Normal-mode Analysis for Collective Neutrino Oscillations

TL;DR

This paper develops a general linear analysis of collective neutrino oscillations, identifying conditions for stable and unstable modes, including fast and slow modes, and clarifies the roles of matter effects and flavor mixing.

Contribution

It derives a comprehensive dispersion relation for collective neutrino modes, including fast and slow oscillations, and clarifies the conditions for their stability and growth.

Findings

Fast modes can grow rapidly, independent of vacuum oscillation frequency.

Matter effects can stabilize flavor evolution by creating fixed points.

Spatial or temporal variations are necessary to seed flavor waves.

Abstract

In an interacting neutrino gas, collective modes of flavor coherence emerge that can be propagating or unstable. We derive the general dispersion relation in the linear regime that depends on the neutrino energy and angle distribution. The essential scales are the vacuum oscillation frequency $\omega=\Delta m^2/(2E)$, the neutrino-neutrino interaction energy $\mu=\sqrt{2}G_{\rm F} n_\nu$, and the matter potential $\lambda=\sqrt{2}G_{\rm F} n_e$. Collective modes require non-vanishing $\mu$ and may be dynamical even for $\omega=0$ ('fast modes'), or they may require $\omega\not=0$ ('slow modes'). The growth rate of unstable fast modes can be fast itself (independent of $\omega$) or can be slow (suppressed by $\sqrt{|\omega/\mu|}$). We clarify the role of flavor mixing, which is ignored for the identification of collective modes, but necessary to trigger collective flavor motion. A large matter effect is needed to provide an approximate fixed point of flavor evolution, while spatial or temporal variations of matter and/or neutrinos are required as a trigger, i.e., to translate the disturbance provided by the mass term to seed stable or unstable flavor waves. We work out explicit examples to illustrate these points.