Abstract
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 sin2θ23msinδm=sin2θ23sinδ 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.
Highlights
Introduction and preliminary remarksIt was shown in 1988 in ref. [1] that in the case of what is currently referred to as the reference 3-neutrino mixing, the magnitude of the CP and T violating (T violating) effects in neutrino oscillations in vacuum are controlled by the rephasing invariant J CP (J m) associated with the Dirac CP violation phase present in the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) [4,5] neutrino mixing matrix: J CP( J m) = Im U (m) e2 (m) μ3 (m) e3 ∗ (m) μ2 (1)
We find the equalities θ2m3 = θ23 and δm = δ, θ23 (θ2m3) and δ being respectively the atmospheric neutrino mixing angle and the Dirac CP violation phase in vacuum 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
The relation (10) between J m and JCP implies, in particular, that we can have J m = 0 only if JCP = 0, i.e., T violation effects can be present in neutrino oscillations taking place in matter with constant density or density distributed symmetrically relative to the middle point only if CP and T violation effects are present in neutrino oscillations taking place in vacuum
Summary
It was shown in 1988 in ref. [1] that in the case of what is currently referred to as the reference 3-neutrino mixing The relation (10) between J m and JCP implies, in particular, that we can have J m = 0 only if JCP = 0, i.e., T violation effects can be present in neutrino oscillations taking place in matter with constant density or density distributed symmetrically relative to the middle point (like in the Earth) only if CP and T violation effects are present in neutrino oscillations taking place in vacuum It was shown in [1] that the presence of matter can enhance somewhat | J m| with respect to its vacuum value | JCP|: in the example considered in [1] the enhancement was by a factor of 3.
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