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Efficient atmospheric, solar, and supernova neutrino propagation through the Earth

Peter B. Denton1,* and Stephen J. Parke2,†

  • *Contact author: pdenton@bnl.gov
  • †Contact author: parke@fnal.gov
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Phys. Rev. D 113, 053002 – Published 10 March, 2026

DOI: https://doi.org/10.1103/jgw1-lnrl

Abstract

Algorithms for computing neutrino oscillation probabilities in sharply varying matter potentials such as the Earth are becoming increasingly important. As the next generation of experiments, DUNE and HyperK as well as the IceCube upgrade and KM3NeT, come online, the computational cost for atmospheric and solar neutrinos will continue to increase. To address these issues, we expand upon our previous algorithm for long-baseline calculations to efficiently handle probabilities through the Earth for atmospheric, nighttime solar, and supernova neutrinos. The algorithm is fast, flexible, and accurate. It can handle arbitrary Earth models with two different schemes for varying density profiles. We also provide a c++ implementation of the code called nufast-earth along with a detailed user manual. The code intelligently keeps track of repeated calculations and only recalculates what is needed on each successive call which can also help provide significant speed-ups.

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References (107)

  1. L. Wolfenstein, Neutrino oscillations in matter, Phys. Rev. D 17, 2369 (1978).
  2. P. B. Denton, Neutrino oscillations in the three flavor paradigm, arXiv:2501.08374.
  3. K. J. Kelly, P. A. Machado, I. Martinez Soler, S. J. Parke, and Y. F. Perez Gonzalez, Sub-GeV atmospheric neutrinos and CP-violation in DUNE, Phys. Rev. Lett. 123, 081801 (2019).
  4. I. Martinez-Soler and H. Minakata, Perturbing neutrino oscillations around the solar resonance, Prog. Theor. Exp. Phys. 2019, 073B07 (2019).
  5. C. A. Argüelles, P. Fernández, I. Martínez-Soler, and M. Jin, Measuring oscillations with a Million atmospheric neutrinos, Phys. Rev. X 13, 041055 (2023).
  6. A. M. Suliga and J. F. Beacom, Distinctive nuclear signatures of low-energy atmospheric neutrinos, Phys. Rev. D 108, 043035 (2023).
  7. A. Abed Abud et al. (DUNE Collaboration), DUNE phase II: Scientific opportunities, detector concepts, technological solutions, J. Instrum. 19, P12005 (2024).
  8. K. Abe et al. (Hyper-Kamiokande Collaboration), Hyper-Kamiokande design report, arXiv:1805.04163.
  9. F. Capozzi, S. W. Li, G. Zhu, and J. F. Beacom, DUNE as the next-generation solar neutrino experiment, Phys. Rev. Lett. 123, 131803 (2019).
  10. A. J. Baltz and J. Weneser, Effect of transmission through the earth on neutrino oscillations, Phys. Rev. D 35, 528 (1987).
  11. J. Bouchez, M. Cribier, J. Rich, M. Spiro, D. Vignaud, and W. Hampel, Matter effects for solar neutrino oscillations, Z. Phys. C 32, 499 (1986).
  12. E. D. Carlson, Terrestrially enhanced neutrino oscillations, Phys. Rev. D 34, 1454 (1986).
  13. M. Cribier, W. Hampel, J. Rich, and D. Vignaud, MSW regeneration of solar νe in the earth, Phys. Lett. B 182, 89 (1986).
  14. K. Abe et al. (Super-Kamiokande Collaboration), Solar neutrino measurements using the full data period of Super-Kamiokande-IV, Phys. Rev. D 109, 092001 (2024).
  15. G. J. Feldman and R. D. Cousins, A unified approach to the classical statistical analysis of small signals, Phys. Rev. D 57, 3873 (1998).
  16. M. Teckenbrock, Fermilab computing experts bolster NOvA evidence, 1 million cores consumed https://news.fnal.gov/2018/07/fermilab-computing-experts-bolster-nova-evidence-1-million-cores-consumed/.
  17. NOvA Collaboration (private communication).
  18. P. B. Denton and S. J. Parke, Fast and accurate algorithm for calculating long-baseline neutrino oscillation probabilities with matter effects, Phys. Rev. D 110, 073005 (2024).
