Three Flavor Neutrino Oscillations in Matter: Flavor Diagonal Potentials, the Adiabatic Basis and the CP phase

We discuss the three neutrino flavor evolution problem with general, flavor-diagonal, matter potentials and a fully parameterized 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 theta_12 and theta_13 are independent of the CP phase, although 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 "non-adiabaticity" 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 mu and tau neutrino potentials are the same, is independent of both 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 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.