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

Gourab Pathak; Mrinal Kumar Das

REVIEW 3 major objections 4 minor 65 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A $T'$ modular model with Froggatt-Nielsen-like weights fits neutrino oscillation data and leptogenesis.

desk verdict A workable FN-like modular T' seesaw fit, with fixable typos and a leptogenesis benchmark that needs to be tied to the model. read the letter →

arxiv 2508.13578 v1 pith:XMHLF3JG submitted 2025-08-19 hep-ph

Gourab Pathak , Mrinal Kumar Das This is my paper · ORCID

classification hep-ph

keywords modularsymmetryFroggatt-NielsenmechanismT-primegroupType-Iseesawneutrinooscillationsneutrinolessdoublebetadecaythermalleptogenesisbaryonasymmetry

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

The paper tries to establish that the Froggatt-Nielsen mechanism can be transplanted into modular symmetry: the modular weights of superfields play the role of the $U(1)_{FN}$ charges, so no extra gauge symmetry is needed to produce Yukawa hierarchies. In a $T'$ modular model with Type-I seesaw, the resulting mass matrices are shown to accommodate all neutrino oscillation parameters within their $3\sigma$ ranges. The same parameter choices predict a narrow normal-ordering window $\sum m_\nu \in [0.0576, 0.0646]$ eV, an effective $0\nu\beta\beta$ mass below current experimental reach, and a baryon asymmetry from thermal leptogenesis consistent with observation. A sympathetic reader would care because this is a single, parameter-lean construction claiming to tie neutrino flavor structure to the matter-antimatter asymmetry.

What carries the argument

The engine of the model is the FN-like modular-weight mechanism: invariance of the superpotential forces total modular weight zero, and the weighton $\phi$ compensates the weights that would otherwise forbid an operator, so each coupling comes with powers of $\Phi=\langle\phi\rangle/\Lambda$. The hierarchy of the mass matrices is set by these powers, while the flavor structure is set by $T'$ tensor contractions of modular forms. The basic building block is the weight-1 doublet $Y_2^{(1)}=(Y_1,Y_2)^T$; all higher-weight Yukawas used here, of weights 2 through 5, are constructed from it by $T'$ multiplication rules, and these forms enter the Dirac and Majorana matrices that feed the seesaw.

What would settle it

Derive $Y_3^{(3)}$ from the explicit weight-1 doublet $Y_2^{(1)}$ using the stated $T'$ multiplication rules and compare the resulting Dirac matrix elements $m_{13},m_{23},m_{33}$ with Eq. (26); any mismatch invalidates the reported fit. Observationally, a cosmological measurement of $\sum m_\nu$ outside $[0.0576,0.0646]$ eV, or a $0\nu\beta\beta$ signal with $|m_{ee}|$ above the model's predicted band, would rule the parameter choice out.

Watch

Extended reading notes

Core claim

The central claim is that modular weights can replace Froggatt-Nielsen charges. With $T'$ modular symmetry, a new singlet "weighton" $\phi$ of modular weight $-1$ acquires a VEV, and each allowed operator is suppressed by $\Phi^n=(\langle\phi\rangle/\Lambda)^n$, reproducing the FN power counting without a $U(1)_{FN}$ gauge group. Using the modular forms $Y_2^{(1)}(\tau)$ and their tensor products up to weight 5 in the charged-lepton, Dirac, and Majorana superpotentials, the Type-I seesaw formula $m_\nu = M_D M_R^{-1} M_D^T$ is numerically scanned over $\tau$, the $g_i$, $\Phi$, and $\Lambda$. The paper reports that the resulting mixing angles and mass-squared differences lie inside the $3\sigma$ oscillation bounds, with $\sum m_\nu\in[0.0576,0.0646]$ eV, $|m_{ee}|$ below current experiment, and $\eta_B$ from thermal leptogenesis matching BBN/CMB values.

Load-bearing premise

The load-bearing premise is that the weight-3 triplet modular form $Y_3^{(3)}$ appearing in the Dirac superpotential has a definite but never-stated form; the reported fit depends on that form, and if its components differ from what is implicitly assumed, the mass matrices and all resulting predictions change.

Editorial extensions

If this is right

If the claim is right, neutrino mass and mixing hierarchies can be generated without adding a $U(1)_{FN}$ gauge symmetry, because modular weights already supply the ordering.
The model makes a sharp cosmological target: normal-ordering neutrino masses summing to $[0.0576,0.0646]$ eV, a range future cosmological surveys could confirm or exclude.
It predicts that $0\nu\beta\beta$ remains unobservable at current exposure, so a future discovery of $|m_{ee}|$ above the model's band would rule the construction out.
It shows thermal leptogenesis from the lightest right-handed neutrino, with $M_1\simeq 1.8\times 10^{10}$ GeV, can work together with a modular flavor symmetry in the single-flavour approximation.
The same FN-like modular setup can be transplanted to other neutrino mass models, as the paper itself suggests.

