$SU(5)$ Grand Unified Theory with $A_4$ Modular Symmetry

TL;DR

This paper develops a supersymmetric $SU(5)$ GUT model with $A_4$ modular symmetry, leading to specific predictions for lepton mixing and CP violation, and introduces a novel orbifold-based vacuum alignment mechanism.

Contribution

It presents the first GUT model with modular symmetry as a family symmetry, explicitly constructing an $A_4$-based supersymmetric $SU(5)$ theory in 6d with novel boundary condition-driven flavon alignments.

Findings

Maximal atmospheric neutrino mixing and leptonic CP violation predicted.

Hierarchical Yukawa matrices for quarks and leptons constructed.

Explicit model with orbifold boundary conditions for flavon vacuum alignment.

Abstract

We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric $SU(5)$ in 6d, where the two extra dimensions are compactified on a $T_2/\mathbb{Z}_2$ orbifold. We have shown that, if there is a finite modular symmetry, then it can only be $A_4$ with an (infinite) discrete choice of moduli, where we focus on $\tau = \omega=e^{i2\pi/3}$, the unique solution with $|\tau|=1$. The fields on the branes respect a generalised CP and flavour symmetry $A_4\ltimes \mathbb{Z}_2$ which is isomorphic to $S_4$ which leads to an effective $\mu-\tau$ reflection symmetry at low energies, implying maximal atmospheric mixing and maximal leptonic CP violation. We construct an explicit model along these lines with two triplet flavons in the bulk, whose vacuum alignments are determined by orbifold boundary conditions, analogous to those used for $SU(5)$ breaking with doublet-triplet splitting. There are two right-handed neutrinos on the branes whose Yukawa couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark Yukawa matrix.