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REVIEW 4 major objections 6 minor 56 references

A $\Gamma_{3}$ modular symmetric approach for two-zero textures in left-right symmetric model

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A single flavon-free A4 modular left-right symmetric model can realize all seven experimentally allowed two-zero neutrino mass textures.

desk verdict The A4 construction is explicit and the seven mass matrices are there, but the paper never diagonalizes the charged leptons, so the physical neutrino texture is not the one being fitted—the central claim is unestablished. read the letter →

arxiv 2608.07623 v1 pith:2SJPUXU7 submitted 2026-08-07 hep-ph

Ankita Kakoti , Happy Borgohain This is my paper · ORCID
classification hep-ph
keywords two-zerotextureleft-rightsymmetricmodelA4modularsymmetryformsneutrinomassmatrixresonantleptogenesisneutrinolessdoublebetadecayseesawmechanism
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 the seven two-zero patterns in the light neutrino mass matrix that survive current neutrino oscillation data can all be realized inside a left-right symmetric model whose flavor sector is an A4 (Gamma3) modular symmetry, with no flavon fields. The realization is achieved by assigning modular weights 4, 8 and 10 to the lepton fields so that modular invariance of the superpotential forbids the entries that become the zeros. If the construction holds, neutrino flavor structure is reduced to a small set of symmetry assignments plus a single complex modulus, and the model makes concrete predictions for the sum of neutrino masses, the Dirac CP phase, the baryon asymmetry from resonant leptogenesis, and the effective mass for neutrinoless double beta decay. The paper's central phenomenological claims are that exactly the seven allowed texture classes arise, that classes B1, B2 and C reproduce the observed baryon asymmetry, and that every class stays below the current neutrinoless double beta decay bound.

What carries the argument

The central machinery is the assignment of $A_4$ ($\Gamma_3$) modular weights and representations to the three left-handed lepton doublets $L_L$ and right-handed singlets $L_R^c$, with Yukawa couplings promoted to modular forms of weight 4, 8 and 10. The modular forms $Y^{(k)}$ are fixed functions of the modulus $\tau$, and the rule that the modular weights in every term of the superpotential sum to zero decides which entries of the Dirac mass matrix $M_D$, the right-handed Majorana matrix $M_R$, and the left-handed Majorana matrix $M_L$ are non-zero. Combining type-I and type-II seesaw contributions, $M_\nu = M_\nu^{\mathrm{I}} + M_\nu^{\mathrm{II}}$, then produces the two-zero textures, each characterized by exactly two vanishing entries among the six independent elements of the symmetric light neutrino mass matrix.

What would settle it

Compute, for each of the seven charge and weight assignments, the lowest modular weight at which an $A_4$-invariant operator contributes to a position that the texture sets to zero; if any such operator enters at a weight that is not suppressed far below the neutrino mass scale set by the non-zero entries, the exact-zero claim fails. On the experimental side, a future neutrinoless double beta decay signal above the largest effective mass the paper predicts for the allowed classes would rule out the framework.

Watch

Extended reading notes

Core claim

Within the generic left-right symmetric model, promoting flavor symmetry to the $A_4$ modular group and choosing modular weights 4, 8 and 10 for the three left-handed lepton doublets and right-handed singlets yields a light neutrino mass matrix $M_\nu = M_\nu^{\mathrm{I}} + M_\nu^{\mathrm{II}}$ that takes exactly the seven experimentally allowed two-zero textures. The zeros are not imposed by hand: a superpotential term is present only if the $A_4$ representations combine to a singlet and the modular weights of the fields sum to zero, so the allowed entries of $M_D$, $M_R$ and $M_L$ are fixed by the symmetry. By permuting the $A_4$ charges and modular weights of the lepton fields, the paper obtains classes B2 and B1 from weight 4, classes B4, B3, A2 and A1 from weight 8, and class C from weight 10. With these mass matrices, resonant leptogenesis gives a baryon asymmetry consistent with observation for B1, B2 and C, and the computed total effective Majorana mass for neutrinoless double $\beta$ decay remains below the current experimental limit for all seven classes.

Load-bearing premise

The construction assumes that the superpotential terms written down are the only contributions to the light neutrino mass matrix, so the zero entries are exact; additional interactions or quantum corrections could fill those entries, and the paper does not estimate how large such corrections could be.