Reading between the lines

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

Extension: because the allowed region concentrates $\tau$ near $\mathrm{Re}\,\tau\simeq 0$ and $\mathrm{Im}\,\tau\simeq 1.45$, the same weighton mechanism could in principle fix charged-lepton hierarchies from the same modulus; the paper does not explore this.
Extension: the leptogenesis plot uses one representative set of Yukawa couplings rather than a scan over the full allowed region, so scanning $\eta_B$ over the entire parameter space would show whether the baryon asymmetry is a generic prediction or a tuned point.
Extension: the leptogenesis calculation uses the single-flavour Boltzmann approximation; flavour-resolved Boltzmann equations could shift $\eta_B$ and sharpen the model's testable range.
Extension: the paper assumes normal ordering and does not discuss inverted ordering; an inverted-ordering scan would test whether the construction is specific to the normal-ordering fit.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a typical modular flavor model paper, but with a genuinely new ingredient — the Froggatt-Nielsen-like mechanism applied to leptons, where modular weights act as FN charges and a 'weighton' field supplies the suppression powers. That was previously done only for quarks (Kuranaga et al. 2021). The specific T' charge assignment and superpotential are original, and the numerical fit does what it claims: a scan over τ, g_i, Φ, and Λ reproduces the neutrino oscillation parameters within 3σ and yields a tight prediction for Σmν (0.0576–0.0646 eV) and |m_ee| below current limits. The correlation plots look reasonable, and the J_CP and δ_CP ranges are in the expected ballpark.

The paper has three real soft spots, none fatal but all needing correction in revision. First, Table I gives the weighton φ a modular weight of 0, while the text and the superpotential powers require −1; that's a typo. Second, the Dirac superpotential in Eq. (24) uses Y_3^{(3)}, which is never defined, and Y_{2''}^{(3)} for the g2 term. The mass matrix entries in Eq. (26) make it clear the g3 term uses Y_3^{(2)} (weight 2) and the g2 term uses Y_{2''}^{(5)} (weight 5), both defined in Sec. II. These notation slips look careless but are easily fixed. Third, and more substantively, the leptogenesis result is not properly tied to the model. The ULYSSES run uses a single Yukawa matrix Y_D (Eq. 44) and masses M1, M2, M3, but the paper does not give the underlying values of τ, Φ, g_i, Λ, nor show that those inputs reproduce the oscillation fit, nor that the M_R of Eq. (28) yields the quoted mass eigenvalues. As written, η_B is an independent input, not a prediction. This is a reproducibility gap rather than a conceptual flaw — the framework could well deliver the asymmetry if the benchmark is real — but the reader cannot check it. A referee should ask for the full benchmark point and a consistency check.

The paper would also benefit from a χ² or best-fit point and from toning down the word 'prediction' for the asymmetry until the benchmark is supplied. On net, the central claim that this FN-like modular T' seesaw accommodates neutrino data is supported. I'd send it to peer review and ask for those corrections.

Referee Report

3 major / 4 minor

Summary. The manuscript constructs a Type-I seesaw neutrino mass model based on the modular double-cover group T', in which an FN-like mechanism is implemented through a new chiral superfield φ ('weighton') whose VEV generates powers of Φ = ⟨φ⟩/Λ that suppress Yukawa couplings. The model assigns modular weights to the lepton and Higgs superfields, uses modular forms of the T' group of weights 2, 3, and 5, and derives charged-lepton, Dirac neutrino, and heavy Majorana mass matrices. The authors scan the free parameters (Re τ, Im τ, g1, g2, g3, Λ, Φ), select points passing 3σ neutrino oscillation constraints, and present correlations for the sum of neutrino masses, the effective double-beta decay mass |m_ee|, the Jarlskog invariant, and the Dirac CP phase. They then feed a single Yukawa benchmark into the ULYSSES package to compute the baryon asymmetry from thermal leptogenesis, claiming consistency with the observed η_B.

