Mass Matrix of Majorana Neutrinos

H. Nishiura (Junior College of Osaka Institute of Technology); T. Fukuyama (Ritsumeikan Univ.)

REVIEW

Mass Matrix of Majorana Neutrinos

T. Fukuyama (Ritsumeikan Univ.), H. Nishiura (Junior College of Osaka Institute of Technology)

High Energy Physics - Phenomenology · Submitted 4 Feb 1997 · 12 references

Pith's verdict: A tidy but thin texture exercise: the algebra is right, the uniqueness claim is asserted rather than derived, and the one physical 'prediction' is the input assumption restated.

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arXiv IDhep-ph/9702253 (version 1)

Pith ID4LP37Y76, Pith's permanent record of this paper and its review

CitationH. Nishiura (Junior College of Osaka Institute of Technology), T. Fukuyama (Ritsumeikan Univ.). "Mass Matrix of Majorana Neutrinos." arXiv:hep-ph/9702253 (1997). https://pith.science/paper/4LP37Y76

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Short version: the matrix algebra is fine, the classification is narrow, and the headline physical claim is circular. Worth a look if you work on texture zeros; not worth much otherwise. The paper takes the observed solar and atmospheric deficits, assumes νe–νµ and νµ–ντ oscillations with maximal 2-3 mixing, and asks what light-neutrino mass matrix the seesaw allows if M_light, M_D and M_R share a structure with vanishing (1,1) entries. The algebra from Eq. (8) through Eq. (12) checks out — I verified the entries A, B, C against U diag(-m1, m2, m3) U^T — and Eq. (15) is a correct inversion of the seesaw formula for the assumed form. The genuinely new bit is the four-item list after Eq. (15): two candidates force m3 = 0 and are dropped, the third is the fourth with the sign of the 2-3 angle flipped. That is a legitimate, if small, exercise, and the authors deserve credit for stating plainly, right after Eq. (15), that their assumptions do not reproduce the mass relations of Eq. (1) that motivated the model in the first place. That kind of honesty is rarer than it should be. Now the soft spots. The stress-test note is right that the enumeration is the load-bearing gap: 'same structure' is defined only as 'the same relationships between the components,' the paper concedes the structure need not be identical to Eq. (14), and the four types are then asserted. Exhaustiveness is never shown, and the m3 = 0 rejection is likewise asserted rather than computed. That is the step that turns a texture guess into a uniqueness claim, so it matters more than anything else here. Second, and sharper: Eq. (11), -c1^2 m1 + s1^2 m2 = 0, is imposed by hand to kill the (1,1) entry, and Eq. (18) for the neutrinoless double beta amplitude is literally that same expression. The 'suppression' is the input, not a consequence. The closing comparison to the experimental bound ('at least 10^-3 times' smaller) also glosses over the fact that the model gives exactly zero by construction. Third, Peccei-Quinn symmetry appears in the abstract and in the Lagrangian's symmetry list but never enters a calculation; either use it or drop the billing. None of this is fatal — the paper is internally consistent and the authors flag their own main limitation — but the uniqueness claim needs a real derivation before it means anything. Send it out if you have a referee who enjoys texture classification; otherwise it is a 1997 letter-length note that has largely been absorbed into later work.

Referee report

Significance

The algebraic observation that the texture of Eq. (12) is mapped onto itself by the seesaw inversion, Eq. (15), is a clean and potentially useful structural statement, and the identification of the right-handed Majorana scale with the PQ scale is an economical idea worth recording. If the uniqueness claim of Eq. (14) held, it would be a genuinely predictive statement about the lepton mass matrix. However, the two headline results as written do not survive scrutiny: the double-beta 'prediction' of Eq. (18) is identically zero by the imposed condition Eq. (11), and the uniqueness of Eq. (14) rests on an enumeration that is asserted rather than derived. The paper also does not perform a quantitative fit to the solar and atmospheric data it invokes, so the abstract's claim that the model is 'constrained from' those experiments overstates what is actually done.

Major objections (5)

