Minimal structure for neutrino mass matrix

Manmohan Gupta; Manoj Kumar; Monika Randhawa; Nikhila Awasthi

REVIEW 3 major objections 4 minor 95 references

Minimal structure for neutrino mass matrix

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A one-phase neutrino mass texture reproduces all measured mixing angles and predicts the unknown ones.

desk verdict A one-phase two-zero neutrino texture with sharp testable predictions, but the 3 sigma box scan needs a likelihood upgrade before the 'accommodates data' claim is fully credible. read the letter →

arxiv 2412.15684 v2 pith:F3SYY3N4 submitted 2024-12-20 hep-ph

Manoj Kumar , Nikhila Awasthi , Monika Randhawa , Manmohan Gupta This is my paper · ORCID

classification hep-ph

keywords neutrinomassmatrixtexturezerosCPviolationleptonmixingMajoranaphasesneutrinolessdoublebetadecaynormalordering

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 proposes that the observed lepton mixing follows from a neutrino mass matrix with only two vanishing entries and a single complex phase, directly transplanting a minimal structure that works for quarks. Taking the charged-lepton masses to be diagonal, the authors show that this texture-2-zero complex symmetric matrix reproduces all three mixing angles within their 3-$\sigma$ ranges for normal neutrino mass ordering. The same matrix yields narrow predictions for quantities not yet measured: the lightest neutrino mass between $3.14$ and $8.13 \times 10^{-3}$ eV, a Jarlskog invariant around $0.031$-$0.036$, and a Dirac CP phase confined to two quadrants. If these predictions survive measurement, they would point to a common minimal texture behind quark and lepton mass matrices.

What carries the argument

The load-bearing object is the complex symmetric neutrino mass matrix of Eq. (3), a texture-2-zero matrix with only one phase parameter, combined with the assumption that the charged-lepton mass matrix is exactly diagonal in the flavor basis. Exact diagonalization of the neutrino mass matrix relates the PMNS mixing angles and the CP invariants to the three free parameters $m_{\nu_1}$, $D_\nu$ and $\phi$. This machinery turns the measured oscillation angles into allowed parameter ranges, and then converts those ranges into the paper's predictions for $m_{\nu_1}$, $J_{\mathrm{CP}}$, $\delta_{\mathrm{CP}}$, $\langle m_{ee}\rangle$ and the Majorana phases.

What would settle it

A measurement of the lightest neutrino mass above $8.13\times 10^{-3}$ eV under normal ordering, or an unambiguous determination of $\delta_{\mathrm{CP}}$ outside the intervals $(0\text{--}90)^\circ$ and $(180\text{--}270)^\circ$, would rule out the flavor-basis texture; establishing inverted mass ordering would also falsify the claim.

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Extended reading notes

Core claim

The paper demonstrates that a texture-2-zero complex symmetric Majorana neutrino mass matrix carrying only one phase, $\phi$, is sufficient to accommodate the current neutrino oscillation data in normal ordering. With the charged-lepton mass matrix taken to be exactly diagonal, the PMNS matrix equals the neutrino diagonalizing matrix, and the three mixing angles follow from just three free parameters: the lightest neutrino mass $m_{\nu_1}$, the $(2,2)$ element $D_\nu$, and $\phi$. The resulting parameter space produces the predictions of Table 2, including $m_{\nu_1} \in (3.14\text{--}8.13)\times 10^{-3}$ eV, $J_{\mathrm{CP}}^{\mathrm{max}}\in (3.07\text{--}3.59)\times 10^{-2}$, $\delta_{\mathrm{CP}}$ in $(0\text{--}90)^\circ$ or $(180\text{--}270)^\circ$, and $\langle m_{ee}\rangle \in (0.0050\text{--}0.0081)$ eV. Further simplifying the matrix by setting $D_\nu = B_\nu = C_\nu$ leaves only one free parameter and narrows the predictions sharply, e.g. $\delta_{\mathrm{CP}}=(257.6\text{--}259.2)^\circ$ and $m_{\nu_1}=(4.84\text{--}5.15)\times 10^{-3}$ eV. The same texture cannot reproduce the mixing angles under inverted mass ordering.

Load-bearing premise

The analysis assumes the charged-lepton mass matrix is exactly diagonal in the flavor basis, so that all lepton mixing comes from the neutrino sector; if charged-lepton corrections are significant, the extracted parameter ranges and predictions would change.