Significance. If the model definition is made internally consistent, the framework is of genuine interest: it offers an FN-like mechanism without an extra U(1)_FN gauge symmetry, and it connects neutrino oscillation data to predictions for 0νββ and leptogenesis within one modular T' setup. The use of the public ULYSSES code is a strength, and the derived predictions for Σm_ν and |m_ee| are of the sort the community can use to discriminate models. However, the manuscript as submitted contains unresolved contradictions in the superpotential and in the modular-weight assignment of the weighton, and the leptogenesis calculation is not demonstrably tied to the fitted model parameter space. These are load-bearing issues, not cosmetic ones.

major comments (3)

[Section III, Eq. (24)] The Dirac superpotential in Eq. (24) does not generate the Dirac mass matrix in Eq. (26). The third term of Eq. (24) uses a weight-3 triplet Y_3^{(3)}, but no such modular form is defined in Section II.B; the only triplet defined there is the weight-2 form Y_3^{(2)} in Eq. (18), and the entries m13, m23, m33 in Eq. (26) are precisely the components of Y_3^{(2)} with a common factor Φ^4. Similarly, the second term of Eq. (24) uses Y_{2''}^{(3)}, but the corresponding entries m11 and m12 are built from the weight-5 doublet Y_{2''}^{(5)} of Eq. (21). Equation (24) also omits the powers of Φ that appear in Eq. (26). As written, Eq. (24) is not modular invariant and is inconsistent with the mass matrix that is actually used in the numerical scan. Please rewrite the superpotential with the correct modular-form labels and with explicit Φ factors.
[Table I and text near Eq. (22)] The text states that the weighton φ carries modular weight −1, but Table I lists k_I(φ) = 0. The Φ powers in Eqs. (26) and (28) are justified only if φ has modular weight −1; with k_I(φ) = 0, the suppression factors Φ^3, Φ^4, Φ^5, Φ^6, Φ^8, and Φ^9 have no modular-invariance rationale, and the superpotential terms are not modular invariant. This is an internal contradiction that must be resolved, since the FN-like mechanism advertised in the title and abstract relies on the modular-weight compensation by φ.
[Section IV.C] The leptogenesis calculation is not shown to follow from the model. The ULYSSES input Y_D in Eq. (44) and the quoted heavy-neutrino masses M1 = 1.8×10^10 GeV, M2 = 6.7×10^10 GeV, M3 = 1.2×10^11 GeV are asserted to 'also satisfy the neutrino oscillation data', but no values of (τ, Φ, g1, g2, g3, Λ) are given, and no check is shown that Y_D equals M_D/v_u with M_D from Eqs. (25)–(26) or that the quoted M_i are the eigenvalues of M_R in Eq. (28). Since M_D and M_R are highly structured functions of τ and Φ, a generic 3×3 Y_D cannot serve as a model prediction. Please provide the benchmark parameter point and explicitly verify both relations; otherwise the claimed consistency of η_B with observation is untethered from the neutrino-sector fit.

minor comments (4)

[Eq. (10)] The Kähler potential is written as K = Σ Φ_i ̄Φ_i / Imτ^{-k_i}, which is ambiguous; the modular-invariant form should be (Im τ)^{-k_i} or, more standardly, (-iτ + īτ)^{k_i}. Please clarify the exponent placement.
[Section IV.A] The scan-and-select procedure is described, but no goodness-of-fit measure (e.g., χ^2 or pulls) or the number of accepted points is reported. This would help the reader judge how robust the displayed allowed regions are and whether the fit is a fine-tuned corner of parameter space.
[Section IV.C] There is a typo: 'mwthod' should be 'method'. Also, 'equillibrium' should be 'equilibrium'. These do not affect the physics but should be corrected.
[Section III] The T' Clebsch–Gordan coefficients used to construct the singlet contractions are not specified. Without them (or a reference containing them), the explicit entries of M_l, M_D, and M_R in Eqs. (23), (26), and (28) cannot be independently checked. Consider adding an appendix with the contraction rules.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the neutrino sector is fit to oscillation data and the reported predictions are derived outputs, while the leptogenesis benchmark and the Y_3^{(3)} notation are gaps rather than circular reductions.

