# 8.1. Bimodal Oscillations¶

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

$\nu_e + \bar{\nu_e} \leftrightarrow \nu_x + \bar{\nu_x}.$

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

$\ddot {\tilde q_+} \approx - \eta (\eta + \mu) \tilde q_+,$

where $$\eta=\pm 1$$ deterimines the hierarchy, $$\mu=2\sqrt{2}G_F \lvert \omega_0 \rvert^{-1} n_{\mathrm {tot}}$$. We find out from the equation that normal hierarchy (NH, $$\eta=1$$) doesn’t have instabilities, but inverted hierarchy (IH, $$\eta=-1$$) has instabilities with growth rate $$\sqrt{\mu-1}$$, if $$\mu>1$$.

## 8.1.1. 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.

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