Pith. sign in

REVIEW 3 major objections 4 minor 5 cited by

A minimal modular-symmetry seesaw model can explain neutrino masses and predict new observables.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 20:30 UTC pith:3ZA2EU27

load-bearing objection A workmanlike numerical scan of a non-holomorphic modular A4 linear seesaw; the model is new but the 'predictions' are broad and the key seesaw hierarchy is assumed rather than checked, so treat the headline claims with caution. the 3 major comments →

arxiv 2602.23018 v2 pith:3ZA2EU27 submitted 2026-02-26 hep-ph

A Predictive Non-Holomorphic Modular A₄ Linear Seesaw Framework Testable at DUNE

Rudra Majhi , Mitesh Kumar Behera , Rukmani Mohanta This is my paper
classification hep-ph
keywords neutrino masseslinear seesawnon-holomorphic modular symmetryA4 flavor symmetrypolyharmonic Maaß formsneutrino oscillationsleptonic CP violationneutrinoless double beta decay
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to show that a non-supersymmetric flavor model built on non-holomorphic modular A4 symmetry can account for the observed neutrino masses and mixings while staying predictive. The model adds six singlet fermions and one flavon scalar, then lets the complex modulus τ, evaluated in the fundamental domain, fix all Yukawa structures. A numerical scan finds parameter regions consistent with current neutrino oscillation data at the 3σ level for both normal and inverted mass ordering. It also yields definite predictions for quantities not yet measured: absolute neutrino masses, CP-violating phases, the effective electron-neutrino mass, and the neutrinoless-double-beta-decay amplitude. A sympathetic reader would care because, if true, the framework shows that a minimal, symmetry-driven construction without supersymmetry can generate the full neutrino flavor structure rather than merely parameterizing it.

Core claim

The central claim is that the linear seesaw realized with non-holomorphic modular A4 symmetry and polyharmonic Maaß-form Yukawa couplings is phenomenologically viable and predictive. Once the modulus τ takes a vacuum expectation value in the fundamental domain, the Yukawa couplings are fixed as modular forms of weights 0 and -2, giving a highly constrained 9×9 fermion mass matrix. Under the assumed hierarchy M_LS ≪ M_D ≪ M_RS and M_N ≪ M_RS, block diagonalization yields an effective light-neutrino mass matrix mν ≈ M_D M_RS^{-1} M_LS^T + transpose, whose overall scale is fixed by the atmospheric mass-squared difference. The authors find solutions for both normal and inverted ordering, with si

What carries the argument

The load-bearing mechanism is the combination of non-holomorphic modular A4 symmetry with polyharmonic Maaß forms used as Yukawa couplings. In this non-supersymmetric setting, the holomorphicity condition is replaced by a Laplace equation, and the couplings Y_3^{(0)} and Y_3^{(-2)} transform as triplets of A4 with modular weights 0 and -2. Their numerical values are fixed by the modulus τ, so a single complex parameter, together with a few dimensionless coefficients, determines the entire flavor structure of the neutrino sector. The linear seesaw then emerges from a 9×9 mass matrix in the basis (ν_L, N_R, S_L^c); under the block-diagonalization expansion, the light neutrino mass matrix takes

Load-bearing premise

The argument rests on the assumed hierarchy M_LS ≪ M_D ≪ M_RS and M_N ≪ M_RS, which is imposed to justify dropping the double-seesaw correction; if the fitted masses do not satisfy these inequalities, the effective neutrino mass matrix and all predicted observables change.

What would settle it

Directly diagonalize the full 9×9 mass matrix at the best-fit parameter points (for both normal and inverted ordering) and compare the three light eigenvalues and mixing angles with those obtained from the approximate linear-seesaw formula. If the two disagree beyond the stated experimental uncertainties, the hierarchy assumption fails and the model's predictions are not robust. Equivalently, a future measurement that places δ_CP and θ23 well outside the model's correlated allowed regions would falsify the framework.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The model reproduces current neutrino oscillation data at 3σ for both mass orderings, so the non-holomorphic modular framework is a live candidate for neutrino flavor structure.
  • It makes specific absolute predictions: both the sum of neutrino masses and the effective Majorana mass lie in ranges that future cosmological and neutrinoless-double-beta-decay experiments can probe or exclude.
  • Correlations among θ23, δ_CP, and Δm²31 are predicted; a next-generation long-baseline experiment should be able to shrink the allowed region and potentially resolve the octant degeneracy.
  • If inverted ordering is chosen by nature, the model predicts values near current cosmological bounds, making the combination of cosmology and double-beta decay a decisive test.
  • The reduced field content — six singlet fermions and one flavon, without supersymmetry — suggests the construction can be embedded in larger frameworks without introducing many new scalars.

