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Matter-antimatter asymmetry in minimal inverse seesaw framework with $A_4$ modular symmetry

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A minimal inverse seesaw model with A4 modular symmetry can fit neutrino mixing, evade charged-lepton flavor violation bounds, and generate the observed baryon asymmetry through resonant leptogenesis.

desk verdict Competent incremental A4 modular inverse seesaw paper; the Z' leptogenesis term is new, but the headline mixing-angle 'predictions' are scan-box artifacts until a wider scan proves otherwise. read the letter →

arxiv 2505.03000 v1 pith:NLXN76GT submitted 2025-05-05 hep-ph

Gourab Pathak , Mrinal Kumar Das This is my paper · ORCID
classification hep-ph
keywords inverseseesawA4modularsymmetryneutrinomixingleptonflavorviolationresonantleptogenesisbaryonasymmetryZ'bosonneutrinolessdoublebetadecay
topics Matter-Antimatter Asymmetry
open problems Matter-Antimatter Asymmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a minimal inverse seesaw model with A4 modular symmetry and claims that one parameter regime can address three puzzles simultaneously: neutrino masses and mixing, the absence of charged lepton flavor violation, and the matter-antimatter asymmetry of the universe. Within the current 3-$\sigma$ allowed region of oscillation data, the model excludes part of the parameter space for the atmospheric angle, predicting $sin^{2}$ θ23 > 0.44, and finds a clear linear decrease of $sin^{2}$ θ12 with $sin^{2}$ θ23. The same parameter points that fit oscillations satisfy the MEG and BaBar bounds on μ→eγ, τ→eγ, and τ→μγ, keep the neutrinoless double $\beta$ decay rate below current sensitivity, and produce a baryon asymmetry near Y_B ~ 8.6×10⁻¹¹ through resonant leptogenesis of a nearly degenerate heavy neutrino pair, including Z′-mediated lepton number conserving scatterings. A sympathetic reader would care because the paper demonstrates that a single modular symmetry can tie low-energy flavor data to the cosmological baryon asymmetry.

What carries the argument

The central object is the 7×7 neutral fermion mass matrix in the flavor basis (ν_L, N_R^c, S), whose block structure gives the inverse seesaw light neutrino mass formula m_ν = M_D $M_R^{{-1}}$ μ (M_R^T)^{-1} M_D^T. The A4 modular symmetry, with a single complex modulus τ controlling all Yukawa couplings, provides the flavor structure without a large flavon sector; the heavy sector splits into two nearly degenerate pairs, and the lightest pair's decay in the resonance regime generates the CP asymmetry for leptogenesis. The same modulus τ sets the Dirac CP phase in the PMNS matrix, so the model connects low-energy CP violation to the CP asymmetry relevant for baryogenesis.

What would settle it

A future measurement of sin² θ23 below 0.44 in the currently allowed 3σ range, or a dedicated scan covering the full fundamental domain of the modulus τ that finds oscillation-allowed points with sin² θ23 < 0.44, would falsify the model's lower-bound prediction. Similarly, a measured deviation from the predicted linear relation between sin² θ12 and sin² θ23 at the precision of upcoming experiments would rule out the correlation.

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Extended reading notes

Core claim

The paper's central claim is that an inverse seesaw extension of the Standard Model based on A4 modular symmetry, with two right-handed neutrinos and two singlet fermions plus a local U(1)_{B−L} symmetry, is enough to reproduce all current neutrino oscillation observables while making three testable predictions: a lower bound of 0.44 on sin² θ23, a linear correlation between sin² θ12 and sin² θ23 in the 3σ region, and a strong link between the Dirac CP phase, the Jarlskog invariant, and the atmospheric angle. Using the same parameter space that fits oscillations, the model respects all three measured charged lepton flavor violating branching ratios, places the effective neutrinoless double beta decay mass |m_ee| below the KamLAND-Zen and projected nEXO sensitivities, and yields the observed baryon asymmetry through resonant leptogenesis from the decay of the lightest nearly degenerate heavy neutrino pair, with the Z′-mediated lepton number conserving scatterings included but not spoiling the final asymmetry.

Load-bearing premise

The predictions depend on the randomly scanned parameter ranges for the modulus and couplings; if those ranges do not cover the model's full viable parameter space, the claimed exclusions and correlations could be artifacts of the scan rather than inherent predictions.

