Trimaximal $\mu \text{−}\tau$ reflection symmetry

The $\mu \text{−}\tau$ reflection symmetry $({\nu }_e,{\nu }_{\mu },{\nu }_{\tau })\to ({\overline{\nu }}_e,{\overline{\nu }}_{\tau },{\overline{\nu }}_{\mu })$ and the TM1 mixing (a Pontecorvo–Maki–Nakagawa–Sakata matrix with the first column fixed to the tribimaximal form) are both well compatible with experiments. If both approaches are simultaneously assumed, all lepton mixing parameters except for ${\theta }_{13}$ are predicted. In particular, one expects maximal $CP$ violation ($|\delta |=90°$), maximal atmospheric mixing (${\theta }_{23}=45°$), a slightly less-than-tribimaximal solar mixing angle (${\theta }_{12}\approx 34°$), as well as values of 0 or $\pi$ for the two Majorana phases. We study the renormalization stability of this highly predictive framework when neutrino mass is described by an effective Weinberg operator and by the type I seesaw mechanism, both in the standard model and with supersymmetry.

Figure 1

Prediction on the effective mass $|M_{ee}|$, according to Eqs. (31) and (32). In the notation $N_{\pm \pm }$ ($I_{\pm \pm }$), $N/I$ standards for normal/inverted mass ordering respectively and the subscripts are the signs of two Majorana phases $e^{i{\alpha }_{21}}$ and $e^{i{\alpha }_{31}}$ (${\alpha }_{21}$, ${\alpha }_{31}$ are always 0 or $\pi$ in this model). The light green region is the bound from the global fit, taken from [76].

Figure 2

RG running of the mixing angles (left panel) and the Dirac/Majorana phases (right panel) in the SM for the normal hierarchy and $m_L=0.05\mathrm{eV}$.

Figure 3

RG corrections for all the 8 cases, $N/I$ for normal/inverted hierarchy and “$+/-$” for $e^{i\alpha }=+1/-1$ where $\alpha$ stands for the two Majorana phases. The result depends on the lightest neutrino mass, which is set at (1,20,40,60) meV for points colored from green to red.

Figure 4

Same as Fig. 3 except that the SM is replaced with the MSSM for $\tan \beta =20$.

Figure 5

The distributions of RG correction in the SM extended by the type I seesaw.

Figure 6

The distributions of RG correction in the MSSM with $\tan \beta =20$ extended by the type I seesaw.