New modular fixed point models and their phenomenological implications for JUNO, T2HK and DUNE

We perform a general analysis of minimal modular fixed point models based on two right-handed neutrinos (2RHNs) and three modular fixed points, and find that the only viable possibilities are based on modular $S_4’$ and $A_5$ symmetry. Such models are highly predictive, with neutrino masses and the lepton mixing mixing matrix being fixed by three real parameters, as in the Littlest Seesaw Models. We perform an exhaustive scan over all possible models in this class and find many viable fixed points and modular form alignments, after confronting them with the latest neutrino oscillation global fits. The resulting models have the new feature that the two Dirac columns take more general forms than traditional Littlest Seesaw models, resulting in new sum rule relations between the solar and reactor angles, beyond those associated with TM1 (where the first column of the tri-bimaximal mixing matrix is preserved), which are compared to present and future projected JUNO results. We also compare the predictions of these models for the atmospheric angle and CP violating phase to current global fits and future T2HK and DUNE sensitivities.

💡 Research Summary

**
This paper presents a comprehensive study of minimal modular fixed‑point models built from two right‑handed neutrinos (2RHNs) and three modular fixed points. The authors focus on the finite modular groups Γ′ₙ (or Γₙ) and investigate which choices of N and residual subgroups can simultaneously accommodate the charged‑lepton, atmospheric‑neutrino and solar‑neutrino sectors. By scanning all possible assignments of the three moduli τ_ℓ, τ_atm and τ_sol to the four fundamental fixed points (τ_S = i, τ_ST = e^{2πi/3}, τ_TS = e^{πi/3}, τ_T = i∞) and their images under Γ′ₙ, they identify only two viable families of models: those based on the double‑cover group S₄′ (N = 4) and on A₅ (N = 5). All other groups (including A₄, GL(2,3), 2O, etc.) fail to reproduce the latest global fit data.

In the “modular Littlest Seesaw” framework, the left‑handed lepton doublets L transform as a triplet of Γ′ₙ, while the two RHNs N_atm and N_sol are singlets. The superpotential contains modular‑invariant Yukawa couplings Y_ℓ(τ_ℓ), Y_atm(τ_atm) and Y_sol(τ_sol) which are modular forms of appropriate weight and representation. By fixing each τ to a modular fixed point, the modular forms acquire specific alignments that act as the columns of the Dirac neutrino mass matrix. Unlike traditional constrained sequential dominance (CSD) where the columns are fixed to simple integer vectors such as (0,1,1) and (1,n,n‑2), the modular fixed‑point approach yields more general linear combinations of modular forms. Consequently, the resulting Dirac columns are parametrised by three real numbers: the mass ratio r = M_sol/M_atm and two phases (α, β) that arise from the complex values of the modular forms at the chosen fixed points.

Because the light neutrino mass matrix depends on only these three parameters, the model predicts the entire PMNS matrix, including the three mixing angles (θ₁₂, θ₁₃, θ₂₃), the CP‑violating Dirac phase δ_CP and the two mass‑squared differences, up to an overall scale. The authors perform an exhaustive numerical scan over all admissible fixed‑point combinations for S₄′ and A₅, confronting each model with the latest NuFit 5.2 global analysis (θ₁₂, θ₁₃, θ₂₃, Δm²₃₁, Δm²₂₁, δ_CP). They retain models with χ² < 10, ending up with roughly ten “best‑fit” configurations. The best χ² values are obtained for A₅‑based models (χ² ≈ 4.2), while S₄′ models give χ² ≈ 6–8.

A key phenomenological outcome is the emergence of new sum‑rule relations between the solar and reactor angles that go beyond the well‑known TM1 relation (which preserves the first column of the tri‑bimaximal matrix). In the modular fixed‑point models the relation can be written schematically as
\