Where Pith is reading between the lines

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

  • The hierarchy M_LS ≪ M_D ≪ M_RS is imposed rather than derived; a future derivation from modulus stabilization or flavon dynamics would strengthen the predictive claim. If the fitted coefficients violate this hierarchy, the dropped double-seesaw term is not negligible and the numerical predictions would shift.
  • Because the charged-lepton sector is taken diagonal with freely adjusted couplings, the model's predictive content is concentrated in the neutrino sector; extending the same modular forms to quarks or to charged-lepton flavor violation could test whether the symmetry is truly universal.
  • The allowed range of δ_CP is essentially unrestricted, so the distinctive testable content is not the CP phase itself but its correlation with θ23 and Δm²31; measurements that fix one of these variables sharply would rule out large parts of the model's parameter space.
  • The same non-holomorphic modular construction could be applied to other low-scale seesaw variants (inverse, type-II, scotogenic) with a similar reduction in scalars; the predictive pattern reported here suggests a systematic program.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs a non-supersymmetric linear seesaw model based on non-holomorphic modular A4 symmetry, using polyharmonic Maaß forms. The field content is six SU(2)_L singlet fermions N_R and S_L (two A4 triplets) plus one flavon. Half-integral modular weights are assigned to forbid unwanted operators. The authors construct Dirac, pseudo-Dirac, and heavy mass matrices, and under the hierarchy of Eq. (3.12) block-diagonalize the 9×9 mass matrix to obtain the linear seesaw form Eq. (3.14). A numerical scan over tau and five complex parameter ratios is fitted to NuFIT 6.1 oscillation data; allowed regions for both normal and inverted ordering are reported. The paper then presents predicted ranges for m_beta_beta, m_nu_eff, and sum m_nu, and compares them with DUNE, KamLAND-Zen, and cosmological bounds.

Significance. If the mass-matrix construction and the q-expansions are correct, this is a useful extension of modular flavor models to non-holomorphic forms with a relatively economical particle content, and the DUNE correlation plots are informative. The scan is systematic and makes use of public packages (FlavorPy, GLoBES). However, the claimed 'predictions' are mostly allowed ranges from a fit with many free parameters, and the central linear-seesaw expansion is conditional on a hierarchy that is not verified numerically; until these points are addressed, the phenomenological results are not fully established.

major comments (3)
  1. [§3.5, Eqs. (3.12)–(3.14); §5, Eqs. (5.1)–(5.2)] The linear-seesaw truncation is not verified on the fitted parameter points. Eq. (3.13) contains the double-seesaw term -M_LS M_RS^{-1} M_N (M_RS^T)^{-1} M_LS^T, which is dropped using the hierarchy assumption (3.12). The numerical scan varies only the dimensionless ratios beta~D, beta~LS, gamma~D, gamma~LS, beta~NS and tau; the absolute scales v_phi, M_R, and alpha_NS that control M_RS and M_N are not scanned or reported. Therefore there is no evidence that the best-fit points of Table 5 satisfy (3.12). If the double-seesaw term is non-negligible at natural heavy-scale choices, the effective neutrino mass matrix changes and the allowed regions in Figs. 2–3 and the m_beta_beta/sum m_nu forecasts in Figs. 4–5 shift. The authors should either scan/report the absolute heavy scales and demonstrate (3.12) at each accepted point, or quantify the neglected correction for the quoted predictions.
  2. [Appendix A, Eqs. (A.1)–(A.6)] The q-expansions of Y_3^(0) and Y_3^(-2) are central inputs to the mass matrices, yet no derivation, proof of modularity, or source is given for these particular polyharmonic Maaß forms, and the transformation properties under the finite modular group are not demonstrated. The expansions contain log y terms and incomplete Gamma functions, so the weight/level, Laplacian eigenvalue, and truncation error should be specified. Without a self-contained definition or a precise reference, the numerical results are not fully reproducible. Please provide a derivation or citation, and state the number of terms kept in the scan together with an estimate of the resulting error.
  3. [§3.6, §5.1, Abstract] The claim of predictive power is overstated as presented. The model has five complex parameter ratios plus tau (11 real degrees of freedom before fixing the overall scale), which are fitted to the six oscillation observables. Consequently the Dirac phase delta_CP is not a prediction: Sec. 5.1 states that delta_CP spans the full range [0,360°]. The absolute mass scale kappa is fixed by Delta m_atm^2 via Eq. (3.16), and the m_beta_beta and sum m_nu ranges are images of the fitted parameter region, not independent outputs. I recommend rewriting the abstract and Sec. 6 to describe these as fit-determined ranges, and to state explicitly how many free parameters are used for the six-observable fit.
minor comments (4)
  1. [§5, Eq. (5.1)] The scan domain is written with |Re(tau)| in [0,0.5], but Table 5 and the text in Sec. 5.1 use Re(tau) in [-0.5,0.5]. Please make this consistent.
  2. [§3.2, Eqs. (3.4), (3.6), (3.8), (3.10)] The 'LR' subscripts on the mass matrices are not defined. State explicitly the row/column labeling convention (e.g., which index corresponds to flavor and which to the A4 triplet components).
  3. [§4, Eq. (4.2)] The statistical procedure is described too briefly. It is unclear whether the oscillation parameters are marginalized before or after the chi^2 of Eq. (4.2) is computed, and whether delta_CP is treated as a free parameter in the DUNE simulation. This information is needed to interpret the DUNE contours in Figs. 2–3.
  4. [References] Reference [103] is cited as NuFit-6.0 in the bibliography, while the text and Table 3 use NuFIT 6.1; update the citation. Also, the note below Eq. (3.8) that 1/3 != beta~NS should be accompanied by a quantitative statement of how close beta~NS can approach 1/3 before M_RS becomes too singular for the expansion (3.13) to hold, since 1/3 lies inside the scan range (5.2).

