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arxiv: 2606.11346 · v2 · pith:M5QEHTRTnew · submitted 2026-06-09 · ✦ hep-ph

Radiative Neutrino Mass in a Nonholomorphic T' Modular Invariant Model

Mohamed Amin Loualidi , Mohamed Miskaoui , Salah Nasri This is my paper

Pith reviewed 2026-06-27 12:20 UTC · model grok-4.3

classification ✦ hep-ph
keywords radiative neutrino massmodular symmetryT' groupdark matterone-loop diagramT4-2-i topologynonholomorphic modular formsZ2 symmetry
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0 comments X

The pith

Nonholomorphic T' modular symmetry realizes the T4-2-i radiative neutrino mass model without tree-level seesaws and with stable dark matter.

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

This paper establishes that modular assignments under the double-cover group T' using both even- and odd-weight polyharmonic Maass forms can implement a one-loop diagram for Majorana neutrino masses while forbidding all tree-level type-I and type-II seesaw contributions. The residual Z2 symmetry near the fixed point tau equals i automatically stabilizes the dark-matter candidate without an ad hoc discrete symmetry. The model accommodates neutrino oscillation data for both mass orderings, satisfies flavor violation and precision bounds, and produces the observed dark-matter relic density through coannihilations, with direct detection naturally suppressed by the loop-induced Higgs portal.

Core claim

The T4-2-i topology is realized in a nonholomorphic modular-invariant framework based on T' where the modular assignments forbid tree-level seesaw operators, the even- and odd-weight Maass forms enlarge the allowed structures, and the residual Z2 from the vicinity of tau=i stabilizes the lightest odd state, allowing both fermionic and scalar dark-matter candidates that fit all experimental constraints with the fermionic case viable for both normal and inverted orderings.

What carries the argument

The nonholomorphic modular-invariant framework based on the double-cover group T' with even- and odd-weight polyharmonic Maass forms that enlarge the space of allowed modular structures while enforcing the residual Z2 symmetry.

If this is right

  • Both normal and inverted neutrino mass orderings remain viable in the allowed parameter space.
  • The relic abundance is largely controlled by coannihilation with the inert scalar partners for the fermionic dark matter candidate N1.
  • The spin-independent direct-detection rate remains naturally suppressed because it arises only through a loop-generated Higgs portal.
  • The model satisfies charged-lepton-flavor-violating bounds, electroweak precision observables, the Higgs diphoton signal strength, and the cosmological bound on the sum of neutrino masses.

Where Pith is reading between the lines

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

  • Similar modular techniques could be used to address tree-level problems in other one-loop mass generation models.
  • The fixed-point stabilization mechanism might generalize to other modular groups for automatic discrete symmetries.
  • Future direct detection experiments could test the loop-suppressed portal by searching for the predicted small rates.
  • Predictions for specific neutrino mixing parameters from the modular forms could be checked against upcoming oscillation data.

Load-bearing premise

The modular weights and field assignments under T' can be chosen such that all tree-level type-I and type-II seesaw operators are forbidden while the one-loop T4-2-i diagram is allowed and the Z2 symmetry remains intact.

What would settle it

An observation of a tree-level neutrino mass contribution or a spin-independent direct-detection rate inconsistent with the loop-generated Higgs portal in the parameter region that reproduces the relic abundance and neutrino data would falsify the construction.

Figures

Figures reproduced from arXiv: 2606.11346 by Mohamed Amin Loualidi, Mohamed Miskaoui, Salah Nasri.

Figure 1
Figure 1. Figure 1: FIG. 1: One-loop [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Annihilation and coannihilation topologies contributing to [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Representative one-loop origin of the effective [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: From left to right, the panels show the relic density versus [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: From left to right the panels show [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Top plots: from left to right, the panels display (∆ [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: From left to right and top to bottom, the panels show radiative decays, three-body decays, and [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Same as figure 4, but for IO [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Same as figure 5, but for IO [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Same as figure 6, but for IO [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Same as figure 7, but for IO [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
read the original 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.

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

1 major / 2 minor

Summary. The paper constructs a nonholomorphic T' modular invariant model realizing the T4-2-i one-loop topology for radiative Majorana neutrino masses. Even- and odd-weight polyharmonic Maass forms are used to forbid tree-level type-I and type-II seesaw operators, while a residual Z2 near the τ=i fixed point stabilizes the DM candidate. The model is confronted with neutrino oscillation data, CLFV bounds, EW precision observables, Higgs diphoton strength, DM relic density, Σmν, and direct-detection limits. For fermionic DM (lightest odd state N1), viable regions exist for both normal and inverted orderings, with relic density set by coannihilation and SI DD suppressed by a loop-induced Higgs portal.

