$(L_e-L_{\mu }-L_{\tau })$ discrete symmetry for heavy right-handed neutrinos and degenerate leptogenesis

The degenerate leptogenesis is studied when the degeneracy in two of the heavy right-handed neutrinos [the third one is irrelevant if $\mu -\tau$ symmetry is assumed] is due to $\bar{L}\equiv (L_e-L_{\mu }-L_{\tau })$ discrete symmetry. It is shown that a sizeable leptogenesis asymmetry $(ϵ\succsim {10}^{-6})$ is possible. The level of degeneracy required also predicts the Majorana phase needed for the asymmetry. Since it is the same phase, which appears in the double $\beta$% -decay and this prediction is testable. Implication of non-zero reactor angle ${\theta }_{13}$ are discussed. It is shown that the contribution from $$ to leptogenesis asymmetry parameter may even dominate.An accurate measument of $\sin ^{2}\theta_{13}$ would have important implications for the mass degeneracy of heavy right-handed neutrinos.