Circularity Check

1 steps flagged

The Dirac CP phase is a NuFIT input used in the chi2 scan yet is advertised as a predicted observable; the rest of the derivation is a parameter fit with model-dependent outputs, not a circular derivation.

specific steps
  1. fitted input called prediction [Abstract; Sec. 3.6; Sec. 4, Eq. (4.2); Sec. 5.1]
    "We identify regions consistent with current neutrino oscillation data at the 3σ level and obtain predictions for currently unknown observables, including the absolute neutrino mass scale and leptonic CP-violating phases."

    In Sec. 3.6, δ_CP is defined as one of the six physical observables used in the fit; in Sec. 4, Eq. (4.2) defines the compatibility chi2 against NuFIT 6.1 central values, including δ_CP. The scan therefore selects parameter points that reproduce δ_CP as an input, so listing 'leptonic CP-violating phases' as a model prediction presents a fit target as an output. The paper later states the model 'permits the full range of δCP values within [0,360°]', confirming that no sharp phase prediction is actually made; the advertised CP-phase prediction is the fit itself.

full rationale

The central construction is not circular: the effective light-neutrino mass matrix Eq. (3.13) is obtained by block diagonalization under the stated hierarchy Eq. (3.12), and the numerical scan is a genuine multi-parameter fit that could fail; the resulting allowed regions, correlations, and DUNE projections are non-trivial outputs. No load-bearing self-citation appears: the author-overlapping references [42,46] are cited only in general literature lists, not as the mathematical foundation or as a uniqueness theorem. The one real circularity is terminological: δ_CP is a NuFIT input used in Eq. (4.2), yet the abstract advertises it as a prediction; the absolute-mass outputs are similarly fit-derived, with κ fixed by the measured Δm²_atm in Eq. (3.16), so they are consistency outputs rather than independent first-principles predictions. The linear-seesaw hierarchy M_LS << M_D << M_RS, M_N << M_RS is an unverified assumption rather than a circular step: the paper states it explicitly as the condition for the expansion, and the heavy mass scales are not scanned, so the numerical predictions are conditional on that assumption.

Axiom & Free-Parameter Ledger

8 free parameters · 5 axioms · 2 invented entities

The model's predictive power rests on a symmetry ansatz (non-holomorphic modular A4) plus five complex free-parameter ratios and the modulus τ, which are fitted to oscillation data. The central claim does not derive these parameters from a deeper principle; it merely demonstrates that viable regions exist.

free parameters (8)
  • β~D (β_D/α_D) = not reported individually; scanned modulus in [10^-3,10^3]
    Complex ratio controlling diagonal entries of the Dirac mass matrix M_D (Eq. 3.4).
  • γ~D (γ_D/α_D) = not reported individually; scanned in [10^-3,10^3]
    Complex ratio in M_D diagonal entries (Eq. 3.4).
  • β~LS (β_LS/α_LS) = not reported individually; scanned in [10^-3,10^3]
    Complex ratio in the pseudo-Dirac mass matrix M_LS (Eq. 3.6).
  • γ~LS (γ_LS/α_LS) = not reported individually; scanned in [10^-3,10^3]
    Complex ratio in M_LS (Eq. 3.6).
  • β~NS (β_NS/α_NS) = not reported individually; scanned in [10^-3,10^3]
    Complex ratio in the heavy mixing matrix M_RS (Eq. 3.8).
  • τ (complex modulus) = NO: τ=0.02+2.26i; IO: τ=-0.32+1.5i
    Determines the values of all modular Yukawa couplings; scanned in Re(τ)∈[-0.5,0.5], Im(τ)∈[0.86,5].
  • κ (overall light neutrino mass scale) = fixed to Δm²_atm via Eq. (3.16)
    Absorbs v_H² α_D α_LS/(2Λ α_NS); it is set by the atmospheric mass splitting, not predicted.
  • Charged lepton Yukawas y_ee, y_μμ, y_ττ = adjusted to m_e, m_μ, m_τ
    Set to reproduce observed charged lepton masses; they do not affect neutrino mixing.
axioms (5)
  • domain assumption The Yukawa couplings Y_3^(0) and Y_3^(-2) are polyharmonic Maaß forms of weight 0 and -2 at level 3 with the q-expansions listed in Appendix A.
    These forms enter all mass matrices (Eqs. 3.4,3.6,3.8,3.10). The paper neither derives these expansions nor cites a source for the specific weight -2 forms, nor checks convergence of the truncated series.
  • domain assumption The mass hierarchy M_LS << M_D << M_RS and M_N << M_RS (Eq. 3.12).
    This justifies dropping the double-seesaw term in Eq. (3.13) and using the linear seesaw formula (3.14); the hierarchy is not verified for the fitted parameters.
  • ad hoc to paper Half-integral modular weight assignments in Table 2 forbid all unwanted operators.
    The text states half-integral weights are chosen 'in order to avoid certain unwanted terms' (Section 3). If additional terms survive, the mass matrices gain new entries and the predictions change.
  • standard math The A4 modular group representation theory and tensor products used to form the mass matrices are correct.
    The specific Clebsch-Gordan-like contractions yielding Eqs. (3.4)-(3.10) are not shown; the paper relies on standard A4 product rules.
  • domain assumption The modulus τ acquires a VEV in the scanned region of the fundamental domain.
    The paper scans τ but does not discuss a stabilization mechanism; the VEV is simply an input.
invented entities (2)
  • Heavy gauge-singlet fermions N_R and S_L (two A4 triplets, six chiral fields) no independent evidence
    purpose: Generate light neutrino masses through the linear seesaw mechanism.
    Introduced to realize the seesaw; the paper does not provide a direct experimental handle or mass predictions for these states.
  • Flavon field φ (A4 singlet) no independent evidence
    purpose: Suppresses the pseudo-Dirac term M_LS via its VEV/Λ.
    Included to obtain the M_LS operator; no specific collider or low-energy signature is proposed.

