Pith. sign in

REVIEW 4 major objections 4 minor 32 references

Testing Texture Two Zero Neutrino Mass Matrices Under Current Experimental Scenario

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Two neutrino mass matrices survive Planck's 0.17 eV limit

desk verdict A plausible update of texture-two-zero neutrino mass matrices under the 2018 Planck bound, but the printed analytic formulas do not reproduce the quoted Σ minima and need correction before the exclusion claim is usable. read the letter →

arxiv 1909.01552 v2 pith:V5TUP7NG submitted 2019-09-04 hep-ph

Madan Singh This is my paper · ORCID
classification hep-ph PACS 14.60.Pq
keywords neutrinomassmatricestexturetwozeroMajorananeutrinosorderingatmosphericmixingangleoctantDiracCPphaseneutrinolessdoublebetadecaysumofmasses
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 re-tests the five remaining “texture two zero” patterns of the neutrino Majorana mass matrix against the current experimental map: Planck's bound $\Sigma<0.17$ eV on the sum of neutrino masses, the updated $2\sigma$ range of the atmospheric angle $\theta_{23}$, and the KamLAND-Zen limit on neutrinoless double $\beta$ decay. It claims that only two patterns, B2 and B4, survive, and only for normal mass ordering; B1, B3, and C are excluded at $2\sigma$ for both orderings. The survivors force a quasi-degenerate spectrum, $\theta_{23}>45^\circ$, a Dirac CP phase near $270^\circ$, and an effective Majorana mass $|M_{ee}|$ close to the KamLAND-Zen sensitivity. This matters because texture two zero matrices are a minimal, symmetry-motivated ansatz for neutrino flavor, so a shortlist of two concrete patterns can be decisively tested by upcoming long-baseline, cosmological, and double-$\beta$ experiments.

What carries the argument

The central object is the texture two zero Majorana mass matrix in the flavor basis: a $3\times3$ symmetric matrix with two independent zero entries, classified into cases B1–B4 and C. The load-bearing identity is the reconstruction of the mass ratios $\alpha=m_1/m_3$, $\beta=m_2/m_3$ from the two vanishing entries together with the mixing angles, and the formula $\Sigma=\sqrt{\delta m^2/(\beta^2-\alpha^2)}\,(\alpha+\beta+1)$ that turns the texture conditions into a definite prediction of $\Sigma$ as a function of $\theta_{23}$. The argument then compares those $\Sigma(\theta_{23})$ curves with the Planck bound and the $2\sigma$ $\theta_{23}$ window, using the leading-order-in-$s_{13}$ approximations to locate the minima analytically.

What would settle it

Run the same $2\sigma$ scan using the exact texture-two-zero reconstruction of the mass matrix instead of the approximate formulas (14), (17), (19), and (20), and read off the minimum $\Sigma$ for B1, B3, and C. If any of those minima falls below $0.17$ eV, the paper's central exclusion fails; if all three stay above $0.17$ eV, the shortlist of B2 and B4 survives.

Watch

Extended reading notes

Core claim

Starting from the five two-zero forms of the flavor-basis Majorana mass matrix allowed at $3\sigma$, the paper uses the texture conditions to write the neutrino masses as $m_3=\sqrt{\delta m^2/(\beta^2-\alpha^2)}$, $m_2=m_3\beta$, $m_1=m_3\alpha$, with $\alpha,\beta$ fixed by the mixing angles, and hence expresses $\Sigma=m_1+m_2+m_3$ as a function of $\theta_{23}$. Over the $2\sigma$ inputs, the leading-order-in-$s_{13}$ formulas give $\Sigma_{\min}\simeq 0.24$ eV for B1 and B3 (both orderings), which exceeds the Planck limit; for B2 and B4 they give $\Sigma_{\min}\simeq 0.154$ eV for normal ordering, inside the limit; for case C the inverted-ordering formula gives $\Sigma_{\min}\simeq 0.174$ eV, marginally above $0.17$ eV. The conclusion is that B2 and B4 are the only viable texture two zero cases, both requiring normal ordering, upper octant $\theta_{23}$, $\delta$ in a narrow band around $270^\circ$, and $|M_{ee}|$ in the range $0.0389$–$0.0422$ eV or above, near the KamLAND-Zen bound. The paper also shows that Majorana phases $\rho$ and $\sigma$ vanish as $\delta$ approaches $270^\circ$, and that distinguishing B2 from B4 at low energies will require sub-degree precision on $\delta$.