Editorial extensions

If this is right

  • The model predicts a lower bound sin² θ23 > 0.44, which current global fits already permit but which next-generation long-baseline experiments can directly test.
  • The linear relation between sin² θ12 and sin² θ23 in the 3σ region means a precise measurement of either angle sharpens the prediction for the other.
  • Parameter points that fit neutrino oscillations automatically satisfy the current LFV bounds from MEG and BaBar, so the model makes the absence of observed charged lepton flavor violation a consequence of the same structure that fixes neutrino mixing.
  • The final baryon asymmetry is essentially independent of the Z′ gauge coupling g_{B−L}, because stronger Z′-mediated scatterings delay the generation of asymmetry but do not suppress it, preserving Y_B ~ 8.6×10⁻¹¹.
  • The model predicts restricted ranges for the CP observables, including J_CP in [-0.07,0.07] and δ_CP in the intervals 0°–85° and 279°–359°, which future CP-violation measurements can check.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the linear θ12–θ23 relation survives a scan over the full fundamental domain of τ rather than the chosen parameter box, it would provide a rare cross-check between two independent oscillation observables that JUNO and DUNE could test at the sub-percent level.
  • The claimed insensitivity of the final baryon asymmetry to g_{B−L} suggests a general lesson: adding lepton number conserving Z′ interactions need not spoil low-scale resonant leptogenesis, a possibility worth testing in other inverse seesaw constructions.
  • The paper's use of a single complex modulus τ as the only flavor source points toward a geometric program: map which regions of the fundamental domain of τ produce each phenomenological signature, turning the current scan box into a full characterization of the model.
  • The combination of neutrino fit, LFV constraints, and leptogenesis within one parameter scan suggests that future data, especially a precise measurement of θ23 or the Dirac phase, could distinguish this A4 modular inverse seesaw from other TeV-scale seesaw frameworks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper constructs a minimal inverse seesaw model with A4 modular symmetry and a local U(1)_{B-L} symmetry. Neutrino masses and mixing are generated through a 7x7 neutral fermion mass matrix depending on the modulus tau and four continuous parameters (alpha_p, beta_p, mu_0, v_phi). The authors scan these parameters, retain points consistent with the 3 sigma neutrino oscillation data, and report a lower bound sin^2(theta23) >= 0.44 and a linear correlation between sin^2(theta12) and sin^2(theta23). They then check charged lepton flavor violating decays, neutrinoless double beta decay, and compute the baryon asymmetry via resonant leptogenesis including Z'-mediated lepton-number-conserving scatterings, obtaining Y_B ~ 8.6e-11.

Significance. If the claimed predictions are robust, the model offers a simultaneous, economical explanation of neutrino oscillation parameters, charged lepton flavor violation, and the baryon asymmetry within a modular-symmetry framework. The paper provides explicit modular forms and covers a broad set of observables, which is a strength. However, because the results are derived from a filtered random scan over a restricted parameter box, the advertised 'predictions' need stronger support to be credible; the current analysis does not yet establish that the lower bound on sin^2(theta23) and the theta12-theta23 correlation are features of the model rather than of the scan.

major comments (4)
  1. [Sec. III.A, Eq. (17); Figs. 2 and 3] The claimed lower bound sin^2(theta23) > 0.44 and the linear correlation between sin^2(theta12) and sin^2(theta23) are derived from a random scan restricted to Re[tau] in [-0.5, 0.5], Im[tau] in [0.5, 1.5], alpha_p in [1e-4, 1e-3], beta_p in [1e-3, 1e-2], mu_0 in [0.1, 10] GeV, and v_phi in [1e5, 1e8] GeV. These intervals, especially for alpha_p, beta_p, mu_0, and v_phi, are not fixed by the A4 modular structure or the charge assignments; they are chosen for numerical convenience. The paper does not report the number of scan points, the sampling density, or any convergence test. Consequently, the reported exclusion of sin^2(theta23) < 0.44 and the narrow band in Fig. 3 could be artifacts of the chosen box. Since these are the headline claims of the abstract and conclusion, the authors must demonstrate robustness by substantially widening the scan, at least for the unconstrained continuous parameters, or provide an analytic argument that the features are inherent to the model.
  2. [Sec. IV.B, Eq. (41)] The Boltzmann equation for Y_{B-L} contains a sum over j = 1, 2 with a factor (Y_chi1/Y_chi1^eq - 1) inside the sum; for the j = 2 term this factor should be (Y_chi2/Y_chi2^eq - 1). As written (or as likely implemented), the decay term of chi2 is weighted by the abundance of chi1, which would spoil the CP-asymmetry contribution from chi2 and make the computed Y_B unreliable. The authors should correct the equation and re-run the numerical integration, or clarify that the printed equation is a typographical error and that the code uses the correct abundance for each species.
  3. [Sec. IV, text before Eq. (24)] The paper states that Delta L = 1 and Delta L = 2 scattering processes 'can be safely neglected in our study as in our work K >> 1' with references [73,74]. In the standard leptogenesis literature, K >> 1 denotes the strong-washout regime, where inverse decays and Delta L = 1 scatterings are typically important and must be included. The reasoning as stated appears to be the opposite of the usual expectation. Please justify the neglect quantitatively, e.g., by comparing the relevant reaction densities to the decay density gamma_D for the scanned parameter points, or modify the Boltzmann equations to include these processes. This is directly relevant to the reliability of the reported baryon asymmetry.
  4. [Sec. V, Conclusion] The conclusion states that the region Re[tau] in (-0.11, 0.98) is excluded by the model, but the scan in Sec. III.A only covers Re[tau] in [-0.5, 0.5]. Values of Re[tau] above 0.5 were never sampled, so the model cannot exclude them on the basis of the presented analysis. This claim should be corrected to the range actually scanned, or the scan must be extended beyond 0.5 to justify the exclusion.
minor comments (4)
  1. [Eq. (41)] The left-hand side of the equation is printed as 'YB-L/dz' rather than 'dY_{B-L}/dz'; the missing differential operator should be corrected.
  2. [Fig. 2, bottom panel] The text claims a lower limit sin^2(theta23) = 0.44, but the figure does not indicate this boundary explicitly; adding a horizontal guidance line would help the reader verify the claim.
  3. [Reference [55]] The table is labeled NuFIT 5.2 (2022), but the cited reference [55] is the 2020 JHEP paper by Esteban et al. Please update the reference to the NuFIT 5.2 publication or confirm that the numbers indeed correspond to the cited paper.
  4. [Sec. IV.B, after Eq. (43)] The notation gamma_D and gamma_z' is introduced without explicit definitions in the text; the reader is directed to Appendix C, but the definitions of gamma_D and gamma_z' in terms of Y_eq and cross sections should be stated at first use for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the theta23 lower bound and theta12-theta23 correlation are scan outputs conditional on NuFIT data, not inputs, and no load-bearing self-citation carries the derivation.

