Origin of symmetric PMNS and CKM matrices

The Pontecorvo–Maki–Nakagawa–Sakata and Cabibbo–Kobayashi–Maskawa matrices are phenomenologically close to symmetric, and a symmetric form could be used as zeroth-order approximation for both matrices. We study the possible theoretical origin of this feature in flavor symmetry models. We identify necessary geometric properties of discrete flavor symmetry groups that can lead to symmetric mixing matrices. Those properties are actually very common in discrete groups such as $A_4$, $S_4$, or $\mathrm{\Delta }(96)$. As an application of our theorem, we generate a symmetric lepton mixing scheme with ${\theta }_{12}={\theta }_{23}=36.21°$; ${\theta }_{13}=12.20°$, and $\delta =0$, realized with the group $\mathrm{\Delta }(96)$.

Figure 1

The geometrical relation of the mirror planes and the bisecting planes. The mirrors are placed on the $y-z$, $z-x$, or $x-y$ plane. The two round disks are called bisecting planes because they bisect all the square mirror planes. The bisecting planes are boundaries of octants.

Figure 2

The complete collection of all six possible bisecting planes and their geometrical relation with the mirror planes. A rotational symmetry with its axis on one of these bisecting planes can give, according to Theorem A, a symmetric mixing $|U|=|U{|}^T$.

Figure 3

Tetrahedron symmetry. The dark blue circles show the bisecting planes. The axes of the 120° rotational symmetries of the tetrahedron are on those planes; therefore, according to Theorem A, the tetrahedral group as a flavor symmetry can produce a symmetric mixing matrix $|U|=|U{|}^T$.

Figure 4

Octahedron symmetry. Similar to Fig. 3, according to Theorem A, the octahedral group can also be used to produce a symmetric mixing matrix $|U|=|U{|}^T$.