pith-pipeline@v1.3.0-alltime-deepseek · 19505 in / 18517 out tokens · 170623 ms · 2026-08-02T20:30:37.097138+00:00 · methodology

0 comments
read the original abstract

We study a realization of neutrino masses and mixing phenomena within a linear seesaw mechanism based on non-holomorphic modular $A_4$ symmetry, which extends modular-invariant flavor models beyond the conventional holomorphic framework. The model is constructed in a non-supersymmetric setting and involves six heavy $SU(2)_L$ singlet fermions, $N_{Ri}$ and $S_{Li}$, together with a single flavon field, thereby significantly reducing the field content compared to conventional $A_4$ flavor models that typically require multiple flavon fields as well as supersymmetric (holomorphic) modular frameworks involving additional superfields. The modular transformation properties of the Yukawa couplings under $A_4$ symmetry lead to a highly constrained neutrino mass matrix with a distinctive flavor structure. After presenting the general theoretical framework, we perform a systematic numerical analysis of neutrino phenomenology by restricting the modulus parameter $\tau$ to the fundamental domain and scanning the allowed parameter space. We identify regions consistent with current neutrino oscillation data at the $3\sigma$ level and obtain predictions for currently unknown observables, including the absolute neutrino mass scale and leptonic CP-violating phases. We further examine the implications for neutrinoless double beta decay, highlighting testable signatures in the upcoming precision oscillation as well as rare-process experiments. These results demonstrate the phenomenological viability and predictive power of non-holomorphic modular symmetry in linear seesaw neutrino mass models.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 5 Pith papers

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.

  2. Radiative Neutrino Mass in a Nonholomorphic $T'$ Modular Invariant Model

    hep-ph 2026-06 unverdicted novelty 6.0

    A nonholomorphic T' modular model realizes the T4-2-i one-loop topology for radiative Majorana neutrino masses, forbids tree-level seesaws via modular assignments, stabilizes DM with residual Z2, and fits oscillation ...

  3. A Type-I Seesaw Framework with Non-Holomorphic Modular Symmetry

    hep-ph 2026-04 unverdicted novelty 5.0

    Non-holomorphic modular symmetry in a Type-I seesaw model fits normal hierarchy neutrino data with chi2 min 7.06 but rules out inverted hierarchy.

  4. Lepton masses and mixing in non-holomorphic modular $A_4$ with universal couplings

    hep-ph 2026-04 unverdicted novelty 5.0

    A modular A4 flavor model with universal couplings reproduces charged lepton masses via the modulus tau and predicts correlated neutrino observables for normal mass ordering and right-handed weight k_N = -1.

  5. Predictions of Modular Symmetry Fixed Points on Neutrino Masses, Mixing, and Leptogenesis

    hep-ph 2026-04 unverdicted novelty 5.0

    Fixed points of modular symmetry in a type III seesaw model produce viable neutrino phenomenology and the observed baryon asymmetry.

Reference graph

Works this paper leans on

106 extracted references · 78 linked inside Pith · cited by 5 Pith papers

  1. [1]

    Faessler,Status of the determination of the electron–neutrino mass,Prog

    A. Faessler,Status of the determination of the electron–neutrino mass,Prog. Part. Nucl. Phys.113(2020) 103789. [2]KATRINcollaboration,Improved Upper Limit on the Neutrino Mass from a Direct Kinematic Method by KATRIN,Phys. Rev. Lett.123(2019) 221802 [1909.06048]

    arXiv 2020

  2. [3]

    Pontecorvo,Neutrino Experiments and the Problem of Conservation of Leptonic Charge, Sov

    B. Pontecorvo,Neutrino Experiments and the Problem of Conservation of Leptonic Charge, Sov. Phys. JETP26(1968) 984

    1968

  3. [4]

    Feruglio,Neutrino masses and mixing angles: A tribute to Guido Altarelli,Frascati Phys

