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REVIEW 4 major objections 5 minor 168 references

This paper argues that a non-holomorphic A4 modular symmetry naturally supplies the quasi-degenerate heavy neutrino masses required for resonant leptogenesis, and predicts a double-peaked gravitational wave spectrum.

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 →

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T0 review · deepseek-v4-flash

2026-08-01 14:18 UTC pith:YK2PKS5Z

load-bearing objection Coherent but overclaimed: the quasi-degenerate RHN spectrum is a scan input, not a prediction of the modular symmetry; still a solid model-building paper worth refereeing. the 4 major comments →

arxiv 2607.18803 v1 pith:YK2PKS5Z submitted 2026-07-21 hep-ph

A Non-Holomorphic Modular A₄ Framework for Resonant Leptogenesis with Gravitational Wave Signatures

Mitesh Kumar Behera , Jaydeb Das , Niloy Mondal This is my paper
classification hep-ph
keywords Non-holomorphic A4 modular symmetryPolyharmonic Maaß formsType-I seesawNeutrino phenomenologyResonant leptogenesisDomain wallPhase transitionGravitational waves
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.

The paper is trying to show that the near-degenerate right-handed neutrino spectrum needed for resonant leptogenesis does not have to be put in by hand. It builds a type-I seesaw model in which the neutrino Yukawa couplings and the heavy-neutrino Majorana mass matrix are fixed by a non-holomorphic A4 modular symmetry, using polyharmonic Maaß forms that can have negative modular weight. After fitting the modulus and a few couplings to current neutrino oscillation data, the heavy neutrino masses N2 and N3 come out nearly degenerate, so the CP asymmetry is resonantly enhanced and the observed baryon asymmetry can be produced with heavy-neutrino masses near 10^6 GeV. The paper further introduces a Z3 scalar whose spontaneous breaking creates domain walls; a tiny bias radiatively induced by the right-handed neutrino Yukawa makes those walls annihilate, producing a low-frequency gravitational wave peak, while a first-order phase transition produces a second, much higher-frequency peak. If the claim is right, the same framework connects neutrino flavor data, the matter-antimatter asymmetry, and a distinctive observable gravitational wave signature.

Core claim

In this model, the right-handed neutrino Majorana mass matrix is built from weight -2 polyharmonic Maaß forms, M_R = M_0[(β_R/3)Y_3^{(-2)} + γ_R Y_1^{(-2)}], where Y_3 is an A4 triplet and Y_1 is an A4 singlet. After unitary diagonalisation, the spectrum naturally satisfies M_N1 > M_N2 ≈ M_N3, because the singlet contribution has a two-fold degeneracy and the fitted parameter region keeps β_R far below γ_R. That tiny splitting, ΔM23, is comparable to the heavy-neutrino decay width, resonantly enhancing the self-energy CP asymmetry in the decays of N2 and N3. The paper shows that, in the strong washout regime, this yields the observed baryon asymmetry with M_N3 around 10^6 GeV, far below the

What carries the argument

The central object is the right-handed neutrino Majorana mass matrix M_R = M_0[(β_R/3)Y_3^{(-2)} + γ_R Y_1^{(-2)}], with Y_1 and Y_3 polyharmonic Maaß forms—smooth modular functions of a complex modulus satisfying a Laplace-type equation, which unlike holomorphic modular forms can carry negative modular weight and therefore build Yukawa couplings in a non-supersymmetric setting. Because the Y_1 singlet term has two degenerate eigenvalues, in the regime β_R << γ_R two heavy neutrinos are quasi-degenerate by structure rather than by fiat; that small splitting feeds the resonant self-energy enhancement in leptogenesis. A second mechanism is the Z3-breaking Yukawa interaction α_R Φ N^c N, whose

Load-bearing premise

The framework's central prediction of two nearly equal heavy neutrino masses comes from the scan imposing a tiny ratio β_R/γ_R, and the gravitational wave signal additionally requires the small coupling α_R ≈ 1e-21; the paper does not show that either value is forced by the modular symmetry or by neutrino data, so if either premise fails the leptogenesis and gravitational wave predictions shift or disappear.

What would settle it

Measure the Dirac CP-violating phase δ_CP in long-baseline neutrino experiments: the fitted region predicts δ_CP near -90 degrees for normal ordering, and a value far from that range would exclude the benchmark points. Separately, if a future gravitational-wave observatory with adequate sensitivity sees no low-frequency domain-wall peak at the frequency tied to a roughly 10^6 GeV right-handed neutrino, the domain-wall annihilation link is falsified.

