Off Diagonal Geometric Phases in Neutrino Mixing

Diagonal and higher order off-diagonal geometric phases are sensitive to the mass ordering and the Dirac CP violating phase, \(\delta \) . The third order off-diagonal geometric phase ( \(\Phi _{\mu e\tau }\) ) is invariant under any cyclic or non-cyclic permutations of flavour indices when the Dirac CP phase is zero. For non-zero \(\delta \) , \(\Phi _{\mu e\tau }(\delta )=\Phi _{e \mu \tau }(-\delta )\) , and the third order off-diagonal geometric phase is sensitive to the sign of \(\delta \) . The diagonal geometric phase is either 0 or \(\pi \) in the MSW resonance region and takes non-trivial values elsewhere for 2 flavour model. The transition between zero and \(\pi \) occurs at the point of complete oscillation inversion called the nodal point, where the diagonal geometric phase is not defined. Finally, in the two flavour model, the only second order off-diagonal geometric phase is a topological invariant and is always \(\pi \) .