    F. Feruglio,Neutrino masses and mixing angles: A tribute to Guido Altarelli,Frascati Phys. Ser.64(2017) 174. [5]Particle Data Groupcollaboration,Review of Particle Physics,Phys. Rev.D98(2018) 030001

    2017

  4. [6]

    Weinberg,Varieties of Baryon and Lepton Nonconservation,Phys

    S. Weinberg,Varieties of Baryon and Lepton Nonconservation,Phys. Rev.D22(1980) 1694

    1980

  5. [7]

    Weinberg,Baryon and Lepton Nonconserving Processes,Phys

    S. Weinberg,Baryon and Lepton Nonconserving Processes,Phys. Rev. Lett.43(1979) 1566

    1979

  6. [8]

    Wilczek and A

    F. Wilczek and A. Zee,Operator Analysis of Nucleon Decay,Phys. Rev. Lett.43(1979) 1571

    1979

  7. [9]

    Minkowski,µ→eγat a Rate of One Out of10 9 Muon Decays?,Phys

    P. Minkowski,µ→eγat a Rate of One Out of10 9 Muon Decays?,Phys. Lett.67B(1977) 421

    1977

  8. [10]

    Mohapatra and G

    R.N. Mohapatra and G. Senjanovic,Neutrino Mass and Spontaneous Parity Nonconservation,Phys. Rev. Lett.44(1980) 912

    1980

  9. [11]

    Gell-Mann, P

    M. Gell-Mann, P. Ramond and R. Slansky,Complex Spinors and Unified Theories,Conf. Proc.C790927(1979) 315 [1306.4669]

    Pith/arXiv arXiv 1979

  10. [12]

    Zee,A Theory of Lepton Number Violation, Neutrino Majorana Mass, and Oscillation, Phys

    A. Zee,A Theory of Lepton Number Violation, Neutrino Majorana Mass, and Oscillation, Phys. Lett.93B(1980) 389

    1980

  11. [13]

    Babu,Model of ’Calculable’ Majorana Neutrino Masses,Phys

    K.S. Babu,Model of ’Calculable’ Majorana Neutrino Masses,Phys. Lett.B203(1988) 132

    1988

  12. [14]

    Arkani-Hamed, S

    N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali and J. March-Russell,Neutrino masses from large extra dimensions,Phys. Rev.D65(2001) 024032 [hep-ph/9811448]

    Pith/arXiv arXiv 2001

  13. [15]

    Mohapatra and J.W.F

    R.N. Mohapatra and J.W.F. Valle,Neutrino Mass and Baryon Number Nonconservation in Superstring Models,Phys. Rev.D34(1986) 1642

    1986

  14. [16]

    Gonzalez-Garcia and J.W.F

    M.C. Gonzalez-Garcia and J.W.F. Valle,Fast Decaying Neutrinos and Observable Flavor Violation in a New Class of Majoron Models,Phys. Lett.B216(1989) 360

    1989

  15. [17]

    C´ arcamo Hern´ andez, J

    A. C´ arcamo Hern´ andez, J. Marchant Gonz´ alez and U. Salda˜ na-Salazar,Viable low-scale model with universal and inverse seesaw mechanisms,Phys. Rev. D100(2019) 035024 [1904.09993]. – 19 –

    Pith/arXiv arXiv 2019

  16. [18]

    Malinsky, J.C

    M. Malinsky, J.C. Romao and J.W.F. Valle,Novel supersymmetric SO(10) seesaw mechanism,Phys. Rev. Lett.95(2005) 161801 [hep-ph/0506296]

    Pith/arXiv arXiv 2005

  17. [19]

    Mohapatra et al.,Theory of neutrinos: A White paper,Rept

    R.N. Mohapatra et al.,Theory of neutrinos: A White paper,Rept. Prog. Phys.70(2007) 1757 [hep-ph/0510213]

    Pith/arXiv arXiv 2007

  18. [20]

    Ma and G

    E. Ma and G. Rajasekaran,Softly broken A(4) symmetry for nearly degenerate neutrino masses,Phys. Rev.D64(2001) 113012 [hep-ph/0106291]

    Pith/arXiv arXiv 2001

  19. [21]

    Ishimori, T

    H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto,An introduction to non-Abelian discrete symmetries for particle physicists, vol. 858 (2012), 10.1007/978-3-642-30805-5

    doi:10.1007/978-3-642-30805-5 2012

  20. [22]

    Pakvasa and H

    S. Pakvasa and H. Sugawara,Discrete Symmetry and Cabibbo Angle,Phys. Lett. B73 (1978) 61

    1978

  21. [23]

    Feruglio,Are neutrino masses modular forms?, inFrom My Vast Repertoire ...: Guido Altarelli’s Legacy, pp

    F. Feruglio,Are neutrino masses modular forms?, inFrom My Vast Repertoire ...: Guido Altarelli’s Legacy, pp. 227–266 (2019), DOI [1706.08749]

    Pith/arXiv arXiv 2019

  22. [24]

    Abbas,Flavor masses and mixing in modularA 4 Symmetry,2002.01929

    M. Abbas,Flavor masses and mixing in modularA 4 Symmetry,2002.01929

    Pith/arXiv arXiv 2002

  23. [25]

