Slow and fast collective neutrino oscillations: Invariants and reciprocity

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 ${\dot{\mathbf{P}}}_{\ddot{I}}‰=(\ddot{I}‰\mathbf{B}+\hat{I}¼\mathbf{P})\times {\mathbf{P}}_{\ddot{I}}‰$, where $\ddot{I}‰=\hat{I}”m^2/2E$, $\hat{I}¼=\sqrt{2}{G}_{\mathrm{F}}(n_{\hat{I}}½+n_{\bar{\hat{I}}½})$, $\mathbf{B}$ is a unit vector in the mass direction in flavor space, and $\mathbf{P}=\int d\ddot{I}‰{\mathbf{P}}_{\ddot{I}}‰$. For an axisymmetric angle distribution, the fast EOMs are ${\dot{\mathbf{D}}}_v=\hat{I}¼({\mathbf{D}}_0-v{\mathbf{D}}_1)\times {\mathbf{D}}_v$, where ${\mathbf{D}}_v$ is the Bloch vector for lepton number, $v=\cos \hat{I}¸$ 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.

16 pages; post-publication corrected treatment of Lax vectors and Gaudin invariants, especially for three-flavor case. Matching the corrections in the Erratum accepted on PRD

Cited by in corpus (4)

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• Sufficient and Necessary Conditions for Collective Neutrino Instability: Fast, Slow, and Mixed
• Single-wave solutions of the neutrino fast flavor system. Part II. Weak instabilities and their resonant behavior
• Neutrino quantum kinetics in three flavors