The ubiquitous flavor pendulum

TL;DR

This work reveals that pendulum-like dynamics are a universal feature of minimal systems of interacting spins representing dense neutrino ensembles. By mapping slow, fast, and single-wave flavor equations onto a gyroscopic pendulum, it shows that three-vector triads with a conserved length inevitably exhibit pendular motion, and that such dynamics can extend to a continuum of modes in integrable cases via Lax vectors and Gaudin invariants. The slow and fast systems admit bimodal coherence and continuum pendula, while the single-wave system generally does not, highlighting the role of integrability. The results provide a unifying framework for understanding nonlinear flavor evolution and offer insights into how pendular behavior might inform practical neutrino transport models, especially in the context of weak instabilities and their saturation.

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. 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.