[Eqs. (11) and (18)] The single physical number reported in the paper is circular. Eq. (11) imposes -c1^2 m1 + s1^2 m2 = 0 by hand, and Eq. (18) then writes the neutrinoless double beta amplitude as <m_nu> = |-c1^2 m1 + s1^2 m2|, which is exactly the left-hand side of Eq. (11). The amplitude is therefore zero by construction, not a consequence of the seesaw, of the texture Eq. (14), or of Peccei-Quinn symmetry. The subsequent sentence, that the estimate is 'lower than the present upper bound by at least 10^-3 times', is either a statement that the amplitude vanishes identically (in which case the numerical comparison is misleading) or an implicit admission that Eq. (11) is only approximately enforced (in which case the size of <m_nu> is set by the tuning and is not predicted). The authors should state explicitly that Eq. (11) is a texture assumption, and if they wish to retain a double-beta discussion they should compute the leading non-zero contribution to <m_nu> generated by departures from Eq. (11) and by the CP phases they have set aside.
[Paragraph following Eq. (15)] The uniqueness claim for Eq. (14) is the paper's central structural result, but the four-type enumeration that produces it is asserted, not derived. The text states that 'running the remaining components of M_D and M_R as free parameters, the seesaw mechanism (7) under the conditions (a) and (b) constrains the allowed mass matrix in the following four types', and then lists them without a derivation. Condition (a), 'same structure', is defined only as 'the same relationships between the components', and the paper itself concedes two sentences earlier that 'in (a) their structure is not necessarily identical to Eq. (14)'. With the defining condition left informal, the reader cannot check that the list of four is exhaustive, and the rejection of types (1) and (2) on the grounds that they 'are enforced to m3 = 0' is likewise not shown. The authors should either give the derivation of the enumeration (for example, by writing the most general symmetric matrix with zero (1,1) entry and imposing the seesaw fixed-point condition explicitly) or downgrade the claim from uniqueness to a consistent example.
[Eq. (1)] Eq. (1) is dimensionally inconsistent: it writes m_nu_e = m_e^2 M_R, whereas the seesaw formula of Eq. (7) gives m_nu ~ m_D^2 / M_R. The displayed relation should read m_e^2 / M_R (and similarly for the other flavours). This is not a cosmetic slip, because the identification M_R ~ v_PQ stated in the sentence after Eq. (1) is never checked numerically: with m_D ~ m_tau and v_PQ ~ 10^10 GeV one obtains m_nu ~ 0.1 eV, which is the right order, but the manuscript does not perform this check and the inverted formula would give a wildly different answer. The authors should correct Eq. (1) and add the one-line numerical consistency check that ties M_R ~ v_PQ to the observed mass scale.
[Eq. (2) and the paragraph following it] The claim that the Lagrangian of Eq. (2) possesses 'global Peccei-Quinn and lepton number symmetries' is not substantiated. No PQ or lepton-number charge assignment for the fields l_L, e_R, nu_R, phi_1, phi_2, phi_3 is given, so the reader cannot verify that the three Yukawa terms are invariant. In particular the third term nu^c_R nu_R phi_3 requires phi_3 to carry lepton number -2, and the spontaneous breaking of that symmetry by <phi_3> produces a Goldstone boson whose relation to the PQ axion (and whose cosmological consequences) is not discussed. Since the identification of the seesaw scale with the PQ scale is one of the paper's stated motivations, the charge assignment and the axion/majoron identification should be made explicit.
[Eqs. (13) and (14)] The choice of the texture Eq. (14) over the diagonal M_D option of Eq. (13) is justified on aesthetic grounds: the text says the matrix in Eq. (13) 'has a rather complicated structure and is unlikely to possess some symmetry', and then adopts the same-structure assumption. 'Unlikely to possess some symmetry' is not a physical criterion, and the diagonal-M_D case is the one that most directly realizes the naive hierarchy of Eq. (1). The authors should either give a symmetry argument (for example, a flavour symmetry under which the textures of Eq. (14) are the most general invariant matrices) or state plainly that Eq. (14) is a model-building assumption chosen for economy, not a consequence of the framework.

Minor objections (7)

[Final section, first sentence] The sentence 'Finally we consider the physical consequences of Mlight in Eq.(11)' refers to Eq. (11), which is the cancellation condition, not the mass matrix; the mass matrix is Eq. (12). The cross-reference should be corrected.
[Paragraph following Eq. (16)] The text writes 's2 = -pi/4 and s2 = pi/4' where it means theta_2 = -pi/4 and theta_2 = pi/4; s2 denotes sin(theta_2), not the angle. The notation should be corrected to avoid confusion with the s_i, c_i shorthand introduced after Eq. (9).
[Paragraph after Eq. (10)] The statement that 'the lightest neutrino mass is considered to be generated only by the flavour mixing' is unclear: m1 is a mass eigenvalue, not a quantity generated by mixing. The intended meaning appears to be that the (1,1) entry of M_light is set to zero by the small-angle solar solution, and the sentence should say so directly.
[Introduction and Eq. (8)] The paper assumes theta_2 ~ pi/4 and theta_3 ~ 0 from the atmospheric and solar deficits but presents no quantitative comparison with the cited data (no chi-squared, no allowed ranges, no discussion of the two solar solutions). Since the abstract frames the paper as constraining the model from those experiments, at least a brief quantitative statement of the ranges used would strengthen the motivation.
[Final paragraph] The estimate m2 ~ O(10^-3 eV) from the solar neutrino experiment is quoted without a reference or a statement of which oscillation solution is assumed. A citation and a one-line justification would be appropriate.
[Fig. 1] Fig. 1 is referenced in the text but the caption is minimal and the figure itself is not reproduced in the manuscript text supplied for review. The caption should identify the particles in the diagram and the origin of the helicity-suppression factor m_j/E mentioned in the caption.
[Throughout] The manuscript contains numerous spacing and typographical errors in the running text (for example 'exp eriments', 's ymmetry', 'di scussed', 'stra tegy', 'trnsformations', 'posess', 'necessarilly', 'constra ints', 'generati on', 'bou d'). A careful copy-edit is needed before publication.

