Naturalness, vacuum stability, and leptogenesis in the minimal seesaw model

The right-handed neutrinos within the type-I seesaw mechanism can induce large radiative corrections to the Higgs mass, and naturalness arguments can then be used to set limits on their mass scale and Yukawa couplings. Driven by minimality, we consider the presence of two degenerate right-handed neutrinos. We compare the limits from naturalness with the ones from the stability of the electroweak vacuum and from lepton flavor violation. Implications from neutrinoless double beta decay are also discussed and renormalization effects for the light neutrino parameters are presented. Adding small perturbations to the degenerate heavy neutrino spectrum allows for successful leptogenesis.

Figure 1

One-loop correction to the Higgs mass from neutrino Yukawa couplings due to the heavy neutrinos. The line inside the loop shows the orientation of lepton number.

Figure 2

Naturalness contours in the $\mathrm{Im}[z]–M_N$ plane. The left (right) plot is for NH (IH). In the colored shaded regions, $\delta {\mu }^2$ is demanded to be below $(p\%\text{of}1\mathrm{TeV}{)}^2$, where $p=500$, 100, 50, 20, 10, 5, 1 (from top to bottom). The unshaded regions are disfavored by naturalness.

Figure 3

Running of the SM Higgs quartic coupling $\lambda$ for NH (left panel) and IH (right panel) in the minimal seesaw model. The red solid line corresponds to the SM running, while the blue dotted line corresponds to the $\mathrm{SM}+\text{heavy}$ neutrino contribution for $M_N=10^5\mathrm{GeV}$. For this mass ${(\mathrm{Tr}[Y_{\nu }^{†}{Y}_{\nu }])}^{1/2}=0.066$ is small and hence it almost overlaps with the SM line. In the inset both lines are shown with magnified view. For $M_N=2\times 10^4\mathrm{GeV}$ the maximal allowed trace value is larger, ${(\mathrm{Tr}[Y_{\nu }^{†}{Y}_{\nu }])}^{1/2}=0.328$, hence the effect is sizable and the corresponding pink line is visibly separated from the SM one.

Figure 4

Bounds on ${(\mathrm{Tr}[Y_{\nu }^{†}{Y}_{\nu }])}^{1/2}$ from metastability (red dashed), naturalness (blue dotted), LFV (dark red dot-dashed) and perturbativity (green solid).

Figure 5

Bounds on $\mathrm{Im}[z]$ as a function of $M_N$, with the same notation as in figure 4.

Figure 6

Running of solar and atmospheric mass-squared differences with the renormalization scale. The plots correspond to the central values of the oscillation parameters, the Majorana phase is chosen to be $\pi /4$.

Figure 7

Running of ${\sin }^2{\theta }_{12}$ with the renormalization scale for NH and IH. The plots correspond to the central value of ${\sin }^2{\theta }_{12}$, all other parameters are varied in their $3\sigma$ ranges.

Figure 8

Running of ${\sin }^2{\theta }_{23}$ with the renormalization scale for NH and IH. The plots correspond to the central value of ${\sin }^2{\theta }_{23}$, all other parameters are varied in their $3\sigma$ ranges.

Figure 9

Running of ${\sin }^2{\theta }_{13}$ with the renormalization scale for NH and IH. The plots correspond to the central value of ${\sin }^2{\theta }_{13}$, all other parameters are varied in their $3\sigma$ ranges.

Figure 10

Leptogenesis constraints on the minimal seesaw parameter space. The region above the brown shaded region cannot reproduce the correct BAU (either in magnitude or in sign). For comparison, the naturalness constraints [cf. Fig. 5] are also shown for $\delta {\mu }^2<(p\mathrm{TeV}{)}^2$, where $p=5$ (orange), 1 (blue) and 0.2 (magenta), the shaded regions being disfavored.

Figure 11

Contours of the correct BAU as a function of the low energy $CP$ phases. Here we have fixed $M_N=1\mathrm{TeV}$ and $\mathrm{Im}[z]=0.3$.