Large non-perturbative effects of small Δm221/Δm231and sin θ13on neutrino oscillation and CP violation in matter

In the framework of three generations, we consider the CP violation inneutrino oscillation with matter effects. At first, we show that thenon-perturbative effects of two small parameters, \Delta m_{21}^2/\Deltam_{31}^2 and \sin \theta_{13}, become more than 50% in certain ranges of energyand baseline length. This means that the non-perturbative effects should beconsidered in detailed analysis in the long baseline experiments. Next, wepropose a method to include these effects in approximate formulas foroscillation probabilities. Assuming the two natural conditions,\theta_{23}=45^\circ and the fact that the matter density is symmetric, a setof approximate formulas, which involve the non-perturbative effects, has beenderived in all channels.Comment: 25 pages, 4 figures, version to appear in JHE