The number of sufficient and necessary conditions for CP conservation with Majorana neutrinos: three or four?

Bingrong Yu; Shun Zhou

REVIEW 3 major objections 4 minor 14 references

The number of sufficient and necessary conditions for CP conservation with Majorana neutrinos: three or four?

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Three vanishing weak-basis invariants do not always guarantee CP conservation for Majorana neutrinos; the paper shows when a fourth is required.

desk verdict The counterexample to the three-invariant sufficiency claim is real and worth knowing; the paper's own positive three-invariant alternatives are only numerically motivated and need more work before they should be relied on. read the letter →

arxiv 1908.09306 v3 pith:PJGAZCSC submitted 2019-08-25 hep-ph

Bingrong Yu , Shun Zhou This is my paper · ORCID

classification hep-ph PACS 14.60.Pq11.30.Er

keywords CPconservationMajorananeutrinosleptonicviolationweak-basisinvariantsphaseslightestneutrinomassordering

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 asks whether CP conservation for three Majorana neutrinos is guaranteed by three vanishing weak-basis invariants — basis-independent combinations of the lepton mass matrices — or whether four are needed. It answers that the number depends on the lightest neutrino mass $m_1$. For $m_1$ above roughly $0.0265$ eV, the standard set $I_1=I_2=I_3=0$ can all hold while the Majorana phases $\rho$ and $\sigma$ are nonzero, so CP is still violated; the paper exhibits an explicit $m_1=0.03$ eV counterexample. Within the experimentally allowed range $m_1<0.04$ eV, two alternative triplets of invariants are shown to suffice, and a quadruplet always suffices regardless of $m_1$. The result matters because experimental searches for leptonic CP violation need to know how many vanishing conditions actually enforce the symmetry.

What carries the argument

The machinery is the set of weak-basis invariants — traces of commutators and products of $H_l=M_l M_l^\dagger$, $H_\nu=M_\nu M_\nu^\dagger$, and $G_{l\nu}=M_\nu H_l^* M_\nu^\dagger$ — whose vanishing is meant to encode CP conservation. The paper evaluates these in the basis where $M_\nu$ is diagonal, where $I_1=0$ forces the Dirac phase $\delta$ to $0$ or $180^\circ$, and the remaining conditions become coupled nonlinear equations for the Majorana phases $\rho$ and $\sigma$. The determinant identity $\mathrm{Det}(A)=h_{12}^2 h_{13}^2 h_{23}^2 m_1^2 m_2^2 m_3^2 \Delta_{21}^2 \Delta_{31}^2 \Delta_{32}^2$ is what makes the four-invariant proof work: it shows the homogeneous linear system for $\sin(2\rho)$, $\sin(2\sigma)$, and $\sin(2\rho-2\sigma)$ has only the trivial solution. The thresholds come from solving pairs of those nonlinear equations numerically and tracking when the solutions pass from complex to real.

What would settle it

Compute $I_2$ and $\hat{I}_2$ on a fine grid of $(\rho,\sigma)$ for $m_1=0.03$ eV with $\theta_{23}$ at the upper edge of its allowed range; a common zero away from the trivial phases $0$ and $90^\circ$ would falsify the paper's claimed sufficiency of $\{I_1,I_2,\hat{I}_2\}$ in the allowed region.

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Extended reading notes

Core claim

The central discovery is that the three invariants proposed earlier are not sufficient for CP conservation when the lightest neutrino mass is large enough. Working in the basis with diagonal neutrino masses and setting $\delta=0$ through $I_1=0$, the paper reduces the vanishing of $I_2$ and $I_3$ to a pair of nonlinear trigonometric equations in $\rho$ and $\sigma$. At $m_1=0.03$ eV with best-fit mixing angles and mass splittings, these equations have real solutions $(\rho,\sigma)=(38.551^\circ,173.146^\circ)$ and $(141.449^\circ,6.854^\circ)$ at which $I_2=I_3=0$, yet both Majorana phases are nontrivial, so CP is violated. Scanning $m_1$ shows that such real solutions exist only for $m_1 > m_* \approx 0.0265$ eV, fixing the threshold. Replacing $I_3$ by the additional invariants $\hat{I}_2$ and $\hat{I}_3$ shifts the threshold to $m'_* \approx 0.0557$ eV for $\{I_1,I_2,\hat{I}_2\}$ and to $m''_* \approx 0.142$ eV for $\{I_1,\hat{I}_2,\hat{I}_3\}$, above the cosmological mass bound, so these triplets are sufficient inside the allowed region; the four-invariant set $\{I_1,I_2,\hat{I}_2,\hat{I}_3\}$ is sufficient for every $m_1$, via a nonzero determinant of the coefficient matrix.

