Three-neutrino mixing and combined vacuum oscillations and MSW transitions of solar neutrinos

Assuming three-flavor neutrino mixing takes place in vacuum, we investigate the possibility that the solar ${\nu }_e$ take part in MSW transitions in the Sun due to $\Delta m_{31}^2\sim {(10\mathrm{}}^{-7}-{10}^{-4})$ eV${}^2$, followed by long wavelength vacuum oscillations on the way to the Earth, triggered by $\Delta m_{21}^2$ (or $\Delta m_{32}^2)$ $\sim {(10\mathrm{}}^{-12}-{10}^{-10})$ eV${}^2$, $\Delta m_{31}^2$ and $\Delta m_{21}^2$ $\left(\Delta m_{32}^2\right)$ being the corresponding neutrino mass squared differences. The solar ${\nu }_e$ survival probability is shown to be described in this case by a simple analytic expression. Depending on whether the vacuum oscillations are due to $\Delta m_{21}^2$ or $\Delta m_{32}^2$ there are two very different types of interplay between the MSW transitions and the vacuum oscillations of the solar ${\nu }_e.$ Performing an analysis of the most recently published solar neutrino data we have found several qualitatively new solutions of the solar neutrino problem of the hybrid MSW transitions + vacuum oscillations type. The solutions differ in the way the $\mathrm{pp}$, ${}^7$Be and ${}^8$B neutrino fluxes are affected by the transitions in the Sun and the oscillations in vacuum. The specific features of the new solutions are discussed. One of the distinctive predictions of the hybrid MSW + vacuum oscillation solutions found is the existence of strong and very characteristic distortions of the spectrum of ${}^8\mathrm{B}$ neutrinos.