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Leptogenesis, $0\nu\beta\beta$ and lepton flavor violation in modular left-right asymmetric model with polyharmonic $Maa\beta$ forms

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Pith's one-line read A non-supersymmetric modular model using polyharmonic Maass forms and an extended inverse seesaw fits neutrino oscillation data within 3σ and keeps neutrinoless double beta decay, lepton flavor violation, and the baryon asymmetry inside…

desk verdict Interesting application of polyharmonic Maass forms to a non-SUSY left-right model, but the BAU claim is internally inconsistent with the stated parameters and the neutrino 'predictions' are circular fits. read the letter →

arxiv 2504.21701 v1 pith:JQUGGZQ3 submitted 2025-04-30 hep-ph

Bhabana Kumar , Mrinal Kumar Das This is my paper · ORCID
classification hep-ph
keywords neutrinomassesandmixingmodularflavorsymmetrypolyharmonicMaassformsleft-rightasymmetricmodelextendedinverseseesawneutrinolessdoublebetadecayleptonviolationresonantleptogenesis
topics Matter-Antimatter Asymmetry
open problems Matter-Antimatter Asymmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a non-supersymmetric extension of the Standard Model built on polyharmonic Maass forms—modular functions that need not be holomorphic in the modulus $\tau$—can explain neutrino masses and mixing without auxiliary scalar fields (flavons) to break the flavour symmetry. The authors construct a left-right asymmetric gauge model, in which the left- and right-handed gauge couplings are not equal, and generate neutrino masses through an extended inverse seesaw mechanism, taking three weight-zero polyharmonic Maass forms, $Y_{3,1}^{(0)}, Y_{3,2}^{(0)}, Y_{3,3}^{(0)}$, as an $A_4$ triplet of Yukawa couplings. They report that the model reproduces the neutrino mixing angles within $3\sigma$ and predicts a sum of neutrino masses below the current cosmological bound. Using the same parameter space, they compute the effective Majorana mass for $0\nu\beta\beta$ decay, the branching ratios for $\mu\to e\gamma$, $\tau\to\mu\gamma$, and $\tau\to e\gamma$, and the baryon asymmetry from resonant leptogenesis; the LFV rates and baryon asymmetry are consistent with current limits, and the effective Majorana mass is predominantly below the experimental bound. The model therefore demonstrates that modular flavour symmetry can operate without supersymmetry and produces testable lepton-number-violating signatures.

What carries the argument

The load-bearing object is the weight-zero polyharmonic Maass form: a modular function of the modulus $\tau$ that satisfies the hyperbolic Laplacian condition $\Delta_k Y=0$ rather than holomorphicity, so it may carry both holomorphic and non-holomorphic components and may have zero or negative modular weight. The paper takes the three forms in Eq. (A7) as an $A_4$ triplet and inserts them into the Dirac, Majorana, $N$–$S$ mixing, and sterile mass matrices of an extended inverse seesaw, so that the flavour structure is fixed by the modulus $\tau$ together with a set of complex couplings. The companion mechanism is resonant leptogenesis: among the six heavy neutral fermions, a nearly degenerate quasi-Dirac pair has mass splitting comparable to its decay width, enhancing the CP asymmetry through the $f_{ik}$ factors in Eq. (6.5), and the resulting baryon asymmetry is obtained after applying a dilution factor.

What would settle it

Evaluate the hyperbolic Laplacian $\Delta_0$ acting on each of the three expansions in Eq. (A7); if any function is not annihilated by $\Delta_0$, or if the triplet $(Y_{3,1}^{(0)}, Y_{3,2}^{(0)}, Y_{3,3}^{(0)})$ fails to transform as an $A_4$ triplet under the modular generators $S$ and $T$, the claimed Yukawa structure and all derived predictions collapse. This check can be done directly from the published expansions.

