Theoretical Aspect of Nonunitarity in Neutrino Oscillation

Nonunitarity can arise in neutrino oscillation when the matrix with elements $\mathbf{U}_{\alpha i}$ which relate the neutrino flavor $\alpha$ and mass $i$ eigenstates is not unitary when sum over the kinematically accessible mass eigenstates or over the three Standard Model flavors. We review how high scale nonunitarity arises after integrating out new physics which is not accessible in neutrino oscillation experiments. In particular, we stress that high scale unitarity violation is only apparent and what happens is that the neutrino flavor states become nonorthogonal due to new physics. Since the flavor space is complete, unitarity has to be preserved in time evolution and that the probabilities of a flavor state oscillates to all possible flavor states always sum up to unity. We highlight the need to modify the expression of probability to preserve unitarity when the flavor states are nonorthogonal. We will continue to call this high scale unitarity violation in reference to a nonunitary $\mathbf{U}$. We contrast this to the low scale nonunitarity scenario in which there are new states accessible in neutrino oscillation experiments but the oscillations involving these states are fast enough such that they are averaged out. We further derive analytical formula for the neutrino oscillation amplitude involving $N$ neutrino flavors without assuming a unitarity $\mathbf{U}$ which allows us to prove a theorem that if $\left(\mathbf{U}\mathbf{U}^{\dagger}\right)_{\alpha\beta}=0$ for all $\alpha\neq\beta$, then the neutrino oscillation probability in an arbitrary matter potential is indistinguishable from the unitarity scenario. Independently of matter potential, while nonunitarity effects for high scale nonunitarity scenario disappear as $\left(\mathbf{U}\mathbf{U}^{\dagger}\right)_{\alpha\beta}\to 0$ for all $\alpha\neq\beta$, low scale nonunitarity effects can remain.