Editorial extensions

If this is right

  • All seven experimentally allowed two-zero textures arise from one flavon-free modular $A_4$ LRSM, so the usual flavon alignment problem is bypassed.
  • The model yields correlated predictions linking $\sum m_\nu$, $\delta_{\mathrm{CP}}$, the solar and atmospheric mixing angles, the baryon asymmetry, and the neutrinoless double beta decay effective mass, so measuring any of these sharpens the others.
  • Resonant leptogenesis works only for classes B1, B2 and C in this framework, so a confirmed baryon asymmetry together with a texture from the other classes would require new ingredients.
  • Every texture class survives the current neutrinoless double beta decay limit, so future experiments with improved sensitivity can distinguish the predicted total effective mass ranges among classes.
  • Several classes disfavor or exclude one of the two mass orderings for particular parameters, giving a route to ordering discrimination.

Reading between the lines

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

  • The paper never estimates how higher-order modular-invariant operators, non-minimal Kähler terms, or loop corrections fill the positions set to zero; an extension would compute the lowest modular weight that can populate each zero and translate it into a bound on the modulus $\tau$.
  • The same weight-assignment scanning could be applied to other modular groups such as $S_3$, $S_4$ or $A_5$; the seven-texture classification is a property of the two-zero condition, not of $A_4$, so the method is transferable.
  • The authors take the type-I and type-II seesaw terms to contribute with equal strength; relaxing this assumption would change the relative sizes of the non-zero entries and could alter which textures reproduce the baryon asymmetry, a testable difference for future leptogenesis and $0\nu\beta\beta$ analyses.
  • The allowed set of seven textures is imported from current global fits; future experiments with sharper oscillation measurements may exclude some textures, which would single out a subset of the modular weight assignments and make the model more predictive.
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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 manuscript constructs an A4 (Gamma3) modular-symmetric left-right symmetric model without flavon fields, assigns modular weights 4, 8 and 10 to the lepton sector, and claims to realize, one by one, the seven phenomenologically allowed two-zero textures of the light neutrino mass matrix (classes A1, A2, B1, B2, B3, B4, C). For each texture the paper scans the modular parameter and the VEV ratio, compares the resulting neutrino observables with NuFit-6.0 ranges, and then studies resonant leptogenesis and neutrinoless double beta decay. The central claims are that classes B1, B2 and C reproduce the observed baryon asymmetry and that all seven classes yield effective 0νββ masses compatible with KamLAND-Zen.

Significance. If correct, the framework would be a compact, flavon-free modular LRSM in which a small set of A4 charge and modular-weight assignments determines the entire neutrino mass matrix and yields correlated predictions for the sum of neutrino masses, Dirac CP phase, baryon asymmetry, and 0νββ. The paper has useful strengths: the mass matrices are written out explicitly, all seven two-zero classes are systematically addressed, and the phenomenological section confronts several independent constraints simultaneously. However, the central derivation contains a load-bearing algebraic inconsistency, and the flavor basis in which the textures are defined is never specified. As it stands, the claimed realization of the seven textures and the resulting phenomenological fits are not established.