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Extended reading notes

Core claim

The paper's central claim is that the three functions $Y_{3,1}^{(0)}, Y_{3,2}^{(0)}, Y_{3,3}^{(0)}$ in Eq. (A7)—asserted to be weight-zero polyharmonic Maass forms for the level-3 principal congruence subgroup $\Gamma(3)$, forming a triplet of the finite modular group $A_4$—can serve as the complete Yukawa sector of a non-supersymmetric left-right asymmetric model. With these couplings, the active neutrino mass matrix takes the extended inverse seesaw form $m_\nu = M_D M_N^{-1} M_S (M_D M_N^{-1})^T$, where the smallness of neutrino mass is set by the keV-scale sterile mass $M_S$ rather than by an extremely heavy right-handed scale. Scanning the three Yukawa couplings and the modulus $\tau$, the authors find regions of parameter space where the mixing angles and mass splittings fall inside the $3\sigma$ ranges for both normal and inverted ordering, and where the sum of neutrino masses satisfies the current cosmological bound. Over the same region, the model gives $\lambda$-diagram contributions to $0\nu\beta\beta$ decay mostly below the experimental limit, LFV branching ratios below current bounds, and a resonant-leptogenesis CP asymmetry large enough to reproduce the observed baryon asymmetry, with viable heavy-neutrino masses of roughly 200 TeV to 2000 TeV for normal ordering and 50 TeV to 100 TeV for inverted ordering.

Load-bearing premise

The load-bearing premise is that the three functions $Y_{3,1}^{(0)}, Y_{3,2}^{(0)}, Y_{3,3}^{(0)}$ written down in Eq. (A7) of Appendix A are genuinely weight-zero polyharmonic Maass forms for $\Gamma(3)$ transforming as an $A_4$ triplet; the paper gives these expansions without derivation, and if they are wrong every mass matrix and every numerical result built on them changes.

Editorial extensions

If this is right

  • The model gives a concrete non-supersymmetric realization in which the same three Yukawa couplings and one modulus control neutrino masses, $0\nu\beta\beta$ decay, LFV, and the baryon asymmetry, so a measurement in any one channel constrains the others.
  • The leptogenesis mechanism fixes the heavy neutrino masses to a specific window—roughly 200 TeV to 2000 TeV for normal ordering and 50 TeV to 100 TeV for inverted ordering—so the scale at which new physics must appear is a prediction of the neutrino fit.
  • The predicted branching ratios for $\tau\to\mu\gamma$ and $\tau\to e\gamma$ can lie just below present limits (about $10^{-8}$ to $10^{-11}$), while $\mu\to e\gamma$ is predicted to be far below them ($10^{-15}$ to $10^{-18}$), giving a distinctive pattern for next-generation LFV searches.
  • The effective Majorana mass for $0\nu\beta\beta$ decay is dominated by right-handed neutrino exchange, with values mostly between about $10^{-6}$ eV and $10^{-1}$ eV, so experiments closing in on that range can confirm or exclude the model.

Reading between the lines

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

  • An immediate check the authors leave implicit: because the expansions in Eq. (A7) are supplied without derivation, one can verify by direct computation whether they satisfy $\Delta_0 Y=0$ and the $A_4$ triplet transformation law; that verification would either cement or invalidate the numerical results that follow.
  • The framework's permission of zero and negative modular weights is not used to its fullest here; applying the same construction to quark-sector mass matrices or to other finite modular groups would test whether the non-holomorphic mechanism can reproduce fermion hierarchies as well as neutrino angles.
  • The distinct LFV pattern—$\tau\to\mu\gamma$ and $\tau\to e\gamma$ near present limits with $\mu\to e\gamma$ far below—means a future observation of tau LFV without muon LFV would single out this kind of inverse-seesaw modular model.
  • The paper's inverted-ordering leptogenesis region is sparse; if future data were to exclude normal ordering, this construction would face tension in explaining the baryon asymmetry even though its oscillation fit would survive.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper constructs a non-supersymmetric left-right asymmetric model with an extended inverse seesaw mechanism, using weight-zero polyharmonic Maass forms for Γ(3) as Yukawa couplings transforming as an A4 triplet. The authors fit the neutrino sector to oscillation data, then compute predictions for 0νββ decay, charged lepton flavor violation, and baryogenesis via resonant leptogenesis, and claim simultaneous consistency with all current experimental bounds.