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

If this is right

  • The observed baryon asymmetry can be produced with right-handed neutrino masses near 10^6 GeV, far below the usual bound for hierarchical thermal leptogenesis, without hand-tuning the heavy-neutrino mass degeneracy.
  • The low-frequency gravitational wave peak from domain-wall annihilation is tied to the right-handed neutrino Yukawa sector, so detecting it would indirectly probe the leptogenesis scale.
  • The high-frequency gravitational wave peak comes from the Z3-breaking first-order phase transition, and the two peaks are predicted to be separated by several orders of magnitude, giving a distinctive double-peaked signature.
  • The fitted parameter region predicts the Dirac CP-violating phase near -90 degrees for normal neutrino mass ordering, a prediction that future long-baseline neutrino experiments can test.
  • The same tiny Z3-breaking coupling that makes domain walls annihilate also suppresses the decay of the pseudo-Goldstone boson, allowing it to be a cosmologically long-lived dark matter candidate.

Where Pith is reading between the lines

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

  • An implication the paper does not spell out is that the quasi-degenerate spectrum is not a standalone symmetry prediction: it emerges because the scan keeps β_R/γ_R tiny. If that ratio were naturally of order one, the resonant enhancement would vanish and the model would revert to ordinary seesaw leptogenesis at a higher mass scale.
  • The gravitational wave observability window effectively selects a narrow range of the Z3-breaking coupling α_R around 1e-21. A future measurement or upper bound on the low-frequency peak could therefore be inverted to constrain this coupling, linking the gravitational wave spectrum directly to the seesaw sector's structure.
  • A natural next step would be to make the ratios β_R/γ_R and α_R dynamical quantities rather than scanned inputs, checking whether the preferred hierarchy survives when derived from a more complete scalar or flavon sector.

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

4 major / 5 minor

Summary. The paper constructs a type-I seesaw model with a non-holomorphic A4 modular symmetry, in which polyharmonic Maass forms determine the lepton Yukawa and Majorana mass structures. A numerical scan over the modulus and the dimensionless couplings is used to fit the neutrino oscillation data (NuFIT 6.1), and the resulting right-handed neutrino spectrum is reported to be naturally quasi-degenerate, enabling resonant leptogenesis at M_N ~ 10^6 GeV. The model is then extended with a Z3-charged complex scalar whose phase transition produces domain walls; a radiatively induced bias from the RHN sector annihilates the walls, producing a low-frequency GW signal, while the first-order phase transition produces a second, high-frequency GW peak. The advertised result is a double-peaked GW spectrum that simultaneously fits neutrino data and reproduces the baryon asymmetry.

Significance. If the central claims were fully supported, the paper would be a valuable step in connecting modular flavor symmetry to resonant leptogenesis and gravitational-wave cosmology. The manuscript is transparent and technically detailed in several respects: the modular-form q-expansions are given explicitly, the neutrino fit is benchmarked against NuFIT, the phase-transition formulas are checked against CosmoTransitions, and the GW spectra are computed for a broad set of proposed detectors. These are real strengths. However, as argued below, the key advertised feature — that the quasi-degenerate RHN spectrum is 'naturally' generated rather than imposed — is not currently demonstrated; the hierarchy β_R ≪ γ_R is put in by the scan ranges, and the successful baryon asymmetry is matched by selecting a single benchmark point. The GW predictions also depend on an extremely small, unexplained coupling α_R ~ 1e-21. The paper is therefore better read as a benchmark construction than as a derivation, and the central naturalness claim needs substantial revision or retraction.