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The paper

The paper claims

Two structural assumptions plus the seesaw formula leave exactly one viable neutrino mass texture, and it predicts a double-beta decay rate far below current bounds.

The reading

The paper asks whether the solar and atmospheric neutrino deficits can be described by a Majorana neutrino mass matrix whose form is fixed by symmetry rather than fitted by hand. The authors build a model in which neutrino masses come from the seesaw mechanism — a heavy right-handed sector dragging the light masses down — with a Peccei-Quinn symmetry setting the heavy scale, the atmospheric deficit carried by near-maximal $2$–$3$ mixing and the solar deficit by small-angle $1$–$2$ mixing. One further condition is imposed, that the lightest mass eigenstate contributes nothing to the $(1,1)$ entry, which reduces the light mass matrix to a three-parameter form with a zero corner and a symmetric $2$–$3$ block. The authors then ask how much freedom is left: if the Dirac and heavy mass matrices must share that same pattern with the same zero corner, only one of four candidate textures survives, because the others either force the heaviest mass to zero or are the same texture with the sign of the mixing angle flipped. The physical payoff they draw is for neutrinoless double beta decay, where the predicted amplitude sits far below the then-current experimental bound.

What carries the argument

The seesaw formula $M_{light} = -M_D M_R^{-1} M_D^{T}$ is the engine: it converts an assumed pattern in the Dirac and heavy Majorana matrices into a pattern in the light matrix that experiments can see. The second piece of machinery is the texture-equality assumption — the same zeros and the same relations among entries in all three matrices — which is what turns the seesaw relation into a small finite classification problem instead of an unconstrained fit. A third piece is the condition $-c_1^2 m_1 + s_1^2 m_2 = 0$, imposed on the grounds that no lower bound on $m_1$ follows from the neutrino anomalies; this is what puts the zero in the $(1,1)$ entry and makes the small-angle solar solution the one being assumed.

Extended reading notes

Core claim

The central claim is that the seesaw relation $M_{light} = -M_D M_R^{-1} M_D^{T}$, combined with two structural assumptions — that $M_{light}$, $M_D$ and $M_R$ share the same texture, and that each has a vanishing $(1,1)$ entry — admits exactly one physically acceptable mass matrix. The authors enumerate four candidate textures and discard them one at a time: cases (1) and (2) drive the heaviest mass $m_3$ to zero, which is rejected because $m_3$ is the heaviest neutrino; cases (3) and (4) differ only by the sign of the $2$–$3$ mixing angle, and since the atmospheric data give $s_2 = -1/√2$ or $+1/√2$ with $s_2$ fixed only through $sin^2 2θ_2 ∼ 1$, they describe the same physics. What remains is $M_{light} = [[0, A, -A], [A, B, C], [-A, C, B]]$, a zero corner, a symmetric $2$–$3$ block, and off-diagonal entries of equal magnitude and opposite sign in the first row. The paper is explicit that this uniqueness does not yet reproduce the naive hierarchy $m_{ν_e} = m_e^2/M_R$, $m_{ν_μ} = m_μ^2/M_R$, $m_{ν_τ} = m_τ^2/M_R$ that motivated the model, since the equations relating $A, B, C$ to the Dirac and heavy parameters cannot fix those parameters on their own.

Load-bearing premise

The load-bearing premise is the hand-imposed condition $-c_1^2 m_1 + s_1^2 m_2 = 0$, justified only by the remark that no lower bound on $m_1$ follows from the neutrino anomalies; this is not a consequence of the seesaw formula or of Peccei-Quinn symmetry, and it is exactly the combination that reappears as the neutrinoless double $\beta$ decay amplitude.