Significance. If correct, the paper would demonstrate that non-holomorphic modular forms can be used in non-supersymmetric flavor models, and that a TeV-scale setup can accommodate neutrino masses, 0νββ bounds, LFV constraints, and the baryon asymmetry. The direction is timely and the paper attempts to go beyond the usual holomorphic modular-invariant framework. However, the central quantitative claims are not supported as written: the resonant leptogenesis result is internally inconsistent with the stated parameter choices, the 0νββ calculation omits essential numerical inputs, and the claimed 'predictions' of the mixing angles are circular because those angles are used as input to fix the Yukawa couplings. These load-bearing problems prevent acceptance in the current form.

major comments (4)
  1. [VII.E, Eqs. (3.7), (3.9), (6.1)] The BAU result is internally inconsistent with the stated parameter choices. With v_R=v'=10 TeV, Re[q_i]=Im[q_i] in [0.01,0.1] (so |q_i|≤0.141), |Y_i|≤1.092 from Table V, and g1=g2 in [10,50] keV, every entry of M_R and M_N in Eqs. (3.7) and (3.9) is at most about 4.5 TeV, while M_S entries are at most about 0.1 MeV. Consequently the 6×6 matrix in Eq. (6.1) has all eigenvalues bounded by a few TeV, for example by the Frobenius norm or Gershgorin disk theorem. Section VII.E's claim that successful BAU requires heavy neutrino masses between 200 TeV and 2000 TeV (Fig. 8c,d) therefore cannot be realized at any point in the stated parameter space, so the central leptogenesis claim fails.
  2. [VII.A and Abstract] The claimed 'predictions' of the neutrino mixing angles are circular. Section VII.A states that the Yukawa couplings are calculated using the 3σ values of the neutrino oscillation parameters from Table I, and then the resulting mixing angles are compared with the same experimental ranges. Since the oscillation parameters are used as input to determine the model parameters, the agreement of the mixing angles is a fit, not a prediction. The abstract's statement that the model 'successfully predicts' the mixing angles and the conclusion's emphasis on 'strong predictive power' are therefore overstated. The sum of neutrino masses may still be a prediction, but the mixing angles cannot be claimed as such without a parameter-count analysis showing that the model has fewer free parameters than the observables it reproduces.
  3. [Section IV, Eqs. (4.1)-(4.2)] The 0νββ results are not reproducible because the inputs M_WR, g_R/g_L, tan ζ, and |p| are never specified. The effective mass shown in Figure 5 depends sensitively on these quantities through the factors (M_WL/M_WR)^2(g_R/g_L)^3 and |p|, so without their values the plot cannot be verified. Moreover, in Eq. (4.1) the heavy right-handed neutrino contribution m_{N,ee}^λ has denominator M_Si^2, which is the sterile neutrino mass; this appears to be a typo for M_Ni^2. The same issue appears in Eq. (4.2). The authors should provide the numerical inputs and correct the denominators.
  4. [Appendix A, Eq. (A7)] The entire model rests on the assertion that Y_{3,1}^{(0)}, Y_{3,2}^{(0)}, Y_{3,3}^{(0)} are weight-zero polyharmonic Maass forms for Γ(3) transforming as an A4 triplet, but no derivation or explicit source is given. The q-expansions in Eq. (A7) are simply stated. If these functions are not the correct weight-zero polyharmonic Maass forms, all mass matrices in Eqs. (3.5)-(3.11) change and the subsequent phenomenology follows only by accident. The authors should either derive these expansions or cite the exact reference (including level, weight, and A4 decomposition) and verify the modular transformation property. There is also a typographical error in the third term of Y_{3,1}^{(0)} ('−−12πy'), which makes the expansion ambiguous.
minor comments (6)
  1. [Eq. (3.7)] In the last row, second column of the second matrix, 'q5−g4Y_{3,1}^{(0)}' should read 'q5−q4Y_{3,1}^{(0)}'.
  2. [Section III, after Eq. (3.11)] The text states that 'q9 and q10 are taken in the keV range,' but q9 and q10 never appear in the Lagrangian or in the mass matrices (Eqs. (3.2)-(3.11)). Presumably the keV-scale parameters are g1 and g2; please clarify or remove q9 and q10.
  3. [Section VII.E] The dilution factor is said to be determined 'using equation (4.1)', but Eq. (4.1) is the 0νββ effective-mass formula; the dilution factor is given in Eq. (6.9).
  4. [Eq. (6.9)] There is a spurious minus sign in the first line, '−d≈ ...'; the dilution factor should be positive.
  5. [Section V, Eq. (5.2)] The text after Eq. (5.2) says 'M_W represents the mass of the right-handed gauge boson respectively,' but the formula uses M_WL (the left-handed W mass); the text should read 'left-handed' or define the notation.
  6. [General] There are numerous typographical errors (e.g., 'RESONENT LEPTOGENSIS' in the Section VI heading, 'Maaβ' spelling inconsistencies, and incomplete figure captions). A careful proofread is needed.