full rationale

The paper's derivation chain is self-contained against external data. Starting from the A4 modular charge assignments (Table I), it constructs the Dirac mass matrix MD (Eq. 5), the heavy mass matrix MR (Eq. 7), and the mu matrix (Eq. 9), then forms the light neutrino mass matrix m_nu = MD MR^{-1} mu (MR^T)^{-1} MD^T (Eq. 12). Neutrino observables are computed by diagonalizing this matrix and filtering against the NuFIT 5.2 3-sigma ranges in Table III. The asserted lower bound sin^2(theta23) > 0.44 and the linear sin^2(theta12)-sin^2(theta23) band are summaries of the accepted scan points; no equation imposes these values as inputs, so the claim is not circular. The LFV branching ratios (Eq. 21), the effective Majorana mass |m_ee| (Eq. 19), and the baryon asymmetry from the Boltzmann system (Eqs. 39-41) are evaluated at the same accepted parameter points and compared with independent experimental and observational limits; none of these outputs is used as a fitting target. Ref. [35] is a self-citation by one of the authors, but it appears only in a general list of inverse-seesaw literature and is not load-bearing for the present derivation. The central modular-form formalism is anchored in external references (Feruglio, NuFIT, MEG, BaBar, Planck, and Iso et al. for resonant leptogenesis), so the load-bearing chain does not reduce to the paper's own prior claims. A robustness caveat remains: the scan-box ranges in Eq. (17) may under-sample the model's full parameter space, making the reported lower bound and correlation dependent on the chosen ranges. That is a sampling or selection-effect concern, not a circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 4 invented entities

The model depends on a continuous modulus tau and several coupling/VEV parameters that are scanned to match neutrino data. The heavy fields and Z' are standard additions, but they are introduced solely for this model. The only genuinely new physical input is the Z'-mediated scattering in the Boltzmann equations.

