Slow and fast collective neutrino oscillations: Invariants and reciprocity

TL;DR

This paper analyzes the dynamics of collective neutrino oscillations, identifying invariants and reciprocity between slow and fast regimes, and demonstrates their integrability and underlying similarities.

Contribution

It reveals the mathematical similarities and differences between slow and fast neutrino oscillation equations, showing their invariants and integrability properties.

Findings

Both systems exhibit pendulum-like instabilities.

They share similar Gaudin invariants.

Fast oscillations can be transformed into an equivalent slow system.

Abstract

The flavor evolution of a neutrino gas can show ''slow'' or ''fast'' collective motion. In terms of the usual Bloch vectors to describe the mean-field density matrices of a homogeneous neutrino gas, the slow two-flavor equations of motion (EOMs) are $\dot{\mathbf{P}}_\omega=(\omega\mathbf{B}+\mu\mathbf{P})\times\mathbf{P}_\omega$, where $\omega=\Delta m^2/2E$, $\mu=\sqrt{2} G_{\mathrm{F}} (n_\nu+n_{\bar\nu})$, $\mathbf{B}$ is a unit vector in the mass direction in flavor space, and $\mathbf{P}=\int d\omega\,\mathbf{P}_\omega$. For an axisymmetric angle distribution, the fast EOMs are $\dot{\mathbf{D}}_v=\mu(\mathbf{D}_0-v\mathbf{D}_1)\times\mathbf{D}_v$, where $\mathbf{D}_v$ is the Bloch vector for lepton number, $v=\cos\theta$ is the velocity along the symmetry axis, $\mathbf{D}_0=\int dv\,\mathbf{D}_v$, and $\mathbf{D}_1=\int dv\,v\mathbf{D}_v$. We discuss similarities and differences between these generic cases. Both systems can have pendulum-like instabilities (soliton solutions), both have similar Gaudin invariants, and both are integrable in the classical and quantum case. Describing fast oscillations in a frame comoving with $\mathbf{D}_1$ (which itself may execute pendulum-like motions) leads to transformed EOMs that are equivalent to an abstract slow system. These conclusions carry over to three flavors.