full rationale

The paper's central derivation is a conventional modular-symmetry fit, not a circular reduction. The charged-lepton parameters (alpha, beta, gamma) are fixed by the charged-lepton masses through the trace equations in Eq. (32), independently of neutrino oscillation data. The neutrino parameters (tau, g_i, Phi, Lambda) are then scanned over the ranges in Eq. (36) and selected by requiring the three mixing angles and two mass-squared differences to lie inside the 3-sigma ranges of Table III; this is parameter fitting, not a prediction from the data. The quantities presented as predictions, namely the sum of light neutrino masses (Fig. 3), the effective double-beta mass |m_ee| (Fig. 5), and the Dirac phase delta_CP (Fig. 4), are computed from the resulting light-neutrino mass matrix and are not fed back into the fit. The leptogenesis section uses ULYSSES with the explicit benchmark Yukawa matrix Y_D in Eq. (44) and masses M1, M2, M3; the text asserts that these 'also satisfy the neutrino oscillation data' but does not display the corresponding values of tau, Phi, g_i, and Lambda, nor verify that Eq. (44) equals M_D/v_u from Eqs. (25)-(26). That is an unsupported reproducibility gap in the chain from the fitted model to the eta_B claim, not a circular step, because eta_B is not used to define the benchmark inputs. Similarly, Eq. (24) labels a coupling Y_3^{(3)} while Eq. (26) uses entries matching the weight-2 triplet Y_3^{(2)}; this is an internal inconsistency or typo, not a circularity, since the explicit mass matrix defines the actual model used. There is also no load-bearing self-citation: the modular-form input Y_2^{(1)} is taken from the external work [34], and the self-citations [8,9] are contextual references for modular leptogenesis and scoto-inverse seesaw, not the source of the T' construction. Overall, the derivation chain is self-contained against the fitted inputs, so no circular step can be exhibited.

Assumptions & free parameters 5 free parameters · 7 assumptions · 1 invented entities

The model's central fit rests on scanning seven continuous parameters (τ, g1-3, Φ, Λ, plus tanβ fixed by hand) and on the unverified existence of the Y_3^{(3)} modular form. The weighton field is an invented hidden scalar with no independent evidence.

free parameters (5)

modulus τ = scan range Re ∈ [-0.5, 0.5], Im ∈ [0.8, 1.5]
The complex modulus is scanned and selected to satisfy the 3σ neutrino oscillation data.
g1, g2, g3 = scan range [0.1, 1]
Dirac Yukawa couplings, fitted to oscillation data.
Φ = ⟨φ⟩/Λ = scan range [0.2, 0.6]
Weighton VEV ratio controlling FN suppression, fitted to oscillation data.
Λ = scan range [1e11, 1e14] GeV
FN/Majorana mass scale, fitted to oscillation data.
tanβ = 5
Chosen ratio v_u/v_d following refs. [4, 43].

assumptions (7)

standard math Γ'_3 ≅ T' is the finite modular double covering group
Used throughout Section II B for the flavor group; multiplication rules and modular forms taken from refs. [16, 34].
standard math The q-expansions and products of T' modular forms in Eqs. (17)-(21) are correct
The paper relies on these expressions from ref. [34] without independent verification.
ad hoc to paper The weighton has modular weight -1, enabling modular-invariant superpotential terms with Φ = ⟨ϕ⟩/Λ
The text states -1 but Table I lists 0; the superpotential powers in Eqs. (22), (24), (27) require -1.
ad hoc to paper A weight-3 triplet modular form Y_3^{(3)} exists with the components implicitly used in Eq. (26)
Y_3^{(3)} is used in Eq. (24) but never defined.
standard math Type-I seesaw with M_R >> M_D yields mν = M_D M_R^{-1} M_D^T
Standard seesaw relation, Eq. (31).
domain assumption The FN cutoff equals the Majorana scale, M = Λ
Stated in Section III, with citation [27].
domain assumption Single-flavor Boltzmann approximation is adequate for the chosen leptogenesis benchmark
Used in Section IV C with ULYSSES; the footnote acknowledges resonant leptogenesis for degenerate masses.

invented entities (1)

weighton φ
purpose: A new scalar singlet that compensates modular weights and provides FN-like suppression through Φ = ⟨ϕ⟩/Λ
No direct experimental handle is given; its VEV and mass scale are free inputs and it is not linked to any observed particle or process.

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Cite this review

Pith. "Pith review of Froggatt-Nielsen like mechanism in the framework of Modular Symmetry for Neutrino Mass, Mixing and Leptogenesis." pith.science (2026). https://pith.science/paper/XMHLF3JG

@misc{pith2026250813578,
  author       = {Pith},
  title        = {Pith review of: Froggatt-Nielsen like mechanism in the framework of Modular Symmetry for Neutrino Mass, Mixing and Leptogenesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XMHLF3JG}},
  note         = {Machine review of arXiv:2508.13578}
}

abstract

We study a neutrino mass model with Froggatt-Nielsen (FN) like modular symmetry. The FN mechanism requires an additional gauge symmetry, $U(1)_{FN}$, which is spontaneously broken at high energies. But in this work, we do not need an extra symmetry as modular weights play the role of the FN charges of the additional $U(1)_{FN}$ symmetry. We have constructed a neutrino mass model using FN-like modular symmetry in the $T'$ group. This model can accommodate neutrino oscillation parameters and also address other phenomena beyond the Standard Model, such as neutrinoless double beta decay and the baryon asymmetry of the universe.

