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We present a comprehensive study of the three-active plus $N$ sterileneutrino model as a framework for constraining leptonic unitarity violationinduced at energy scales much lower than the electroweak scale. We formulate aperturbation theory with expansion in small unitarity violating matrix element$W$ while keeping (non-$W$ suppressed) matter effect to all orders. We showthat 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 typicalaccelerator based long-baseline neutrino oscillation experiment, one can derivea very simple form of the oscillation probability which consists only ofzeroth-order terms with the unique exception of probability leaking term$\mathcal{C}_{\alpha \beta}$ of $\mathcal{O} (W^4)$. We argue, based on ourexplicit computation to fourth-order in $W$, that all the other terms arenegligibly small after taking into account the suppression due to the masscondition for sterile states, rendering the oscillation probability {\emsterile-sector model independent}. Then, we identify a limited energy region inwhich this suppression is evaded and the effects of order $W^2$ corrections maybe observable. Its detection would provide another way, in addition todetecting $\mathcal{C}_{\alpha \beta}$, to distinguish between low-scale andhigh-scale unitarity violation. We also solve analytically the zeroth-ordersystem in matter with uniform density to provide a basis for numericalevaluation of non-unitary neutrino evolution.

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