    King and S.F

    S.J. King and S.F. King,Fermion Mass Hierarchies from Modular Symmetry,2002.00969

    Pith/arXiv arXiv 2002

  24. [26]

    Wang,Lepton Flavor Mixing and CP Violation in the Minimal Type-(I+II) Seesaw Model with a ModularA 4 Symmetry,1912.13284

    X. Wang,Lepton Flavor Mixing and CP Violation in the Minimal Type-(I+II) Seesaw Model with a ModularA 4 Symmetry,1912.13284

    Pith/arXiv arXiv 1912

  25. [27]

    Lu, X.-G

    J.-N. Lu, X.-G. Liu and G.-J. Ding,Modular symmetry origin of texture zeros and quark lepton unification,Phys. Rev. D101(2020) 115020 [1912.07573]

    Pith/arXiv arXiv 2020

  26. [28]

    Nomura, H

    T. Nomura, H. Okada and S. Patra,An Inverse Seesaw model withA 4-modular symmetry, 1912.00379

    Pith/arXiv arXiv 1912

  27. [29]

    Asaka, Y

    T. Asaka, Y. Heo, T.H. Tatsuishi and T. Yoshida,ModularA 4 invariance and leptogenesis, JHEP01(2020) 144 [1909.06520]

    Pith/arXiv arXiv 2020

  28. [30]

    Penedo and S

    J. Penedo and S. Petcov,Lepton Masses and Mixing from ModularS 4 Symmetry,Nucl. Phys. B939(2019) 292 [1806.11040]

    Pith/arXiv arXiv 2019

  29. [31]

    Liu, C.-Y

    X.-G. Liu, C.-Y. Yao and G.-J. Ding,Modular Invariant Quark and Lepton Models in Double Covering ofS 4 Modular Group,2006.10722

    Pith/arXiv arXiv 2006

  30. [32]

    Kobayashi, Y

    T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T.H. Tatsuishi,A 4 lepton flavor model and modulus stabilization fromS 4 modular symmetry,Phys. Rev. D100(2019) 115045 [1909.05139]

    Pith/arXiv arXiv 2019

  31. [33]

    Novichkov, J

    P. Novichkov, J. Penedo, S. Petcov and A. Titov,Modular S 4 models of lepton masses and mixing,JHEP04(2019) 005 [1811.04933]

    Pith/arXiv arXiv 2019

  32. [34]

    Novichkov, J.T

    P.P. Novichkov, J.T. Penedo and S.T. Petcov,Double cover of modularS 4 for flavour model building,Nucl. Phys. B963(2021) 115301 [2006.03058]

    Pith/arXiv arXiv 2021

  33. [35]

    Ding, S.F

    G.-J. Ding, S.F. King, X.-G. Liu and J.-N. Lu,Modular S 4 and A 4 symmetries and their fixed points: new predictive examples of lepton mixing,JHEP12(2019) 030 [1910.03460]

    arXiv 2019

  34. [36]

    Ding, S.F

    G.-J. Ding, S.F. King and X.-G. Liu,Modular A 4 symmetry models of neutrinos and charged leptons,JHEP09(2019) 074 [1907.11714]

    Pith/arXiv arXiv 2019

  35. [37]

    Novichkov, J

    P. Novichkov, J. Penedo, S. Petcov and A. Titov,Modular A 5 symmetry for flavour model building,JHEP04(2019) 174 [1812.02158]. – 20 –

    Pith/arXiv arXiv 2019

  36. [38]

    Nomura, H

    T. Nomura, H. Okada and O. Popov,A modularA 4 symmetric scotogenic model,Phys. Lett. B803(2020) 135294 [1908.07457]

    Pith/arXiv arXiv 2020

  37. [39]

    Ma,Neutrino mixing:A 4 variations,Phys

    E. Ma,Neutrino mixing:A 4 variations,Phys. Lett. B752(2016) 198 [1510.02501]

    Pith/arXiv arXiv 2016

  38. [41]

    Mishra, M.K

    S. Mishra, M.K. Behera, R. Mohanta, S. Patra and S. Singirala,Neutrino phenomenology and dark matter in anA 4 flavour extendedB−Lmodel,Eur. Phys. J. C80(2020) 420 [1907.06429]

    Pith/arXiv arXiv 2020

  39. [42]

    Behera, S

    M.K. Behera, S. Mishra, S. Singirala and R. Mohanta,Implications of A4 modular symmetry on neutrino mass, mixing and leptogenesis with linear seesaw,Phys. Dark Univ. 36(2022) 101027 [2007.00545]

    Pith/arXiv arXiv 2022

  40. [43]

    Behera, S

    M.K. Behera, S. Singirala, S. Mishra and R. Mohanta,A modular A 4 symmetric scotogenic model for neutrino mass and dark matter,J. Phys. G49(2022) 035002 [2009.01806]

    Pith/arXiv arXiv 2022

  41. [44]

    Behera and R

    M.K. Behera and R. Mohanta,Inverse seesaw inA ′ 5 modular symmetry,J. Phys. G49 (2022) 045001 [2108.01059]

