76. T. Kamae, T. Abe, and T. Koi, Astrophys. J. 620, 244 (2005).
  77. R. P. Feynman, Phys. Rev. Lett. 23, 1415 (1969).
  78. W. Jiang et al., Chin. Phys. Lett. 37, 119601 (2020).
  79. A. Bolshakova et al. (HARP-CDP Collaboration), Eur. Phys. J. C 70, 543 (2010).
  80. V. Uzhinsky, arXiv:1109.6768.
  81. J. M. Paley et al. (MIPP Collaboration), Phys. Rev. D 90, 032001 (2014).
  82. O. Adriani et al. (LHCf Collaboration), Phys. Rev. D 94, 032007 (2016).
  83. S. Griffith, Angular distribution of gamma rays produced in proton-proton collisions (2024) https://github.com/skgriffith/Gamma-Ray-Angular-Distribution.
  84. M. Ackermann et al. (Fermi-LAT Collaboration), Astrophys. J. 787, 15 (2014).
  85. M. Pesce-Rollins, N. Omodei, V. Petrosian, W. Liu, F. R. da Costa, A. Allafort, and Q. Chen, Astrophys. J. Lett. 805, L15 (2015).
  86. A. Kouloumvakos, A. P. Rouillard, G. H. Share, I. Plotnikov, R. Murphy, A. Papaioannou, and Y. Wu, Astrophys. J. 893, 76 (2020).
  87. A. Para and M. Szleper, arXiv:hep-ex/0110032.

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Phys. Rev. D 112, 043009– Published 8 August, 2025
  • Received 20 December 2024
  • Accepted 28 July 2025
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