Tetrahedral Symmetry A${}_4$ in Anti-SU(5) GUT

Abstract

We construct a flavor model in an anti-SU(5) GUT with a tetrahedral symmetry $A_4$. We choose a basis where $Q_{textem}=-\frac{1}{3}$ quarks and charged leptons are already mass eigenstates. This choice is possible from the $A_4$ symmetry. Then, matter representation ${\overline{10}}_{-1}^{\mathrm{matter}}$ contains both a quark doublet and a heavy neutrino $N$, which enables us to use the $A_4$ symmetry to both $Q_{textem}=+\frac{2}{3}$ quark masses and neutrino masses (through the see-saw via $N$). This is made possible because the anti-SU(5) breaking is achieved by the Higgs fields transforming as anti-symmetric representations of SU(5), ${\overline{10}}_{-1}^H\oplus 10_{+1}^H$, reducing the rank-5 anti-SU(5) group down to the rank-4 standard model group \smg. For possible mass matrices, the $A_4$ symmetry predictions on mass matrices at field theory level are derived. Finally, an illustration from string compactification is presented.

14 pages of LaTeX file with 2 figures