Modular-symmetry-protected seesaw

TL;DR

The paper proposes a modular flavour symmetry framework in which a small modulus deviation $|\epsilon|\ll 1$ acts as the sole source of both charged-lepton mass hierarchies and lepton-number breaking in a symmetry-protected low-scale seesaw, generating active neutrino masses and a pseudo-Dirac splitting of heavy neutral leptons. By embedding the L-symmetric limit into residual modular symmetries and restricting to triplet representations of $\Gamma_3'\simeq T'$ or $\Gamma_4'\simeq S_4'$, the authors construct four benchmark models (A–D) that fit charged-lepton masses and neutrino oscillation data, with distinct Heavy Neutral Lepton (HNL) flavor structures. The HNL phenomenology is elaborated through mixing parameters $\Theta_{lj}$, showing model-dependent hierarchies that can be probed by future experiments, while current cLFV probes remain challenging. Overall, the work links modular symmetry to lepton number and CP violation in a testable, predictive low-scale seesaw framework, offering concrete targets for HNL searches and neutrino experiments.

Abstract

In the presence of a finite modular flavour symmetry, fermion mass hierarchies may be generated by a slight deviation of the modulus from a symmetric point. We point out that this small parameter governing charged-lepton mass hierarchies may also be responsible for the breaking of lepton number in a symmetry-protected low-scale seesaw, sourcing active neutrino masses and the mass splitting of a pseudo-Dirac pair of heavy neutrinos. We discuss the phenomenological implications of this mechanism, including the possibility to test the considered models at future planned and proposed heavy neutral lepton searches.

Figures (4)

• Figure 1: Viable regions for the modulus VEV $\tau$ within the fundamental domain $\mathcal{D}$, for the benchmark modular-symmetry-protected models described in \ref{['sec:modelsubsec']}. Green, yellow and red fills correspond to the 1$\sigma$, 2$\sigma$ and 3$\sigma$ credible regions. These regions are symmetric under the gCP transformation that flips the sign of $\mathop{\mathrm{Re}}\nolimits \tau$. The panel on the right shows a zoomed-in view near $\tau_\text{sym} = \omega$.
• Figure 2: Correlation between the solar and atmospheric mixing angles obtained in the assessment of benchmark model C (left) and between the solar angle and the sum of neutrino masses for benchmark model D (right). The colour coding of credible regions is the same as in \ref{['fig:tau_regions']}. For comparison, we show the prospective $1\sigma$ sensitivities of future long-baseline and reactor experiments HyperKamiokande Hyper-Kamiokande:2025fci and JUNO JUNO:2022mxj, taking into account the corresponding maximum-likelihood values for $\sin^2\theta_{23}$ and $\sin^2\theta_{12}$, respectively. The predicted values of $\sum_i m_i$ in model D (right) respect the "aggressive" cosmological bound identified in Ref. Capozzi:2025wyn.
• Figure 3: The ratios $\Theta^2_e/\Theta^2$ -- $\Theta^2_\mu/\Theta^2$ -- $\Theta^2_\tau/\Theta^2$ associated to the considered models. The coloured points -- top-left panel in turquoise for model A, top-right in brown for model B, bottom-left in yellow for model C, and bottom-right in blue for model D -- are those for which $\chi^2 \leq 10$, with the stars marking the point of maximum posterior probability (minimum Gaussian $\chi^2$). The orange regions correspond to the full parameter space of the type-I seesaw, in the cases with either two (lighter colour) or three (darker colour) RH$\nu$s, with the oscillation data varied within the $3\sigma$ regions obtained in the NuFit 6.0 global analysis Esteban:2024eli. To obtain the region associated to generic type-I scenario with 3 RH$\nu$s, the lightest neutrino mass is varied randomly in the range allowed by the corresponding model when compared against oscillation data.
• Figure 4: The parameter space in the $\Theta^2$ -- $M_\text{av}$ plane of the type-I seesaw scenario for the discussed benchmark models: models C and D with $\Theta_e^2$-dominance in the top panels, model A with $\Theta_\mu^2$-dominance in the middle panel, models A, B and D with $\Theta^2_\tau$-dominance in the bottom panels. The black solid curve represents the seesaw limit. The darker gray regions are excluded by several experiments on HNL production via meson decays (PS191 Bernardi:1985nyBernardi:1987ek, BEBC Barouki:2022bkt, PIENU PIENU:2017wbj, E949 E949:2014gsn, NA62 NA62:2020mcvNA62:2021bji, T2K T2K:2019jwa, NuTeV NuTeV:1999kej, MicroBooNE MicroBooNE:2022ctmMicroBooNE:2023eef CHARM CHARM:1985nkuBoiarska:2021yho, searches at KEK Hayano:1982wu), tau lepton decays (BELLE Belle:2013ytxBelle:2022tfoBelle:2024wyk), and at colliders (DELPHI DELPHI:1996qcc, CMS CMS:2022futCMS:2023jqiCMS:2024xdq, ATLAS ATLAS:2019kpxATLAS:2022atqTastet:2021vwp). The lighter gray region is excluded by BBN Sabti:2020yrtBoyarsky:2020dzc. The dashed curves represent the sensitivities of the upcoming, planned and proposed experiments PIONEER PIONEER:2022yag (cyan), Hyper-K T2K:2019jwa (blue), DUNE Breitbach:2021gvv (pink), MATHUSLA MATHUSLA:2020uve (orange), SHiP SHiP:2018xqw (purple), and searches at HL-LHC Drewes:2019fou (red), FCC-ee Blondel:2022qqo and CEPC CEPCStudyGroup:2018ghi (green). Current constraints and future sensitivities are only indicative, as they are given for $\Theta_e^2 : \Theta^2_\mu:\Theta^2_\tau = 1:0:0$ (upper panel), $0:1:0$ (middle panel) and $0:0:1$ (lower panels) -- such ratios hold only approximately in the scenarios considered here -- and in the case of a single HNL.