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arxiv: 2605.11097 · v1 · submitted 2026-05-11 · ✦ hep-ph

Recognition: no theorem link

Two-loop neutrino mass model with modular S₄ symmetry

A. E. C\'arcamo Hern\'andez , J. Echeverria-Puentes , Vishnudath K. N. , Sergey Kovalenko , Daniel Salinas-Arizmendi , Carlos A. Vaquera-Araujo
Authors on Pith no claims yet

Pith reviewed 2026-05-13 02:33 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino massmodular symmetryS4two-loopdark matterlepton flavor violationflavor symmetry
0
0 comments X

The pith

A modular S4 symmetry generates neutrino masses only at two loops while stabilizing dark matter candidates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper constructs a neutrino mass model in which the modular S4 symmetry, combined with a Z3 symmetry, breaks spontaneously to leave a Z2 remnant. The Z2 symmetry prevents neutrino masses from appearing at tree level or one loop, forcing them to arise at two loops instead. The same Z2 stabilizes both scalar and fermionic dark matter candidates. The model fits the observed charged lepton masses and neutrino oscillation data for the normal ordering, while also predicting charged lepton flavor violation rates that could be seen in upcoming experiments.

Core claim

The model is based on the modular Γ4 ≃ S4 flavour symmetry supplemented by a discrete Z3 symmetry. Spontaneous breaking of the modular symmetry produces a remnant Z2 symmetry that guarantees the radiative origin of active neutrino masses at two loops and stabilizes dark matter candidates. The construction reproduces charged lepton masses and neutrino oscillation data for normal ordering, predicts observable rates for charged lepton flavour violation, and identifies parameter regions consistent with relic density, LFV constraints, and direct detection limits for both scalar and fermionic dark matter.

What carries the argument

The remnant Z2 symmetry that emerges after spontaneous breaking of the modular S4 symmetry, which both enforces two-loop neutrino masses and protects dark matter particles from decaying.

If this is right

  • Charged lepton flavor violation processes occur at observable rates.
  • Viable scalar dark matter candidates arise from singlet-doublet mixing.
  • A fermionic dark matter candidate is strongly linked to lepton flavor violation.
  • The model accommodates neutrino data specifically for normal ordering.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extensions to other modular symmetries could generate different loop orders for neutrino masses.
  • The correlation between fermionic dark matter and LFV rates offers a way to test the model in combined searches.
  • Similar remnant symmetries might apply to other flavor problems such as the hierarchy of quark masses.

Load-bearing premise

The modular S4 symmetry must break in a way that leaves a Z2 remnant capable of forbidding all lower-order contributions to neutrino masses.

What would settle it

Non-observation of the predicted charged lepton flavor violation signals at the sensitivity of experiments like MEG-II or Mu3e would rule out the viable parameter regions.

Figures

Figures reproduced from arXiv: 2605.11097 by A. E. C\'arcamo Hern\'andez, Carlos A. Vaquera-Araujo, Daniel Salinas-Arizmendi, J. Echeverria-Puentes, Sergey Kovalenko, Vishnudath K. N..

Figure 1
Figure 1. Figure 1: Scalar box diagram that induces the effective [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two-loop radiative seesaw mechanism for the active neutrino sector. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Allowed parameter space compatible with the CMS 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Color map for the mixture of the lightest inert neutral scalar in the (∆ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left panel : Allowed values of the real and imaginary parts of the modular field [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Correlation of Br(µ → eγ) with the mass of the lightest heavy neutral lepton mN1 (left panel) and with the charged scalar mass mη± (right panel). The horizontal magenta line indicates the current MEG-II sensitivity [110]. 10−20 10−18 10−16 10−14 10−12 10−10 10−8 Br(µ → eγ) 10−22 10−20 10−18 10−16 10−14 10−12 10−10 10−8 Br( τ → eγ ) MEG-II BaBar superKEKB/Belle II 10−20 10−18 10−16 10−14 10−12 10−10 10−8 Br… view at source ↗
Figure 7
Figure 7. Figure 7: Correlations of Br(τ → eγ) and Br(τ → µγ) against Br(µ → eγ) [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Feynman diagrams for dark matter (DM = χ, N1, ΩR) annihilation channels into Standard Model species (SM = W±, Z, h, ℓ, ν, q) at tree-level. Diagram (a) corresponds to a contact interaction, whereas diagrams (b), (c), and (d) are mediated by a portal particle (PP), which may belong either to the SM or to the dark sector. DM N PP DM N [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Feynman diagram for elastic scattering between dark matter (DM = [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Dependence of DM relic density and DM-nucleon elastic scattering cross section on the lightest inert-scalar mass [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left panel: relic abundance Ωh 2 as a function of the fermionic dark matter mass mN1 , with the color code indicating Br(τ → µγ). Right panel: correlation between Ωh 2 and Br(τ → µγ), with the color code showing Br(µ → eγ). The observed relic abundance is achieved in the region where leptophilic annihilation of N1 is sufficiently efficient, while the electronic Yukawa entries remain suppressed enough to s… view at source ↗
Figure 12
Figure 12. Figure 12: Relic abundance Ωh 2 as a function of the fermionic dark matter mass mΩR in the metastable ΩR scenario. The observed Planck value is reached only in the narrow region where annihilation into Higgs bosons is efficient. abundance, whereas heavier ΩR masses typically lead to overabundance. Regarding direct detection, we can conclude the same behaviour as in the N1 case. Since the interaction of ΩR with quark… view at source ↗
read the original abstract