major comments (4)
  1. [Eq. (2.8); Eq. (3.6); Table 3] The claim that the quasi-degenerate RHN spectrum is 'naturally generated' is not supported. From Eq. (2.8), the γ_R term alone gives eigenvalues {Y1, Y1, -Y1}, so the β_R term is the only source of the physical splitting; in the fitted region ΔM23/MN3 ~ O(β_R/γ_R). The scan priors in Eq. (3.6) take β_R ∈ [1e-13, 1e-9] and γ_R ∈ [1e-10, 1e-2], and all four benchmarks in Table 3 have β_R/γ_R ≤ 2.5e-5. The text in Sec. 3.1 states that the scan is performed to 'facilitate resonant leptogenesis by creating a small mass splitting.' The paper does not show that NuFIT-consistent points would disappear if β_R/γ_R ~ O(1). Thus the quasi-degeneracy is an input from the scan priors, not an output of the modular symmetry. I ask for a scan over β_R/γ_R over a wide range, reporting whether the neutrino-fit region requires the hierarchy, or for an explicit symmetry argument that fixes it.
  2. [Sec. 4; Table 3; Figs. 6–7] The observed baryon asymmetry is matched by selecting one benchmark point (BP3) after the scan; BP1, BP2, and BP4 do not reproduce the observed Y_B. Since Y_B depends sharply on the ratio ΔM23/ΓN3 and on the strong-washout parameter K, this is an existence proof, not a prediction. The paper should show the distribution of the final Y_B over the full neutrino-consistent parameter region, including the fraction of points that fall within, say, a factor of 2–3 of the observed value, and should demonstrate that BP3 is not an isolated fine-tuned point. Without this, the phrase in the abstract and Sec. 4 that successful leptogenesis is achieved as a consequence of the framework is premature.
  3. [Eq. (5.10); Eq. (B.3)–(B.5)] There is an internal inconsistency in the high-temperature bias formula. Starting from Eq. (5.10) with K_T = Tr[J'_F(M_R†M_R/T^2) M_R†Y] and using J'_F(0) = -π^2/24, the high-T expansion gives a linear term with coefficient T^2 vφ/(24√2) cos(θ+δ_T) (with the sign convention depending on the phase definition), not the denominator 12√2 in Eq. (B.4). The two formulas differ by a factor of 2. Since this bias sets T_ann through Eq. (5.14), the discrepancy changes the predicted DW peak frequency and amplitude, as well as the allowed regions in Fig. 9. Please reconcile the expressions and recompute the affected GW curves, or provide the missing step that eliminates the factor.
  4. [Sec. 5.1; Sec. 5.4; Eq. (5.2)] The GW and DM phenomenology is controlled by α_R ≈ 1e-21, which is introduced as a free input with no naturalness or symmetry justification. This same coupling is responsible for (i) the radiatively induced bias that annihilates the domain walls and (ii) the suppression of χ decay that makes it a viable DM candidate. The paper does not quantify the allowed range of α_R for which DWs annihilate after formation but before BBN/domination while χ remains cosmologically stable. Without such a range, the 'characteristic double-peaked spectrum' is a benchmark phenomenon, not a robust prediction. I request a quantitative scan in α_R and a discussion of the implied tuning.
minor comments (5)
  1. [App. B, Eq. (B.1)] Eq. (B.1) is labelled as the fermionic thermal function but displays the bosonic function J_B(x) with ln(1 - e^{-...}). This is likely a typo (should be J_F or the expression should be changed).
  2. [Sec. 4, after Eq. (4.1)] 'takaji autotune diagonalization' should be 'Takagi diagonalization'.
  3. [Eq. (5.19)–(5.24)] The portal coupling is written λHΦ in Eqs. (5.1), (5.20), and (5.21), but λHS appears in Eq. (5.19). Please unify the notation.
  4. [Fig. 8 caption; Sec. 5.1] Minor typos: 'Nef' should be 'N_eff', and 'relavistic' should be 'relativistic'.
  5. [Sec. 5.4] The dark-matter candidate is only discussed qualitatively; the relic abundance is not computed. If χ is claimed as a viable DM candidate, the thermal history and abundance should be addressed or the claim explicitly deferred.

Circularity Check

0 steps flagged

No circular derivation: the flavor fit and GW/leptogenesis outputs are computed from stated free parameters; the 'natural quasi-degeneracy' language is an overstatement of parameter selection, not a by-construction reduction.

full rationale

The paper's derivation chain is not circular in the formal sense. The lepton sector is anchored to external oscillation data: the charged-lepton and neutrino mass matrices follow from the A4 assignments and the polyharmonic Maaß forms of Ref. [86], and the parameters (tau, alpha_D, beta_D, gamma_D, beta_R, gamma_R, M0) are scanned against NuFIT 6.1 via Eq. (3.4). The claims for leptogenesis are then computed with standard CP-asymmetry formulas (Eqs. (4.6)-(4.7)) and Boltzmann equations (Eq. (4.8)); the benchmark points in Table 3 are selected a posteriori and checked to reproduce the observed baryon asymmetry. The GW spectra are likewise outputs of standard formulas for DW annihilation and a first-order phase transition, evaluated at stated benchmarks. The only real weakness is rhetorical: Eq. (2.8) shows that the quasi-degeneracy of the RHN spectrum is controlled by the free hierarchy beta_R/gamma_R, and the scan ranges in Eq. (3.6) already separate beta_R (1e-13 to 1e-9) from gamma_R (1e-10 to 1e-2), while Sec. 3.1 says it selects the region to 'facilitate resonant leptogenesis by creating a small mass splitting.' Calling this an 'intrinsic prediction' (Sec. 4) therefore overstates the symmetry's constraining power. But this is a predictivity/fine-tuning concern, not circularity: beta_R/gamma_R is not defined in terms of Delta M_23 or Y_B, the splitting is computed rather than imposed as a target, and no self-citation or uniqueness theorem carries the argument. The cited prior work by the authors appears only for standard BEQ/phase-transition techniques and is not load-bearing. Hence no step reduces to its input by construction; the appropriate score is within the 0-2 'no significant circularity' band.

