Abstract
We construct a flavor model in an anti-SU(5) GUT with a tetrahedral symmetry $A_4$. We choose a basis where $Q_{text{em}}=-\frac13$ quarks and charged leptons are already mass eigenstates. This choice is possible from the $A_4$ symmetry. Then, matter representation $\overline{10}_{-1}^{\rm\, matter}$ contains both a quark doublet and a heavy neutrino $N$, which enables us to use the $A_4$ symmetry to both $Q_{text{em}}=+\frac23$ 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.
Highlights
We pointed out analytically how the tetrahedral discrete symmetry A4 results from the permutation symmetry S4 [1]
2 3 quark masses and neutrino masses. This is made possible because the anti-SU(5) breaking is achieved by the Higgs fields transforming as antisymmetric representations of SU(5), 10H−1 ⊕ 10Hþ1, reducing the rank-5 anti-SU(5) group down to the rank-4 standard model group SUð3ÞC × SUð2ÞW × Uð1ÞY
The underlying permutation symmetry is useful in model building and it can be accommodated to string compactification
Summary
We pointed out analytically how the tetrahedral discrete symmetry A4 results from the permutation symmetry S4 [1]. String compactification, an ultraviolet completion of the GG SU(5), needs an adjoint representation for breaking the GG SU(5) down to the SM gauge group without changing the rank. The Georgi-Jarlskog quark mass relations [21] need another representation 45 beyond a quintet of Higgs fields. The need for this additional representation makes it difficult for it to be realized in the string compactification. SU(5) is a subgroup of SO(10) [16], but here we consider it an independent GUT since string compactification may not go through an intermediate SO(10) which needs an adjoint representation for spontaneous symmetry breaking to obtain Barr’s flipped SU(5).
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