Load-bearing premise

The exclusion of cases B1, B3, and C relies on leading-order-in-$s_{13}$ approximations for $\Sigma$ as a function of $\theta_{23}$; these formulas diverge as $\theta_{23}$ approaches $45^\circ$, so if the neglected higher-order terms are not negligible at the quoted $2\sigma$ minima, the minimum $\Sigma$ values — and hence which cases are ruled out — could change.

Editorial extensions

If this is right

  • Only the B2 and B4 textures remain viable at $2\sigma$, and only for normal mass ordering; both force $\theta_{23}>45^\circ$, $\delta\simeq 270^\circ$, and a quasi-degenerate spectrum.
  • B1, B3, and C are excluded for both orderings under $\Sigma<0.17$ eV; the older $0.23$ eV limit would have kept them alive.
  • For B2 and B4, $|M_{ee}|$ is bounded below by $0.0389$–$0.0422$ eV, placing the textures within reach of next-generation neutrinoless double beta decay searches and near the KamLAND-Zen $90\%$ limit.
  • The lightest neutrino mass is constrained to $m_1\lesssim 0.06$ eV for the viable cases, sharpening the quasi-degenerate prediction.
  • Measuring the Dirac phase $\delta$ with accuracy better than about $1^\circ$ near $270^\circ$ would discriminate between B2 and B4.

Reading between the lines

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

  • If the surviving textures are correct, the neutrino sector has no room for maximal $\theta_{23}$: the singular behavior of the mass formulas at $45^\circ$ means any future hint of maximal atmospheric mixing would kill all five cases at once.
  • The B2/B4 shortlist creates a sharp target for cosmology: the same models predict $\Sigma$ above $0.141$–$0.151$ eV, so a future cosmological limit pushing below about $0.14$ eV would eliminate them without any oscillation data.
  • Because the exclusion of case C rests on $\Sigma_{\min}\simeq 0.174$ eV versus $0.17$ eV, a small correction by higher-order terms in $s_{13}$ or a slightly looser cosmological bound could reverse that judgment; the $2\sigma$ margin is not large.
  • The vanishing of Majorana phases near $\delta\simeq 270^\circ$ suggests that, if these textures are realized, the dominant CP violation in the lepton sector is Dirac-like, with negligible extra phases in neutrinoless double beta decay.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper re-examines five viable two-zero texture Majorana neutrino mass matrices (B1-B4 and C) using updated oscillation parameters, the Planck bound on the sum of neutrino masses (Sigma < 0.17 eV at 95% CL), and the KamLAND-Zen bound on neutrinoless double beta decay. The author claims that only B2 and B4 remain viable for normal mass ordering at 2 sigma CL, while B1, B3, and C are excluded for both orderings. The viable cases are reported to prefer the upper octant of theta23, a Dirac phase delta near 270 degrees, a quasi-degenerate spectrum, and |Mee| close to the KamLAND-Zen limit. The paper combines approximate analytical formulas for Sigma(theta23) with correlation plots to support these conclusions.

Significance. If the conclusions were correct, the paper would provide a sharp experimental discrimination among texture-two-zero neutrino mass matrices, which is useful for model building. The author also connects the texture analysis to cosmology and neutrinoless double beta decay, which are timely topics. However, the central quantitative support is not reproducible: the key formulas use the solar mass-squared difference where the atmospheric scale is required, the quoted minima are not obtainable from the stated equations, and the numerical scan is not documented. These issues affect the main viability claims, so the significance cannot be assessed without a corrected and reproducible analysis.