Axiom & Free-Parameter Ledger

7 free parameters · 7 axioms · 1 invented entities

The model's predictive power comes mostly from the modular flavor structure, but every phenomenological target (neutrino masses, BAU, observable GW) is reached by scanning or fitting free parameters; the GW sector adds an extra hand-tuned α_R.

free parameters (7)
  • τ (complex modulus) = Re τ ≈ -0.470, Im τ ∈ [2.3,2.7]
    Sets all polyharmonic Maass form values; fitted to neutrino oscillation data.
  • α_D, β_D, γ_D (Dirac Yukawa coefficients) = BP3: (1.38e-5, 2.33e-5, 8.41e-6)
    Scanned over [1e-8,1e-3] and fitted to oscillation data.
  • β_R, γ_R (Majorana mass coefficients) = BP3: (1.55e-12, 1.70e-7)
    Their hierarchy β_R ≪ γ_R produces the quasi-degenerate RHN spectrum; scanned ranges in Eq. (3.6).
  • M0 (overall seesaw scale) = BP3: 1.40e12 GeV
    Fitted to atmospheric mass splitting via κ = v_h²/(2M0).
  • α_R = α'_R (Z3-breaking Yukawa) = 1e-21 for the GW benchmark
    Controls the radiative bias; chosen by hand so the DW GW peak is observable.
  • μ3, vφ (scalar potential parameters) = BP4: μ3 = 1.27e6 GeV, vφ = 2e6 GeV
    Benchmark choices controlling FOPT strength and DW tension; not derived from symmetry.
  • renormalization scale μ = 10^13 GeV
    Fixed by hand in the Coleman-Weinberg logarithms.
axioms (7)
  • standard math Polyharmonic Maass forms Y(-2) exist with the q-series in App. A and transform as A4 representations
    Invoked from Ref. [86]; central to all mass matrices.
  • domain assumption Type-I seesaw with three RHNs and these modular weights is the complete low-energy theory
    No derivation; defines the model in Sec. 2.2.
  • ad hoc to paper β_R/γ_R ≪ 1 is an acceptable rather than fine-tuned input
    The quasi-degeneracy essential for leptogenesis comes from this hierarchy, which is imposed by scan ranges in Eq. (3.6).
  • domain assumption Boltzmann equations with only inverse-decay washout (no ΔL=1 or off-shell ΔL=2 scatterings) are adequate
    Stated in Sec. 4 following Refs. [92,127-130], but not justified for the resonant parameter region.
  • domain assumption Axionic domain-wall simulation constants A=1.10±0.20, C_an=5.01±0.44 apply to Z3 walls
    Taken from Ref. [140] and used in Eq. (5.14).
  • domain assumption No inflation or significant entropy production after DW annihilation dilutes the GW signal
    Standard cosmological evolution is assumed throughout Sec. 5.
  • ad hoc to paper CP symmetry makes χ stable while tiny α_R permits decays that are still slow enough for DM
    Sec. 5.4 relies on both claims simultaneously; this is not a rigorous stability proof.
invented entities (1)
  • Complex scalar Φ (and its CP-odd component χ) no independent evidence
    purpose: Spontaneously breaks Z3, forms domain walls, drives the first-order phase transition, and provides a DM candidate
    Introduced for the phenomenology; its GW signal is tunable through α_R, vφ, μ3, so it has no sharp falsifiable prediction.

pith-pipeline@v1.3.0-alltime-deepseek · 32718 in / 21063 out tokens · 196716 ms · 2026-08-01T14:18:15.491839+00:00 · methodology

0 comments
read the original abstract

We study a type-I seesaw framework based on non-holomorphic $A_4$ modular symmetry, where polyharmonic Maa\ss\ forms construct the Yukawa couplings and right-handed neutrino (RHN) Majorana mass matrix. The use of non-holomorphic modular forms yields highly constrained neutral lepton mass matrices with a more restrictive lepton-sector structure and naturally generates a quasi-degenerate RHN mass spectrum, enabling resonant leptogenesis at an intermediate scale with RHN masses of $\mathcal{O}(10^3)$ TeV without requiring an ad hoc mass degeneracy. We further extend the model by introducing a complex scalar field $(\Phi)$ charged under $\mathbb{Z}_3$ symmetry. The spontaneous breaking of the discrete symmetry after the phase transition associated with $\Phi$ leads to domain-wall (DW) formation. A radiatively induced bias term associated with the RHN sector triggers DW annihilation, resolving the cosmological DW problem, and producing a stochastic gravitational wave (GW) signal that indirectly probes the RHN mass scale. The accompanying first-order phase transition produces a second GW peak, yielding a characteristic double-peaked spectrum with frequencies separated by several orders of magnitude and potentially observable by complementary future GW detectors.

