8.1. Bipolar Model¶

Bimodal in this context means two frequencies [Samuel1996]. With neutrino coherent scattering, the neutrino state consists of two frequencies.

An example of such intability happens in a system composed of equal amounts of neutrinos and antineutrinos. Flavour transform occurs due to

ν_{e} + \bar{ν_{e}} \leftrightarrow ν_{x} + \bar{ν_{x}} .

Vacuum mixing angle triggers the flavour instability.

Neutrino oscillations has a small amplitude inside a SN core (suppressed by matter effects) [Wolfenstein1978], which basically pins down the flavour transformation. As the neutrinos reaches a furthur distance, matter effect could drop out. Neutrino self-interaction becomes more important. [Samuel1996] considers a system of neutrinos and antineutrinos with only vacuum and neutrino self-interactions. The neutrinos and antineutrino forms a bipolar vector in flavor isospin space. The flavor isospin of neutrinos and that of antineutrinos are coupled.

[Duan2013] decomposed the system into “normal modes” of the flavor isospin. The bipolar system is discussed in details in this paper. In a two beam model, the length of one of the perturbations can be discribed using an equation

\ddot{{\tilde{q}}_{+}} \approx - η (η + μ) {\tilde{q}}_{+},

where $η = \pm 1$ deterimines the hierarchy, $μ = 2 \sqrt{2} G_{F} | ω_{0} |^{- 1} n_{tot}$ . We find out from the equation that normal hierarchy (NH, $η = 1$ ) doesn’t have instabilities, but inverted hierarchy (IH, $η = - 1$ ) has instabilities with growth rate $\sqrt{μ - 1}$ , if $μ > 1$ .

8.1.1. Linear Stability Analysis¶

The equation of motion is

\begin{aligned} i \partial_{t} ρ = & [- \frac{ω_{v}}{2} \cos 2 θ σ_{3} + \frac{ω_{v}}{2} \sin 2 θ σ_{1} - μ α \bar{ρ}, ρ] \\ i \partial_{t} \bar{ρ} = & [\frac{ω_{v}}{2} \cos 2 θ σ_{3} - \frac{ω_{v}}{2} \sin 2 θ σ_{1} + μ ρ, \bar{ρ}] . \end{aligned}

For the purpose of linear stability analysis, we assume that

\begin{aligned} ρ = & \frac{1}{2} (\begin{array}{c} 1 & ϵ \\ ϵ^{*} & - 1 \end{array}) \\ \bar{ρ} = & \frac{1}{2} (\begin{array}{c} 1 & \bar{ϵ} \\ {\bar{ϵ}}^{*} & - 1 \end{array}) . \end{aligned}

Plug them into equation of motion and set $θ = 0$ , we have the linearized ones,

\begin{array}{r} i \partial_{t} (\begin{array}{c} ϵ \\ \bar{ϵ} \end{array}) = \frac{1}{2} (\begin{array}{c} - α μ - ω_{v} & α μ \\ - μ & μ + ω_{v} \end{array}) (\begin{array}{c} ϵ \\ \bar{ϵ} \end{array}) . \end{array}

To have real eigenvalues, we require

(- 1 + α)^{2} μ^{2} + 4 (1 + α) μ ω_{v} + 4 ω_{v}^{2} < 0,

which is reduced to

\frac{- 2 ω_{v} (1 + α) - 4 \sqrt{α} | ω_{v} |}{(1 - α)^{2}} < μ < \frac{- 2 ω_{v} (1 + α) + 4 \sqrt{α} | ω_{v} |}{(1 - α)^{2}},

which is simplified to

\sqrt{- 2 ω_{v}} (1 - \sqrt{α})^{2} < μ < \sqrt{- 2 ω_{v}} (1 + \sqrt{α})^{2},

assuming normal hierarchy, i.e., $ω_{v} > 0$ . We immediately notice that this can not happen.

For inverted hierachy, we have $ω_{v} < 0$ , so that

\sqrt{2 | ω_{v} |} (1 + \sqrt{α})^{2} < μ < \sqrt{2 | ω_{v} |} (1 - \sqrt{α})^{2},

Within this region, we have exponential growth.

8.1.2. Refs & Notes¶

Samuel1996(1,2): Samuel, S. (1996). Bimodal coherence in dense self-interacting neutrino gases. Physical Review D, 53(10), 5382–5393. doi:10.1103/PhysRevD.53.5382
Duan2013: Duan, H. (2013). Flavor oscillation modes in dense neutrino media. Physical Review D - Particles, Fields, Gravitation and Cosmology, 88.
Wolfenstein1978: Wolfenstein, L. Neutrino oscillations in matter. Phys. Rev. D 17, 23692374 (1978). Or check papers of MSW effect such as Wick Haxton’s excellent review.