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. We contrast this to the low scale nonunitary 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. Then we derive analytical formula for the neutrino oscillation probability amplitude for an arbitrary flavor of neutrinos without assuming unitarity. This result 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 unitary scenario. The main implication is that nonunitary effects are proportional $\left(\mathbf{U}\mathbf{U}^{\dagger}\right)_{\alpha\beta}$ with $\alpha\neq\beta$ and disappearance experiments $\nu_{\beta}\to\nu_{\alpha}$ are necessary for their discovery. Independently of matter potential, while nonunitary effects for high scale nonunitary scenario disappear as $\left(\mathbf{U}\mathbf{U}^{\dagger}\right)_{\alpha\beta}\to0$ for all $\alpha\neq\beta$, low scale nonunitary effects remain.