major comments (4)
  1. [Section 3, Eqs. (9), (13), (16), (14), (17)] The notation delta m^2 is defined in Eq. (9) as the solar mass-squared difference, but the leading-order relations m1 ~ m2 in Eqs. (12) and (15) imply that the solar splitting is a next-order effect. The mass scale that fixes m3 at leading order is the atmospheric splitting |Delta m31^2|, not delta m^2. Consequently, evaluating Eq. (17) with delta m^2 = 7.55e-5 eV^2 and theta23 = 49.8 degrees gives Sigma of order 0.03 eV for B2 NO, not the quoted Sigma_min ~ 0.154 eV. The text therefore does not demonstrate the claimed exclusions and survivals from the equations as written. In addition, Eq. (13) is not real-valued for theta23 > 45 degrees, which is the region the text associates with inverted ordering, so a separate treatment for IO is needed.
  2. [Section 3, Eqs. (14), (17) and Table 1] Even if delta m^2 were replaced by |Delta m31^2|, the quoted minima do not match a direct evaluation over the stated 2 sigma ranges. For B2 NO, the 2 sigma upper end is theta23 = 49.8 degrees; using |Delta m31^2| = 2.50e-3 eV^2 in Eq. (17) gives Sigma ~ 0.17 eV, not 0.154 eV. Since the expression diverges as theta23 approaches 45 degrees and the allowed range extends close to the singularity, the leading-order approximation is not controlled, and the paper provides no estimate of the neglected next-order s13 corrections. The claimed survival of B2 and B4 under the Planck bound is therefore not established.
  3. [Section 3, computational scan] The author states that input parameters are scanned within their 3 sigma ranges, but the scanning procedure is not specified: there is no description of the number of points, the parameter sampling, the acceptance criteria, or the treatment of the texture-zero conditions. No code or data are provided. Since the analytic formulas above are used as the explicit justification for the exclusions, and the numerical plots cannot be independently reproduced, the central claims in Table 2 are not verifiable from the manuscript.
  4. [Section 3, Eq. (18)] The relation |Mee| ~ (Sigma/3) tan^2 theta23 does not follow from Eq. (11) combined with the leading-order mass relations for B2 and B4 in Eq. (15). Using m1 = m2 = m3 cot^2 theta23 gives |Mee|/Sigma ~ cot^2 theta23/(1 + 2 cot^2 theta23) in the leading order, which differs from the quoted expression by a factor of order tan^2 theta23. The derived bound |Mee| < 0.0684 eV is therefore not supported by the stated equations.
minor comments (4)
  1. [Throughout] There are numerous typographical errors, including 'superkomiokande' for Super-Kamiokande, '2700' and '450' for 270 degrees and 45 degrees, and stray brackets such as 'Table 1]' and 'Fig.[1, 2' in Section 3.
  2. [Figure captions, Figs. 1 and 2] The captions state that the solid line indicates Planck's limit Sigma = 0.23 eV, but the abstract and Section 1 state the current limit is Sigma < 0.17 eV and that 0.23 eV is the older comparison value; the captions should be corrected for clarity.
  3. [Section 3, Eq. (20)] The formula for case C in the inverted ordering case contains an undefined notation c_delta and t2(23), and the expression is not derived or explained; the reader cannot verify the quoted Sigma_min ~ 0.174 eV from the stated equation.
  4. [References] Reference [15] is the 2016 Planck paper, but the abstract describes it as the latest Planck data; the authors should update the reference to the relevant 2018 Planck release or clarify the date.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the texture-zero conditions are imposed independently of the Planck, KamLAND-Zen, and oscillation constraints, and the surviving-case conclusions are not fitted to those constraints.

full rationale

The paper's central claim—that only cases B2 and B4 remain viable for normal ordering at 2σ—is obtained by imposing the texture-two-zero conditions on the Majorana mass matrix and then comparing the resulting sums of neutrino masses and effective Majorana masses with external limits from Planck and KamLAND-Zen. The inputs are the standard neutrino oscillation parameters and mass-squared differences taken from global fits; no fitted quantity is renamed as a prediction. The approximate analytic expressions in Eqs. (12)–(17) are quoted from prior external work (Frampton/Xing, Meloni et al.) and merely reproduce the same texture-zero relations in leading-order form; they are not the target observables themselves. The only self-citation, Ref. [27], is an earlier phenomenological survey and is not load-bearing for the exclusions. The apparent 'predictions' of δ ≈ 270° and θ23 > 45° arise from intersecting the texture-zero parameter space with the scanned oscillation parameters, not from fitting those quantities to the bounds being 'predicted.' The KamLAND-Zen bound is used as an external comparison, not as an input that forces the |Mee| values. Any numerical or domain concern about the leading-order formulas (e.g., behavior near θ23 = 45° or the use of δm² versus Δm²) is a correctness or robustness issue, not a circularity issue under the specified criteria. The paper's own statement that 'no specific conclusion can be made' at 3σ is an honest limitation, not evidence of circular reasoning.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. It relies on the assumed texture-zero patterns (from prior literature), the Majorana nature of neutrinos, and approximate analytic formulas that are used to derive exclusion bounds. These approximations are the most fragile part of the analysis.

assumptions (4)
  • domain assumption The texture two zero mass matrix patterns are viable candidates for the neutrino mass matrix in the flavor basis
    The paper restricts to the five cases B1-B4 and C, motivated by prior work [23-27]; this defines the model space under test.
  • domain assumption Neutrinos are Majorana particles
    The mass matrix is parametrized as Mν = U P diag(m_i) P^T U^T with complex eigenvalues, enabling texture zeros to be imposed on a symmetric Majorana matrix.
  • ad hoc to paper The leading-order approximations in s13 (Eqs. 12-20) are accurate for the θ23 ranges used
    These approximations are used to compute Σ_min and to rule out cases; their validity is not established quantitatively, and they diverge at θ23 = 45°.
  • domain assumption The oscillation parameter ranges in Table 1 and the Planck bound Σ < 0.17 eV are the correct external inputs
    The analysis adopts ranges from updated global fits [9,10] and the Planck 2018 limit; the exclusion conclusions depend on these choices.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Testing Texture Two Zero Neutrino Mass Matrices Under Current Experimental Scenario." pith.science (2026). https://pith.science/paper/V5TUP7NG

