Radiative Neutrino Mass in a Nonholomorphic $T'$ Modular Invariant Model

Abstract

The T4-2-i topology provides a one-loop realization of Majorana neutrino mass and may be viewed as a radiative extension of the type-II seesaw, with a scalar triplet, two inert scalar doublets, and singlet fermions propagating in the loop. A central difficulty in realizing this topology lies in the simultaneous presence of tree-level type-I and type-II seesaw contributions arising from the same particle content. In addition, the stability of the dark-matter candidate typically requires the introduction of an ad hoc discrete symmetry. In this work, we revisit the T4-2-i topology within a nonholomorphic modular-invariant framework based on the double-cover group $T'$. The presence of both even- and odd-weight polyharmonic Maa{\ss} forms considerably enlarges the space of allowed modular structures, while the residual $\mathbb{Z}_2$ symmetry associated with the vicinity of the fixed point $\tau=i$ naturally stabilizes the lightest odd state. The modular assignments forbid the dangerous tree-level contributions, determine the flavor structure of the lepton sector, and allow both fermionic and scalar dark-matter candidates. We confront the model with neutrino-oscillation data, charged-lepton-flavor-violating bounds, electroweak precision observables, the Higgs diphoton signal strength, the observed dark-matter relic abundance, the cosmological bound on the sum of neutrino masses, and direct-detection limits. Focusing on the fermionic dark-matter candidate, in which the lightest odd state is the Majorana fermion $N_1$, we find that both normal and inverted neutrino mass orderings remain viable. In the allowed region, the relic abundance is largely controlled by coannihilation with the inert scalar partners, while the spin-independent direct-detection rate remains naturally suppressed because it arises only through a loop-generated Higgs portal.