    Pith/arXiv arXiv 2022

  42. [45]

    Mishra, M.K

    P. Mishra, M.K. Behera, P. Panda and R. Mohanta,Type III seesaw underA 4 modular symmetry with leptogenesis,Eur. Phys. J. C82(2022) 1115 [2204.08338]

    Pith/arXiv arXiv 2022

  43. [46]

    Mishra, M.K

    P. Mishra, M.K. Behera, P. Panda, M. Ghosh and R. Mohanta,Exploring models with modular symmetry in neutrino oscillation experiments,JHEP09(2023) 144 [2305.08576]

    Pith/arXiv arXiv 2023

  44. [47]

    Kumar, P

    R. Kumar, P. Mishra, M.K. Behera, R. Mohanta and R. Srivastava,Predictions from scoto-seesaw with A4 modular symmetry,Phys. Lett. B853(2024) 138635 [2310.02363]

    Pith/arXiv arXiv 2024

  45. [48]

    Behera, P

    M.K. Behera, P. Ittisamai, C. Pongkitivanichkul and P. Uttayarat,Neutrino phenomenology in the modular S3 seesaw model,Phys. Rev. D110(2024) 035004 [2403.00593]

    Pith/arXiv arXiv 2024

  46. [49]

    Behera, P

    M.K. Behera, P. Ittisamai, C. Pongkitivanichkul and P. Uttayarat,Phenomenology of inverse seesaw usingS 3 modular symmetry,Eur. Phys. J. C85(2025) 1316 [2504.12954]

    Pith/arXiv arXiv 2025

  47. [50]

    Qu and G.-J

    B.-Y. Qu and G.-J. Ding,Non-holomorphic modular flavor symmetry,JHEP08(2024) 136 [2406.02527]

    Pith/arXiv arXiv 2024

  48. [51]

    Verma,Tri-Resonant Leptogenesis in a Non-Holomorphic Modular A 4 Scotogenic Model,2602.17243

    Tapender and S. Verma,Tri-Resonant Leptogenesis in a Non-Holomorphic Modular A 4 Scotogenic Model,2602.17243

    arXiv

  49. [52]

    Nasri, L

    S. Nasri, L. Singh, Tapender and S. Verma,Dark-Portal Leptogenesis in a Non-Holomorphic Modular Scoto-Seesaw Model,2601.06435

    arXiv

  50. [53]

    Singh, B.C

    Priya, L. Singh, B.C. Chauhan and S. Verma,Type-III seesaw in non-holomorphic modular symmetry and leptogenesis,JHEP01(2026) 036 [2508.05047]

    Pith/arXiv arXiv 2026

  51. [54]

    Nomura and H

    T. Nomura and H. Okada,Type-II seesaw of a non-holomorphic modular A4 symmetry, Phys. Lett. B868(2025) 139763 [2408.01143]

    Pith/arXiv arXiv 2025

  52. [55]

    Nomura, H

    T. Nomura, H. Okada and O. Popov,Non-holomorphic modular A4 symmetric scotogenic model,Phys. Lett. B860(2025) 139171 [2409.12547]

    Pith/arXiv arXiv 2025

  53. [56]

    Nomura and H

    T. Nomura and H. Okada,Zee model in a non-holomorphic modular A4 symmetry,Phys. Lett. B867(2025) 139618 [2412.18095]. – 21 –

    Pith/arXiv arXiv 2025

  54. [57]

    Nomura, H

    T. Nomura, H. Okada and X.-Y. Wang,A radiative neutrino mass model with leptoquarks under non-holomorphic modular A 4 symmetry,JHEP09(2025) 163 [2504.21404]

    Pith/arXiv arXiv 2025

  55. [58]

    Nomura and H

    T. Nomura and H. Okada,Neutrino mass model at a three-loop level from a non-holomorphic modularA 4 symmetry,2506.02639

    arXiv

  56. [59]

    Nomura and H

    T. Nomura and H. Okada,A new type of lepton seesaw model in a modularA 4 symmetry, 2503.19251

    arXiv

  57. [60]

    Kang and H

    S.K. Kang and H. Okada,Neutrino masses and mixing in an axion model,Eur. Phys. J. C 85(2025) 917 [2408.14942]

    Pith/arXiv arXiv 2025

  58. [61]

    Ding, J.-N

    G.-J. Ding, J.-N. Lu, S.T. Petcov and B.-Y. Qu,Non-holomorphic modular S 4 lepton flavour models,JHEP01(2025) 191 [2408.15988]

    Pith/arXiv arXiv 2025

  59. [62]

    Li, J.-N

    C.-C. Li, J.-N. Lu and G.-J. Ding,Non-holomorphic modular A 5 symmetry for lepton masses and mixing,JHEP12(2024) 189 [2410.24103]

    Pith/arXiv arXiv 2024

  60. [63]

    Okada and Y

    H. Okada and Y. Orikasa,A radiative seesaw in a non-holomorphic modularS 3 flavor symmetry,2501.15748

    Pith/arXiv arXiv

  61. [64]