@misc{pith2026190901552,
  author       = {Pith},
  title        = {Pith review of: Testing Texture Two Zero Neutrino Mass Matrices Under Current Experimental Scenario},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V5TUP7NG}},
  note         = {Machine review of arXiv:1909.01552}
}
abstract

The latest data from Planck collaboration has presented an improved sensitivity limit of the sum of neutrino masses ($\Sigma$), $\Sigma <0.17eV$ at 95$\%$ confidence level (CL). On the other hand, the updated global fits of neutrino oscillation have shown a refined range of atmospheric mixing angle $\theta_{23}$ at the same CL. In the light of these observations, we have re-investigated the five viable cases ( $B_{1, 2, 3, 4}$ and $C$) of texture two zero Majorana mass matrix in the flavor basis. Using the present Planck's data, we have demonstrated that only cases $B_{2}$ and $B_{4}$ are now viable for normal mass ordering, while remaining three cases does not meet the present experimental constraints at 2$\sigma$ CL. The viable cases are also found to be in agreement with the latest T2K, Super-Kamiokande and NO$\nu$A results, indicating the preference for normal neutrino mass ordering ($m_{1} < m_{2}<m_{3}$), a maximal Dirac CP-violating phase ($\delta \simeq 270^{0}$), and upper octant of neutrino mixing angle ($\theta_{23} > 45^{0}$). In addition, the implication of $\Sigma$ on neutrinoless double beta decay is studied for viable cases.

Figures

Figures reproduced from arXiv: 1909.01552 by the authors.

Figure 1
Figure 1. Correlation plots between Sum of neutrino masses, Σ (eV) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Correlation plots between Sum of neutrino masses, Σ (eV) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. (a) Correlation plots between Sum of neutrino masses, Σ ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Correlation plots between Majorana phases, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Correlation plots between Sum of neutrino masses, Σ (eV) [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 11 canonical work pages

  1. [1]

    Abe et al

    K. Abe et al. [T2K collaboration], Phys. Rev. Lett. 107, 041801 (2 011), arXiv: 1106.2822 [hep-ex]

    arXiv

  2. [2]

    Adamson et al

    P. Adamson et al. [MINOS collaboration], Phys. Rev. Lett. 107, 18 1802 (2011), arXiv: 1108.0015 [hep-ex]

    arXiv 2011

  3. [3]

    Abe et al., [Double Chooz collaboration], Phys

    Y. Abe et al., [Double Chooz collaboration], Phys. Rev. Lett. 108, 1 31801 (2012), arXiv: 1112.6353 [hep-ex]

    arXiv 2012

  4. [4]

    F. P. An et al., [Daya Bay collaboration], Phys. Rev. Lett. 108, 171 803 (2012), arXiv: 1203.1669 [hep-ex]. 11

    arXiv 2012

  5. [5]

    Soo-Bong Kim, for RENO collaboration, Phys. Rev. Lett. 108, 19 1802 (2012), arXiv: 1204.0626[hep-ex]

    arXiv 2012

  6. [6]

    Bellini et al., Phys

    G. Bellini et al., Phys. Rev. Lett. 107 (2011) 141302, arXiv:1104.1 816 [hep-ex]

    work page 2011

  7. [7]

    Adamson et al

    P. Adamson et al. (NOvA), Phys. Rev. Lett. 118, 231801 (2017 ), arXiv:1703.03328 [hep-ex]

    arXiv 2017

  8. [8]

    P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola, J. W. F. V alle, Phys. Lett. B, 782, 633-640 (2018), arXiv:1708.01186 [hep-ph]

    arXiv 2018

Show all 32 references
  1. [9]

    Capozzi, E

    F. Capozzi, E. Lisi, A. Marrone, A. Palazzo, arXiv: 1804.09678 [he p-ph]

    arXiv

  2. [10]

    Esteban, M

    I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo , et al., JHEP 01, 106 (2019), arXiv: 1811.05487 [hep-ph]

    2019 arXiv

  3. [11]

    Radovic, ”Latest Neutrino Oscillation Results from NOvA,” sem inar at Fer- milab (12 January 2018), available at https://www-nova.fnal.gov

