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.

💡 Research Summary

The authors propose a novel, parametrization‑independent strategy to test the unitarity of the lepton mixing (PMNS) matrix using only long‑baseline neutrino‑oscillation experiments. They focus on the combination of the upcoming Hyper‑Kamiokande (T2HK) experiment and a future νₑ beam from a J‑PARC neutrino factory. The key idea is to exploit the full energy dependence of appearance probabilities (ν_μ→νₑ, (\barν_μ→\barνₑ), and νₑ→ν_μ) rather than relying on a reduced set of oscillation parameters.

In the standard three‑generation picture the 3×3 PMNS matrix U is assumed unitary, leading to nine unitarity constraints that can be visualized as triangles in the complex plane (the “unitarity triangle”). If additional sterile states exist, the full (3+M)×(3+M) mixing matrix V remains unitary, but its 3×3 sub‑matrix U is generally non‑unitary. Existing global fits treat U as unitary, so they cannot directly test this assumption.

The paper expands the ν_μ→νₑ appearance probability in vacuum (matter effects are neglected for simplicity) up to second order in Δm²₂₁/Δm²₃₁ and |Uₑ₃|. The probability can be written as a linear combination of four known energy‑dependent functions: P(E)=C₁ sin²Δ₃₁ + C₂ Δ₂₁ sin2Δ₃₁ + C₃ Δ₂₁² + C₄ Δ₂₁ sin²Δ₃₁, where the coefficients C₁–C₄ are independent combinations of the matrix elements: C₁=4|U_{μ3}U_{e3}*|², C₂=4 Re