- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Ubiquitous flavor pendulum

Damiano F. G. Fiorillo1,2 and Georg G. Raffelt3

PDF

Phys. Rev. D 113, 123033 – Published 12 June, 2026

DOI: https://doi.org/10.1103/tkx8-jpj6

Abstract

A system of classical interacting spins can develop collective instabilities, which, in the nonlinear regime, mimic the motion of a gyroscopic pendulum. Known as the flavor pendulum, this behavior appears among the collective modes of a dense neutrino plasma after a strong reduction of phase space through symmetry assumptions. It has been identified in homogeneous slow and fast flavor systems and, most recently, in single-wave solutions of the fast system. For the first time, we explain the reasons for its ubiquitous appearance. We show that a system of three classical spins must always be pendular, or only two in the presence of an external field. Furthermore, such a system always defines a continuum of vectors with time-independent length. If these are identified as interacting spins, they immediately lead to the continuum cases of slow and fast flavor pendula. As another new insight, any of these spins can be chosen as the pendulum, periodically exchanging flavor with the rest of the system.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (48)

  1. J. T. Pantaleone, Neutrino oscillations at high densities, Phys. Lett. B 287, 128 (1992).
  2. S. Samuel, Neutrino oscillations in dense neutrino gases, Phys. Rev. D 48, 1462 (1993).
  3. S. Samuel, Bimodal coherence in dense selfinteracting neutrino gases, Phys. Rev. D 53, 5382 (1996).
  4. H. Duan, G. M. Fuller, J. Carlson, and Y.-Z. Qian, Simulation of coherent nonlinear neutrino flavor transformation in the supernova environment: Correlated neutrino trajectories, Phys. Rev. D 74, 105014 (2006).
  5. H. Duan and J. P. Kneller, Neutrino flavour transformation in supernovae, J. Phys. G 36, 113201 (2009).
  6. H. Duan, G. M. Fuller, and Y.-Z. Qian, Collective neutrino oscillations, Annu. Rev. Nucl. Part. Sci. 60, 569 (2010).
  7. I. Tamborra and S. Shalgar, New developments in flavor evolution of a dense neutrino gas, Annu. Rev. Nucl. Part. Sci. 71, 165 (2021).
  8. M. C. Volpe, Neutrinos from dense environments: Flavor mechanisms, theoretical approaches, observations, and new directions, Rev. Mod. Phys. 96, 025004 (2024).
  9. L. Johns, S. Richers, and M.-R. Wu, Neutrino oscillations in core-collapse supernovae and neutron star mergers, Annu. Rev. Nucl. Part. Sci. 75, 399 (2025).
  10. G. G. Raffelt, H.-T. Janka, and D. F. G. Fiorillo, Neutrinos from core-collapse supernovae, arXiv:2509.16306 [to be published in the Encyclopedia of Particle Physics (2026), https://shop.elsevier.com/books/encyclopedia-of-particle-physics/fischer/978-0-443-26598-3].
  11. A. D. Dolgov, Neutrinos in the early universe, Yad. Fiz. 33, 1309 (1981) [Sov. J. Nucl. Phys. 33, 700 (1981)].
  12. M. A. Rudzsky, Kinetic equations for neutrino spin- and type-oscillations in a medium, Astrophys. Space Sci 165, 65 (1990).
  13. G. Sigl and G. Raffelt, General kinetic description of relativistic mixed neutrinos, Nucl. Phys. B406, 423 (1993).
  14. D. F. G. Fiorillo, G. G. Raffelt, and G. Sigl, Inhomogeneous kinetic equation for mixed neutrinos: Tracing the missing energy, Phys. Rev. Lett. 133, 021002 (2024).
  15. D. F. G. Fiorillo, G. G. Raffelt, and G. Sigl, Collective neutrino-antineutrino oscillations in dense neutrino environments?, Phys. Rev. D 109, 043031 (2024).
  16. D. F. G. Fiorillo and G. G. Raffelt, Theory of neutrino fast flavor evolution. Part I. Linear response theory and stability conditions, J. High Energy Phys. 08 (2024) 225.
  17. D. F. G. Fiorillo and G. G. Raffelt, Theory of neutrino fast flavor evolution. Part II. Solutions at the edge of instability, J. High Energy Phys. 12 (2024) 205.
  18. D. F. G. Fiorillo and G. G. Raffelt, Theory of neutrino slow flavor evolution. Part I. Homogeneous medium, J. High Energy Phys. 04 (2025) 146.
  19. D. F. G. Fiorillo and G. G. Raffelt, Theory of neutrino slow flavor evolution. Part II. Space-time evolution of linear instabilities, J. High Energy Phys. 06 (2025) 146.
  20. D. F. G. Fiorillo, M. Goimil-García, and G. G. Raffelt, Fast flavor pendulum: Instability condition, Phys. Rev. D 111, 083028 (2025).
