On neutrino mixing in matter and CP and T violation effects in neutrino oscillations

Aspects of 3-neutrino mixing and oscillations in vacuum and in matter with constant density are investigated working with a real form of the neutrino Hamiltonian. We find the (approximate) equalities θ23m=θ23 and δm=δ, θ23 (θ23m) and δ (δm) being respectively the atmospheric neutrino mixing angle and the Dirac CP violation phase in vacuum (in matter) of the neutrino mixing matrix, which are shown to represent excellent approximations for the conditions of the T2K (T2HK), T2HKK, NOνA and DUNE neutrino oscillation experiments. A new derivation of the known relation sin⁡2θ23msin⁡δm=sin⁡2θ23sin⁡δ is presented and it is used to obtain a correlation between the shifts of θ23 and δ due to the matter effect. A derivation of the relation between the rephasing invariants which determine the magnitude of CP and T violating effects in 3-flavour neutrino oscillations in vacuum, JCP, and of the T violating effects in matter with constant density, JTm≡Jm, reported in [1] without a proof, is presented. It is shown that the function F which appears in this relation, Jm=JCPF, and whose explicit form was given in [1], coincides with the function F˜ in the similar relation Jm=JCPF˜ derived in [2], although F and F˜ are expressed in terms of different sets of neutrino mass and mixing parameters and have completely different forms.