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Reference graph

Works this paper leans on

168 extracted references · 139 linked inside Pith

  1. [5]

    Esteban, M.C

    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,JHEP09(2020) 178 [2007.14792]. [6]Planckcollaboration,Planck 2018 results. VI. Cosmological parameters,Astron. Astrophys.641(2020) A6 [1807.06209]. [7]Particle Data Groupcollaboration,Review of particle phys...

    Pith/arXiv arXiv 2020

  2. [8]

    Sakharov,Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe,Pisma Zh

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

    1967

  3. [9]

    Kuzmin, V.A

    V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov,On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe,Phys. Lett.155B(1985) 36

    1985

  4. [10]

    Cohen, D.B

    A.G. Cohen, D.B. Kaplan and A.E. Nelson,Baryogenesis at the weak phase transition, Nucl. Phys. B349(1991) 727

    1991

  5. [11]

    Turok and J

    N. Turok and J. Zadrozny,Dynamical generation of baryons at the electroweak transition, Phys. Rev. Lett.65(1990) 2331

    1990

  6. [12]

    M. Dine, P. Huet, R.L. Singleton, Jr and L. Susskind,Creating the baryon asymmetry at the electroweak phase transition,Phys. Lett. B257(1991) 351

    1991

  7. [13]

    Cohen, D.B

    A.G. Cohen, D.B. Kaplan and A.E. Nelson,Spontaneous baryogenesis at the weak phase transition,Phys. Lett. B263(1991) 86

    1991

  8. [14]

    Trodden,Electroweak baryogenesis,Rev

    M. Trodden,Electroweak baryogenesis,Rev. Mod. Phys.71(1999) 1463 [hep-ph/9803479]

    Pith/arXiv arXiv 1999

  9. [15]

    Affleck and M

    I. Affleck and M. Dine,A New Mechanism for Baryogenesis,Nucl. Phys. B249(1985) 361

    1985

  10. [16]

    Thomas and D

    A.M. Thomas and D. Choudhury,Neutron oscillation and baryogenesis from six dimensions,Phys. Rev. D106(2022) L031701 [2205.03846]

    Pith/arXiv arXiv 2022

  11. [17]

    Fukugita and T

    M. Fukugita and T. Yanagida,Baryogenesis Without Grand Unification,Phys. Lett.B174 (1986) 45

    1986

  12. [18]

    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 [hep-ph/9707235]

    Pith/arXiv arXiv 1997

  13. [19]

    Pilaftsis and T.E.J

    A. Pilaftsis and T.E.J. Underwood,Resonant leptogenesis,Nucl. Phys. B692(2004) 303 [hep-ph/0309342]

    Pith/arXiv arXiv 2004

  14. [20]

    B. Dev, M. Garny, J. Klaric, P. Millington and D. Teresi,Resonant enhancement in leptogenesis,Int. J. Mod. Phys. A33(2018) 1842003 [1711.02863]

    Pith/arXiv arXiv 2018

  15. [21]

    Alanne, T

    T. Alanne, T. Hugle, M. Platscher and K. Schmitz,Low-scale leptogenesis assisted by a real scalar singlet,JCAP03(2019) 037 [1812.04421]

    Pith/arXiv arXiv 2019

  16. [22]

    Hugle, M

    T. Hugle, M. Platscher and K. Schmitz,Low-Scale Leptogenesis in the Scotogenic Neutrino Mass Model,Phys. Rev. D98(2018) 023020 [1804.09660]

    Pith/arXiv arXiv 2018

  17. [23]

    Kusenko, K

    A. Kusenko, K. Schmitz and T.T. Yanagida,Leptogenesis via Axion Oscillations after Inflation,Phys. Rev. Lett.115(2015) 011302 [1412.2043]

    Pith/arXiv arXiv 2015

  18. [24]

    Hambye and D

    T. Hambye and D. Teresi,Higgs doublet decay as the origin of the baryon asymmetry,Phys. Rev. Lett.117(2016) 091801 [1606.00017]

    Pith/arXiv arXiv 2016

  19. [25]

    Datta, R

    A. Datta, R. Roshan and A. Sil,Effects of Reheating on Charged Lepton Yukawa Equilibration and Leptogenesis,Phys. Rev. Lett.132(2024) 061802 [2206.10650]

    Pith/arXiv arXiv 2024

  20. [26]

    Bhandari, A

    D. Bhandari, A. Datta and A. Sil,Leptogenesis from a phase transition in a dynamical vacuum,Phys. Rev. D110(2024) 115008 [2312.13157]

    Pith/arXiv arXiv 2024

  21. [27]

    King, S.K

    S.F. King, S.K. Manna, R. Roshan and A. Sil,Leptogenesis with Majoron dark matter, Phys. Rev. D111(2025) 095008 [2412.14121]