Significance. If the modular assignments correctly enforce the operator structure, the work offers a symmetry-based alternative to ad-hoc discrete symmetries in radiative seesaw models and enlarges the space of allowed modular forms. The comprehensive numerical confrontation with multiple datasets (oscillation, LFV, EWPO, Higgs, relic, Σmν, DD) is a strength, showing that coannihilation dominates the relic density while DD remains naturally small. This could inform further modular constructions linking neutrino mass and DM.

major comments (1)
  1. [Abstract and modular assignments section] Abstract and the section describing the modular assignments: the central claim that the chosen T' representations together with even/odd-weight polyharmonic Maass forms identically forbid all tree-level type-I and type-II seesaw operators (while permitting the T4-2-i loop and preserving the residual Z2) is load-bearing. Explicit verification—e.g., tables of allowed vs. forbidden modular forms for the relevant Yukawa and triplet couplings, or equations showing the vanishing of the forbidden structures—is required to confirm that no non-vanishing form reintroduces tree-level contributions.
minor comments (2)
  1. [Phenomenological analysis section] The numerical scan results (e.g., allowed regions in the mN1–mΔ plane or relic-density contours) would benefit from explicit statements of the prior ranges chosen for the Maass-form coefficients and scalar potential parameters.
  2. [Model setup] Notation for the polyharmonic Maass forms (even vs. odd weight) should be introduced with a brief reminder of their transformation properties under T' to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need for explicit verification of the modular assignments. We agree that this is a load-bearing claim and will strengthen the presentation accordingly.

read point-by-point responses

  1. Referee: [Abstract and modular assignments section] Abstract and the section describing the modular assignments: the central claim that the chosen T' representations together with even/odd-weight polyharmonic Maass forms identically forbid all tree-level type-I and type-II seesaw operators (while permitting the T4-2-i loop and preserving the residual Z2) is load-bearing. Explicit verification—e.g., tables of allowed vs. forbidden modular forms for the relevant Yukawa and triplet couplings, or equations showing the vanishing of the forbidden structures—is required to confirm that no non-vanishing form reintroduces tree-level contributions.

    Authors: We agree that explicit verification of the operator structure is essential for clarity. The T' assignments (with the specified even- and odd-weight polyharmonic Maass forms) were chosen so that the tree-level type-I Yukawa coupling L H N and the type-II triplet coupling L L Δ both carry total modular weights that cannot be compensated by any allowed form of the required representation, while the one-loop T4-2-i operators are permitted. In the revised manuscript we will add (i) a table enumerating the modular weight and representation of every relevant operator, (ii) the explicit list of allowed versus forbidden Maass forms for each coupling, and (iii) a short derivation showing that the forbidden structures have vanishing modular forms. These additions will be placed in the modular-assignments section immediately after the field-content table. The residual Z2 near τ = i is preserved by the same weight assignments and will be noted explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; modular symmetry provides independent operator selection

full rationale

The paper selects T' representations, weights, and even/odd polyharmonic Maass forms to determine which operators are allowed or forbidden. This symmetry-based filtering is a standard construction that stands independently of the neutrino oscillation data, relic density, or direct-detection limits. Numerical scans then identify viable parameter regions consistent with those observables, but no claimed result (viability of orderings, coannihilation dominance, loop-suppressed DD) reduces by construction to a fit or self-citation. The central mechanism—simultaneous forbidding of tree-level type-I/II while permitting the T4-2-i loop and residual Z2—follows from the group-theoretic assignments rather than from re-labeling inputs. No load-bearing self-citation chains or ansatze smuggled via prior work appear in the abstract or described structure.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger is estimated from the described construction; full paper would list exact modular weights, Yukawa coefficients, and scalar potential parameters.

free parameters (2)
  • modular weights and Maass form coefficients
    Coefficients in the polyharmonic Maass forms and the specific modular weights assigned to fields are chosen to produce the desired operator structure and are expected to be adjusted when fitting data.
  • scalar potential parameters and mass scales
    Masses and couplings of the inert doublets, triplet, and singlet fermions are free parameters scanned to satisfy relic density and direct-detection bounds.
axioms (2)
  • domain assumption The T' modular assignments together with even- and odd-weight polyharmonic Maass forms forbid all tree-level type-I and type-II seesaw operators while permitting the one-loop T4-2-i diagram.
    This is the central group-theoretic assumption invoked to eliminate dangerous contributions without additional discrete symmetries.
  • domain assumption The residual Z2 symmetry near the tau=i fixed point is sufficient to stabilize the lightest odd state against decay.
    Standard feature of modular models at fixed points, here used to replace an ad-hoc Z2.
invented entities (1)
  • polyharmonic Maass forms of both even and odd weight no independent evidence
    purpose: To enlarge the allowed modular structures beyond holomorphic forms and realize the desired operator selection rules.
    New functional building blocks introduced to increase flexibility in the lepton sector.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Non-holomorphic $S^{\prime}_{4}$ modular symmetry for leptons and leptogenesis

    hep-ph 2026-06 unverdicted novelty 6.0

    36 viable non-holomorphic S'4 modular models for leptons are identified via numerical scans, with two yielding successful unflavored thermal leptogenesis from the real part of τ while fitting neutrino data.

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