Circularity Check

1 steps flagged · score 6.0 of 10

Mixing-angle "predictions" are refits of the Table I oscillation inputs, and the Σmν bound is used as a selection cut, so the central neutrino observables reduce to inputs.

  1. fitted input called prediction [Section VII A (Parameter space of Yukawa coupling), cf. Eq. (2.3), Table I, and Fig. 1 caption.]
    "Utilizing the 3σ values of the neutrino oscillation parameters which is given in Table I, we have calculated the Yukawa couplings associated with the model. After determining the values of the three Yukawa couplings, we computed the Pmν and mixing angles and compared these calculated values with the experimental ranges."

    The Yukawa couplings Y1, Y2, Y3 are free parameters entering MD, MR, MN, and MS in Eqs. (3.5)-(3.11), and the light neutrino mass matrix mν is obtained from them via Eq. (2.3). The stated procedure uses the 3σ ranges of sin^2θ12, sin^2θ23, sin^2θ13, δCP, Δm^2_21, and Δm^2_3l from Table I to determine the Yukawa couplings, then recomputes the same oscillation observables from those fitted couplings and reports agreement as a successful prediction. This is a fit renamed as a prediction: the output mixing angles are, by construction, the inputs used to fix the free parameters. The Σmν claim is likewise weakened because Fig.

full rationale

The main circularity is in Section VII A: the neutrino oscillation parameters are not independent predictions but are used to fix the Yukawa couplings, after which the same quantities are displayed as predictions. The Σmν result is also partly circular because the Planck bound is used to select the allowed Yukawa parameter space before being quoted as a successful prediction. The 0νββ, LFV, and BAU values are computed at the same fitted parameter points, so they are postdictions at those points rather than independent tests, though they are not themselves the direct inputs of the fit. Separate concerns not counted as circularity: the unproven Appendix A q-expansions for Y3,1, Y3,2, Y3,3 are a provenance/correctness risk, and the claimed 200-2000 TeV heavy-neutrino masses in Section VII.E appear inconsistent with v_R = v' = 10 TeV and the quoted |q_i| and |Y_i| ranges, which is a correctness rather than circularity issue. Overall, the central oscillation predictions reduce to the input data, giving a circularity score of 6.

Assumptions & free parameters 6 free parameters · 5 assumptions · 4 invented entities

The neutrino sector is fit with three modular Yukawa couplings, eight complex flavor couplings, two sterile couplings, and hand-chosen VEV scales before the BSM observables are computed. The 0νββ analysis additionally depends on unstated inputs such as M_WR, g_R/g_L, tan ζ, and |p|. No formal verification or external check is provided.

