Renormalization group evolution in the type I and type II seesaw model

We carefully analyze the renormalization group equations in the $\mathrm{\text{type}}\mathrm{I}+\mathrm{II}$ seesaw scenario in the extended standard model and the minimal supersymmetric standard model. Furthermore, we present analytic formulas of the mixing angles and phases and discuss the renormalization group effect on the different mixing parameters in the type II seesaw scenario. The renormalization group equations of the angles are independent of the Majorana phases. In addition, there is a contribution which is proportional to the mass squared difference for a hierarchical spectrum. This is in contrast to the inverse proportionality to the mass squared difference in the effective field theory case.

Figure 1

Feynman diagrams, which are not considered in 64.

Figure 2

Supergraphs, which contribute to the wave function renormalization.

Figure 3

As input values, we have chosen tribimaximal mixing at the GUT scale, $m_1=0\mathrm{eV}$, $\Delta m_{\mathrm{atm}}^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$, $\Delta m_{\mathrm{sol}}^2=1.2\times {10}^{-4}{\mathrm{eV}}^2$, $M_{\Delta }={10}^{10}\mathrm{GeV}$, and ${\Lambda }_6=2.5\times {10}^{-5}{M}_{\Delta }$, corresponding to $⟨\Delta ⟩=0.15\mathrm{eV}$. As we are only interested in showing the generic feature of the RG evolution, we choose the Higgs self-couplings to be ${\Lambda }_{1,2,4,5}=0.5$ for simplicity, since they only indirectly influence the RG evolution of the angles and the flavor-dependent part of the RG equations of the masses. The shadowed area indicates the region where the Higgs triplet is present. It is integrated out at the border between the shadowed and the white area.

Figure 4

Plot showing the evolution of the leptonic mixing angles in the SM. As input values, we have chosen tribimaximal mixing at the GUT scale, $m_1=0\mathrm{eV}$, $\Delta m_{\mathrm{atm}}^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$, $\Delta m_{\mathrm{sol}}^2=1.2\times {10}^{-4}{\mathrm{eV}}^2$, $M_{\Delta }={10}^{10}\mathrm{GeV}$, and ${\Lambda }_6=2.5\times {10}^{-5}{M}_{\Delta }$, corresponding to $⟨\Delta ⟩=0.15\mathrm{eV}$. As we are only interested in showing the generic feature of the RG evolution, we choose the Higgs self-couplings to be ${\Lambda }_{1,2,4,5}=0.5$ for simplicity (In a realistic model, the parameters ${\Lambda }_i$ have to satisfy certain relations to produce the desired vevs.), since they only indirectly influence the RG evolution of the angles and the flavor-dependent part of the RG equations of the masses. In the MSSM with small $\tan \beta$, the plot looks similar due to $C_{\Delta }$ being the same in both theories. The shadowed area indicates the region where the Higgs triplet is present. It is integrated out at the border between the shadowed and the white area.

Figure 5

In the full type II seesaw case, there is a complicated interplay between the two contributions to the neutrino mass matrix. Here, we just plot an example for the following initial values at the GUT scale: $M_{\Delta }={10}^{10}\mathrm{GeV}$, ${\Lambda }_6=4.56\times {10}^9\mathrm{GeV}$, $m_1=0.02\mathrm{eV}$, $\Delta m_{\mathrm{sol}}^2=1.5\times {10}^{-4}{\mathrm{eV}}^2$, $\Delta m_{\mathrm{atm}}^2=5.5\times {10}^{-3}{\mathrm{eV}}^2$, ${\theta }_{12}={\theta }_{23}=\frac{\pi }{4}$, ${\theta }_{13}=0$, $\delta ={\phi }_1={\phi }_2=0$, $Y_{\nu }=0.37\mathrm{diag}({10}^{-2},{10}^{-1},1)$, where $Y_{\Delta }$ is chosen diagonal $Y_{\Delta }=\mathrm{diag}(1.3\times {10}^{-5},1.5\times {10}^{-5},5.1\times {10}^{-5})$ and $M$ is chosen appropriately to produce bimaximal mixing. The differently shaded areas indicate the different energy ranges of the various effective field theories. At each border, a particle, either a right-handed neutrino or the Higgs triplet, is integrated out.