# 7.7. Fast Modes¶

In a paper by Chakraborty et al [chakraborty2016] they found that neutrino instability can grow with a rate that is proportional to the neutrino density, which has much faster oscillation frequencies than vacuum oscillations. For such a fast growth to happen, the author considered head on cliding neutrino beams.

 [chakraborty2016] Chakraborty, S., Hansen, R. S., Izaguirre, I., & Raffelt, G. (2016). Self-induced neutrino flavor conversion without flavor mixing, (10), 17.

As an estimation, the frequencies of vacuum oscillation is

$\begin{split}\omega_{\mathrm v} = \frac{\Delta m^2}{2E}\sim& 6.3\times 10^{-3} \mathrm{m}^{-1} \frac{\Delta m^2_{32}}{2.5\times 10^{-3} \mathrm{eV}^2 } \frac{1MeV}{E} \\ \sim & 1.90\times 10^{-4} \mathrm{m}^{-1} \frac{\delta m^2}{7.5\times 10^{-5}\mathrm{eV}^2} \frac{1\mathrm{MeV}}{E},\end{split}$

where $$E$$ is the neutrino energy. The corresponding oscillation wavelength is simply give by

$\begin{split}\lambda_{12} = & 2\pi/\omega_{12} \sim 1 \mathrm{km}\\ \lambda_{32} = & 2\pi/\omega_{32} \sim 33.1 \mathrm{km}.\end{split}$

The fast modes instability grows with a rate proportional to the neutrino potential $$\mu=\sqrt{2}G_F n_\nu$$, which is very large in dense neutrino media. A large growth rate indicates a faster flavor transformation than vacuum oscillations.

In the past over simplified models have been used and this new instability didn’t show up.

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