  19. P. B. Denton and S. J. Parke, nufast-lbl: A fast code for long-baseline neutrino oscillation probabilities in matter, https://github.com/PeterDenton/NuFast-LBL (2024).
  20. V. D. Barger, K. Whisnant, S. Pakvasa, and R. J. N. Phillips, Matter effects on three-neutrino oscillations, Phys. Rev. D 22, 2718 (1980).
  21. H. W. Zaglauer and K. H. Schwarzer, The mixing angles in matter for three generations of neutrinos and the MSW mechanism, Z. Phys. C 40, 273 (1988).
  22. G. Cardano, Ars Magna (1545).
  23. C. Jarlskog, Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation, Phys. Rev. Lett. 55, 1039 (1985).
  24. S. M. Bilenky and S. T. Petcov, Massive neutrinos and neutrino oscillations, Rev. Mod. Phys. 59, 671 (1987); 61, 169(E) (1989); 60, 575(E) (1988).
  25. V. A. Naumov, Three neutrino oscillations in matter, CP violation and topological phases, Int. J. Mod. Phys. D 1, 379 (1992).
  26. P. F. Harrison and W. G. Scott, Neutrino matter effect invariants and the observables of neutrino oscillations, Phys. Lett. B 535, 229 (2002).
  27. P. B. Denton and S. J. Parke, Simple and precise factorization of the Jarlskog invariant for neutrino oscillations in matter, Phys. Rev. D 100, 053004 (2019).
  28. S. J. Parke, Interplay between the factorization of the Jarlskog invariant and location of the solar and atmospheric resonances for neutrino oscillations in matter, Phys. Rev. D 103, 033003 (2021).
  29. P. B. Denton, S. J. Parke, and X. Zhang, Rotations versus perturbative expansions for calculating neutrino oscillation probabilities in matter, Phys. Rev. D 98, 033001 (2018).
  30. P. B. Denton, S. J. Parke, and X. Zhang, Neutrino oscillations in matter via eigenvalues, Phys. Rev. D 101, 093001 (2020).
  31. P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra, Bull. Am. Math. Soc. 59, 31 (2022).
  32. P. B. Denton, S. J. Parke, and X. Zhang, Fibonacci fast convergence for neutrino oscillations in matter, Phys. Lett. B 807, 135592 (2020).
  33. A. M. Abdullahi and S. J. Parke, Neutrino oscillations in matter using the adjugate of the Hamiltonian, Eur. Phys. J. C 84, 707 (2024).
  34. J. Kopp, Efficient numerical diagonalization of Hermitian 3×3 matrices, Int. J. Mod. Phys. C 19, 523 (2008).
  35. G. Barenboim, P. B. Denton, S. J. Parke, and C. A. Ternes, Neutrino oscillation probabilities through the looking glass, Phys. Lett. B 791, 351 (2019).
  36. S. K. Agarwalla, Y. Kao, and T. Takeuchi, Analytical approximation of the neutrino oscillation matter effects at large θ13, J. High Energy Phys. 04 (2014) 047.
  37. H. Minakata and S. J. Parke, Simple and compact expressions for neutrino oscillation probabilities in matter, J. High Energy Phys. 01 (2016) 180.
  38. P. B. Denton, H. Minakata, and S. J. Parke, Compact perturbative expressions for neutrino oscillations in matter, J. High Energy Phys. 06 (2016) 051.
  39. P. B. Denton and S. J. Parke, Addendum to Compact perturbative expressions for neutrino oscillations in matter, J. High Energy Phys. 06 (2018) 109(A).
  40. T. Ohlsson and H. Snellman, Three flavor neutrino oscillations in matter, J. Math. Phys. (N.Y.) 41, 2768 (2000); 42, 2345(E) (2001).
  41. P. Huber, M. Lindner, and W. Winter, Simulation of long-baseline neutrino oscillation experiments with GLoBES (General Long Baseline Experiment Simulator), Comput. Phys. Commun. 167, 195 (2005).
