Three flavor long-wavelength vacuum oscillation solution to the solar neutrino problem

We investigate the current status of the long-wavelength vacuum oscillation solution to the solar neutrino problem and to what extent the presence of a third neutrino can affect and modify it. Assuming that the smaller mass squared difference that can induce such oscillations $\Delta m_{12}^2$ is in the range ${10}^{-11}-{10}^{-8}\hspace{0.5em}{\mathrm{eV}}^2$ and the larger one, $\Delta m_{23}^2,$ in the range relevant to atmospheric neutrino observations, we analyze the most recent solar neutrino data coming from Homestake, SAGE, GALLEX, GNO, and Super-Kamiokande experiments in the context of three neutrino generations. We include in our vacuum oscillation analysis the Mikheyev-Smirnov-Wolfenstein (MSW) effect in the Sun, which is relevant for some of the parameter space scrutinized. We have also performed, as an extreme exercise, the fit without Homestake data. While we found that the MSW effect basically does not affect the best fitted parameters, it significantly modifies the allowed parameter space for $\Delta m_{12}^2$ larger than $\sim 3\times {10}^{-10}\hspace{0.5em}{\mathrm{eV}}^2,$ in good agreement with the result obtained by Friedland in the case of two generations. Although the presence of a third neutrino does not essentially improve the quality of the fit, the solar neutrino data alone can give an upper bound on ${\theta }_{13},$ which is constrained to be less than $\sim 60°$ at 95 % C.L.