Deviations from tribimaximal mixing: Charged lepton corrections and renormalization group running

We analyze the effects of charged lepton corrections and renormalization group (RG) running on the low energy predictions of theories which accurately predict tri-bimaximal neutrino mixing at the high energy scale. In particular we focus on grand unified theory (GUT) inspired seesaw models with accurate tri-bimaximal neutrino mixing at the GUT scale, in which the charged lepton corrections are Cabibbo-like and give rise to sum rules valid at the GUT scale. We study numerically the RG corrections to a variety of such neutrino mixing sum rules in order to assess their accuracy and reliability when comparing them to future low energy neutrino oscillation experiments. Our results indicate that the RG corrections to neutrino mixing sum rules are typically small (less than one degree), at least in the examples studied with hierarchical neutrinos.

Figure 1

Evolution of sum rules ${\Gamma }_1$, ${\Gamma }_2$, ${\Gamma }_3$ for Cabibbo-like charged lepton corrections for large $\tan (\beta )=50$. Panel (a) shows the running at ${\theta }_{12}^E=5°$ and ${\lambda }_{12}^E=0°$, while panel (b) shows the evolution at ${\theta }_{12}^E=8°$ and ${\lambda }_{12}^E=0°$. In panel (c), the running is at ${\theta }_{12}^E=5°$ and ${\lambda }_{12}^E=15°$ while panel (d) shows the evolution at ${\theta }_{12}^E=5°$, ${\lambda }_{12}^E=30°$. Note how the graphs for ${\Gamma }_1$ and ${\Gamma }_2$ completely overlap in (a) and (b).

Figure 2

Evolution of sum rules ${\Gamma }_1$, ${\Gamma }_3$ for Cabibbo-like charged lepton corrections for various $\tan (\beta )$. Running of ${\Gamma }_1$ is shown in panel (a) and ${\Gamma }_3$ is shown in panel (b) at ${\theta }_{12}^E=5°$, ${\lambda }_{12}^E=0°$ and different values of $\tan (\beta )$. Note the expanded (and different) vertical scales used in these two figures; in all cases the corrections are less than 1°.

Figure 3

Evolution of the deviation parameters $r$, $s$, $a$ from the GUT scale to the electroweak scale without and with charged lepton corrections for large $\tan (\beta )=50$. Panel (a) shows the running of the deviation parameters without any charged lepton corrections, ${\theta }_{12}^E=0°$, ${\lambda }_{12}^E=0°$. Panel (b) shows the running of the deviation parameters in the presence of Cabibbo-like charged lepton corrections, ${\theta }_{12}^E=5°$, ${\lambda }_{12}^E=15°$.

Figure 4

Evolution of the sum rules ${\sigma }_1$ and ${\sigma }_2$ from the GUT scale to the electroweak scale without and with charged lepton corrections for large $\tan (\beta )=50$. Panel (a) shows the running without any charged lepton corrections, ${\theta }_{12}^E=0°$ and ${\lambda }_{12}^E=0°$. In panel (b) the evolution is in the presence of Cabibbo-like charged lepton corrections, ${\theta }_{12}^E=5°$ and ${\lambda }_{12}^E=30°$.

Figure 5

The evolution of the third row deviation parameters ${\xi }_1$, ${\xi }_2$, and ${\xi }_3$ from the GUT scale to the electroweak scale at $\tan (\beta )=50$ for Cabibbo-like and more general charged lepton corrections. Panel (a) shows the result with Cabibbo-like charged lepton corrections with ${\theta }_{23}^E=0°$, ${\lambda }_{23}^E=0°$, ${\theta }_{12}^E=5°$, and ${\lambda }_{12}^E=30°$. Panel (b) is for the case of more general charged lepton corrections with ${\theta }_{23}^E=2°$, ${\lambda }_{23}^E=30°$, ${\theta }_{12}^E=5°$, and ${\lambda }_{12}^E=30°$.

Figure 6

Running of the TB deviation parameters and their related sum rules from the GUT scale to the electroweak scale in the presence of more general charged lepton corrections, ${\theta }_{12}^E=5°$, ${\lambda }_{12}^E=30°$, ${\lambda }_{23}^E=30°$, ${\theta }_{23}^E=2°$, and $\tan (\beta )=50$. Panel (a) shows the evolution of the TB deviation parameters. Panel (b) displays the sum rules ${\sigma }_1$ and ${\sigma }_2$. Note that ${\sigma }_1=0$ at the GUT scale even in the presence of the more general charged lepton corrections.

Figure 7

Sum rules with Majorana phases. The running of ${\sigma }_i$ sum rules and TB deviation parameters are for ${\theta }_{12}^E=0°$, ${\lambda }_{12}^E=0°$, $\tan (\beta )=50$, ${\delta }_1={120}^{°}$, and ${\delta }_2={60}^{°}$.

Figure 8

Sum rules with heavy sequential dominance. This figure shows the evolution of the ${\Gamma }_i$ sum rules for $\tan \beta =50$ [panel (a)], the running of ${\Gamma }_1$ and ${\Gamma }_3$ in terms of $\tan \beta$ [panels (b), (c)] from the GUT scale to the electroweak scale for the case of Cabibbo-like charged lepton corrections with ${\theta }_{12}^E=5°$, ${\lambda }_{12}^E=0°$.