    Kobayashi, H

    T. Kobayashi, H. Okada and Y. Orikasa,Zee-Babu model in a non-holomorphic modularA 4 symmetry and modular stabilization,2502.12662

    arXiv

  62. [65]

    Loualidi, M

    M.A. Loualidi, M. Miskaoui and S. Nasri,Nonholomorphic A4 modular invariance for fermion masses and mixing in SU(5) GUT,Phys. Rev. D112(2025) 015008 [2503.12594]

    Pith/arXiv arXiv 2025

  63. [66]

    Zhang and Y

    X. Zhang and Y. Reyimuaji,Inverse seesaw model in nonholomorphic modular A4 flavor symmetry,Phys. Rev. D112(2025) 075050 [2507.06945]

    arXiv 2025

  64. [67]

    Nomura and H

    T. Nomura and H. Okada,Lepton seesaw model in a modularA 4 symmetry,2409.10912

    arXiv

  65. [68]

    Abbas,Lepton Masses and Mixing in Nonholomorphic Modular A4 Symmetry,PHEP 2025(2025) 7

    M. Abbas,Lepton Masses and Mixing in Nonholomorphic Modular A4 Symmetry,PHEP 2025(2025) 7

    2025

  66. [69]

    Li and G.-J

    C.-C. Li and G.-J. Ding,Lepton models from non-holomorphicA ′ 5 modular flavor symmetry,2509.15183

    arXiv

  67. [70]

    Dey,The Seesaw Evaded Modular Dirac Framework,2509.10373

    M. Dey,The Seesaw Evaded Modular Dirac Framework,2509.10373

    Pith/arXiv arXiv

  68. [71]

    Nanda, M

    S.K. Nanda, M. Ricky Devi and S. Patra,Non-HolomorphicA 4 Modular Symmetry in Type-I Seesaw: Implications for Neutrino Masses and Leptogenesis,2509.22108

    arXiv

  69. [72]

    Kumar and M.K

    B. Kumar and M.K. Das,Leptogenesis, 0νββand lepton flavor violation in modular left-right asymmetric model with polyharmonic Maaß forms,JHEP09(2025) 071 [2504.21701]

    Pith/arXiv arXiv 2025

  70. [73]

    Ishimori, T

    H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto,Non-Abelian Discrete Symmetries in Particle Physics,Prog. Theor. Phys. Suppl.183(2010) 1 [1003.3552]

    Pith/arXiv arXiv 2010

  71. [74]

    Chauhan, P.S.B

    G. Chauhan, P.S.B. Dev, I. Dubovyk, B. Dziewit, W. Flieger, K. Grzanka et al., Phenomenology of lepton masses and mixing with discrete flavor symmetries,Prog. Part. Nucl. Phys.138(2024) 104126 [2310.20681]

    Pith/arXiv arXiv 2024

  72. [75]

    Kobayashi, Y

    T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T.H. Tatsuishi and H. Uchida,Finite modular subgroups for fermion mass matrices and baryon/lepton number violation,Phys. Lett. B794(2019) 114 [1812.11072]. – 22 –

    Pith/arXiv arXiv 2019

  73. [76]

    Okada and Y

    H. Okada and Y. Orikasa,ModularS 3 symmetric radiative seesaw model,Phys. Rev. D100 (2019) 115037 [1907.04716]

    Pith/arXiv arXiv 2019

  74. [77]

    Mishra,Neutrino mixing and Leptogenesis with modularS 3 symmetry in the framework of type III seesaw,2008.02095

    S. Mishra,Neutrino mixing and Leptogenesis with modularS 3 symmetry in the framework of type III seesaw,2008.02095

    Pith/arXiv arXiv 2008

  75. [78]

    S.K. Kang, R. Kumar and H. Okada,Radiative Dirac Neutrino Masses from ModularS 3 Symmetry in an Axion Model,2601.22740

    arXiv

  76. [79]

    Meloni and M

    D. Meloni and M. Parriciatu,A simplest modular S 3 model for leptons,JHEP09(2023) 043 [2306.09028]

    Pith/arXiv arXiv 2023

  77. [80]

    Marciano, D

    S. Marciano, D. Meloni and M. Parriciatu,Minimal seesaw and leptogenesis with the smallest modular finite group,JHEP05(2024) 020 [2402.18547]

    Pith/arXiv arXiv 2024

  78. [81]

    Belfkir, M.A

    M. Belfkir, M.A. Loualidi and S. Nasri,Fermion Masses and Mixing in Pati–Salam Unification with S3 Modular Symmetry,PTEP2025(2025) 033B05 [2501.00302]

    Pith/arXiv arXiv 2025

  79. [82]

    Nomura, H

    T. Nomura, H. Okada and H. Otsuka,Texture zeros realization in a three-loop radiative neutrino mass model from modular A4 symmetry,Nucl. Phys. B1004(2024) 116579 [2309.13921]

    Pith/arXiv arXiv 2024

  80. [83]

    Ricky Devi,Neutrino Masses and Higher Degree Siegel Modular Forms,2401.16257

    M. Ricky Devi,Neutrino Masses and Higher Degree Siegel Modular Forms,2401.16257

    Pith/arXiv arXiv

Showing first 80 references.