Slow and fast collective neutrino oscillations: Invariants and reciprocity

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 $\omega =\mathrm{\Delta }{m}^2\mathrm{/}2E$, $\mu =\sqrt{2}{G}_{\mathrm{F}}(n_{\nu }+n_{\bar{\nu }})$, $\mathbf{B}$ 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 ${\mathbf{D}}_1=\int dvv{\mathbf{D}}_v$. 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.