major comments (4)
  1. [Section III, Eq. (3.8)] The light neutrino matrix in Eq. (3.8) does not follow from the preceding matrices by the stated seesaw formula. Writing a = Y_1^4 and b = Y_{1'}^4 in Eqs. (3.3), (3.5) and (3.7), a direct multiplication of M_D M_R^{-1} M_D^T gives (2,2) entry b^2/a and (2,3) entry [a^2b^2 - b^4 + a^4 + ab^3]/[ab(a+b)], whereas Eq. (3.8) has b and a respectively. The matrix in Eq. (3.8) is instead simply proportional to M_L, which would require M_D M_R^{-1} M_D^T ∝ M_L; this proportionality does not hold for the displayed matrices. Since the same computation underlies the other texture matrices in Eqs. (3.15), (3.20), (3.23), (3.26), (3.29) and (3.33), the central claim that all seven two-zero textures are realized is unsupported as written.
  2. [Section III, Tables III-VIII and Eqs. (3.3)-(3.33)] The charged-lepton mass matrix is never constructed. In this model the same bidoublet and the same A4 assignments generate the charged-lepton and Dirac neutrino Yukawa couplings, so the charged-lepton mass matrix is generically non-diagonal. The physical light neutrino mass matrix in the mass basis of charged leptons is U_e^T Mν U_e with a non-trivial U_e, and this rotation generically fills two-zero entries. The paper compares Mν directly with NuFit oscillation data in Figs. 1-7 without specifying or deriving such a basis, so the claimed two-zero textures are defined only in an unspecified flavor basis and the oscillation fits are not justified.
  3. [Section IV, Figs. 1-11 and Table IX] The numerical analysis is not documented in a reproducible way. No chi-square or likelihood function is defined, the scan ranges for the modular parameter τ, the VEV ratio v_L/v_R, and the overall mass scale are not given, and the red 'best-fit' points in the figures are not defined by any stated criterion. The text repeatedly states that data points lie inside the 3σ NuFit ranges, but without a statistical measure and a description of the sampling, the viability claims and the summary in Table IX cannot be verified or reproduced.
  4. [Section IV.A, Eqs. (4.1)-(4.5)] The resonant leptogenesis analysis lacks the necessary quantitative ingredients. The heavy right-handed neutrino mass spectrum is never presented, and the degeneracy condition M_i - M_j ~ Γ_i needed for resonant enhancement is asserted rather than demonstrated for the scanned parameters. Moreover, Eq. (4.2) as written is not the standard CP asymmetry formula for quasi-degenerate heavy neutrinos, since the denominator does not have the usual product (Y†Y)_ii (Y†Y)_jj structure, and the efficiency parametrization in Eq. (4.5) is introduced without derivation. Consequently the claim that classes B1, B2 and C reproduce the observed baryon asymmetry is not quantitatively supported.
minor comments (6)
  1. [Section II, Eq. (2.4)] The type-I seesaw term is written with a plus sign; the conventional expression is M_I = -M_D M_R^{-1} M_D^T. Please state the sign convention, since it can affect cancellations with the type-II term even when texture zeros are unchanged.
  2. [Section III.B, after Eq. (3.21)] The text says the assignments are 'shown in table V' a second time, but the table in question is Table VI.
  3. [Section III.C, Eq. (3.31)] The superpotential appears to contain a duplicated term: the same L^c_R2 iσ2 Δ_R L^c_R3 Y_1^6 contribution is written twice inside one parenthesis. Please check whether one of the fields should be L^c_R3 iσ2 Δ_R L^c_R2.
  4. [Section V, bullet on Class B1] The text states that sin^2θ23 data fall in the range 0.5 to 0.6 'radians'; sin^2θ23 is dimensionless, so the unit should be removed.
  5. [Figures 1-11] The figures would benefit from explicit axis labels, legends identifying normal versus inverted ordering in every panel, and a statement of how many scan points were used; several panels currently rely on color alone.
  6. [Introduction and Section III] Reference [27] realizes two-zero textures with modular A4 symmetry; the present manuscript should state more explicitly what is new relative to that work, beyond embedding the same ideas in the LRSM.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the two-zero textures are group-theoretic constructions, and the numerical checks are parameter-space scans against external data.

full rationale

The central derivation chain is not circular. Tables III to VIII fix A4 representations and modular weights, and the allowed superpotential terms, hence the positions of zeros in MD, MR, and ML, are determined by A4 invariant contractions; for example, the zeros in Eq. (3.8) follow from the absence of the corresponding singlet contractions in Eqs. (3.2) to (3.7). The seven-texture set is an external phenomenological input, not a model output, and the text explicitly says it constructs rather than predicts this set. The numerical analysis scans the free modular parameter and VEV ratio and tests the resulting points against NuFit, Planck, and KamLAND-Zen; several classes fail these tests, so the agreement is not enforced by construction. The self-citations to prior LRSM work provide standard mass-matrix and leptogenesis formulas and are not load-bearing for the modular texture claim. One correctness concern, but not a circularity, is that the charged-lepton mass matrix and the basis rotation U_e are never constructed, so Eqs. (3.8) to (3.33) are compared to NuFit in an unspecified flavor basis.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The construction is honest but carries many hand-chosen inputs: the seven textures are taken from prior phenomenology, the modular weights and A4 representations are chosen per texture, the numerical outputs come from scanning τ and the VEV ratio, and the leptogenesis and 0νββ inputs are assumed. These choices explain why the model appears to accommodate the known seven patterns rather than predicting them independently.

