Large Changes of Atmospheric Mixing Angle ${\theta }_{23}$ with Four Flavor Mixing from Planck Scale Effects

Abstract

We consider the effects of Planck scale on four flavour neutrino mixings. The gravational interaction at $M_x=M_{\mathrm{planck}}$, we find that for degenerate neurino mass order, the Planck scale effects changes the mixing angle ${\theta }_{23}^{\prime }$, ${\theta }_{12}^{\prime }$ values and ${\theta }_{13}^{\prime }$, ${\theta }_{14}^{\prime }$, ${\theta }_{34}^{\prime }$, ${\theta }_{24}^{\prime }$ are unchanged above the GUT scale. In this paper, we study neutrino mixing in four flavor above the GUT scale.

Comments: 10 pages