    Pith/arXiv arXiv 2025

  22. [28]

    Bhattacharya, N

    S. Bhattacharya, N. Mondal, R. Roshan and D. Vatsyayan,Leptogenesis, dark matter and gravitational waves from discrete symmetry breaking,JCAP06(2024) 029 [2312.15053]. – 33 –

    Pith/arXiv arXiv 2024

  23. [29]

    Bhattacharya, D

    S. Bhattacharya, D. Mahanta, N. Mondal and D. Pradhan,Two-component dark matter and low scale thermal Leptogenesis,JCAP09(2025) 032 [2412.21202]

    Pith/arXiv arXiv 2025

  24. [30]

    Bhattacharya, N

    S. Bhattacharya, N. Mondal and A. Sil,Exploring Leptogenesis, WIMP Dark Matter, and Gravitational Waves in an extended Scalar Framework,2512.02672

    arXiv

  25. [31]

    Choudhury, J

    D. Choudhury, J. Das and T. Srivastava,Solving Cosmological Puzzles using Finite Temperature˚SMEFT,2604.21492

    Pith/arXiv arXiv

  26. [32]

    Barman, A

    B. Barman, A. Basu, D. Borah and N. Das,Leptogenesis with sub-electroweak-scale reheating temperature,2607.09282

    Pith/arXiv arXiv

  27. [33]

    Rubakov and M.E

    V.A. Rubakov and M.E. Shaposhnikov,Electroweak baryon number nonconservation in the early universe and in high-energy collisions,Usp. Fiz. Nauk166(1996) 493 [hep-ph/9603208]

    Pith/arXiv arXiv 1996

  28. [34]

    Klinkhamer and N.S

    F.R. Klinkhamer and N.S. Manton,A Saddle Point Solution in the Weinberg-Salam Theory,Phys. Rev. D30(1984) 2212

    1984

  29. [35]

    Manton,Topology in the Weinberg-Salam Theory,Phys

    N.S. Manton,Topology in the Weinberg-Salam Theory,Phys. Rev. D28(1983) 2019

    1983

  30. [36]

    Bertone, D

    G. Bertone, D. Hooper and J. Silk,Particle dark matter: Evidence, candidates and constraints,Phys. Rept.405(2005) 279 [hep-ph/0404175]

    Pith/arXiv arXiv 2005

  31. [37]

    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]. [38]Daya Baycollaboration,Observation of electron-antineutrino disappearance at Daya Bay, Phys. Rev. Lett.108(2012) 171803 [1203.1669]. [39]RENOcollaboration,Observation of Reactor Electron Antineutrino Disappearance in ...

    Pith/arXiv arXiv 2001

  32. [40]

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

    F. Feruglio,Are neutrino masses modular forms?, inFrom My Vast Repertoire ...: Guido Altarelli’s Legacy, A. Levy, S. Forte and G. Ridolfi, eds., pp. 227–266 (2019), DOI [1706.08749]

    Pith/arXiv arXiv 2019

  33. [41]

    Cremades, L.E

    D. Cremades, L.E. Ibanez and F. Marchesano,Computing Yukawa couplings from magnetized extra dimensions,JHEP05(2004) 079 [hep-th/0404229]

    Pith/arXiv arXiv 2004

  34. [42]

    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

  35. [43]

    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

  36. [44]

    Nomura and H

    T. Nomura and H. Okada,Neutrino mass model at a three-loop level from a non-holomorphic modular A 4 symmetry,Chin. Phys. C50(2026) 023108 [2506.02639]

    arXiv 2026

  37. [45]

    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

  38. [46]

    Nomura and H

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

    Pith/arXiv arXiv 2025

  39. [47]

    D.W. Kang, J. Kim, T. Nomura and H. Okada,Natural mass hierarchy among three heavy Majorana neutrinos for resonant leptogenesis under modular A 4 symmetry,JHEP07 (2022) 050 [2205.08269]. – 34 –

    Pith/arXiv arXiv 2022

  40. [48]

    Borah, M.K

    D. Borah, M.K. Das and A. Mukherjee,Common origin of nonzeroθ 13 and baryon asymmetry of the Universe in a TeV scale seesaw model withA 4 flavor symmetry,Phys. Rev. D97(2018) 115009 [1711.02445]

    Pith/arXiv arXiv 2018

  41. [49]

    Branco, R

    G.C. Branco, R. Gonzalez Felipe, M.N. Rebelo and H. Serodio,Resonant leptogenesis and tribimaximal leptonic mixing with A(4) symmetry,Phys. Rev. D79(2009) 093008 [0904.3076]

    Pith/arXiv arXiv 2009

  42. [50]

