Equation of Motion

The equation of motion for neutrino oscillations with self-interaction is ¹ $$ \begin{equation} \ri \frac{\dd}{\dd t}\rho = [ \mathsf H,\rho], \label{chap:collective-sec:collective-eqn:equation-of-motion-general} \end{equation} $$ where the total derivative is $$ \begin{equation} \frac{\dd}{\dd t} = \partial_t + \mathbf v\cdot \boldsymbol{\nabla}, \end{equation} $$ and the Hamiltonian is composed of three different terms: $$ \begin{equation} \mathsf H = \mathsf H_{\mathrm v} +\mathsf H_{\nu\nu}. \end{equation} $$ In the above equation, the vacuum Hamiltonian is $$ \begin{align*} \mathsf H_\vv =& \begin{cases} -\frac{1}{2}\eta \omega_\vv \sigma_3 & \text{for neutrinos},\\ \frac{1}{2}\eta \omega_\vv \sigma_3 & \text{for antineutrinos}, \end{cases} \end{align*} $$ where $\eta = +1$ for the normal neutrino mass hierarchy and $-1$ for the inverted neutrino mass hierarchy, $\omega_\vv$ is the vacuum oscillation frequency of the neutrino or antineutrino. The neutrino self-interaction is more complicated and can be written as $$ \begin{equation} \mathsf H_{\nu\nu} = \sqrt{2}G_{\mathrm F} \int \frac{E'^2\dd E'}{8\pi^3} \dd\Gamma_{\mathbf v'} \left[ n(E',\mathbf v')\rho(E',\mathbf v') - \bar n(E',\mathbf v')\bar\rho(E',\mathbf v') \right] (1-\mathbf v \cdot \mathbf v'), \label{chap:collective-sec:collective-eqn:equation-of-motion-self-interaction} \end{equation} $$ where $\dd \Gamma_\nu$ is the differential solid angle, $n(E,\mathbf v)$ and $\rho(E, \mathbf v)$ are the number density and the flavor density matrix of the neutrino with energy $E$ and velocity $\mathbf v$, and $\bar n(E, \mathbf v)$ and $\bar \rho(E,\mathbf v)$ are the corresponding quantities of the antineutrino.

The presence of the neutrino self-interaction potential makes the equation of motion \eqref{chap:collective-sec:collective-eqn:equation-of-motion-general} nonlinear, and many interesting phenomena arise because of it. For example, a dense neutrino medium can experience synchronized oscillations during which all the neutrinos and antineutrinos oscillate with the same frequency ^{2 3 4 5}. To see this, I will consider an isotropic and homogeneous neutrino gas and use the flavor isospin picture discussed in Sec. Flavor Isospin Formalism. The flavor isospin $\vec s$ of the neutrino is defined by $$ \begin{equation} \rho = \frac{1}{2} + \vec s \cdot \vec \sigma. \end{equation} $$ The equation of motion of the flavor isospin is $$ \begin{equation} \dot{\vec s} = \vec s \times \left(\vec H_\vv + \vec H_{\nu\nu} \right), \label{chap:collective-eqn:flavor-isospin-eom} \end{equation} $$ where $$ \begin{equation} \vec H_{\mathrm v} = \omega_{\mathrm v}\begin{pmatrix} -\sin 2\theta_{\mathrm v}\\ 0 \\ \cos 2\theta_{\mathrm v} \end{pmatrix} = \omega_\vv \vec B, \end{equation} $$ and $$ \begin{equation} \vec H_{\nu\nu} = \frac{\sqrt{2}G_{\mathrm F}}{2\pi^2} \int E'^2\dd E' n(E') \vec s(E'). \end{equation} $$ Here for simplicity I have assumed $n_{\mathrm e}=0$ and $\bar n=0$. When the neutrino density is very large, $\vec H_{\nu\nu}$ dominates over $\vec H_\vv$ in Eqn.~\eqref{chap:collective-eqn:flavor-isospin-eom}, and $\vec s$ precesses about $\vec H_{\nu\nu}$ rapidly. Meanwhile, the total flavor isospin $$ \begin{equation} \vec S = \int \dd E' f(E') \vec s(E') \end{equation} $$ precesses about $\vec H_\vv$ slowly with a mean oscillation frequency $\langle \omega_\vv \rangle$, where $$ \begin{equation} f(E) = \frac{ E^2 n(E) }{ \int E'^2 \dd E' n(E') } \end{equation} $$ is the energy distribution of the neutrino. To see this, one can multiply Eqn.~\eqref{chap:collective-eqn:flavor-isospin-eom} by $f(E)$ and integrate over $E$ which gives $$ \begin{align} \dot{\vec S} &= \int \dd E f(E)\vec s(E) \times \omega_\vv \vec B \\ &\to \int \dd E f(E) \left[ \frac{\vec s(E) \cdot \vec S}{ \lvert \vec S \rvert^2 } \vec S \right] \times \omega_\vv \vec B \\ &= \langle \omega_\vv \rangle \vec S \times \vec B, \end{align} $$ where $$ \begin{equation} \langle \omega_\vv \rangle = \int \dd E f(E) \left[ \frac{\vec s(E) \cdot \vec S}{ \lvert \vec S \rvert^2 } \right] \omega_\vv. \end{equation} $$ In the above equation I have replaced flavor isospin $\vec s$ with its projection along the direction of $\vec S$ because its precession about $\vec S$ is much faster than the precession of $\vec S$ about $\vec B$.

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