Figures

Figures reproduced from arXiv: 2508.13578 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** FIG. 3: Correlation of the mixing angles [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: FIG. 5: Effective neutrino mass [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7: Variation of lepton number density for single flavor and baryon asymmetry parameter with [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

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Works this paper leans on

65 extracted references · 12 canonical work pages

[1]

Feruglio,Are neutrino masses modular forms?, pp

F. Feruglio,Are neutrino masses modular forms?, pp. 227–266. 2019.arXiv:1706.08749

arXiv 2019
[2]

Meloni and M

D. Meloni and M. Parriciatu,A simplest modular S3 model for leptons, JHEP 09 (2023) 043, [arXiv:2306.09028]

arXiv 2023
[3]

Okada and Y

H. Okada and Y. Orikasa,ModularS3 symmetric radiative seesaw model, Phys. Rev. D100 (2019), no. 11 115037, [arXiv:1907.04716]

arXiv 2019
[4]

Kashav and S

M. Kashav and S. Verma,Broken scaling neutrino mass matrix and leptogenesis based on A4 modular invariance, JHEP 09 (2021) 100, [arXiv:2103.07207]

arXiv 2021
[5]

In this work, we consider a modular symmetric model based on Γ′, and all the matter fields are represented by chiral superfields

In the double covering of the modular group, we can define modular forms of odd modular weights as well. In this work, we consider a modular symmetric model based on Γ′, and all the matter fields are represented by chiral superfields. These fields transform as the modular forms of level N and weight k as follows: γ : Φ i→ (cτ +d)kiρΦ(γ)i,jΦj, (9) where Φi...

work page 2022
[6]

Nomura and H

T. Nomura and H. Okada,A modularA4 symmetric model of dark matter and neutrino, Phys. Lett. B797 (2019) 134799, [arXiv:1904.03937]

arXiv 2019
[7]

M. K. Behera, S. Singirala, S. Mishra, and R. Mohanta,A modular A4 symmetric scotogenic model for neutrino mass and dark matter, J. Phys. G49 (2022), no. 3 035002, [arXiv:2009.01806]

arXiv 2022
[8]

Altarelli and F

G. Altarelli and F. Feruglio,Tri-bimaximal neutrino mixing, A(4) and the modular symmetry, Nucl. Phys. B741 (2006) 215–235, [hep-ph/0512103]

arXiv 2006

Show all 65 references

[9]

Pathak and M

G. Pathak and M. K. Das,Matter-antimatter asymmetry in minimal inverse seesaw framework withA4 modular symmetry, arXiv:2505.03000

arXiv
[10]

Pathak, P

G. Pathak, P. Das, and M. K. Das,Neutrino mass genesis in scoto-inverse seesaw with modularA4, Eur. Phys. J. C85 (2025), no. 5 569, [arXiv:2411.13895]

2025 arXiv
[11]

Kobayashi, Y

T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi,A4 lepton flavor model and modulus stabilization fromS4 modular symmetry, Phys. Rev. D100 (2019), no. 11 115045, [arXiv:1909.05139]. [Erratum: Phys.Rev.D 101, 039904 (2020)]

2019 arXiv
[12]

J. T. Penedo and S. T. Petcov,Lepton Masses and Mixing from ModularS4 Symmetry, Nucl. Phys. B939 (2019) 292–307, [arXiv:1806.11040]

2019 arXiv
[13]

Zhang and S

X. Zhang and S. Zhou,Inverse seesaw model with a modular S 4 symmetry: lepton flavor mixing and warm dark matter, JCAP 09 (2021) 043, [arXiv:2106.03433]. 19

2021 arXiv
[14]

Wang and S

X. Wang and S. Zhou,The minimal seesaw model with a modular S4 symmetry, JHEP 05 (2020) 017, [arXiv:1910.09473]

2020 arXiv
[15]

P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov,Modular A5 symmetry for flavour model building, JHEP 04 (2019) 174, [arXiv:1812.02158]

2019 arXiv
[16]

G.-J. Ding, S. F. King, and X.-G. Liu,Neutrino mass and mixing withA5 modular symmetry, Phys. Rev. D100 (2019), no. 11 115005, [arXiv:1903.12588]

2019 arXiv
[17]

Liu and G.-J

X.-G. Liu and G.-J. Ding,Neutrino Masses and Mixing from Double Covering of Finite Modular Groups, JHEP 08 (2019) 134, [arXiv:1907.01488]

2019 arXiv
[18]

Mishra, M

P. Mishra, M. K. Behera, and R. Mohanta,Neutrino phenomenology, W-mass anomaly, and muon (g-2) in a minimal type-III seesaw model using a T’ modular symmetry, Phys. Rev. D 107 (2023), no. 11 115004, [arXiv:2302.00494]