  42. H. Nunokawa, S. J. Parke, and R. Zukanovich Funchal, Another possible way to determine the neutrino mass hierarchy, Phys. Rev. D 72, 013009 (2005).
  43. P. Huber, J. Kopp, M. Lindner, M. Rolinec, and W. Winter, New features in the simulation of neutrino oscillation experiments with GLoBES 3.0: General long baseline experiment simulator, Comput. Phys. Commun. 177, 432 (2007).
  44. S. Parke, What is Δmee2 ?, Phys. Rev. D 93, 053008 (2016).
  45. P. B. Denton and S. J. Parke, The effective Δmee2 in matter, Phys. Rev. D 98, 093001 (2018).
  46. X. Wang and S. Zhou, Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter, J. High Energy Phys. 05 (2019) 035.
  47. C. A. Argüelles, J. Salvado, and C. N. Weaver, nusquids: A toolbox for neutrino propagation, Comput. Phys. Commun. 277, 108346 (2022).
  48. M. Maltoni, From ray to spray: Augmenting amplitudes and taming fast oscillations in fully numerical neutrino codes, J. High Energy Phys. 11 (2023) 033.
  49. S. Luo and Z.-Z. Xing, A Pythagoras-like theorem for CP violation in neutrino oscillations, Phys. Lett. B 845, 138142 (2023).
  50. J. Page, Fast exact algorithm for neutrino oscillation in constant matter density, Comput. Phys. Commun. 300, 109200 (2024).
  51. T. E. Gonzalo and M. Lucente, PEANUTS: A software for the automatic computation of solar neutrino flux and its propagation within Earth, Eur. Phys. J. C 84, 119 (2024).
  52. P. B. Denton and S. J. Parke, Smallness of matter effects in long-baseline muon neutrino disappearance, Phys. Rev. D 109, 053002 (2024).
  53. P. B. Denton and S. J. Parke, nufast-earth: A fast code for neutrino oscillation probabilities in matter through the Earth, https://github.com/PeterDenton/NuFast-Earth (2025).
  54. H. TkalČić and B. L. N. Kennett, Core structure and heterogeneity: A seismological perspective, Australian J. Earth Sci. 55, 419 (2008).
  55. A. Morelli, A. M. Dziewonski, and J. H. Woodhouse, Anisotropy of the inner core inferred from PKIKP travel times, Geophys. Res. Lett. 13, 1545 (1986).
  56. L. Vinnik, B. Romanowicz, and L. Breger, Anisotropy in the center of the inner core, Geophys. Res. Lett. 21, 1671 (1994).
  57. S. Tanaka, Possibility of a low p-wave velocity layer in the outermost core from global SMKS waveforms, Earth Planet. Sci. Lett. 259, 486 (2007).
  58. X. Song and D. V. Helmberger, Seismic evidence for an inner core transition zone, Science 282, 924 (1998).
  59. S.-i. Karato, A. Forte, R. Liebermann, G. Masters, and L. Stixrude, Earth’s deep interior: mineral physics and tomography from the atomic to the global scale, Washington DC American Geophysical Union Geophysical Monograph Series 117 (2000).
  60. M. Ishii and A. M. Dziewoński, The innermost inner core of the earth: Evidence for a change in anisotropic behavior at the radius of about 300 km, Proc. Natl. Acad. Sci. U.S.A. 99, 14026 (2002).
  61. D. A. Frost, M. Lasbleis, B. Chandler, and B. Romanowicz, Dynamic history of the inner core constrained by seismic anisotropy, Nat. Geosci. 14, 531 (2021).
  62. A. Nicolaidis, M. Jannane, and A. Tarantola, Neutrino tomography of the earth, J. Geophys. Res. 96, 21811 (1991).
  63. W. Winter, Neutrino tomography—Learning about the Earth’s interior using the propagation of neutrinos, Earth Moon Planets 99, 285 (2006).
  64. E. K. Akhmedov, M. Maltoni, and A. Y. Smirnov, 1-3 leptonic mixing and the neutrino oscillograms of the Earth, J. High Energy Phys. 05 (2007) 077.
  65. S. K. Agarwalla, T. Li, O. Mena, and S. Palomares-Ruiz, Exploring the Earth matter effect with atmospheric neutrinos in ice, arXiv:1212.2238.
