Three flavor neutrino oscillations in matter: Flavor diagonal potentials, the adiabatic basis, and theCPphase

Abstract

We discuss the three neutrino flavor evolution problem with general, flavor-diagonal, matter potentials and a fully parametrized mixing matrix that includes $CP$ violation, and derive expressions for the eigenvalues, mixing angles, and phases. We demonstrate that, in the limit that the mu and tau potentials are equal, the eigenvalues and matter mixing angles ${\stackrel{\texttildelow{}}{\ensuremath{\theta}}}_{12}$ and ${\stackrel{\texttildelow{}}{\ensuremath{\theta}}}_{13}$ are independent of the $CP$ phase, although ${\stackrel{\texttildelow{}}{\ensuremath{\theta}}}_{23}$ does have $CP$ dependence. Since we are interested in developing a framework that can be used for $S$ matrix calculations of neutrino flavor transformation, it is useful to work in a basis that contains only off-diagonal entries in the Hamiltonian. We derive the ``nonadiabaticity'' parameters that appear in the Hamiltonian in this basis. We then introduce the neutrino $S$ matrix, derive its evolution equation and the integral solution. We find that this new Hamiltonian, and therefore the $S$ matrix, in the limit that the $\ensuremath{\mu}$ and $\ensuremath{\tau}$ neutrino potentials are the same, is independent of both ${\stackrel{\texttildelow{}}{\ensuremath{\theta}}}_{23}$ and the $CP$ violating phase. In this limit, any $CP$ violation in the flavor basis can only be introduced via the rotation matrices, and so effects which derive from the $CP$ phase are then straightforward to determine. We then show explicitly that the electron neutrino and electron antineutrino survival probability is independent of the $CP$ phase in this limit. Conversely, if the $CP$ phase is nonzero and mu and tau matter potentials are not equal, then the electron neutrino survival probability cannot be independent of the $CP$ phase.