# The dispersion relation of the fast neutrino oscillation wave

## TL;DR

This paper investigates the complex dispersion relation branches of neutrino oscillation waves in dense media, proposing a method to identify flavor instabilities through critical points of the dispersion relation, with implications for understanding collective neutrino behavior.

## Contribution

It introduces a novel approach linking complex dispersion relation branches to critical points, enhancing the understanding of neutrino flavor instabilities in symmetric media.

## Key findings

- Complex DR branches are bounded by critical points.
- Flavor instabilities can be identified by tracing critical points.
- Method applies to neutrino media with axial symmetry.

## Abstract

A dense neutrino medium can support flavor oscillation waves which are coherent among different momentum modes of the neutrinos. The dispersion relation (DR) branches of such a wave with complex frequencies and/or wave numbers can lead to the exponential growth of the wave amplitude which in turn will engender a collective flavor transformation in the neutrino medium. In this work we propose that the complex DR branches of the neutrino oscillation wave should be bound by the critical points of the DR. We demonstrate how this theory can be applied to the neutrino medium with an (approximate) axial symmetry about the propagation direction of the neutrino oscillation wave. We also show how the flavor instabilities in this medium can be identified by tracing the critical points of the DR as the electron lepton number distribution of the neutrino medium is changed continuously.