  66. C. Rott, A. Taketa, and D. Bose, Spectrometry of the Earth using neutrino oscillations, Sci. Rep. 5, 15225 (2015).
  67. W. Winter, Atmospheric neutrino oscillations for Earth tomography, Nucl. Phys. B908, 250 (2016).
  68. J. C. D’Olivo, J. A. Herrera Lara, I. Romero, O. A. Sampayo, and G. Zapata, Earth tomography with atmospheric neutrino oscillations, Eur. Phys. J. C 80, 1001 (2020).
  69. A. Kumar and S. K. Agarwalla, Validating the Earth’s core using atmospheric neutrinos with ICAL at INO, J. High Energy Phys. 08 (2021) 139.
  70. K. J. Kelly, P. A. N. Machado, I. Martinez-Soler, and Y. F. Perez-Gonzalez, DUNE atmospheric neutrinos: Earth tomography, J. High Energy Phys. 05 (2022) 187.
  71. P. B. Denton and R. Pestes, Neutrino oscillations through the Earth’s core, Phys. Rev. D 104, 113007 (2021).
  72. F. Capozzi and S. T. Petcov, Neutrino tomography of the Earth with ORCA detector, Eur. Phys. J. C 82, 461 (2022).
  73. J. C. D’Olivo Saez, J. A. H. Lara, I. Romero, and O. A. Sampayo, Oscillation tomografy study of Earth’s composition and density with atmospheric neutrinos, Eur. Phys. J. C 82, 614 (2022).
  74. L. Maderer, E. Kaminski, J. A. B. Coelho, S. Bourret, and V. Van Elewyck, Unveiling the outer core composition with neutrino oscillation tomography, Front. Earth Sci. 11, 1008396 (2023).
  75. A. K. Upadhyay, A. Kumar, S. K. Agarwalla, and A. Dighe, Locating the core-mantle boundary using oscillations of atmospheric neutrinos, J. High Energy Phys. 04 (2023) 068.
  76. D. Raikwal and S. Choubey, Earth tomography with the ICAL detector at INO, Phys. Rev. D 109, 073011 (2024).
  77. A. K. Upadhyay, A. Kumar, S. K. Agarwalla, and A. Dighe, Constraining the core radius and density jumps inside Earth using atmospheric neutrino oscillations, arXiv:2405.04986.
  78. S. T. Petcov, Neutrino tomography of the earth: The earth total mass, moment of inertia and hydrostatic equilibrium constraints, Eur. Phys. J. C 84, 991 (2024).
  79. C. Jesús-Valls, S. T. Petcov, and J. Xia, Neutrino oscillation tomography of the Earth with the Hyper-Kamiokande detector, Eur. Phys. J. C 85, 703 (2025).
  80. S. Chattopadhyay, K. J, and A. K. Upadhyay (IceCube Collaboration), Demonstrating the ability of IceCube DeepCore to probe Earth’s interior with atmospheric neutrino oscillations, Eur. Phys. J. Special Topics 234, 5055 (2025).
  81. K. J, A. K. Upadhyay, A. Kumar, and S. K. Agarwalla, Probing Earth’s core using atmospheric neutrino oscillations in the presence of NSI at INO-ICAL, J. High Energy Phys. 10 (2025) 213.
  82. S. Chattopadhyay, J. Krishnamoorthi, and A. K. Upadhyay (IceCube Collaboration), Demonstrating the ability of IceCube DeepCore to probe Earth’s interior with atmospheric neutrino oscillations, Eur. Phys. J. Special Topics 234, 5055 (2025).
  83. E. K. Akhmedov, M. A. Tortola, and J. W. F. Valle, Geotomography with solar and supernova neutrinos, J. High Energy Phys. 06 (2006) 053.
  84. P. Bakhti and A. Y. Smirnov, Oscillation tomography of the Earth with solar neutrinos and future experiments, Phys. Rev. D 101, 123031 (2020).
  85. M. Lindner, T. Ohlsson, R. Tomas, and W. Winter, Tomography of the earth’s core using supernova neutrinos, Astropart. Phys. 19, 755 (2003).
