Neutrino Oscillation Probabilities in Matter with Direct and Indirect Unitarity Violation in the Lepton Mixing Matrix

In the presence of both direct and indirect unitarity violation in the lepton mixing matrix, we derive a complete set of series expansion formulas for neutrino oscillation probabilities in matter of constant density. Expansions in the mass hierarchy parameter $\alpha \equiv \Delta m_{21}^{2} / \Delta m_{31}^{2}$ and those unitarity violation parameters $s^{2}_{ij}$ (for i = 1, 2, 3 and j = 4, 5, 6) up to the first order are studied in this paper. We analyse the accuracy of the analytical series expansion formulas in different regions of L / E. A detailed numerical analysis is also performed, of which the different effects of the direct and the indirect unitarity violation are particularly emphasized. We also study in this paper the summed $\nu^{}_{\alpha} \rightarrow \nu^{}_{e, \nu, \tau}$ probabilities, whose deviation from the unity provides a definite signal of the unitarity violation.