2023 arXiv
[19]

P. P. Novichkov, J. T. Penedo, and S. T. Petcov,Double cover of modularS4 for flavour model building, Nucl. Phys. B963 (2021) 115301, [arXiv:2006.03058]

2021 arXiv
[20]

Liu, C.-Y

X.-G. Liu, C.-Y. Yao, and G.-J. Ding,Modular invariant quark and lepton models in double covering ofS4 modular group, Phys. Rev. D103 (2021), no. 5 056013, [arXiv:2006.10722]

2021 arXiv
[21]

X. Wang, B. Yu, and S. Zhou,Double covering of the modularA5 group and lepton flavor mixing in the minimal seesaw model, Phys. Rev. D103 (2021), no. 7 076005, [arXiv:2010.10159]

2021 arXiv
[22]

Yao, X.-G

C.-Y. Yao, X.-G. Liu, and G.-J. Ding,Fermion masses and mixing from the double cover and metaplectic cover of theA5 modular group, Phys. Rev. D103 (2021), no. 9 095013, [arXiv:2011.03501]

2021 arXiv
[23]

M. K. Behera and R. Mohanta,Inverse seesaw inA′ 5 modular symmetry, J. Phys. G49 (2022), no. 4 045001, [arXiv:2108.01059]

2022 arXiv
[24]

C. D. Froggatt and H. B. Nielsen,Hierarchy of Quark Masses, Cabibbo Angles and CP Violation, Nucl. Phys. B147 (1979) 277–298

1979
[25]

M. Ibe, S. Shirai, and K. Watanabe,Comprehensive Bayesian exploration of Froggatt-Nielsen mechanism, JHEP 03 (2025) 150, [arXiv:2412.19484]

2025 arXiv
[26]

Kamikado, T

H. Kamikado, T. Shindou, and E. Takasugi,Froggatt-Nielsen hierarchy and the neutrino mass matrix, arXiv:0805.1338

arXiv
[27]

Qiu, J.-W

Y.-C. Qiu, J.-W. Wang, and T. T. Yanagida,Predictions of mee and neutrino mass from a consistent Froggatt-Nielsen model, Phys. Rev. D108 (2023), no. 11 115021, 20 [arXiv:2307.16470]

2023 arXiv
[28]

Cornella, D

C. Cornella, D. Curtin, G. Krnjaic, and M. Mellors,Testing the Froggatt-Nielsen Mechanism with Lepton Violation, arXiv:2501.00629
[29]

Brdar, A

V. Brdar, A. J. Helmboldt, S. Iwamoto, and K. Schmitz,Type-I Seesaw as the Common Origin of Neutrino Mass, Baryon Asymmetry, and the Electroweak Scale, Phys. Rev. D100 (2019) 075029, [arXiv:1905.12634]

2019 arXiv
[30]

M. K. Behera, P. Ittisamai, C. Pongkitivanichkul, and P. Uttayarat,Exploring type-I seesaw under S 3 modular symmetry, EPJ Web Conf.312 (2024) 02010

2024
[31]

Minkowski,µ→ eγ at a rate of one out of 109 muon decays?, Physics Letters B67 (1977), no

P. Minkowski,µ→ eγ at a rate of one out of 109 muon decays?, Physics Letters B67 (1977), no. 4 421–428

1977
[32]

Schechter and J

J. Schechter and J. W. Valle,Neutrino masses insu(2)⊗u(1) theories, Physical Review D22 (1980), no. 9 2227

1980
[33]

R. N. Mohapatra and G. Senjanović,Neutrino mass and spontaneous parity nonconservation, Physical Review Letters44 (1980), no. 14 912

1980
[34]

S. J. D. King and S. F. King,Fermion mass hierarchies from modular symmetry, JHEP 09 (2020) 043, [arXiv:2002.00969]

2020 arXiv
[35]

Kuranaga, H

H. Kuranaga, H. Ohki, and S. Uemura,Modular origin of mass hierarchy: Froggatt-Nielsen like mechanism, JHEP 07 (2021) 068, [arXiv:2105.06237]

2021 arXiv
[36]

B. J. P. Jones,The Physics of Neutrinoless Double Beta Decay: A Primer, inTheoretical Advanced Study Institute in Elementary Particle Physics: The Obscure Universe: Neutrinos and Other Dark Matters, 8, 2021. arXiv:2108.09364

2021 arXiv
[37]

M. J. Dolinski, A. W. P. Poon, and W. Rodejohann,Neutrinoless Double-Beta Decay: Status and Prospects, Ann. Rev. Nucl. Part. Sci.69 (2019) 219–251, [arXiv:1902.04097]

