# Background Matter Basis

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

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

In this chapter, I will focus on the transition probability between the background matter eigenstates $$$$\begin{pmatrix} \ket{\nu_{\mathrm L}} \\ \ket{\nu_{\mathrm H}} \end{pmatrix} = \mathsf U^\dagger \begin{pmatrix} \ket{\nu_{\mathrm e}} \\ \ket{\nu_{\mathrm x}} \end{pmatrix}.$$$$ 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) . ↩︎