    Adhikary, M

    B. Adhikary, M. Chakraborty and A. Ghosal,Flavored leptogenesis with quasidegenerate neutrinos in a broken cyclic symmetric model,Phys. Rev. D93(2016) 113001 [1407.6173]

    Pith/arXiv arXiv 2016

  43. [51]

    Karmakar and A

    B. Karmakar and A. Sil,Spontaneous CP violation in lepton-sector: A common origin for θ13, the Dirac CP phase, and leptogenesis,Phys. Rev. D93(2016) 013006 [1509.07090]

    Pith/arXiv arXiv 2016

  44. [52]

    Datta, B

    A. Datta, B. Karmakar and A. Sil,Flavored leptogenesis and neutrino mass with A 4 symmetry,JHEP12(2021) 051 [2106.06773]

    Pith/arXiv arXiv 2021

  45. [53]

    Singh, M

    L. Singh, M. Kashav and S. Verma,Minimal type-I Dirac seesaw and leptogenesis under A4 modular invariance,Nucl. Phys. B1007(2024) 116666 [2405.07165]

    Pith/arXiv arXiv 2024

  46. [54]

    Parriciatu,A minimalistic perspective on neutrino CP-violation and Leptogenesis: Modular Invariance,PoSICHEP2024(2025) 244

    M. Parriciatu,A minimalistic perspective on neutrino CP-violation and Leptogenesis: Modular Invariance,PoSICHEP2024(2025) 244

    2025

  47. [55]

    Pathak and M.K

    G. Pathak and M.K. Das,Matter-antimatter asymmetry in minimal inverse seesaw framework with A 4 modular symmetry,J. Phys. G53(2026) 025004 [2505.03000]

    Pith/arXiv arXiv 2026

  48. [56]

    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

  49. [57]

    Tavartkiladze,Minimal modular flavor symmetry and lepton textures near fixed points, Phys

    Z. Tavartkiladze,Minimal modular flavor symmetry and lepton textures near fixed points, Phys. Rev. D113(2026) 095023 [2512.24804]

    Pith/arXiv arXiv 2026

  50. [58]

    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

  51. [59]

    Chauhan, D

    Priya, B.C. Chauhan, D. Kumar and T. Nomura,Predictions of Modular Symmetry Fixed Points on Neutrino Masses, Mixing, and Leptogenesis,2604.04585

    Pith/arXiv arXiv

  52. [60]

    Singh, B.C

    Priya, L. Singh, B.C. Chauhan and S. Verma,Radiative Lifting ofZ 3 Domain-Wall Degeneracy in a Type-III Seesaw Model: Implications for Leptogenesis and Gravitational Waves,2606.22642

    Pith/arXiv arXiv

  53. [61]

    Batra, R

    C. Batra, R. Mandal, K. Rawat and T. Tong,Connecting Flavor and Baryon Asymmetry via Leptogenesis in Effective Froggatt-Nielsen Theory,2603.05372

    Pith/arXiv arXiv

  54. [62]

    Wei and Y

    D. Wei and Y. Jiang,Domain wall networks from first-order phase transitions and gravitational waves,Phys. Rev. D110(2024) 123505 [2208.07186]

    Pith/arXiv arXiv 2024

  55. [63]

    Borboruah and U.A

    Z.A. Borboruah and U.A. Yajnik,Left-right symmetry breaking and gravitational waves: A tale of two phase transitions,Phys. Rev. D110(2024) 043016 [2212.05829]

    Pith/arXiv arXiv 2024

  56. [64]

    Fornal, D

    B. Fornal, D. Polynice and L. Thompson,Probing the neutrino seesaw scale with gravitational waves,Phys. Rev. D110(2024) 095013 [2406.16463]

    Pith/arXiv arXiv 2024

  57. [65]

    Roshan and I

    R. Roshan and I. Saha,Twin-peaked gravitational wave signal from a dark sector phase transition,2603.15829

    arXiv

  58. [66]

    Saikawa,A review of gravitational waves from cosmic domain walls,Universe3(2017) 40 [1703.02576]

    K. Saikawa,A review of gravitational waves from cosmic domain walls,Universe3(2017) 40 [1703.02576]. – 35 –

    Pith/arXiv arXiv 2017

  59. [67]

    Roshan and G

    R. Roshan and G. White,Using gravitational waves to see the first second of the Universe, Rev. Mod. Phys.97(2025) 015001 [2401.04388]

    Pith/arXiv arXiv 2025

  60. [68]

    Kamionkowski, A

    M. Kamionkowski, A. Kosowsky and M.S. Turner,Gravitational radiation from first order phase transitions,Phys. Rev. D49(1994) 2837 [astro-ph/9310044]

    Pith/arXiv arXiv 1994

  61. [69]