2019 arXiv
[38]

S. M. Bilenky and C. Giunti,Neutrinoless double-beta decay: A brief review, Mod. Phys. Lett. A 27 (2012) 1230015, [arXiv:1203.5250]

2012 arXiv
[39]

J. J. Gomez-Cadenas, J. Martin-Albo, M. Sorel, P. Ferrario, F. Monrabal, J. Munoz-Vidal, P. Novella, and A. Poves,Sense and sensitivity of double beta decay experiments, JCAP 06 (2011) 007, [arXiv:1010.5112]

2011 arXiv
[40]

A. D. Sakharov,Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Pisma Zh. Eksp. Teor. Fiz.5 (1967) 32–35

1967
[41]

G.-J. Ding, F. R. Joaquim, and J.-N. Lu,Texture-zero patterns of lepton mass matrices from 21 modular symmetry, JHEP 03 (2023) 141, [arXiv:2211.08136]

2023 arXiv
[42]

Hochmuth, S

K. Hochmuth, S. Petcov, and W. Rodejohann,uPMNS =u† luν, Physics Letters B654 (2007), no. 5-6 177–188

2007
[43]

Esteban, M

I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou,The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178, [arXiv:2007.14792]

2020 arXiv
[44]

Antusch and V

S. Antusch and V. Maurer,Running quark and lepton parameters at various scales, JHEP 11 (2013) 115, [arXiv:1306.6879]

2013 arXiv
[45]

Aghanim et al.,Planck 2018 results

Planck Collaboration, N. Aghanim et al.,Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys.641 (2020) A6, [arXiv:1807.06209]. [Erratum: Astron.Astrophys. 652, C4 (2021)]

2020 arXiv
[46]

Abe et al.,Search for the Majorana Nature of Neutrinos in the Inverted Mass Ordering Region with KamLAND-Zen, Phys

KamLAND-Zen Collaboration, S. Abe et al.,Search for the Majorana Nature of Neutrinos in the Inverted Mass Ordering Region with KamLAND-Zen, Phys. Rev. Lett.130 (2023), no. 5 051801, [arXiv:2203.02139]

2023
[47]

nEXO Collaboration, J. B. Albert et al.,Sensitivity and Discovery Potential of nEXO to Neutrinoless Double Beta Decay, Phys. Rev. C97 (2018), no. 6 065503, [arXiv:1710.05075]

2018 arXiv
[48]

Marciano, D

S. Marciano, D. Meloni, and M. Parriciatu,Minimal seesaw and leptogenesis with the smallest modular finite group, JHEP 05 (2024) 020, [arXiv:2402.18547]

2024 arXiv
[49]

Davidson, E

S. Davidson, E. Nardi, and Y. Nir,Leptogenesis, Phys. Rept.466 (2008) 105–177, [arXiv:0802.2962]

2008 arXiv
[50]

Buchmuller, P

W. Buchmuller, P. Di Bari, and M. Plumacher,Leptogenesis for pedestrians, Annals Phys. 315 (2005) 305–351, [hep-ph/0401240]

2005 arXiv
[51]

Buchmuller, R

W. Buchmuller, R. D. Peccei, and T. Yanagida,Leptogenesis as the origin of matter, Ann. Rev. Nucl. Part. Sci.55 (2005) 311–355, [hep-ph/0502169]

2005 arXiv
[52]

Pilaftsis,CP violation and baryogenesis due to heavy Majorana neutrinos, Phys

A. Pilaftsis,CP violation and baryogenesis due to heavy Majorana neutrinos, Phys. Rev. D 56 (1997) 5431–5451, [hep-ph/9707235]

1997 arXiv
[53]

Blanchet, P

S. Blanchet, P. S. B. Dev, and R. N. Mohapatra,Leptogenesis with TeV Scale Inverse Seesaw in SO(10), Phys. Rev. D82 (2010) 115025, [arXiv:1010.1471]

2010 arXiv
[54]

Plumacher,Baryogenesis and lepton number violation, Z

M. Plumacher,Baryogenesis and lepton number violation, Z. Phys. C74 (1997) 549–559, [hep-ph/9604229]

1997 arXiv
[55]

Barbieri, P

R. Barbieri, P. Creminelli, A. Strumia, and N. Tetradis,Baryogenesis through leptogenesis, 22 Nucl. Phys. B575 (2000) 61–77, [hep-ph/9911315]

2000 arXiv
[56]

Fileviez Perez, C

P. Fileviez Perez, C. Murgui, and A. D. Plascencia,Baryogenesis via leptogenesis: Spontaneous B and L violation, Phys. Rev. D104 (2021), no. 5 055007, [arXiv:2103.13397]

