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 eV(2) ae(2) m (J) (2) ae(2) (1-10) GeV2 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 (alpha beta) of (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 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 (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.