Editorial extensions

If this is right

If normal ordering is the true neutrino mass hierarchy, the lightest neutrino mass must lie in the range $(3.14\text{--}8.13)\times 10^{-3}$ eV.
The Dirac CP phase $\delta_{\mathrm{CP}}$ is predicted to be either in the first or third quadrant; the further simplified $D_\nu=B_\nu=C_\nu$ case predicts a very narrow band around $258^\circ$.
The Jarlskog invariant $J_{\mathrm{CP}}$ is predicted to be large enough that next-generation long-baseline experiments could observe leptonic CP violation.
The effective Majorana mass $\langle m_{ee}\rangle$ is predicted in a narrow range near $5\text{--}8$ meV, within reach of upcoming neutrinoless double-beta-decay searches.
Inverted mass ordering is excluded by this minimal texture, so confirming the ordering also tests the structure.

Reading between the lines

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

The near-linear correlation between $A_\nu$ and $m_{\nu_1}$ means that a model-independent measurement of the absolute neutrino mass scale would sharply constrain the allowed matrix-element ranges.
If future long-baseline data place $\delta_{\mathrm{CP}}$ outside the predicted quadrants, the whole flavor-basis texture would be ruled out even before the full neutrino mass matrix is reconstructed.
The one-phase structure invites a symmetry justification that the paper does not provide; identifying the discrete symmetry that forces the texture zeros and the phase relation would turn the phenomenological ansatz into a model.
A direct test of the extreme $D_\nu=B_\nu=C_\nu$ simplification would come from combining a precise $\delta_{\mathrm{CP}}$ measurement with the predicted tight correlation between $\delta_{\mathrm{CP}}$ and $\theta_{23}$.

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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

The genuinely new thing here is not the zero pattern—(1,1) and (1,3) zeros are already in the two-zero classifications—but the restriction to a single phase and the D_nu = B_nu = C_nu subcase. Those give concrete numbers: m_nu1 around 5 meV, delta_CP near 258 degrees, J_CP around 3.3 to 3.5 percent, and around 7 meV. That is a target list DUNE, Hyper-K, KATRIN, and 0νββ experiments can actually shoot at.

What the paper does well is honesty. The non-flavor basis is reported as failing at 1 sigma for theta23, inverted ordering is shown not to work, and the paper is clear about what is input and what is predicted. The scan is simple, but the resulting parameter ranges are narrow, so the model has real predictive content if the fit is right.

The soft spots are real but not fatal. The biggest one is the 3 sigma box acceptance. Treating theta12, theta23, theta13, and the mass splittings as independent boxes ignores the NuFIT likelihood correlations, especially the theta23–delta_CP correlation. A model can pass all individual 3 sigma boxes while sitting in a low-likelihood corner. The stress-test note is right: without a chi-square scan or code, the statistical weight of the accommodation is not established. The exact diagonalization is also not shown, so I cannot check the scan independently. Second, the D=B=C simplification is post-hoc—they choose it after seeing the allowed bands overlap. That is a legitimate model-building move, but it means the sharp predictions in Table 3 are partly a product of the selection, not an independent prediction. Third, the diagonal charged-lepton assumption is standard but unquantified; a few sentences on how charged-lepton corrections would shift the ranges would help. None of these break the central claim; they just bound what the paper can assert.

The predictions are not circular: delta_CP, m_nu1, and are not used to select the parameter sets, so they are genuine outputs.

Who is this for? People working on texture zeros and flavor model building. It is a useful phenomenological reference with concrete experimental targets. It deserves a serious referee, but I would ask the authors to replace the box scan with a likelihood-based fit and to make the diagonalization code or at least the exact expressions available. That would turn a plausible accommodation into a testable one.

Referee Report

3 major / 4 minor

Summary. This paper proposes a texture-2-zero complex symmetric Majorana neutrino mass matrix with a single phase parameter, motivated by a previously successful texture-4-zero quark mass matrix. In the flavor basis with diagonal charged leptons, the authors scan the free parameters (mν1, Dν, ϕ) over 3σ ranges of the neutrino mass-squared differences and require the resulting mixing angles to lie in their 3σ intervals, finding that normal ordering can reproduce the three mixing angles. They then derive predictions for the lightest neutrino mass, J_CP, δ_CP, the effective Majorana mass, and the Majorana phases, and further propose a simplified texture with Dν = Bν = Cν that yields narrower predictions. A non-flavor basis analysis in which the charged lepton mass matrix also has texture two-zero is included, and inverted ordering is found to be disfavored in both bases.