    Ellis, M

    J. Ellis, M. Lewicki and J.M. No,On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal,JCAP04(2019) 003 [1809.08242]

    Pith/arXiv arXiv 2019

  62. [70]

    Croon, V

    D. Croon, V. Sanz and G. White,Model Discrimination in Gravitational Wave spectra from Dark Phase Transitions,JHEP08(2018) 203 [1806.02332]

    Pith/arXiv arXiv 2018

  63. [71]

    Beniwal, M

    A. Beniwal, M. Lewicki, M. White and A.G. Williams,Gravitational waves and electroweak baryogenesis in a global study of the extended scalar singlet model,JHEP02(2019) 183 [1810.02380]

    Pith/arXiv arXiv 2019

  64. [72]

    Mazumdar and G

    A. Mazumdar and G. White,Review of cosmic phase transitions: their significance and experimental signatures,Rept. Prog. Phys.82(2019) 076901 [1811.01948]

    Pith/arXiv arXiv 2019

  65. [73]

    Kobakhidze, A

    A. Kobakhidze, A. Manning and J. Yue,Gravitational waves from the phase transition of a nonlinearly realized electroweak gauge symmetry,Int. J. Mod. Phys. D26(2017) 1750114 [1607.00883]

    Pith/arXiv arXiv 2017

  66. [74]

    Z. Kang, P. Ko and T. Matsui,Strong first order EWPT&strong gravitational waves in Z3-symmetric singlet scalar extension,JHEP02(2018) 115 [1706.09721]

    Pith/arXiv arXiv 2018

  67. [75]

    Kannike, K

    K. Kannike, K. Loos and M. Raidal,Gravitational wave signals of pseudo-Goldstone dark matter in theZ 3 complex singlet model,Phys. Rev. D101(2020) 035001 [1907.13136]

    Pith/arXiv arXiv 2020

  68. [76]

    Chakrabarty, H

    N. Chakrabarty, H. Roy and T. Srivastava,Single-step first order phase transition and gravitational waves in a SIMP dark matter scenario,Nucl. Phys. B998(2024) 116392 [2212.09659]

    Pith/arXiv arXiv 2024

  69. [77]

    Ellis, M

    J. Ellis, M. Lewicki, M. Merchand, J.M. No and M. Zych,The scalar singlet extension of the Standard Model: gravitational waves versus baryogenesis,JHEP01(2023) 093 [2210.16305]

    Pith/arXiv arXiv 2023

  70. [78]

    J. Das, S. Niyogi and T. Srivastava,Revisiting singlet fermion dark matter with a scalar portal: connecting Higgs phenomenology and strong electroweak phase transition,JCAP06 (2026) 018 [2601.13147]

    Pith/arXiv arXiv 2026

  71. [79]

    Srivastava, J

    T. Srivastava, J. Das, A. Ghosh and A. Chaudhuri,Electroweak phase transition, gravitational waves and collider probes in multi-scalar dark matter scenarios,JCAP02 (2026) 032 [2507.05917]

    arXiv 2026

  72. [80]

    Chaudhuri and J

    A. Chaudhuri and J. Das,Study of entropy production due to electroweak phase transition in Z2 symmetric extension of the Standard Model,Phys. Rev. D106(2022) 095016 [2206.08699]

    Pith/arXiv arXiv 2022

  73. [81]

    Chaudhuri, P

    A. Chaudhuri, P. Mishra and R. Mohanta,Gravitational Wave Signatures of U(1) X Breaking and Right-Handed Neutrino Dynamics,JCAP07(2026) 061 [2508.09835]

    Pith/arXiv arXiv 2026

  74. [82]

    Borah, P

    P. Borah, P. Ghosh, S. Roy and A.K. Saha,Electroweak phase transition in a right-handed neutrino superfield extended NMSSM,JHEP08(2023) 029 [2301.05061]

    Pith/arXiv arXiv 2023

  75. [83]

    Borah, P

    P. Borah, P. Ghosh and A.K. Saha,Prospecting bipartite dark matter through gravitational waves,JCAP05(2025) 035 [2412.17141]. – 36 –

    Pith/arXiv arXiv 2025

  76. [84]

    Borah, S

    D. Borah, S. Jyoti Das and I. Saha,Dark matter from phase transition generated PBH evaporation with gravitational waves signatures,Phys. Rev. D110(2024) 035014 [2401.12282]

    Pith/arXiv arXiv 2024

  77. [85]

    Bhattacharyya, S

    A. Bhattacharyya, S. Biswas and S. Niyogi,Electroweak phase transitions in aU(1) D extension of the standard model with dimension-six operators: Gravitational waves and LHC signatures,2603.18583

    Pith/arXiv arXiv

  78. [86]

    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

  79. [87]

    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]

    Pith/arXiv arXiv 2019

  80. [88]

    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

Showing first 80 references.