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 ${\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\mathbf{P}}}_{\ensuremath{\omega}}=(\ensuremath{\omega}\mathbf{B}+\ensuremath{\mu}\mathbf{P})\ifmmode\times\else\texttimes\fi{}{\mathbf{P}}_{\ensuremath{\omega}}$, where $\ensuremath{\omega}=\mathrm{\ensuremath{\Delta}}{m}^{2}/2E$, $\ensuremath{\mu}=\sqrt{2}{G}_{\mathrm{F}}({n}_{\ensuremath{\nu}}+{n}_{\overline{\ensuremath{\nu}}})$, $\mathbf{B}$ is a unit vector in the mass direction in flavor space, and $\mathbf{P}=\ensuremath{\int}d\ensuremath{\omega}{\mathbf{P}}_{\ensuremath{\omega}}$. For an axisymmetric angle distribution, the fast EOMs are ${\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\mathbf{D}}}_{v}=\ensuremath{\mu}({\mathbf{D}}_{0}\ensuremath{-}v{\mathbf{D}}_{1})\ifmmode\times\else\texttimes\fi{}{\mathbf{D}}_{v}$, where ${\mathbf{D}}_{v}$ is the Bloch vector for lepton number, $v=\mathrm{cos}\ensuremath{\theta}$ is the velocity along the symmetry axis, ${\mathbf{D}}_{0}=\ensuremath{\int}dv{\mathbf{D}}_{v}$, and ${\mathbf{D}}_{1}=\ensuremath{\int}dvv{\mathbf{D}}_{v}$. We discuss similarities and differences between these generic cases. Both systems can have pendulumlike 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 pendulumlike motions) leads to transformed EOMs that are equivalent to an abstract slow system. These conclusions carry over to three flavors.