We propose a two-loop radiative neutrino mass model based on the modular $\Gamma_4 \simeq S_4$ flavour symmetry supplemented by a discrete $Z_3$ symmetry. After spontaneous modular symmetry breaking, a remnant $Z_2$ symmetry guarantees both the radiative origin of active neutrino masses and stabilizes dark matter candidates. The model successfully reproduces charged lepton masses and neutrino oscillation data for normal ordering. It also predicts observable rates for charged lepton flavour violation (LFV). Due to the singlet-doublet mixing the model provides a viable scalar dark matter candidate. A fermionic dark matter candidate, strongly linked to LFV, is also present. We identify parameter regions consistent with relic density, LFV constraints, and direct detection limits, providing testable benchmark configurations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a two-loop radiative neutrino mass model based on modular S4 (Gamma4) flavor symmetry supplemented by a Z3 discrete symmetry. Spontaneous breaking of the modular symmetry is claimed to leave a remnant Z2 that both enforces the two-loop origin of neutrino masses and stabilizes dark matter candidates. The model is stated to reproduce charged lepton masses and neutrino oscillation data for normal ordering, while predicting observable charged lepton flavor violation (LFV) rates. Viable scalar (singlet-doublet mixed) and fermionic DM candidates are identified in regions consistent with relic density, LFV bounds, and direct detection limits.

Significance. If the remnant Z2 protection is rigorously verified, the construction provides a unified modular-symmetry explanation for radiative neutrino masses, flavor mixing, and DM stability, with testable LFV predictions. The modular S4 approach is current and the two-loop structure is a non-trivial feature. However, the extensive fitting of Yukawa couplings, VEVs, and other parameters to neutrino data, relic density, and LFV constraints introduces significant circularity, making the LFV and DM 'predictions' largely post-fit rather than independent outputs. This reduces the model's falsifiability and overall impact unless the symmetry protection is shown to be parameter-independent.

major comments (3)
  1. [Model construction / symmetry breaking] Model construction and symmetry breaking section: The central claim that the remnant Z2 (arising after the modulus tau acquires its VEV) forbids all one-loop neutrino mass operators must be demonstrated explicitly. List the S4 representations, Z3 charges, and modular weights of the lepton doublets, right-handed neutrinos, and scalars; then enumerate all possible dimension-5 and one-loop effective operators allowed before and after breaking to confirm none survive. Without this explicit check, the two-loop suppression and DM stabilization rest on an unverified assumption.
  2. [Neutrino mass generation] Neutrino mass generation section (likely around the two-loop diagrams): The abstract asserts reproduction of oscillation data for normal ordering, but the fitting procedure for the free Yukawa couplings and VEVs must be shown to leave the two-loop diagrams as the dominant contribution without additional fine-tuning or cancellation. Specify which parameters are fixed by the modular weights versus those adjusted post-hoc.
  3. [DM and LFV phenomenology] DM and LFV phenomenology section: The claimed link between the fermionic DM candidate and LFV processes requires quantitative demonstration that the parameter regions satisfying the relic density (and direct detection) automatically produce LFV rates within current bounds without further adjustment. If the regions are selected after fitting, the predictivity for future LFV searches is weakened.
minor comments (2)
  1. [Model construction] Clarify the notation for modular weights and the specific form of the modular forms used in the Yukawa couplings; ensure consistency between text and any tables of field assignments.
  2. [Phenomenology] In the numerical results, provide a table or plot showing the range of predicted LFV branching ratios for the benchmark points that satisfy all constraints, rather than only stating 'observable rates'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below with clarifications and indicate revisions where they strengthen the manuscript without altering its core results.

read point-by-point responses

  1. Referee: Model construction and symmetry breaking section: The central claim that the remnant Z2 (arising after the modulus tau acquires its VEV) forbids all one-loop neutrino mass operators must be demonstrated explicitly. List the S4 representations, Z3 charges, and modular weights of the lepton doublets, right-handed neutrinos, and scalars; then enumerate all possible dimension-5 and one-loop effective operators allowed before and after breaking to confirm none survive. Without this explicit check, the two-loop suppression and DM stabilization rest on an unverified assumption.

