Atmospheric neutrinos at Super-Kamiokande and parametric resonance in neutrino oscillations

We consider the three neutrino oscillations of atmospheric neutrinos in the scheme with a mass hierarchy and a small admixture of the electron neutrino in the heavy mass eigenstate characterized by the mixing angle $\theta_{13}$. We show that for $\Delta m^2 = (1 - 3)\times 10^{-3}$ eV$^2$ preferred by the SuperKamiokande data and $\sin^2 2\theta_{13} \sim 0.03 - 0.15$ the oscillations of multi-GeV neutrinos in the subdominant $\nu_{\mu} \leftrightarrow \nu_e$ mode are parametrically enhanced. The resonance matter effects and in particular, the parametric enhancement, lead to an observable excess of the e-like events with a specific zenith angle dependence. The up-down asymmetry of the multi-GeV e-like events can reach 15 % for $|\cos \Theta_e| > 0.2 $ and up to 30 % for $|\cos \Theta_e| > 0.6 $ where $\Theta_e$ is the zenith angle of the electron. The parametric effect may be relevant for the interpretation of the SuperKamiokande data. Its identification would allow one to further restrict the neutrino masses and lepton mixing.