  86. R. Hajjar, O. Mena, and S. Palomares-Ruiz, Earth tomography with supernova neutrinos at future neutrino detectors, Phys. Rev. D 108, 083011 (2023).
  87. A. M. Dziewonski and D. L. Anderson, Preliminary reference earth model, Phys. Earth Planet. Inter. 25, 297 (1981).
  88. I. Mocioiu and R. Shrock, Matter effects on neutrino oscillations in long baseline experiments, Phys. Rev. D 62, 053017 (2000).
  89. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  90. P. B. Denton and R. Pestes, The impact of different parameterizations on the interpretation of CP violation in neutrino oscillations, J. High Energy Phys. 05 (2021) 139.
  91. H. Yokomakura, K. Kimura, and A. Takamura, Overall feature of CP dependence for neutrino oscillation probability in arbitrary matter profile, Phys. Lett. B 544, 286 (2002).
  92. M. Blennow and A. Y. Smirnov, Neutrino propagation in matter, Adv. High Energy Phys. 2013, 972485 (2013).
  93. S.-F. Ge, K. Hagiwara, and C. Rott, A novel approach to study atmospheric neutrino oscillation, J. High Energy Phys. 06 (2014) 150.
  94. S. P. Mikheyev and A. Y. Smirnov, Resonance amplification of oscillations in matter and spectroscopy of solar neutrinos, Sov. J. Nucl. Phys. 42, 913 (1985).
  95. P. O. Lagage, M. Cribier, J. Rich, and D. Vignaud, MSW effect and (anti)-neutrinos from SN1987A, Phys. Lett. B 193, 127 (1987).
  96. J. Arafune, M. Fukugita, T. Yanagida, and M. Yoshimura, Neutrino burst From SN1987A and the solar neutrino puzzle, Phys. Rev. Lett. 59, 1864 (1987).
  97. D. Notzold, MSW effect analysis for SN1987A sets severe restrictions on neutrino masses and mixing angles, Phys. Lett. B 196, 315 (1987).
  98. H. Minakata, H. Nunokawa, K. Shiraishi, and H. Suzuki, Neutrinos from supernova explosion and the Mikheev-Smirnov-Wolfenstein effect, Mod. Phys. Lett. A 02, 827 (1987).
  99. I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pinheiro, and T. Schwetz, NuFit-6.0: Updated global analysis of three-flavor neutrino oscillations, J. High Energy Phys. 12 (2024) 216.
  100. S. T. Petcov, Diffractive—like (or parametric resonance—like?) enhancement of the earth (day—night) effect for solar neutrinos crossing the earth core, Phys. Lett. B 434, 321 (1998).
  101. E. K. Akhmedov, A. Dighe, P. Lipari, and A. Y. Smirnov, Atmospheric neutrinos at Super-Kamiokande and parametric resonance in neutrino oscillations, Nucl. Phys. B542, 3 (1999).
  102. E. Lisi and D. Montanino, Earth regeneration effect in solar neutrino oscillations: An analytic approach, Phys. Rev. D 56, 1792 (1997).
  103. E. K. Akhmedov, M. A. Tortola, and J. W. F. Valle, A simple analytic three flavor description of the day night effect in the solar neutrino flux, J. High Energy Phys. 05 (2004) 057.
  104. J. Coelho, R. Pestes, A. Domi, lvmcerisy, D. Stocco, MatteoLouppp, S. Bourret, Ushakrhmn, lmaderer, and V. Carretero, joaoabcoelho/oscprob: v2.2.0, https://github.com/joaoabcoelho/OscProb (2025).
  105. S. J. Parke, Nonadiabatic level crossing in resonant neutrino oscillations, Phys. Rev. Lett. 57, 1275 (1986).
  106. G. L. Fogli, E. Lisi, and D. Montanino, Matter enhanced three flavor oscillations and the solar neutrino problem, Phys. Rev. D 54, 2048 (1996).
  107. A. N. Khan, H. Nunokawa, and S. J. Parke, Why matter effects matter for JUNO, Phys. Lett. B 803, 135354 (2020).

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Phys. Rev. D 113, 053002– Published 10 March, 2026
  • Received 17 November 2025
  • Accepted 23 February 2026
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