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It is shown that it is possible to obtain the analytical expression for theeffective mixing angle in matter using the formalism which was developed byKimura, Takamura and Yokomakura for the neutrino oscillation probability inmatter with constant density. If we assume that the imaginary part of theintegral of the difference of the energy eigenvalues of the two levels at eachlevel-crossing is given by the ratio $\gamma$ of the difference of the energyeigenvalues of the two levels to the derivative of the effective mixing angleat the level-crossing, then the nonadiabatic contribution to the oscillationprobability can be expressed analytically by this formalism. We give oneexample in which the energy eigenvalues cannot be expressed as roots of aquadratic equation and we show that our assumption is correct in theapproximation of the small mixing angle.

On nonadiabatic contributions to the neutrino oscillation probability and the formalism by Kimura, Takamura and Yokomakura