2021 arXiv
[57]

Granelli, K

A. Granelli, K. Moffat, Y. F. Perez-Gonzalez, H. Schulz, and J. Turner,ULYSSES: Universal LeptogeneSiS Equation Solver, Comput. Phys. Commun.262 (2021) 107813, [arXiv:2007.09150]

2021 arXiv
[58]

Granelli, C

A. Granelli, C. Leslie, Y. F. Perez-Gonzalez, H. Schulz, B. Shuve, J. Turner, and R. Walker, ULYSSES, universal LeptogeneSiS equation solver: Version 2, Comput. Phys. Commun.291 (2023) 108834, [arXiv:2301.05722]

2023 arXiv
[59]

Asaka and T

T. Asaka and T. Yoshida,Resonant leptogenesis at TeV-scale and neutrinoless double beta decay, JHEP 09 (2019) 089, [arXiv:1812.11323]

2019 arXiv
[60]

S. Iso, N. Okada, and Y. Orikasa,Resonant Leptogenesis in the Minimal B-L Extended Standard Model at TeV, Phys. Rev. D83 (2011) 093011, [arXiv:1011.4769]

2011 arXiv
[61]

Granelli, K

A. Granelli, K. Moffat, and S. T. Petcov,Flavoured resonant leptogenesis at sub-TeV scales, Nucl. Phys. B973 (2021) 115597, [arXiv:2009.03166]

2021 arXiv
[62]

P. Das, M. K. Das, and N. Khan,Phenomenological study of neutrino mass, dark matter and baryogenesis within the framework of minimal extended seesaw, JHEP 03 (2020) 018, [arXiv:1911.07243]

2020 arXiv
[63]

Chakraborty and H

I. Chakraborty and H. Roy,Type-I thermal leptogenesis inZ3-symmetric three Higgs doublet model, Eur. Phys. J. C80 (2020), no. 11 1038, [arXiv:1909.07790]

2020 arXiv
[64]

Buchmuller, P

W. Buchmuller, P. Di Bari, and M. Plumacher,Some aspects of thermal leptogenesis, New J. Phys. 6 (2004) 105, [hep-ph/0406014]

2004 arXiv
[65]

Blanchet, P

S. Blanchet, P. Di Bari, D. A. Jones, and L. Marzola,Leptogenesis with heavy neutrino flavours: from density matrix to Boltzmann equations, JCAP 01 (2013) 041, [arXiv:1112.4528]. 23

2013 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Feruglio,Are neutrino masses modular forms?, pp

F. Feruglio,Are neutrino masses modular forms?, pp. 227–266. 2019.arXiv:1706.08749

arXiv 2019

[2] [2]

Meloni and M

D. Meloni and M. Parriciatu,A simplest modular S3 model for leptons, JHEP 09 (2023) 043, [arXiv:2306.09028]

arXiv 2023

[3] [3]

Okada and Y

H. Okada and Y. Orikasa,ModularS3 symmetric radiative seesaw model, Phys. Rev. D100 (2019), no. 11 115037, [arXiv:1907.04716]

arXiv 2019

[4] [4]

Kashav and S

M. Kashav and S. Verma,Broken scaling neutrino mass matrix and leptogenesis based on A4 modular invariance, JHEP 09 (2021) 100, [arXiv:2103.07207]

arXiv 2021

[5] [5]

In this work, we consider a modular symmetric model based on Γ′, and all the matter fields are represented by chiral superfields

In the double covering of the modular group, we can define modular forms of odd modular weights as well. In this work, we consider a modular symmetric model based on Γ′, and all the matter fields are represented by chiral superfields. These fields transform as the modular forms of level N and weight k as follows: γ : Φ i→ (cτ +d)kiρΦ(γ)i,jΦj, (9) where Φi...

work page 2022

[6] [6]

Nomura and H

T. Nomura and H. Okada,A modularA4 symmetric model of dark matter and neutrino, Phys. Lett. B797 (2019) 134799, [arXiv:1904.03937]

arXiv 2019

[7] [7]

M. K. Behera, S. Singirala, S. Mishra, and R. Mohanta,A modular A4 symmetric scotogenic model for neutrino mass and dark matter, J. Phys. G49 (2022), no. 3 035002, [arXiv:2009.01806]

arXiv 2022

[8] [8]

Altarelli and F

G. Altarelli and F. Feruglio,Tri-bimaximal neutrino mixing, A(4) and the modular symmetry, Nucl. Phys. B741 (2006) 215–235, [hep-ph/0512103]

arXiv 2006