Unifying Residual $\mathbb Z^{23}_2 \otimes \mathbb Z^{12}_2$ Symmetries and Quark-Lepton Complementarity

Abstract

The residual $\mathbb Z^{23}_2$ ($\mathbb Z^{\mu\tau}_2$) and $\mathbb Z^{12}_2$ ($\mathbb Z^s_2$ or $\overline{\mathbb Z}^s_2$) symmetries can not only apply to the lepton sector, but also explain the mainly $1$--$2$ mixing in the quark sector. With both up and down quarks subject to the same $\mathbb Z^{23}_2$ symmetry, which interchanges the second and third generations, the $1$--$3$ and $2$--$3$ mixing angles in CKM vanish while the Cabibbo angle is dictated by two residual $\mathbb Z^{12}_2$ symmetries. It turns out that the quark-lepton complementarity conditions are natural consequences of unifying residual $\mathbb Z^{23}_2 \otimes \mathbb Z^{12}_2$ symmetries in both lepton and quark sectors.