Significance. If the numerical results are reliable, the paper adds to the class of predictive texture-zero models: the single-phase texture is extremely economical, and the predicted absolute neutrino mass scale (mν1 ~ 3–8 meV), δ_CP, and ⟨m_ee⟩ are testable in upcoming experiments. The authors also make falsifiable predictions for the Majorana phases and present the non-flavor basis as a robustness check. However, the statistical validation is based on a 3σ box scan rather than a full likelihood, and no code or exact diagonalization is provided, so the quantitative claims are not yet independently checkable. The correlation analysis (e.g., mν1–δ_CP, θ23–mν1) is a useful phenomenological guide, but its significance is overstated without a proper goodness-of-fit measure.

major comments (3)

[Section 2 (acceptance criterion)] The compatibility claim rests on a 3σ-box scan in which the mass-squared differences are used as inputs and only the three mixing angles are required to fall inside their intervals. Because the model has three free parameters (mν1, Dν, ϕ) and each observable is tested independently, this procedure has no statistical power to establish that the texture "accommodates" the global oscillation data; correlations in the NuFIT likelihood (notably θ23–δCP) are ignored. A likelihood-based analysis that yields a χ² or a posterior profile is necessary to support the central claim, and to interpret the allowed ranges in Eqs. (10)–(12) as genuine predictions rather than a parametrization of the data.
[Section 2.3 (Dν = Bν = Cν simplification)] The simplified texture in Eq. (19) is introduced after inspecting the allowed common range of Dν, Bν, and Cν in the left panel of Fig. 1, which is obtained from the same fit that the model then claims to predict. This data-dependent selection means the narrow predictions in Table 3 are not an independent test of the simplified ansatz; the equality Dν = Bν = Cν should be imposed from the outset or justified by a symmetry, and then the fit should be redone without using the overlap of the earlier allowed ranges as a motivation.
[Section 2 (exact diagonalization)] The authors state that the exact expressions for Vν are "very long and can not be presented here" and that they used them in the analysis, but no code, algorithm, or even the diagonalization method is supplied. The numerical results in Eqs. (10)–(12) and Tables 2–4 are therefore not independently reproducible. The paper should provide the exact diagonalization procedure or a companion code, since the entire phenomenological output depends on this step.

minor comments (4)

[Section 2.1, Table 2] The predicted δCP range includes (0–90) degrees, which lies outside the NuFIT 3σ interval [139, 350] degrees quoted in Table 1; the statement in the text that "The phase δCP predicted by our model is within this range" is therefore incorrect for part of the parameter space and should be qualified branch by branch.
[Section 3, Eq. (20)] Assuming the charged lepton mass matrix to be complex symmetric with the same phase ϕ as the neutrino matrix is an ad hoc restriction; the paper should explain why this is natural, and discuss how this assumption affects the failure to reproduce θ23 at 1σ in the non-flavor basis.
[Table 1 and Eq. (10)] There are minor formatting issues: the unit "10−5eV 2" should be "10^{-5} eV^2", and Eq. (10) has inconsistent spacing. Also, the phrase "well within their experimental limits" should be "within their 3σ limits", since the acceptance is defined by the 3σ box.
[Abstract and Section 1] The motivation from the quark sector is heuristic; a brief discussion of how a common texture for quarks and leptons could arise in a concrete model would strengthen the physical case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the texture fit uses measured inputs and derives independent unknowns; the simplified submodel is an openly data-inspired restriction, not a fitted parameter relabeled as a prediction.

full rationale

The central derivation chain is self-contained. The paper fixes the charged-lepton mass matrix to be diagonal (Eq. 3), inputs the measured neutrino mass-squared differences from Table 1, and scans the free parameters m_nu1, D_nu, and phi, retaining sets whose mixing angles fall inside the 3 sigma intervals. The predicted quantities in Table 2 (m_nu1, J_CP, delta_CP, <m_ee>, Majorana phases) are not used as inputs to select the parameter sets; delta_CP in particular is deliberately not constrained even though Table 1 lists it. Thus these outputs are genuine consequences of the texture ansatz plus the input mass splittings and angle constraints. The later simplification D_nu = B_nu = C_nu is motivated by the common viable range of those matrix elements found in the earlier scan, which is a data-inspired model restriction rather than a parameter fitted to the final predicted observable; the paper states this explicitly ('This inspires us to simplify our model further'). That is postdiction/model selection, not circularity. The main self-citation (Ref. [38]) provides only quark-sector motivation and is not load-bearing for the lepton-sector results. The absence of a full likelihood treatment and the use of independent 3 sigma boxes is a statistical-correctness concern, not a circularity, because it does not make any derived quantity equal to an input by construction.