    A. Radovic, ”Latest Neutrino Oscillation Results from NOvA,” sem inar at Fer- milab (12 January 2018), available at https://www-nova.fnal.gov. Se e also the subsequent NOvA talks, up to March 2018

    2018

  4. [12]

    ”T2K and NOvA collaborations to produce joint neutrino oscillatio n analy- sis by 2021.” Full announcement (30 January 2018) available at http ://t2k- experiment.org/2018/01/ t2k-nova-announce/

    2021

  5. [13]

    Abe et al

    K. Abe et al. [T2K Collaboration], Phys. Rev. D 96 (2017) no.9, 092 006, arXiv: 1707.01048 [hep-ex]

    2017 arXiv

  6. [14]

    Super-Kamiokande Collaboration: χ 2 maps derived from full 3 σ oscillations analysis, http://www-sk.icrr.u-tokyo.ac.jp/sk/publications/result-e.html

  7. [15]

    P. A. R. Ade et al., [Planck Collaboration], Astronomy and Astroph ysics 594, A13 (2016), arXiv: 1502.01589 [astro-ph.CO]

    2016 arXiv

  8. [16]

    KamLAND-Zen Collaboration, Phys. Rev. Lett., 117, 082503 (2 016), arXiv: 1605.02889

    arXiv

  9. [17]

    J. B. Albert et. al [Exo-200 Collaboration], Nature 510, 229-23 4 (2014), arXiv: 1402.6956 [nucl-ex]

    2014 arXiv

  10. [18]

    Dell’oro, S

    S. Dell’oro, S. Marcocci, M. Viel and F. Vissani, Adv. High Energy P hys., 2162659 (2016), arXiv: 1601.07512

    2016 arXiv

  11. [19]

    Rodejohann, Int

    W. Rodejohann, Int. J. Mod. Phys. E, 20, 1833 (2011), arXiv : 1106.1334 [hep- ph]

    2011 arXiv

  12. [20]

    F. T. Avignone III, S. R. Elliott, J. Engel, Rev. Mod. Phys. 80, 4 81 (2008), arXiv: 0708.1033 [nucl-ex]; J. J. Gomez-Cadenas, J. Martin-Albo, M . Mezzetto, F. Monrabal, M. Sorel, Riv. Nuovo Cim. 35 (2012) 29-98, arXiv: 1109 .5515 [hep-ex]. 12

    2008 arXiv

  13. [21]

    Gupta, G

    M. Gupta, G. Ahuja, Int. J. Mod. Phys. A 26, 2973 (2011) and references therein

    2011

  14. [22]

    Rohit Verma, Advances in High Energy Physics, Volume 2016, Art icle ID 2094323, arXiv: 1607.00958 [hep-ph]

    2016 arXiv

  15. [23]

    Frampton, Sheldon L

    Paul H. Frampton, Sheldon L. Glashow and Danny Marfatia, Phy s. Lett. B 536, 79 (2002), hep-ph/0201008; Zhi-zhong Xing, Phys. Lett. B 530, 159 (2002), hep-ph/0201151

    2002 arXiv

  16. [24]

    Fritzsch, Z

    H. Fritzsch, Z. Z. Xing, S. Zhou, JHEP 1109, 083 (2011), arXiv : 1108.4534v2 [hep-ph]

    2011 arXiv

  17. [25]

    Meloni, A

    D. Meloni, A. Meroni, E. Peinado, Phys. Rev. D 89 (2014) 053009 , arXiv: 1401.3207 [hep-ph]

    2014 arXiv

  18. [26]

    Shun Zhou, Chin. Phys. C 40 (2016) no.3, 033102, arXiv: 1509.0 5300 [hep-ph]

    2016

  19. [27]

    Madan Singh, Gulsheen Ahuja, Manmohan Gupta, Prog. Theor. Exp. Phys. (PTEP) 2016 (12): 123B 08, arXiv: 1603.08083 [hep-ph]

    2016 arXiv

  20. [28]

    P. H. Frampton, M. C. Oh and T. Yoshikawa, Phys. Rev. D 66, 03 3007 (2002)

    2002

  21. [29]

    Walter Grimus and Luis Lavoura, J. Phys. G 31, 693-702 (2005)

    2005

  22. [30]

    Grimus, A.S

    W. Grimus, A.S. Joshipura, L. Lavoura, and M. Tanimoto, Eur. P hys. J. C 36, 227 (2004)

    2004

  23. [31]

    H. C. Goh, R. N. Mohapatra and Siew-Phang Ng, Phys. Rev. D 68 , 115008 (2003)

    2003

  24. [32]

    P. A. R. Ade et al. [Planck Collaboration], arXiv: 1303.5076 [astro- ph.CO]. 13

    arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.