Minimal Type-I Dirac seesaw and Leptogenesis under $A_{4}$ modular invariance

We present a Dirac mass model based on $A_{4}$ modular symmetry within Type-I seesaw framework. This extension of Standard Model requires three right-handed neutrinos and three heavy Dirac fermions superfields, all singlet under $SU(2)_{L}$ symmetry. The scalar sector is extended by the inclusion of a $SU(2)_{L}$ singlet superfield, $\chi$. Here, the modular symmetry plays a crucial role as the Yukawa couplings acquire modular forms, which are expressed in terms of Dedekind eta function $\eta(\tau)$. Therefore, the Yukawa couplings follow transformations akin to other matter fields, thereby obviating the necessity of additional flavon fields. The acquisition of $vev$ by complex modulus $\tau$ leads to the breaking of $A_{4}$ modular symmetry. We have obtained predictions on neutrino oscillation parameters, for example, the normal hierarchy for the neutrino mass spectrum. Furthermore, we find that heavy Dirac fermions, in our model, can decay to produce observed baryon asymmetry of the Universe through Dirac leptogenesis.