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Single-wave solutions of the neutrino fast flavor system. Part I. Mechanical properties

Damiano F. G. Fiorillo, Georg G. Raffelt

TL;DR

The paper analyzes single-wave (SW) solutions in the fast neutrino flavor system by recasting the nonlinear, inhomogeneous problem as a mechanical system of interacting spins. It establishes that, unlike the homogeneous slow/fast cases, SW is not integrable due to the absence of Gaudin invariants, which precludes a straightforward extension to a continuum of modes. The authors show SW reduces to a single-wave, time-only spin dynamics in a corotating frame and connect SW to coprecession and nonlinear flavor waves, while confirming that a two-beam flavor pendulum inevitably arises but cannot generally be extended to multi-angle distributions. They further derive dispersion relations for instabilities, analyze weak-instability regimes, and discuss the limits of pendulum-like behavior in continuous ensembles, setting the stage for a forthcoming treatment of weak instabilities in Paper II.

Abstract

A dense neutrino plasma can exhibit collective flavor evolution caused by neutrino--neutrino refraction. Recently, a new class of exact nonlinear inhomogeneous solutions was discovered: single-wave (SW) solutions of the fast flavor system. The key property is that the flavor occupation numbers remain homogeneous, whereas the field of flavor coherence varies spatially with a single wave vector. The equations of motion for this structure resemble those of a collection of classical spins, in analogy with the homogeneous slow and fast flavor cases. In contrast, the SW system is not integrable (it does not possess Gaudin invariants) so that, while two-beam pendulum solutions are inevitable, they do not extend to a multi-angle system. We develop a taxonomy of all known nonlinear collective flavor solutions, explaining the overlap between categories and their differences.

Single-wave solutions of the neutrino fast flavor system. Part I. Mechanical properties

TL;DR

The paper analyzes single-wave (SW) solutions in the fast neutrino flavor system by recasting the nonlinear, inhomogeneous problem as a mechanical system of interacting spins. It establishes that, unlike the homogeneous slow/fast cases, SW is not integrable due to the absence of Gaudin invariants, which precludes a straightforward extension to a continuum of modes. The authors show SW reduces to a single-wave, time-only spin dynamics in a corotating frame and connect SW to coprecession and nonlinear flavor waves, while confirming that a two-beam flavor pendulum inevitably arises but cannot generally be extended to multi-angle distributions. They further derive dispersion relations for instabilities, analyze weak-instability regimes, and discuss the limits of pendulum-like behavior in continuous ensembles, setting the stage for a forthcoming treatment of weak instabilities in Paper II.

Abstract

A dense neutrino plasma can exhibit collective flavor evolution caused by neutrino--neutrino refraction. Recently, a new class of exact nonlinear inhomogeneous solutions was discovered: single-wave (SW) solutions of the fast flavor system. The key property is that the flavor occupation numbers remain homogeneous, whereas the field of flavor coherence varies spatially with a single wave vector. The equations of motion for this structure resemble those of a collection of classical spins, in analogy with the homogeneous slow and fast flavor cases. In contrast, the SW system is not integrable (it does not possess Gaudin invariants) so that, while two-beam pendulum solutions are inevitable, they do not extend to a multi-angle system. We develop a taxonomy of all known nonlinear collective flavor solutions, explaining the overlap between categories and their differences.
Paper Structure (30 sections, 47 equations, 1 figure)

This paper contains 30 sections, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kinetic equations
  3. General case
  4. Flavor isospin convention
  5. Axial symmetry
  6. Slow flavor system
  7. Fast flavor system
  8. Systems of interacting spins
  9. Homogeneous slow and fast systems
  10. Single-wave fast system
  11. Single-wave slow system
  12. Integrability and Gaudin invariants
  13. Gaudin invariants of the single-wave system?
  14. Single-coordinate solutions
  15. Coprecession solutions and nonlinear flavor waves
  16. ...and 15 more sections

Figures (1)

  • Figure 1: Classification of the known exact nonlinear solutions of collective flavor evolution. As usual, slow refers to flavor evolution driven by neutrino mass differences, whereas fast to the limit of vanishing masses. The inhomogeneous solutions (green and yellow boxes) are axisymmetric. The green block represents the single-coordinate solutions discussed in Sec. \ref{['sec:Single-Coordinate']} which arise from homogeneous or static ones through a Lorentz boost.