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

No new particles, forces, or symmetries are introduced. The model is a purely phenomenological mass matrix texture, and its free parameters are the scanned neutrino mass scale, the (2,2) element, and the single phase. The main implicit inputs are standard neutrino-physics assumptions about Majorana neutrinos, diagonal charged leptons, and normal ordering.

free parameters (3)

m_nu1 (lightest neutrino mass) = 3.14-8.13 x 10^-3 eV for Eq. (3); 4.84-5.15 x 10^-3 eV for Eq. (19)
Scanned and selected by requiring the resulting mixing angles to fall within the 3 sigma ranges of Table 1.
D_nu (2,2) matrix element = 0.022-0.031 eV
Scanned free parameter in the flavor-basis fit; the quoted range comes from 3 sigma compatibility.
Phase phi = -14.6 to 14.6 degrees
Scanned over [-pi, pi] and kept only where the mixing angles match data; the text notes that phi = 0 is allowed at 3 sigma.

assumptions (5)

domain assumption Neutrinos are Majorana particles, so the neutrino mass matrix is complex symmetric.
Stated in Section 2; this is needed for the texture and for interpreting <m_ee> as a neutrinoless double-beta decay observable.
domain assumption The charged lepton mass matrix is diagonal in the flavor basis.
Eq. (3) sets M_l = diag(m_e, m_mu, m_tau), making U_PMNS = V_nu; off-diagonal charged-lepton mixing is neglected.
domain assumption Normal ordering is assumed because inverted ordering fails to reproduce the mixing data.
Section 3 states that the given mass matrices cannot accommodate inverted ordering, so all predictions apply only to normal ordering.
ad hoc to paper The texture zeros at (1,1) and (1,3) are exact and have no symmetry justification.
The zeros are borrowed from the quark texture of ref. [38]; no underlying symmetry is proposed to enforce them.
domain assumption A 3 sigma compatibility criterion is sufficient to call the texture viable.
The scan accepts any parameter set reproducing the angles within 3 sigma; no chi-square or likelihood is used, so viability is defined by existence in a broad range.

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Cite this review

Pith. "Pith review of Minimal structure for neutrino mass matrix." pith.science (2026). https://pith.science/paper/F3SYY3N4

@misc{pith2026241215684,
  author       = {Pith},
  title        = {Pith review of: Minimal structure for neutrino mass matrix},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3SYY3N4}},
  note         = {Machine review of arXiv:2412.15684}
}

abstract

Taking clue from the minimal structure of texture 4-zero hermitian mass matrices, which are very successful in accommodating quark mixing data, we propose a form of texture 2-zero complex symmetric neutrino mass matrix with only one phase parameter. This minimal mass matrix not only accommodates the available neutrino oscillation data, but also makes interesting predictions for the unknown parameters like the lightest neutrino mass $m_{\nu_1}$ (for normal ordering(NO)), Jarlskog's rephasing invariant $J_{CP}$, Dirac type CP violating phase $\delta_{CP}$ and effective neutrino mass $\left< m_{ee} \right>$. We also explore the correlations between the parameters of the model, that lead us to minimizing the parameters further.

Figures

Figures reproduced from arXiv: 2412.15684 by the authors.

**Figure 2.** Variation of Aν w.r.t. mν1 for the case of neutrinos following NO and having the structure of mass matrices as given in Eq.(3). of this phenomenological scenario. A careful scrutiny of the left panel of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Allowed region for Majorana phases, η1 and η2, for the case of neutrinos following NO and having the structure of mass matrices as given in Eq.(3). mν1 = (4.84 − 5.15) × 10−3 eV J max CP = 0.033 − 0.035 δCP = (257.6 − 259.2)◦ ⟨mee⟩ = (0.0065 − 0.0070) eV η1 = (122 − 125)◦ η2 = (23 − 26)◦ [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: It is clear that a precise determination of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 4.** Figure 4: Allowed region for lightest neutrino mass ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Correlations of δCP with η1, η2 and θ23 for the case of neutrinos following NO and having mass matrix structure as given in Eq.(19). Further, in the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

Discussion (0). Continue with ORCID to comment.

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

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