Dispersion Relations

In the two-beam model, I have assumed the neutrino flavor density matrix $\rho(z)$ depends on $z$ only. For the neutrino medium in a real physical environment, $\rho(t,\mathbf r)$ is a function of both time $t$ and spatial coordinate $\mathbf r$. The collective mode $a$ of neutrino oscillations takes the form $$ \begin{equation} \epsilon_i(t,\mathbf r) = Q_{i}^{(a)} e^{\ri (\Omega_a t - \mathbf K_a \cdot \mathbf r) } \label{chap:collective-sec:dispersion-relation-eqn:collective-mode-epsilon} \end{equation} $$ where index $i$ denotes the species and the propagation direction of the neutrino, and $\Omega_a$ and $\mathbf K_a$ are the oscillation frequency and wavevector of the corresponding collective mode, respectively. In this section, I will review the dispersion relations between $\Omega_a$ and $\mathbf K_a$ discussed by I. Izaguirre, G. Raffelt, and I. Tamborra in reference ¹.

The equation of motion for the flavor transformation of a dense neutrino medium in flavor basis is $$ \begin{equation} \ii (\partial_t + \mathbf v\cdot \mathbf{\nabla}) \rho = \left[ \frac{\lambda}{2} \sigma_3 + \mathsf H_{\nu\nu}, \rho \right], \label{eqn-liouville-eqn} \end{equation} $$ where I have ignored the vacuum Hamiltonian. This is because the vacuum oscillation frequency of the neutrino $$ \begin{align*} \omega_{\mathrm v} = \frac{\Delta m^2}{2E} \sim& \frac{2\pi}{ 10 \mathrm{km} } \left(\frac{\Delta m^2_{32}}{2.5\times 10^{-3} \mathrm{eV}^2 } \right) \left( \frac{10 \mathrm{MeV}}{E} \right) \\ \sim & \frac{2\pi}{ 330 \mathrm{km} } \left( \frac{\Delta m_{12}^2}{7.5\times 10^{-5}\mathrm{eV}^2} \right) \left( \frac{10 \mathrm{MeV}}{E} \right) \end{align*} $$ is much smaller than the neutrino potential $$ \begin{equation*} \mu = \sqrt{2}G_{\mathrm F} n \sim \frac{1}{0.01 \mathrm{km}} \left(\frac{100\mathrm{km}}{R}\right)^2 \left(\frac{1\mathrm{MeV}}{E}\right) \left( \frac{ L }{ 10^{50}\mathrm{erg\cdot s^{-1}} } \right), \end{equation*} $$ where $E$ is the typical energy of the supernova neutrino, $n$ is the total number density of the neutrino, $R$ is the distance from the center of the supernova, and $L$ is the total luminosity of the neutrino. In Eqn.~\eqref{eqn-liouville-eqn}, $$ \begin{equation} \mathsf H_{\nu\nu} = \sqrt{2} G_{\mathrm F} \iint \dd \Gamma' v^\mu v'_\mu \int \frac{E'^2 \mathrm d E'}{8\pi^3} \left( (n_{\nu_{\mathrm e}}' - n_{\nu_{\mathrm x}}' )\rho' - (n_{\bar\nu_{\mathrm e}}' - n_{\bar\nu_{\mathrm x}}' ) \bar\rho' \right), \end{equation} $$ where $$ \begin{equation} v^\mu = (1, \mathbf v)^{\mathrm T} = ( 1, \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta )^{\mathrm T} \end{equation} $$ is the four velocity of the neutrino and the primed quantities such as $\rho'=\rho_{\mathbf v'} (t, \mathbf r)$ depend on the primed physical variable $\mathbf v'$. Without the vacuum Hamiltonian, the equation of motion for the antineutrino is the same as Eqn.~\eqref{eqn-liouville-eqn}.

