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

TL;DR

This work extends the modular Littlest Seesaw framework to two finite modular groups, $S'_4$ and $A_5$, by exploiting three fixed points in the modulus to constrain the Dirac neutrino structure via triplet modular forms. The authors perform an exhaustive scan of 4 viable $S'_4$ cases (A–D) and 14 $A_5$ cases (I–XIV), obtaining highly predictive neutrino mass matrices with a massless lightest neutrino ($m_1=0$) and a fixed first column of the PMNS matrix, yielding new sum rules that relate solar and reactor angles beyond TM1. Confronting these models with the latest global neutrino fits and JUNO results, several cases are already favored, while JUNO’s precision will decisively test many remaining scenarios; DUNE and T2HK will further discriminate among them via $\sin^2\theta_{23}$ and $\delta_{CP}$. The framework predicts a normal hierarchy with $m_{\beta\beta}$ in the few meV range, and a largely testable structure for future neutrinoless double-beta decay experiments. Overall, modular fixed-point models offer a highly predictive and testable path to leptonic flavor, with clear experimental benchmarks for upcoming neutrino facilities.

Abstract

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.

Figures (5)

• Figure 1: The contour plots of $\sin^2\theta_{12}$, $\sin^2\theta_{13}$, $\sin^{2}\theta_{23}$ and $\delta m^2/\Delta m^2$ in the plane of $r\equiv m_s/m_a$ versus $\eta$ for the four possible modular Littlest seesaw models in $S'_4$ modular symmetry. The panels in the upper-left, upper-right, lower- left, lower-right are for case A, case B, case C and case D respectively. The cyan, red, green and blue areas denote the $3\sigma$ regions of $\sin^{2}\theta_{23}$, $\sin^{2}\theta_{13}$ and $\delta m^2/\Delta m^2$ respectively. The solid lines denote the $3\sigma$ upper bounds, the thin lines denote the $3\sigma$ lower bounds and the dashed lines refer to their best fit values, as adopted from latest neutrino global fit Capozzi:2025wyn. The red circle indicates the phenomenologically viable region of parameter space.
• Figure 2: The contour plots of $\sin^2\theta_{12}$, $\sin^2\theta_{13}$, $\sin^{2}\theta_{23}$ and $\delta m^2/\Delta m^2$ in the $\eta/\pi-r$ plane for the four possible modular Littlest seesaw models in $A^\prime_5$ modular symmetry. The panels in the upper-left, upper-right, lower-left, lower-right are for case I, case II, case X and case XI respectively. The cyan, red, green and blue areas denote the $3\sigma$ regions of $\sin^{2}\theta_{23}$, $\sin^{2}\theta_{13}$ and $m_{2}^{2}/m_{3}^{2}$ respectively. The solid lines denote the $3\sigma$ upper bounds, the thin lines denote the $3\sigma$ lower bounds and the dashed lines refer to their best fit values, as adopted from latest neutrino global fit Capozzi:2025wyn. The red circle indicates the phenomenologically viable region of parameter space.
• Figure 3: Correlation between $\sin^2\theta_{12}$ and $\sin^2\theta_{13}$ in the modular Littlest seesaw models based on $S'_4$ and $A_5$ modular symmetries, and the small dots indicate the best fit values of the models. The gray regions denote the $1\sigma$, $2\sigma$ and $3\sigma$ regions of $\sin^2\theta_{12}$ and $\sin^2\theta_{13}$ for the normal mass hierarchy Capozzi:2025wyn, and the red pentagram stands for the best-fit point of the neutrino global analysis Capozzi:2025wyn. The left and right panels are for the global analysis without and with JUNO respectively.
• Figure 4: Correlation between $\sin^2\theta_{12}$ and $\sin^2\theta_{13}$ in the modular Littlest seesaw models based on $S'_4$ and $A_5$ modular symmetries, and the small dots indicate the best fit values of the models. The left and right panels are for JUNO 59.1 days and JUNO 6 years of data taking respectively. The central value and uncertainty of $\sin^2\theta_{12}$ are taken from the latest JUNO measurement JUNO:2025gmd, its $1\sigma$ uncertainty is expected to be reduced to $0.5\%$ after six years of data collection JUNO:2022mxj. The gray regions denote the $1\sigma$, $2\sigma$ and $3\sigma$ regions of $\sin^2\theta_{12}$ and $\sin^2\theta_{13}$ for the normal mass hierarchy JUNO:2025gmdCapozzi:2025wyn, and the red pentagram stands for their central values JUNO:2025gmdCapozzi:2025wyn.
• Figure 5: Correlation between $\delta_{CP}$ and $\sin^2\theta_{23}$ in the modular Littlest seesaw models based on $S'_4$ and $A_5$ modular symmetries, and the small dots and triangles indicate the best fit values of the models. The top-left panel is for the modular Littlest seesaw models based on $S'_4$ modular symmetry, and the other three panels are for $A_5$ modular group. The gray regions denote the $1\sigma$, $2\sigma$ and $3\sigma$ regions of $\delta_{CP}$ and $\sin^2\theta_{23}$ for the normal mass hierarchy Capozzi:2025wyn, and the red pentagram stands for the best-fit point of the neutrino global analysis Capozzi:2025wyn.