$$ \newcommand\Tr{\mathrm{Tr}} \newcommand{\braket}[2]{\langle #1 \mid #2 \rangle} \newcommand\I{\mathbb{I}} \newcommand{\avg}[1]{\left< #1 \right>} \newcommand{\RD}{D} \newcommand{\ri}{\mathrm{i}} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Sign}{Sign} \newcommand{\ii}{\mathrm i} \newcommand{\vv}{\mathrm v} \newcommand{\ff}{\mathrm f} \newcommand{\mm}{\mathrm m} \newcommand{\ee}{\mathrm e} \newcommand{\xx}{\mathrm x} \newcommand{\RR}{\mathrm R} \newcommand{\dd}{\mathrm d} \newcommand{\FF}{\mathrm F} \newcommand{\BB}{\mathrm B} \newcommand{\vph}{v_{\mathrm{ph}}} $$

Background Matter Basis

For a uniform matter density $\lambda(r) = \lambda_0$, one can define a matter basis in which the Hamiltonian is diagonalized: $$ \begin{equation} \mathsf H^{(\mm)} = \mathsf U^\dagger \mathsf H^{(\ff)} \mathsf U = -\frac{\omega_\mm}{2} \sigma_3, \label{chap:matter-sec:background-eqn:hamiltonian-matter-basis} \end{equation} $$ where $$ \begin{equation} \mathsf U = \begin{pmatrix} \cos \theta_\mm & \sin \theta_\mm \\ \sin\theta_\mm & \cos \theta_\mm \end{pmatrix} \end{equation} $$ with $$ \begin{equation} \theta_{\mathrm{m}}= \frac{1}{2} \arctan\left( \frac{\sin 2\theta_{\mathrm v}}{ \cos 2\theta_{\mathrm v} - \lambda_0/\omega_{\mathrm v} } \right), \label{chap:matter-sec:background-eqn:thetam-expression} \end{equation} $$ and $$ \begin{equation} \omega_{\mathrm{m}} = \omega_{\mathrm{v}} \sqrt{ ( \lambda_0/\omega_{\mathrm{v}} - \cos (2\theta_{\mathrm{v}}) )^2 + \sin^2(2\theta_{\mathrm{v}}) } \label{chap:matter-sec:background-eqn:omegam} \end{equation} $$ is the neutrino oscillation frequency in matter.

In the rest of the chapter, I will consider the matter profiles of the form $$ \begin{equation} \lambda(r) = \lambda_0 + \delta \lambda(r), \label{eq-general-matter-profile} \end{equation} $$ where $\delta \lambda(r)$ describes the fluctuation of the matter density. I will use the background matter basis defined in Eqn. \ref{chap:matter-sec:background-eqn:hamiltonian-matter-basis}, Eqn. \eqref{chap:matter-sec:background-eqn:thetam-expression} and Eqn. \eqref{chap:matter-sec:background-eqn:omegam}. In this basis, the Hamiltonian reads $$ \begin{equation} \mathsf H^{(\mathrm{m})} = -\frac{\omega_\mm}{2} \sigma_3 + \frac{1}{2} \delta\lambda(r) \cos 2\theta_{\mathrm m} \sigma_3 - \frac{1}{2} \delta\lambda(r) \sin 2\theta_{\mathrm m} \sigma_1. \label{eq-hamiltonian-bg-matter-basis-general} \end{equation} $$

In this chapter, I will focus on the transition probability between the background matter eigenstates $$ \begin{equation} \begin{pmatrix} \ket{\nu_{\mathrm L}} \\ \ket{\nu_{\mathrm H}} \end{pmatrix} = \mathsf U^\dagger \begin{pmatrix} \ket{\nu_{\mathrm e}} \\ \ket{\nu_{\mathrm x}} \end{pmatrix}. \end{equation} $$ Given this transition probability, it is trivial to calculate the conversion between flavors.

All the numerical examples in this chapter are calculated with $\sin^2(2\theta_{\mathrm v}) = 0.093$ and $\omega_{\vv} = 1.3\times 10^{-16}\mathrm{MeV}$ 1.

  1. C. Patrignani, "Review of particle physics", Chinese Physics C 40, 100001 (2016) . ↩︎

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