Geometric phases in neutrino mixing

Abstract

Neutrinos can acquire both dynamic and geometric phases due to the nontrivial mixing between mass and flavor eigenstates. In this paper, we derive the general expressions for all plausible gauge invariant diagonal and off-diagonal geometric phases in the three- flavor neutrino model using the kinematic approach. We find that diagonal and higher-order off-diagonal geometric phases are sensitive to the mass ordering and the Dirac CP violating phase [Formula: see text]. We show that, third-order off-diagonal geometric phase [Formula: see text] is invariant under any cyclic or non-cyclic permutations of flavor indices when the Dirac CP phase is zero. For nonzero [Formula: see text], we find that [Formula: see text]. We also prove that, only the third-order off-diagonal geometric phase is sensitive to the sign of [Formula: see text]. Further, we explore the effects of matter background using a two-flavor neutrino model and show that the diagonal geometric phase is either [Formula: see text] or [Formula: see text] in the MSW resonance region and takes nontrivial values elsewhere. The transition between zero and [Formula: see text] occurs at the point of complete oscillation inversion called the nodal point, where the diagonal geometric phase is not defined. Also, in two-flavor approximations, two distinct diagonal geometric phases are co-functions with respect to the mixing angle. Finally, in the two-flavor model, we show that the only second-order off-diagonal geometric phase is a topological invariant quantity and is always [Formula: see text].