Editorial extensions

If this is right

If the argument holds, the light neutrino mass matrix is not arbitrary: its zero corner and its symmetric $2$–$3$ block are consequences of the seesaw relation plus the requirement that all three mass matrices share a pattern.
The atmospheric deficit is fixed at maximal $ν_μ$–$ν_τ$ mixing and the solar deficit at small-angle $ν_e$–$ν_μ$ mixing, so any solar solution requiring large mixing would fall outside the construction.
The neutrinoless double beta decay amplitude $\langle m_ν \rangle = |-c_1^2 m_1 + s_1^2 m_2|$ is driven to zero by the same condition that produces the zero corner, so the model predicts no observable signal at the several-eV sensitivity then available.
With $m_2 \sim O(10^{-3}\,\mathrm{eV})$, the predicted $\langle m_ν \rangle$ is at least $10^{-3}$ times below the experimental upper bound, so a positive signal would count against the texture rather than for it.

Reading between the lines

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

The quantity that the paper reports as its double-beta prediction is literally the same expression it set to zero by hand a few equations earlier, so the null result is a restatement of the tuning rather than an independent output of the seesaw mechanism.
Because the solar mixing angle has since been measured to be large rather than small, the small-angle premise behind the zero corner is the part of the construction most exposed to oscillation data.
The classification argument is conditional on the texture-equality assumption; relaxing it reopens the four-way list, so the uniqueness claim is best read as a statement about a restricted family rather than about seesaw models in general.
A texture with a vanishing $(1,1)$ entry and a symmetric $2$–$3$ block is the leptonic analogue of the zero-corner quark textures of the 1970s; comparing the two would test whether the same structural principle is at work in both sectors or whether the leptonic case is coincidence.

Original abstract

We present a massive Majorana neutrino model and see how it is constrained from the solar and atmospheric neutrino deficit experiments. This model incorporates the seesaw mechanism and Peccei-Quinn symmetry. Its consequence to the neutrinoless double beta decay is also discussed.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,822 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.20 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a massive Majorana neutrino model and see how it is constrained from the solar and atmospheric neutrino deficit experiments. This model incorporates the seesaw mechanism and Peccei-Quinn symmetry. Its consequence to the neutrinoless double beta decay is also discussed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a massive Majorana neutrino model and see how it is constrained from the solar and atmospheric neutrino deficit experiments.	contribution 1.00	0.85	0.13	asserted 0.94
s1	This model incorporates the seesaw mechanism and Peccei-Quinn symmetry.	method 0.81	0.41	0.04	asserted 1.00
s2	Its consequence to the neutrinoless double beta decay is also discussed.	contribution 0.88	0.61	0.08	asserted 0.70

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from analysis of existing data 0.68.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** ———— The amplitude of this process is proportional to < mν > defined by [12] < mν >≡ |X 3 j=1 U 2 1jmj | (17) Here Uij is in our theory given by Eq.(8) and < mν >= | − c 2 1m1 + s 2 1m2| (18) From the solar neutrino experiment m2 is estimated to be m2 ∼ O(10−3 eV ), whereas the experimental upper bound of < mν > is of the several eV order. Therefore our estimation is lower than the present upper boud by at least 10−3 times. Acknowledgements We are grateful to K. Ueda and S. Yu at Ritsumeikan University for useful comments. 6

Figures reproduced from arXiv:hep-ph/9702253 by the authors.

References

Works cited (12)

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R. Davis Jr., Proc. of the 6’th International Workshop on Neutrino Tele- scope, Venezia, March 1994, edited by M. Baldo Ceolin

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M. Fukugita and T. Yanagida, Physics of Neutrinos in Physics and Astro- physics of Neutrinos ed. by M. Fukugita and A. Suzuki. (Springer-Verlag 1994)

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How the literature uses it

88 arXiv papers cite this in 97 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 58 sentences without a confident label

Contests it 1 paper

In particular, we note that popular flavor symmetry frameworks with $μ- τ$ symmetry, such as those in [ref] (see Ref. [ref] for a recent review and references) usually predict $θ_13=0$ (zero reactor mixing), and also $θ_23, δ_CP$ values at odds with current global neutrino oscillation studies [ref] .

Testing generalized CP symmetries with precision studies at DUNE · Introduction · contest 0.78

Cites it as background 37 papers

Similarly the $μ$-$τ$ interchange symmetry [ref] can be expressed as [equation] The so-called bipair mixing [ref] assigns [equation] as case (1), and [equation] as case (2).

Examination of pairs in neutrino mixing matrix · Introduction · 0.92
The simplest of such flavour symmetries is the symmetry [ref] which exchanges the mu and tau fields.

Fermion Masses and Mixings in a $\mu$-$\tau$ symmetric SO(10) · Introduction · 0.88
The subindex $μ-τ$ in Eq. ([ref]) refers to the so-called $μ-τ$ symmetry [ref] since it satisfies the relations $|U_μi| = |U_τi|$.

Deviations to Tri-Bi-Maximal mixing in the limit of $\mu-\tau$ symmetry · Introduction · 0.91

Discussion

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[1] [1]

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