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

\[\nu_e + \bar{\nu_e} \leftrightarrow \nu_x + \bar{\nu_x}.\]

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. Linear Stability Analysis

The equation of motion is

\[\begin{split}i\partial_t \rho =& \left[ -\frac{\omega_v}{2} \cos2\theta \sigma_3 + \frac{\omega_v}{2}\sin 2\theta \sigma_1 - \mu \alpha \bar \rho , \rho\right] \\ i\partial_t \bar\rho =& \left[ \frac{\omega_v}{2} \cos2\theta \sigma_3 - \frac{\omega_v}{2}\sin 2\theta \sigma_1 + \mu \rho , \bar\rho\right].\end{split}\]

For the purpose of linear stability analysis, we assume that

\[\begin{split}\rho =& \frac{1}{2}\begin{pmatrix} 1 & \epsilon \\ \epsilon^* & -1 \end{pmatrix} \\ \bar\rho =& \frac{1}{2}\begin{pmatrix} 1 & \bar\epsilon \\ \bar \epsilon^* & -1 \end{pmatrix}.\end{split}\]

Plug them into equation of motion and set \(\theta=0\), we have the linearized ones,

\[\begin{split}i\partial_t \begin{pmatrix} \epsilon \\ \bar\epsilon \end{pmatrix} = \frac{1}{2}\begin{pmatrix} -\alpha \mu - \omega_v & \alpha \mu \\ -\mu & \mu + \omega_v \end{pmatrix}\begin{pmatrix} \epsilon \\ \bar\epsilon \end{pmatrix}.\end{split}\]

To have real eigenvalues, we require

\[(-1+\alpha)^2 \mu^2 + 4(1+\alpha)\mu \omega_v + 4 \omega_v^2 < 0,\]

which is reduced to

\[\frac{ -2\omega_v (1+\alpha) - 4\sqrt{ \alpha } \lvert \omega_v \rvert }{ (1-\alpha)^2 } < \mu < \frac{ -2\omega_v (1+\alpha) + 4\sqrt{ \alpha } \lvert \omega_v \rvert }{ (1-\alpha)^2 },\]

which is simplified to

\[\sqrt{ -2\omega_v }{ (1-\sqrt{\alpha})^2 } < \mu < \sqrt{ -2\omega_v }{ (1+\sqrt{\alpha})^2 },\]

assuming normal hierarchy, i.e., \(\omega_v > 0\). We immediately notice that this can not happen.

For inverted hierachy, we have \(\omega_v < 0\), so that

\[\sqrt{ 2\lvert\omega_v\rvert }{ (1+\sqrt{\alpha})^2 } < \mu < \sqrt{ 2\lvert\omega_v\rvert }{ (1-\sqrt{\alpha})^2 },\]

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.

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