Multiple modular symmetries as the origin of flavor

Abstract

We develop a general formalism for multiple moduli and their associated modular symmetries. We apply this formalism to an example based on three moduli with finite modular symmetries ${S}_{4}^{A}$, ${S}_{4}^{B}$, and ${S}_{4}^{C}$, associated with two right-handed neutrinos and the charged lepton sector, respectively. The symmetry is broken by two bitriplet scalars to the diagonal ${S}_{4}$ subgroup. The low energy effective theory involves the three independent moduli fields ${\ensuremath{\tau}}_{A}$, ${\ensuremath{\tau}}_{B}$, and ${\ensuremath{\tau}}_{C}$, which preserve the residual modular subgroups ${Z}_{3}^{A}$, ${Z}_{2}^{B}$, and ${Z}_{3}^{C}$, in their respective sectors, leading to trimaximal ${\mathrm{TM}}_{1}$ lepton mixing, consistent with current data, without flavons.