free parameters (7)
  • alpha_p = scanned in [1e-4, 1e-3]
    Yukawa coefficient in the Dirac mass matrix; scanned to fit neutrino oscillation data.
  • beta_p = scanned in [1e-3, 1e-2]
    Yukawa coefficient for the N_R-S-phi coupling; scanned to fit oscillation data.
  • mu_0 = scanned in [0.1, 10] GeV
    Majorana mass parameter for sterile singlets; scanned to set the light neutrino mass scale.
  • v_phi = scanned in [1e5, 1e8] GeV
    VEV of the U(1)_{B-L} breaking scalar; determines Z' mass and gauge coupling range.
  • tau (modulus) = Re[tau] in [-0.5, 0.5], Im[tau] in [0.5, 1.5]
    Modulus VEV controls all modular form Yukawa couplings; scanned, not predicted.
  • tan(beta) = 5
    Ratio of Higgs doublet VEVs, fixed by hand following refs. [6, 56].
  • m_Z' = 4 TeV
    Z' boson mass fixed by hand in the numerical leptogenesis study.
assumptions (6)
  • domain assumption A4 modular symmetry with a single modulus tau determines the Yukawa couplings.
    The central framework assumption; taken from the modular flavor paradigm of ref. [1].
  • domain assumption U(1)_{B-L} gauge symmetry with the charge assignment in Table I.
    Introduced to eliminate unwanted terms and to generate the Z' mass via spontaneous breaking.
  • domain assumption Inverse seesaw hierarchy M_R >> M_D, mu.
    Required for the light neutrino mass formula Eq. (12); standard ISS assumption.
  • domain assumption Both Higgs doublets survive at sphaleron freeze-out, giving Y_B = 8/23 Y_{B-L}.
    Used in Eq. (23); the prefactor would change if one Higgs doublet decouples.
  • ad hoc to paper Delta L = 1 and Delta L = 2 scattering processes are negligible because K >> 1.
    Assumed in Sec. IV to keep only decays and Z'-mediated scatterings; if K were not large, the final asymmetry would change.
  • ad hoc to paper Superparticle contributions are negligible with mSUSY = 10^14 GeV and F = 10^18 GeV.
    Stated in Sec. IV; the heavy neutrino masses are 10^5-10^7 GeV, so the claim that they lie above mSUSY is not consistent.
invented entities (4)
  • Z' gauge boson independent evidence
    purpose: Mediates new lepton number conserving scatterings in leptogenesis and contributes to LFV processes.
    A massive Z' at 4 TeV from U(1)_{B-L} breaking would be testable at colliders, though the paper does not discuss LHC limits.
  • Two right-handed neutrinos N_Ri
    purpose: Generate light neutrino masses via inverse seesaw and provide heavy states for resonant leptogenesis.
    Standard ISS fields with modular weights -3; no direct experimental signature beyond the model predictions.
  • Two sterile singlet fermions S_i
    purpose: Provide the small lepton number breaking parameter mu in the inverse seesaw.
    Part of the ISS setup; no distinct observable predicted outside the model.
  • Modulus tau
    purpose: Source of flavor structure and CP violation in the Yukawa sector.
    The VEV of tau is scanned, not predicted; there is no independent handle on its value outside the model.

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Cite this review

Pith. "Pith review of Matter-antimatter asymmetry in minimal inverse seesaw framework with $A_4$ modular symmetry." pith.science (2026). https://pith.science/paper/NLXN76GT

@misc{pith2026250503000,
  author       = {Pith},
  title        = {Pith review of: Matter-antimatter asymmetry in minimal inverse seesaw framework with $A_4$ modular symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NLXN76GT}},
  note         = {Machine review of arXiv:2505.03000}
}
abstract

We propose a minimal inverse seesaw framework based on $A_4$ modular symmetry. We have studied the neutrino oscillation parameters in our work and our model excludes some $3 \sigma$ values of the mixing angle $\theta_{23}$. Also, there is a clear linear relation between the mixing angles $\theta_{12}$ and $\theta_{23}$ found in the allowed $3 \sigma$ region. We also examine whether the parameter points consistent with neutrino oscillation data simultaneously comply with the experimental limits on lepton flavor violating (LFV) decays, specifically: $\mu \longrightarrow e \gamma$, $\tau \longrightarrow e \gamma$, and $\tau \longrightarrow \mu \gamma$. We have also investigated the matter-antimatter asymmetry of our universe via the resonant leptogenesis mechanism. Here, we present the contribution of lepton number conserving scattering processes mediated by the $ Z' $ boson in the context of leptogenesis.

Figures

Figures reproduced from arXiv: 2505.03000 by the authors.

Figure 1
Figure 1. FIG. 1: ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Variation of sum of the neutrino masses [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Correlation between mixing angle [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4: ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Effective neutrino mass [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Relationship between [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: ( [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Diagram contributing to CP asymmetry [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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Forward citations

Cited by 2 Pith papers

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

  1. A Non-Holomorphic Modular $A_4$ Framework for Resonant Leptogenesis with Gravitational Wave Signatures

    hep-ph 2026-07 conditional novelty 5.0 of 10

    A non-holomorphic modular A4 seesaw model yields quasi-degenerate right-handed neutrinos, enabling resonant leptogenesis at ~10^6 GeV and a double-peaked gravitational-wave signature.

  2. Froggatt-Nielsen like mechanism in the framework of Modular Symmetry for Neutrino Mass, Mixing and Leptogenesis

    hep-ph 2025-08 conditional novelty 5.0 of 10

    A T' modular-symmetry model with a 'weighton' scalar reproduces neutrino oscillation data within 3σ and gives predictions for neutrinoless double beta decay and leptogenesis.

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