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We discuss the transition and survival probabilities in $3+1$ neutrino flavormixing scenario in presence of matter effects. We adopt the well-known OMSD(OneMass Scale Dominance) approximation to carry out our analysis. After that weperform series expansion about $\sin \theta_{13}$ term upto second order. Wefind that our results are consistent with the already existing $\alpha - \sin\theta_{13}$ approximated relations in the limit of vanishing $\alpha$ andphases involving sterile neutrinos. We also figure out that survival transitionprobability becomes independent of the fundamental and sterile CP phases underour formalism. Hence, it provides us a new way to look at only matter effectscontribution to oscillation probability. Also, the transition probability atthe same time gives an independent study of CP-violation arising from thesterile phases, in the vicinity of fundamental CP violation phase. We providethe relation for the atmospheric probability in the presence of matter byperforming the series expansion upto linear order about parameter $A (= 2EV)$,with V being the effective matter potential under OMSD approximation.

Study of matter effects in the presence of sterile neutrino using OMSD approximation