Non-unitary evolution of neutrinos in matter and the leptonic unitarity test

We present a comprehensive study of the three-active plus $N$ sterile neutrino model as a framework for constraining leptonic unitarity violation induced at energy scales much lower than the electroweak scale. We formulate a perturbation theory with expansion in small unitarity violating matrix element $W$ while keeping (non-$W$ suppressed) matter effect to all orders. We show that under the same condition of sterile state masses $0.1\, \text{eV}^2 \lesssim m^2_{J} \lesssim (1-10)\, \text{GeV}^2$ as in vacuum, assuming typical accelerator based long-baseline neutrino oscillation experiment, one can derive a very simple form of the oscillation probability which consists only of zeroth-order terms with the unique exception of probability leaking term $\mathcal{C}_{\alpha \beta}$ of $\mathcal{O} (W^4)$. We argue, based on our explicit computation to fourth-order in $W$, that all the other terms are negligibly small after taking into account the suppression due to the mass condition for sterile states, rendering the oscillation probability {\em sterile-sector model independent}. Then, we identify a limited energy region in which this suppression is evaded and the effects of order $W^2$ corrections may be observable. Its detection would provide another way, in addition to detecting $\mathcal{C}_{\alpha \beta}$, to distinguish between low-scale and high-scale unitarity violation. We also solve analytically the zeroth-order system in matter with uniform density to provide a basis for numerical evaluation of non-unitary neutrino evolution.