Electroweak naturalness in the three-flavor type I seesaw model and implications for leptogenesis

In the type I seesaw 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, $M_{N_1}\lesssim 3\times 10^7\mathrm{GeV}$. In this paper we derive generic bounds applicable in any three-flavor type I seesaw model. We find $M_{N_1}\lesssim 4\times 10^7\mathrm{GeV}$ and $M_{N_2}\lesssim 7\times 10^7\mathrm{GeV}$. In the limit of one massless neutrino, there is no naturalness bound on $M_{N_3}$ in the Poincaré protected decoupling limit. Our results confirm that no type I seesaw 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.

Figure 1

Loop diagram leading to $\delta {\mu }^2$.

Figure 2

As a function of the lightest neutrino mass in normal ordering (NO), shown as darker/hatched/lighter (red/blue-hatched/green color) is the region of attainable values for the $B_3\ge B_2\ge B_1$ upper bounds on right-handed neutrino masses, by requiring that the corrections to the electroweak $\mu$ parameter be no greater than 1 TeV [Eq. (9)]. The regions assume the orthogonal matrix $R$ is real. Thick solid lines show the case when $R=\mathbb{I}$. The case for complex $R$ is similar, except there is no lower limit to the $B_j$ (see text).

Figure 3

As in Fig. 2 but as a function of the lightest neutrino mass in inverted ordering (IO). Note that the thick blue line is obscured by the thick green line.