Neutrino Self-interactions
Bipolar Model
The nature of this section is to provide the linear stability analysis of bipolar model. Bipolar model is a model of neutrino oscillations with the presence of neutrino and antineutrinos. It is also called bimodal oscillations 1, which means two frequencies in the context. An example of such instability happens in a system composed of equal amounts of neutrinos and antineutrinos.
Neutrino oscillations has a small amplitude inside a SN core (suppressed by matter effects) 2, which basically pins down the flavour transformation. As the neutrinos reaches a further distance, matter effect could drop out. Neutrino self-interaction becomes more important. S. Samuel considers a system of neutrinos and antineutrinos with only vacuum and neutrino self-interactions 1. The neutrinos and antineutrino forms a bipolar vector in flavor isospin space. The flavor isospin of neutrinos and that of antineutrinos are coupled.
The equation of motion is $$ \begin{align*} i\partial_t \rho =& \left[ -\frac{\omega_\vv}{2} \cos2\theta \sigma_3 + \frac{\omega_\vv}{2}\sin 2\theta \sigma_1 - \mu \alpha \bar \rho , \rho\right] \\ i\partial_t \bar\rho =& \left[ \frac{\omega_\vv}{2} \cos2\theta \sigma_3 - \frac{\omega_\vv}{2}\sin 2\theta \sigma_1 + \mu \rho , \bar\rho\right]. \end{align*} $$ For the purpose of linear stability analysis, we assume that $$ \begin{align*} \rho =& \frac{1}{2}\begin{pmatrix} 1 & \epsilon \\ \epsilon^* & -1 \end{pmatrix} \\ \bar\rho =& \frac{1}{2}\begin{pmatrix} 1 & \bar\epsilon \\ \bar \epsilon^* & -1 \end{pmatrix}. \end{align*} $$ Plug them into equation of motion and set $\theta=0$, we have the linearized ones, $$ \begin{equation*} i\partial_t \begin{pmatrix} \epsilon \\ \bar\epsilon \end{pmatrix} = \frac{1}{2}\begin{pmatrix} -\alpha \mu - \omega_\vv & \alpha \mu \\ -\mu & \mu + \omega_v \end{pmatrix}\begin{pmatrix} \epsilon \\ \bar\epsilon \end{pmatrix}. \end{equation*} $$ To have real eigenvalues, we require $$ \begin{equation*} (-1+\alpha)^2 \mu^2 + 4(1+\alpha)\mu \omega_\vv + 4 \omega_v^2 < 0, \end{equation*} $$ which is reduced to $$ \begin{equation*} \frac{ -2\omega_\vv (1+\alpha) - 4\sqrt{ \alpha } \lvert \omega_\vv \rvert }{ (1-\alpha)^2 } < \mu < \frac{ -2\omega_\vv (1+\alpha) + 4\sqrt{ \alpha } \lvert \omega_\vv \rvert }{ (1-\alpha)^2 }. \end{equation*} $$ It is simplified to $$ \begin{equation*} \sqrt{ -2\omega_\vv }{ (1-\sqrt{\alpha})^2 } < \mu < \sqrt{ -2\omega_\vv }{ (1+\sqrt{\alpha})^2 }, \end{equation*} $$ assuming normal hierarchy, i.e., $\omega_\vv > 0$. We immediately notice that this can not happen.
For inverted hierarchy, we have $\omega_\vv < 0$, so that $$ \begin{equation*} \sqrt{ 2\lvert\omega_v\rvert }{ (1+\sqrt{\alpha})^2 } < \mu < \sqrt{ 2\lvert\omega_v\rvert }{ (1-\sqrt{\alpha})^2 }, \end{equation*} $$ Within this region, neutrinos experience exponential growth.
For completeness, we also write down the formalism in flavor isospin picture. $$ \begin{align} i\partial_t \vec s &= \vec s \times ( \eta \vec H_\vv + \alpha \mu \bar{\vec s} )\\ i\partial_t \bar{\vec s} &= \bar{\vec s} \times ( \eta \vec H_\vv + \mu \vec s ), \label{chap:app-sec:bipolar-eqn:flavor-isospin-eom} \end{align} $$ where $\eta$ is the hierarchy, and $\alpha$ is the ratio of neutrino number density and antineutrino number density.