In the supernova, the fluxes of non-electron-flavor neutrinos are approximately the same. It is convenient to define the electron lepton number (ELN) as ¹ $$ \begin{equation} G({\mathbf v}) = \sqrt{2}G_{\mathrm F} \int \frac{E'^2 \mathrm d E'}{8\pi^3} \left[ n_{\nu_{\mathrm e}}(\cos\theta',\phi',E') - n_{\bar\nu_{\mathrm e}}(\cos\theta',\phi',E') \right]. \end{equation} $$ I will perform the linear stability analysis as in Sec.~\ref{chap:collective-sec:two-beams}. For this purpose, I will assume that the density matrices of the neutrino and antineutrino have the form $$ \begin{equation} \rho_{\mathbf v} (t,\mathbf r) = \bar \rho_{\mathbf v} (t,\mathbf r) = \begin{pmatrix} 1 & \epsilon_{\mathbf v}(t,\mathbf r) \\ \epsilon^*_{\mathbf v}(t,\mathbf r) & 0 \end{pmatrix}, \end{equation} $$ where $\lvert \epsilon_{\mathbf v}(t,\mathbf r) \rvert \ll 1$. Plugging this into the equation of motion, I obtain the linearized equation of motion $$ \begin{equation} \ri ( \partial_t + \mathbf v \cdot \boldsymbol{\nabla} ) \epsilon = v^\mu \Phi_\mu \epsilon - \int \dd \Gamma' v^\mu v'_\mu G(\mathbf v') \epsilon', \label{chap:collective-sec:fast-mode-eqn:eom-continuous-general-linearized} \end{equation} $$ where $$ \begin{equation} \Phi_\mu = \left( -\int \dd \Gamma' G(\mathbf v'), \int \dd \Gamma' \mathbf v' G(\mathbf v') \right)^{\mathrm T}. \end{equation} $$ Assuming the solution in Eqn.~\eqref{chap:collective-sec:dispersion-relation-eqn:collective-mode-epsilon}, I obtain $$ \begin{equation} v^\mu k_\mu Q_{\mathbf v} = - \int \dd \Gamma' v^\mu v'_\mu G(\mathbf v') Q_{\mathbf v'}, \label{chap:collective-sec:dispersion-relation-eqn:eom-general-collective-modes-form} \end{equation} $$ where $$ \begin{equation} k_\mu = \begin{pmatrix} \omega \\ \mathbf k \end{pmatrix} = \begin{pmatrix} \Omega - \Phi_0 \\ \mathbf K - \boldsymbol{\Phi} \end{pmatrix}. \end{equation} $$ From Eqn.~\eqref{chap:collective-sec:dispersion-relation-eqn:eom-general-collective-modes-form} I have the formal solution $$ \begin{equation} Q_{\mathbf v} = - \frac{ v^\mu a_\mu }{ v^\alpha k_\alpha}, \label{chap:collective-sec:dispersion-relations-eqn:q-expression-collective-mode} \end{equation} $$ where $$ \begin{equation} a^\nu = \int \dd \Gamma_{\mathbf v} G({\mathbf v}) v^\nu Q_{\mathbf v}. \end{equation} $$ Substituting Eqn.~\eqref{chap:collective-sec:dispersion-relations-eqn:q-expression-collective-mode} into Eqn.~\eqref{chap:collective-sec:dispersion-relation-eqn:eom-general-collective-modes-form}, I obtain $$ \begin{equation} v_\mu \Pi^{\mu}_{\phantom{\mu}\nu} a^\nu = 0, \end{equation} $$ where $$ \begin{equation} \Pi^\mu_{\phantom{\mu}\nu} = \delta^\mu_{\phantom{\mu}\nu} + \int \dd \Gamma G(\mathbf v) \frac{v^\mu v_\nu}{\omega- {\mathbf k}\cdot {\mathbf v} }. \end{equation} $$ The dispersion relation between $\omega$ and $\mathbf k$ is obtained by solving the characteristic equation ¹ $$ \begin{equation} \operatorname{det}\left( \Pi^\mu_{\phantom{\mu}\nu} \right) = 0 \label{eqn-dr-determinant-equation} \end{equation} $$ since we are looking for non-trivial solutions of $a^\nu$.

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