Constraining the minimal type-III seesaw model with naturalness, lepton flavor violation, and electroweak vacuum stability

We study the minimal type-III seesaw model in which we extend the standard model by adding two $SU(2{)}_L$ triplet fermions with zero hypercharge to explain the origin of the nonzero neutrino masses. We show that the naturalness conditions and the limits from lepton flavor violating decays provide very stringent bounds on the model parameters along with the constraints from the stability/metastability of the electroweak vacuum. We perform a detailed analysis of the model parameter space including all the constraints for both normal as well as inverted hierarchies of the light neutrino masses. We find that most of the regions that are allowed by lepton flavor violating decays and naturalness fall in the stable/metastable region depending on the values of the standard model parameters.

Figure 1

Naturalness contours in the $\mathrm{Im}[z]\text{−}{M}_{\mathrm{\Sigma }}$ plane. Plot (a) is for NH, and plot (b) is for IH. In the shaded regions, $\delta {\mu }^2$ is less than $p\%$ 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 2

Bounds on $z$ from lepton flavor violation (blue dotted line) and naturalness (purple, magenta, and brown solid lines). Plot (a) is for NH, and plot (b) is for IH. The unshaded region is allowed by both LFV as well as naturalness bounds.

Figure 3

RG evolution of the Higgs quartic coupling. The plot on the left side shows the running of $\lambda$ for different values of $M_t$ with fixed $M_{\mathrm{\Sigma }}$ and $\mathrm{Tr}[Y_{\mathrm{\Sigma }}^{†}{Y}_{\mathrm{\Sigma }}{]}^{\frac{1}{2}}$, whereas the plot on the right side shows the running of $\lambda$ for different values of $\mathrm{Tr}[Y_{\mathrm{\Sigma }}^{†}{Y}_{\mathrm{\Sigma }}{]}^{\frac{1}{2}}$ with $M_{\mathrm{\Sigma }}$ and $M_t$ fixed. For both plots, we have taken $M_{\mathrm{\Sigma }1}=M_{\mathrm{\Sigma }2}=M_{\mathrm{\Sigma }}={10}^7\mathrm{GeV}$.

Figure 4

The phase diagram in the $\mathrm{Tr}[Y_{\mathrm{\Sigma }}^{†}{Y}_{\mathrm{\Sigma }}{]}^{\frac{1}{2}}\text{−}{M}_{\mathrm{\Sigma }}$ plane for NH. Here, we have used the central values of $M_t$, $M_h$, and ${\alpha }_s$. The color coding of the lines (blue, purple, magenta, and brown) is the same as in Fig. 2. The horizontal red solid line separates the unstable and the metastable regions of the EW vacuum.

Figure 5

The phase diagram in the $\mathrm{Tr}[Y_{\mathrm{\Sigma }}^{†}{Y}_{\mathrm{\Sigma }}{]}^{\frac{1}{2}}\text{−}{M}_{\mathrm{\Sigma }}$ plane for NH. The plot on the left (right) side gives the most liberal (stringent) bound from vacuum stability with minimum (maximum) value of $M_t$ and maximum (minimum) values of $M_h$ and ${\alpha }_s$. The color coding of the lines (blue, purple, magenta, and brown) is the same as in Fig. 2. The horizontal red solid line separates the unstable and the metastable regions of the EW vacuum.

Figure 6

The phase diagram in the $M_t\text{−}\mathrm{Tr}[Y_{\mathrm{\Sigma }}^{†}{Y}_{\mathrm{\Sigma }}{]}^{\frac{1}{2}}$ plane for the NH for the central values of $M_h$ and ${\alpha }_s$. The dashed lines separate the metastable and the unstable regions, whereas the solid lines separate the stable and the metastable regions. The three colors are for three different values of $M_{\mathrm{\Sigma }}$. The two vertical lines give the LFV and naturalness bounds for $M_{\mathrm{\Sigma }}={10}^4\mathrm{GeV}$, and the region on the left of the LFV line (red) is allowed by both.

Figure 7

The phase diagram in the $M_t\text{−}{M}_h$ plane for two different values of $\mathrm{Tr}[Y_{\mathrm{\Sigma }}^{†}{Y}_{\mathrm{\Sigma }}{]}^{\frac{1}{2}}$ and $M_{\mathrm{\Sigma }}={10}^4\mathrm{GeV}$. The ellipses correspond to the allowed values of $M_t$ and $M_h$ at $1\sigma$, $2\sigma$, and $3\sigma$.

Figure 8

Dependence of confidence level at which the EW vacuum stability is excluded/allowed on $\mathrm{Tr}[Y_{\mathrm{\Sigma }}^{†}{Y}_{\mathrm{\Sigma }}{]}^{\frac{1}{2}}$ for different values of ${\alpha }_s$ and $M_{\mathrm{\Sigma }}$.