free parameters (3)
  • modular parameter τ = not stated; red-dot best fits shown in figures
    The scan over τ generates all quoted oscillation parameter values, but the range, grid size and best-fit value are never reported.
  • VEV ratio v_L/v_R and overall mass scale v^2/v_R = not stated
    The ratio controls the relative type-I and type-II seesaw strength and the overall scale sets the absolute neutrino masses; both are adjusted to place points inside NuFit 3σ regions.
  • per-texture modular weight and A4 representation assignments = discrete: Tables III-VIII, weights -2, -4, -5, 0 etc.
    The A4 charges and modular weights are hand-chosen for each texture so that the desired zeros appear in Mν, effectively encoding the target textures into the model.
assumptions (5)
  • standard math A4/Γ3 modular-form decompositions for weights 4, 6, 8 and 10 are as stated in Eqs. (3.1), (3.16), (3.17) and (3.30).
    Every mass matrix construction depends on these decomposition rules for the modular Yukawa forms; an error here would change the textures.
  • domain assumption The seven two-zero textures A1, A2, B1, B2, B3, B4 and C are the complete set of experimentally allowed two-zero textures.
    The paper imports this classification from earlier phenomenology in Section I and does not re-derive it; the entire goal is to realize exactly these seven.
  • ad hoc to paper Type-I and type-II seesaw terms are both included with equal footing and f_L = f_R, with no other contributions to Mν.
    Section V states 'we have considered equal contributions for both the seesaw mass terms'; this choice fixes the seesaw balance and is not forced by the left-right symmetry alone.
  • ad hoc to paper The superpotential terms written in Section III are the complete origin of the light neutrino mass matrix; no higher-order operators or Kähler corrections fill the texture zeros.
    The zeros in Eqs. (3.8) to (3.33) are exact only if no additional modular-invariant terms contribute, which the paper does not discuss.
  • domain assumption The 0νββ calculation assumes fixed inputs M_WR, M_ΔR = 3 TeV, momentum p = 180 MeV and nuclear matrix element values from Refs. [43, 53].
    Section IV.B uses these inputs without propagating uncertainties; changing them changes the exclusion plots and the claimed compatibility with KamLAND-Zen.

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Pith. "Pith review of A $\Gamma_{3}$ modular symmetric approach for two-zero textures in left-right symmetric model." pith.science (2026). https://pith.science/paper/2SJPUXU7

@misc{pith2026260807623,
  author       = {Pith},
  title        = {Pith review of: A $\Gamma_3$ modular symmetric approach for two-zero textures in left-right symmetric model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SJPUXU7}},
  note         = {Machine review of arXiv:2608.07623}
}
abstract

The observed pattern for neutrino masses and mixing provides compelling evidence for Beyond Standard Model physics which further motivates the search for predictive frameworks that can simultaneously address flavor structure and its phenomenological consequences. This work particularly investigates the realization of all possible seven two-zero neutrino mass textures within the generic left-right symmetric model with $A_{4}$ modular symmetry. By considering modular weights 4,8 and 10, we systematically construct all the possible classes of 2-0 textures without the introduction of any flavon fields which enhances the predictive power of the framework. In this work, we also identify the texture classes capable of simultaneously accommodating current neutrino data, reproducing the observed baryon asymmetry and also yielding experimentally testable results for the effective Majorana neutrino mass for new physics contributions of neutrinoless double beta decay.

Figures

Figures reproduced from arXiv: 2608.07623 by the authors.

Figure 1
Figure 1. FIG. 1: Neutrino oscillation parameters with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Neutrino oscillation parameters with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Neutrino oscillation parameters with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Neutrino oscillation parameters with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Neutrino oscillation parameters with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Neutrino oscillation parameters with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Neutrino oscillation parameters with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Baryon asymmetry parameter with sum of neutrino masses, where the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Effective mass with sum of neutrino masses, where region above the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Effective mass with sum of neutrino masses, where region above the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Effective mass with sum of neutrino masses, where region above the black horizontal [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.