  21. D. F. G. Fiorillo and G. G. Raffelt, Dispersion relation of the neutrino plasma: Unifying fast, slow, and collisional instabilities, J. High Energy Phys. 01 (2026) 147.
  22. D. F. G. Fiorillo, H.-T. Janka, and G. G. Raffelt, Neutrino-mass-driven instabilities as the earliest flavor conversion in supernovae, Phys. Rev. Lett. 135, 231003 (2025).
  23. D. F. G. Fiorillo and G. G. Raffelt, Lepton-number crossings are insufficient for flavor instabilities, Phys. Rev. Lett. 136, 051001 (2026).
  24. D. F. G. Fiorillo and G. G. Raffelt, Fast flavor conversions at the edge of instability in a two-beam model, Phys. Rev. Lett. 133, 221004 (2024).
  25. D. F. G. Fiorillo and G. G. Raffelt, Collective flavor conversions are interactions of neutrinos with quantized flavor waves, Phys. Rev. Lett. 134, 211003 (2025).
  26. S. Hannestad, G. G. Raffelt, G. Sigl, and Y. Y. Y. Wong, Self-induced conversion in dense neutrino gases: Pendulum in flavour space, Phys. Rev. D 74, 105010 (2006); 76, 029901(E) (2007).
  27. H. Duan, G. M. Fuller, J. Carlson, and Y.-Z. Qian, Analysis of collective neutrino flavor transformation in supernovae, Phys. Rev. D 75, 125005 (2007).
  28. L. Johns, H. Nagakura, G. M. Fuller, and A. Burrows, Neutrino oscillations in supernovae: Angular moments and fast instabilities, Phys. Rev. D 101, 043009 (2020).
  29. I. Padilla-Gay, I. Tamborra, and G. G. Raffelt, Neutrino flavor pendulum reloaded: The case of fast pairwise conversion, Phys. Rev. Lett. 128, 121102 (2022).
  30. D. F. G. Fiorillo and G. G. Raffelt, Slow and fast collective neutrino oscillations: Invariants and reciprocity, Phys. Rev. D 107, 043024 (2023).
  31. J. Liu, L. Johns, H. Nagakura, M. Zaizen, and S. Yamada, Dynamical equilibria of fast neutrino flavor conversion, arXiv:2509.26418.
  32. D. F. G. Fiorillo and G. G. Raffelt, Single-wave solutions of the neutrino fast flavor system. Part I. Mechanical properties, arXiv:2601.15372.
  33. D. F. G. Fiorillo and G. G. Raffelt, Single-wave solutions of the neutrino fast flavor system. Part II. Weak instabilities and their resonant behavior, arXiv:2601.18880.
  34. G. G. Raffelt, N-mode coherence in collective neutrino oscillations, Phys. Rev. D 83, 105022 (2011); 104, 089902(E) (2021).
  35. M. Gaudin, Diagonalisation d’une classe d’Hamiltniens de spin, J. Phys. (Paris) 37, 1087 (1976).
  36. Y. Pehlivan, A. B. Balantekin, T. Kajino, and T. Yoshida, Invariants of collective neutrino oscillations, Phys. Rev. D 84, 065008 (2011).
  37. R. F. Sawyer, The multi-angle instability in dense neutrino systems, Phys. Rev. D 79, 105003 (2009).
  38. A. Banerjee, A. Dighe, and G. Raffelt, Linearized flavor-stability analysis of dense neutrino streams, Phys. Rev. D 84, 053013 (2011).
  39. I. Izaguirre, G. Raffelt, and I. Tamborra, Fast pairwise conversion of supernova neutrinos: A dispersion-relation approach, Phys. Rev. Lett. 118, 021101 (2017).
  40. B. Dasgupta and A. Dighe, Collective three-flavor oscillations of supernova neutrinos, Phys. Rev. D 77, 113002 (2008).
  41. L. Johns, Collisional flavor instabilities of supernova neutrinos, Phys. Rev. Lett. 130, 191001 (2023).
  42. Z. Xiong, L. Johns, M.-R. Wu, and H. Duan, Collisional flavor instability in dense neutrino gases, Phys. Rev. D 108, 083002 (2023).
  43. Y.-C. Lin and H. Duan, Collision-induced flavor instability in dense neutrino gases with energy-dependent scattering, Phys. Rev. D 107, 083034 (2023).
  44. D. F. G. Fiorillo, I. Padilla-Gay, and G. G. Raffelt, Collisions and collective flavor conversion: Integrating out the fast dynamics, Phys. Rev. D 109, 063021 (2024).
  45. L. Johns and S. Rodriguez, Collisional flavor pendula and neutrino quantum thermodynamics, arXiv:2312.10340.
  46. I. Padilla-Gay, I. Tamborra, and G. G. Raffelt, Neutrino fast flavor pendulum. II. Collisional damping, Phys. Rev. D 106, 103031 (2022).
  47. F. Klein and A. Sommerfeld, The Theory of the Top. Volume I: Introduction to the Kinematics and Kinetics of the Top, (Birkäuser Boston, MA, 2008), 10.1007/978-0-8176-4721-6.
  48. D. F. G. Fiorillo and G. G. Raffelt, Flavor solitons in dense neutrino gases, Phys. Rev. D 107, 123024 (2023).

Outline

Information

Phys. Rev. D 113, 123033– Published 12 June, 2026
  • Received 11 February 2026
  • Accepted 8 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation