Modular Flavor Symmetries and Fermion Mass Hierarchies

Abstract: We investigate fermion mass hierarchies in models with modular flavor symmetries. Several key conclusions arise from the observation that the determinants of mass matrices transform as 1-dimensional vector-valued modular forms. We demonstrate that, under some fairly general assumptions, achieving hierarchical fermion masses requires the vacuum expectation value of the modulus $\tau$ to be located near one of the critical points, $i$, $i\infty$, or $\omega$. We also revisit the universal near-critical behavior around these points and classify the resulting mass hierarchies for the critical points $i$ and $\omega$. We compare the traditional Froggatt--Nielsen mechanism with its modular variant. The knowledge and boundedness of Fourier and Taylor coefficients are crucial to the predictive power of modular flavor symmetries.