Flavor Structures of Quarks and Leptons from Flipped SU(5) GUT with $A_4$ Modular Flavor Symmetry

We propose to generate the flavor structures of the Standard Model plus neutrinos from flipped SU(5) GUT with $A_4$ modular flavor symmetry. Possible way to assign different moduli values for quarks and leptons in modular GUT scheme is discussed. We propose to reduce the multiple modular symmetries to a single modular symmetry in the low energy effective theory with proper boundary conditions. We classify all possible scenarios in this scheme according to the assignments of the modular $A_4$ representations for matter superfields and give the expressions of the quark and lepton mass matrices predicted by our scheme at the GUT scale. After properly selecting the modular weights for various superfields that can lead to better fitting, we can obtain the best-fit points with the corresponding $\chi^2$ values for the sample subscenarios. We find that the flavor structures of the Standard Model plus neutrinos can be fitted perfectly in such a $A_4$ modular flavor GUT scheme with single or two modulus fields. Especially, the $\chi^2_{total}$ of our fitting can be as low as $1.558$ for sample ${\bf IX^\prime}$ of scenario ${\bf III}$ even if only a single common modulus field for both quark and lepton sectors is adopted. The most predictive scenario ${\bf III}$, in which all superfields transform as triplets of $A_4$, can be fitted much better with two independent moduli fields $\tau_q,\tau_l$ for quark sector and lepton sector ($\chi^2_{total}\approx 95$) than that with the single modulus case ($\chi^2_{total}\approx 282.4$).