    Authors: We agree that an explicit operator enumeration will make the remnant Z2 protection fully transparent. In the revised manuscript we will add a dedicated table listing the S4 representations, Z3 charges and modular weights of all lepton doublets, right-handed neutrinos and scalar fields. We will then enumerate every dimension-5 and one-loop effective operator allowed by the unbroken symmetries and show that each is forbidden once the modulus VEV breaks S4 to the remnant Z2, while the two-loop diagrams remain permitted. This addition confirms the protection is symmetry-based and parameter-independent. revision: yes

  2. Referee: Neutrino mass generation section (likely around the two-loop diagrams): The abstract asserts reproduction of oscillation data for normal ordering, but the fitting procedure for the free Yukawa couplings and VEVs must be shown to leave the two-loop diagrams as the dominant contribution without additional fine-tuning or cancellation. Specify which parameters are fixed by the modular weights versus those adjusted post-hoc.

    Authors: The modular weights and S4 assignments fix the allowed Yukawa structures and modular forms, leaving only the modulus tau and a small set of scalar VEVs as free parameters. By construction the unbroken symmetries forbid all lower-order neutrino mass operators, so the two-loop diagrams are the sole contribution; no cancellations are imposed by hand. The numerical fit to normal-ordering data is performed within these symmetry-allowed ranges without additional tuning. We will insert a clarifying table that distinguishes symmetry-fixed quantities from the fitted parameters. revision: partial

  3. Referee: DM and LFV phenomenology section: The claimed link between the fermionic DM candidate and LFV processes requires quantitative demonstration that the parameter regions satisfying the relic density (and direct detection) automatically produce LFV rates within current bounds without further adjustment. If the regions are selected after fitting, the predictivity for future LFV searches is weakened.

    Authors: The scan first fits the symmetry-allowed parameters to neutrino and charged-lepton data. Only afterward are relic-density and direct-detection constraints applied to identify viable regions for the fermionic DM candidate. These same regions automatically satisfy current LFV bounds because the DM annihilation channels share the same Yukawa couplings that mediate LFV. No extra tuning is performed to enforce LFV compliance. The resulting benchmark points therefore constitute genuine predictions for future LFV searches. We will add explicit statements and reference the existing figures to emphasize this sequential, adjustment-free consistency. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained model construction

full rationale

The paper defines a Lagrangian with modular S4 and Z3 symmetries, assigns representations and weights to fields, invokes spontaneous breaking to a remnant Z2, and states that this Z2 forbids tree-level and one-loop neutrino mass terms while allowing two-loop ones. It then numerically fits a finite set of free parameters (Yukawa couplings, VEVs, modulus tau) to reproduce charged-lepton masses and neutrino oscillation data, after which it computes LFV branching ratios and DM relic density for the same parameter points. None of these steps reduces by construction to its own inputs: the symmetry-allowed operators are fixed by group theory before any fit, the two-loop suppression is a direct consequence of the charge assignments rather than a redefinition of the data, and the LFV/DM outputs are distinct observables computed from the fitted parameters. No self-citation is used to justify uniqueness or to smuggle an ansatz; the central claims rest on explicit symmetry analysis and numerical reproduction of external data. The structure is therefore a standard, non-circular model-building exercise.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The central claim rests on the assumption that modular S4 forms and the chosen Z3 charges produce the required remnant Z2 after breaking, plus a large number of Yukawa couplings and vacuum expectation values that are adjusted to data.

free parameters (1)
  • Yukawa couplings and VEVs
    Multiple couplings and vacuum expectation values are adjusted to reproduce charged lepton masses, neutrino oscillation parameters, and dark matter relic density.
axioms (2)
  • domain assumption Modular S4 symmetry with specific modular forms and weights
    Invoked to organize the flavor structure of the Yukawa matrices.
  • ad hoc to paper Spontaneous breaking leaves a remnant Z2 symmetry
    Assumed to enforce the two-loop suppression of neutrino masses and DM stability.
invented entities (2)
  • scalar DM candidate from singlet-doublet mixing no independent evidence
    purpose: Provide a viable scalar dark matter particle
    Introduced via the field content and mixing; no independent evidence outside the model.
  • fermionic DM candidate linked to LFV no independent evidence
    purpose: Provide a second DM candidate whose interactions are tied to flavor violation
    Introduced as part of the particle content; no independent evidence outside the model.

pith-pipeline@v0.9.0 · 5460 in / 1529 out tokens · 56613 ms · 2026-05-13T02:33:46.866997+00:00 · methodology

discussion (0)

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Reference graph

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