Influence of second-order corrections to the energy-dependence of neutrino flavor conversion formulae

We discuss the {\em intermediate} wave-packet formalism for analytically quantifying the energy dependence of the two-flavor conversion formula that is usually considered for analyzing neutrino oscillations and adjusting the focusing horn, target position and/or detector location of some flavor conversion experiments. Following a sequence of analytical approximations where we consider the second-order corrections in a power series expansion of the energy, we point out a {\em residual} time-dependent phase which, in addition to some well known wave-packet effects, can subtly modify the oscillation parameters and limits. In the present precision era of neutrino oscillation experiments where higher precision measurements are required, we quantify some small corrections in neutrino flavor conversion formulae which lead to a modified energy-dependence for $\nu_{\mu}\leftrightarrow\nu_{e}$ oscillations.