Electroweak naturalness in three-flavour Type I see-saw and implications for leptogenesis

In the Type I see-saw model, the naturalness requirement that corrections to the electroweak $\mu$ parameter not exceed 1 TeV results in a rough bound on the lightest right-handed neutrino mass, $M_{N_1}\lesssim 3\times {10}^7$ GeV. In this letter we derive generic bounds applicable in any three-flavour Type I see-saw model. We find $M_{N_1}\lesssim 4\times {10}^7$ GeV and $M_{N_2}\lesssim 7\times 10^7$ GeV. In the limit of one massless neutrino, there is no naturalness bound on $M_{N_3}$ in the Poincare protected decoupling limit. Our results confirm that no Type I see-saw model can explain the observed neutrino masses and baryogenesis via hierarchical ($N_1$-, $N_2$-, or $N_3$-dominated) thermal leptogenesis while remaining completely natural.