# Two-Beam Model and Flavor Instabilities

During the synchronized flavor transformation, the whole neutrino medium behaves as if there was a single neutrino oscillating with the mean frequency $\langle \omega_\vv \rangle$ in vacuum. Under certain conditions, the neutrino self-interaction potential can cause the neutrino medium to evolve in a way completely different than vacuum oscillations. I will illustrate this phenomenon using the two-beam neutrino model.

In the two-beam model, neutrinos and antineutrinos are constantly emitted in two different directions from a plane which we assume to be the $x$-$y$ plane. I define the emission angles of the neutrino beams $\theta_1$ and $\theta_2$ to be inside the plane spanned by the velocity vectors of the two neutrino beams (see Fig. Two-Beam Line Model). I will assume that the neutrino emission plane is homogeneous and that the neutrino and antineutrino have the same density $n$ in the emission plane. I will also assume that the matter density vanishes everywhere in space.

The Hamiltonian of the $i$th neutrino beam in the vacuum basis is \begin{align} \mathsf H_i &= -\frac{\omega_i}{2} \sigma_3 + \mu ( \rho_1 - \rho_2), \end{align} where $\omega_1 = \eta\omega_\vv$ for the neutrino beam and $\omega_2=-\eta\omega_\vv$ for the antineutrino beam, \begin{align} \mu = \sqrt{2} G_{\mathrm F} (1 - \cos(\theta_1 - \theta_2)) n \end{align} is the neutrino self-interaction potential.

I will assume that the neutrino and antineutrino are emitted in the pure electron flavor at $z=0$. I will also assume the vacuum mixing angle $\theta_\vv \ll 1$. Therefore, the flavor density matrix of the $i$th neutrino beam in the vacuum basis at $z=0$ is $$$$\rho_i (z=0) \approx \begin{pmatrix} 1 & \epsilon_i (0) \\ \epsilon_i^* (0) & 0 \end{pmatrix}, \label{chap:collective-sec:two-beams-eqn:density-matrix-perturbed}$$$$ where $$$$\epsilon_i(0) = \sin (2\theta_\vv) \ll 1.$$$$

I solve the two-beam model numerically with $\epsilon_i = 10^{-3}$ and $\eta=-1$. The solution is shown in Fig. Numerical Solution to Two-beam Line Model. When $\omega_\vv z \ll 1$, the flavor isospin of the neutrino stays in the electron flavor state, i.e., in the direction of third axis $\vec e_3$ in the flavor space (see the left panel of Fig. Numerical Solution to Two-beam Line Model). Accordingly, the electron flavor survival probability $P_{\nu_\ee}\approx 1$ at small $z$ but falls down rapidly at a certain value of $z$ (see the right panel of Fig. Numerical Solution to Two-beam Line Model). Then both the flavor isospin and $P_{\nu_\ee}$ come back to their original positions until they fall down again. The flavor isospin of the antineutrino follows a similar pattern but is in a different direction. This flavor transformation phenomenon is clearly different from the vacuum oscillations shown in Fig. Two-Flavor Oscillations in Vacuum.

To gain a deeper understanding of the collective oscillation phenomenon, I will focus on the linear regime where $\lvert \epsilon_i(z) \rvert \ll 1$. The equation of motion is linearized in this regime and becomes $$$$\ri \partial_z \begin{pmatrix} \epsilon_1 \\ \epsilon_2 \end{pmatrix} = \begin{pmatrix} \mu + \eta\omega_\vv & - \mu \\ \mu & - \eta\omega_\vv - \mu \end{pmatrix} \begin{pmatrix} \epsilon_1 \\ \epsilon_2 \end{pmatrix}. \label{chap:collective-sec:bipolar-linearized-eom}$$$$ This equation has two normal modes which correspond to the collective modes of neutrino oscillations: $$$$\begin{pmatrix} \epsilon_1 (z) \\ \epsilon_2 (z) \end{pmatrix} = \begin{pmatrix} Q_{1\pm} \\ Q_{2\pm} \end{pmatrix} e^{i K_{z\pm} z}, \label{chap:collective-sec:two-beams-eqn:equation-of-motion-collective-mode-assumption}$$$$ where $(Q_{1\pm}, Q_{2\pm})^{\mathrm T}$ are the eigenvectors of the two normal modes, and $$$$K_{z\pm} = \pm \sqrt{ \omega_\vv (\omega_\vv + 2 \eta \mu) }$$$$ are the wavenumbers of the corresponding modes. For the normal mass hierarchy, $\eta = 1$, and $K_{z\pm}$ are always real. In this case, the electron flavor probability $P_{\nu_\ee}$ is always approximately 1. However, for the inverted neutrino mass hierarchy, $\eta=-1$, and $K_{z\pm}$ are imaginary when $\mu > \omega_\vv/2$. The normal mode with wavenumber $K_{z-} = -\ri \sqrt{ \omega_\vv (2 \mu-\omega_\vv) }$ is a runaway solution which explains the rapid decrease of $P_{\nu_\ee}$ in Fig. Numerical Solution to Two-beam Line Model.

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