Slow and fast collective neutrino oscillations: Invariants and reciprocity

TL;DR

The paper establishes a reciprocity between slow and fast collective neutrino oscillations by recasting their Bloch-vector EOMs and showing that fast dynamics map to a constrained slow system in a comoving frame. It develops Lax-vector formalisms and Gaudin invariants to prove classical and quantum integrability for both regimes, and connects normal-mode spectra to the roots of Lax vectors. A key constraint, R=0 (or its moving-frame analogue), governs when a slow system can be mapped to a fast one, providing a unified view of pendulum-like instabilities and spectral crossings. The analysis is extended to three flavors, where SU(3) Bloch vectors yield analogous invariants and integrability, offering a broad, rigorous framework for understanding collective neutrino flavor evolution in dense environments.

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 $\dot{\mathbf{P}}_ω=(ω\mathbf{B}+μ\mathbf{P})\times\mathbf{P}_ω$, where $ω=Δm^2/2E$, $μ=\sqrt{2} G_{\mathrm{F}} (n_ν+n_{\barν})$, $\mathbf{B}$ is a unit vector in the mass direction in flavor space, and $\mathbf{P}=\int dω\,\mathbf{P}_ω$. For an axisymmetric angle distribution, the fast EOMs are $\dot{\mathbf{D}}_v=μ(\mathbf{D}_0-v\mathbf{D}_1)\times\mathbf{D}_v$, where $\mathbf{D}_v$ is the Bloch vector for lepton number, $v=\cosθ$ 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.