free parameters (6)
  • Y1, Y2, Y3 (modular-form Yukawa couplings) = NH: 0.003-1.056, 0.003-0.92, 0.003-1.092; IH: 0.01-0.99, 0.004-1.014, 0.006-1.006 (Table V)
    Computed directly from the 3σ neutrino oscillation parameters in Section VII A, so the neutrino observables advertised as predictions are fit inputs.
  • q1-q8 (complex flavor couplings) = Scanned with Re and Im in [0.01, 0.1]
    Chosen by hand so the model can satisfy all experimental bounds; they appear in the Dirac, Majorana, N-S, and sterile mass matrices.
  • g1, g2 (sterile Yukawa couplings) = g1=g2 in [10, 50] keV
    Picked to give keV-scale sterile masses in the inverse seesaw; not derived from data.
  • vR and v' (VEVs of ΔR and χR) = 10 TeV
    Set to 10 TeV so that the right-handed scale is TeV-scale; this choice controls the heavy neutrino mass range and the 0νββ normalization.
  • q9, q10 (undefined couplings) = keV range, as stated
    Mentioned as taken in the keV range but never defined or used in any equation; their role is unclear.
  • M_WR, g_R/g_L, tan ζ, |p| (0νββ inputs) = Not stated
    Equation (4.1) cannot be evaluated without these quantities, so the plotted effective masses in Fig. 5 rest on unstated inputs.
assumptions (5)
  • domain assumption The functions in Eq. (A7) are weight-zero polyharmonic Maass forms for Γ(3) transforming as an A4 triplet and can act as Yukawa couplings in a non-supersymmetric theory.
    Assumed in Appendix A, Eqs. (A4)-(A7); no derivation or source is given in this paper, and the entire mass matrix construction depends on it.
  • domain assumption The block-diagonalization in Eq. (2.3) is valid for the 9x9 extended inverse seesaw at TeV scale.
    Used to define mν, mS, and mR; the perturbative ordering is asserted rather than derived.
  • domain assumption The new gauge extension is anomaly-free with the stated particle content and breaking chain.
    No anomaly cancellation check appears anywhere in the manuscript.
  • standard math The resonant leptogenesis formulas and dilution factors from refs [34-38] apply to the six heavy eigenstates.
    The formulas are cited rather than derived, and the numerical BAU result depends on them.
  • standard math The A4 multiplication and contraction rules used in Eqs. (3.4)-(3.11) are the standard ones.
    Standard group theory, but the specific contractions are not spelled out, making independent reproduction harder.
invented entities (4)
  • Three sterile fermions S_i
    purpose: Enable the extended inverse seesaw and generate light neutrino masses through the MS term.
    No predicted mass or mixing is provided that could be observed independently; only broad keV-scale ranges for g1 and g2 are given.
  • Scalar doublet χR
    purpose: Generate the N-S mixing term through its vacuum expectation value v'.
    No mass or coupling prediction is given beyond v'=10 TeV.
  • Scalar triplet ΔR
    purpose: Break SU(2)_R and generate the Majorana mass matrix MR through its vacuum expectation value vR.
    No direct collider or low-energy observable is tied to this particle in the paper.
  • Right-handed gauge boson WR and the associated U(1)R gauge structure
    purpose: Mediate the WL-WR current contributions in 0νββ decay and carry right-handed interactions.
    vR=10 TeV implies a TeV-scale WR, but the paper never states M_WR or g_R/g_L, so no sharp falsifiable prediction is provided.

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Pith. "Pith review of Leptogenesis, $0\nu\beta\beta$ and lepton flavor violation in modular left-right asymmetric model with polyharmonic $Maa\beta$ forms." pith.science (2026). https://pith.science/paper/JQUGGZQ3

@misc{pith2026250421701,
  author       = {Pith},
  title        = {Pith review of: Leptogenesis, $0\nu\beta\beta$ and lepton flavor violation in modular left-right asymmetric model with polyharmonic $Maa\beta$ forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQUGGZQ3}},
  note         = {Machine review of arXiv:2504.21701}
}
abstract

In the absence of supersymmetry, modular forms need not be holomorphic functions of the modulus $\tau$. Using this idea, we construct a non-supersymmetric framework using polyharmonic $Maa\beta$ forms. In this approach, the Yukawa coupling is no longer strictly holomorphic in $\tau$ but instead incorporates both holomorphic and non-holomorphic components. We realize a non-supersymmetric, left-right asymmetric model based on the $\Gamma_3$ modular group, where the active neutrino masses are generated via an extended inverse seesaw mechanism. The model successfully predicts the sum of neutrino masses below the current experimental bound and accommodates neutrino mixing angles within the $3\sigma$ range. Given its strong predictive power in neutrino oscillation parameters, we further explore its implications for beyond Standard Model (BSM) phenomena, including neutrinoless double beta ($0\nu\beta\beta$) decay, lepton flavor violation (LFV), and baryogenesis via leptogenesis (BAU). Our findings indicate that the model predicts an effective Majorana mass and LFV branching ratios consistent with experimental constraints while also providing a viable explanation for the observed baryon asymmetry through resonant leptogenesis.

Figures

Figures reproduced from arXiv: 2504.21701 by the authors.

Figure 1
Figure 1. FIG. 1: Figure shows the parameter space of Yukawa couplings that satisfy the Planck bound on the [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Density plots illustrating the parameter space of Yukawa couplings that satisfy the 3 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Density plots illustrating the parameter space of Yukawa couplings that satisfy the 3 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Figure show the parameter space of Re( [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Figure shows the variation of effective Majorana mass with the lightest neutrino mass [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Figure shows the parameter space lightest neutrino mass that satisfies the bound on [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Figure shows the parameter space of CP asymmetry [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The top two figures illustrate the parameter space of the lightest neutrino mass that satisfies [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

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