Unitarity test of lepton mixing via energy dependence of neutrino oscillation

We study the method to test the unitarity of the lepton mixing matrix by using only the long baseline neutrino oscillation experiments, such as the combination of the T2HK experiment and the one with the $ν_e$ beam from a future neutrino factory at J-PARC. Without a specific parametrization, one can directly extract the elements of the lepton mixing matrix by observing the energy dependence of the oscillation probabilities. A non-trivial test of the unitarity under the three-generation assumption can thus be made possible by examining the orthogonality in a similar manner to the unitarity triangle in the quark sector. As the first trial, we perform the analysis based on the simplified situation where the matter effects in the neutrino oscillation can be neglected. Under this simplified analysis, we demonstrate the observation of the unitarity violation in the $3\times3$ part of the lepton mixing matrix for a parameter set in the four-generation model. The statistically most significant measurement can be provided by the energy dependences of the combination of the CP conjugate modes, $ν_μ\to ν_e$ and $\bar ν_μ\to \bar ν_e$, at T2HK and, independently, by the T conjugate modes, $ν_μ\to ν_e$ and $ν_e \to ν_μ$, with the latter measured at the neutrino factory experiments.