  The neutrino oscillation probabilities in vacuum and matter are discussed,
considering the framework of three active and one light sterile neutrinos. We
study in detail the rephasing invariants and CP asymmetry observables, and
investigate the four-neutrino oscillations in long-baseline neutrino
experiments, such as DUNE, NO$\nu$A and T2HK. Our results show that the matter
effect enhances quite a significantly the oscillation probabilities of electron
neutrino and electron antineutrino appearance channels within a certain energy
range, while no considerable change arises in the CP asymmetry analysis due to
the matter effect. Moreover, separation between the results with and without
the sterile neutrino is not so significant and that is also affected by
CP-violating phases. Comparing the results for these three experiments, all of
them have similar features, nevertheless, sizes and separations of the
oscillation probabilities in DUNE are bit larger.

High Energy Physics - Phenomenology