Load-bearing premise

The decisive assumption is numerical rather than analytic: the paper's scan finds no real CP-violating solutions below the claimed critical masses, but this is checked along a scan in $m_1$ at fixed best-fit mixing parameters, not proven over the full experimentally allowed parameter space.

Editorial extensions

If this is right

If the lightest neutrino mass is measured above about $0.0265$ eV, the older three-condition test $I_1=I_2=I_3=0$ can wrongly certify CP conservation; a fourth invariant must be included.
For $m_1$ below about $0.0265$ eV, the original three conditions remain sufficient, so the failure of the old criterion is confined to heavier, more degenerate neutrino spectra.
The alternative triplets $\{I_1,I_2,\hat{I}_2\}$ and $\{I_1,\hat{I}_2,\hat{I}_3\}$ are sufficient within the current cosmological bound, giving three-condition tests that work until $m_1$ is better constrained.
The quadruplet $\{I_1,I_2,\hat{I}_2,\hat{I}_3\}$ is sufficient and necessary for any $m_1$, so a four-condition test is the robust choice regardless of future mass measurements.
For $m_1=0$, only one Majorana phase remains, so any one of $I_2$, $\hat{I}_2$, or $\hat{I}_3$ vanishing is enough to enforce CP conservation.

Reading between the lines

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

Editorial extension: a shift in the allowed ranges of $\theta_{23}$ or the mass splittings would move the thresholds $m_*$, $m'_*$, and $m''_*$; repeating the same two-equation scan with updated global-fit inputs would show whether the proposed triplets remain safe.
Editorial extension: the counterexample suggests that any three nonlinear invariant equations can in principle admit simultaneous nontrivial zeros, so a general criterion for which triplets are 'safe' up to a given mass would be a natural next step.
Editorial extension: for inverted mass ordering, the analogous thresholds should be computed with the lightest of the two lighter masses playing the role of $m_1$; the same scanning method applies.

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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

Colleague,

The thing to know about this paper is that it settles one question and leaves a second one open. The settled question: the old three weak-basis invariants I1,I2,I3 from Dreiner et al. are not sufficient for CP conservation in the Majorana sector. The explicit counterexample at m1 = 0.03 eV, with rho = 38.551 deg and sigma = 173.146 deg, checks out. I verified the logic: I1 = 0 forces delta to be trivial, and then the two equations I2 = 0 and I3 = 0 genuinely have a common real solution in the physical range of the Majorana phases. The figures showing the zero crossings are convincing, and the authors correctly stress that these are nonlinear equations, so the old "trivial phases only" argument was just wrong. This is a solid, useful result.

The four-invariant sufficiency is also fine; the determinant calculation in Eq. (23) is clean, and it properly restores the Branco-Lavoura-Rebelo conclusion. The paper is honest about citing earlier work here.

The soft spot is the paper's own alternative: two new sets of three invariants that are claimed to be sufficient and necessary within the experimentally allowed region. That claim is not established. The evidence is a numerical scan at fixed best-fit values of the angles and mass splittings, and the paper reports critical masses m'_* ~ 0.056 eV and m''_* ~ 0.142 eV. The stress-test point is fair: the solver returning complex roots for m1 below those values is not a proof that no real roots exist, and the critical masses would move if theta23 or the mass splittings were allowed to vary within their ranges. It is plausible that these sets work across the whole allowed region, but the paper would need an analytic elimination argument or at least a dense scan over the full parameter ranges to make that claim stick. As written, it is a conjecture supported by examples.

A minor blemish: the text converts the Planck sum bound of 0.12 eV into m1 < 0.04 eV for normal ordering. That is a bit high; with the quoted splittings the sum bound gives m1 < about 0.03 eV. The counterexample at 0.03 eV still sits right at the edge, so this does not damage the main point, but the bound should be corrected.

Who should read this: anyone who uses weak-basis invariants in model building or who cites the Dreiner claim without checking the lightest neutrino mass. The counterexample is the takeaway; the new three-invariant sets are a useful suggestion rather than a demonstrated theorem.

Recommendation: this deserves peer review. A referee should press for a parameter-range scan or a monotonicity argument for the new sufficiency claims, and the paper should present the three-invariant results as conditional on that numerical evidence. But the counterexample alone justifies publication.

Best, [You]

Referee Report

3 major / 4 minor

Summary. The paper asks whether three weak-basis (WB) invariants are sufficient and necessary conditions for CP conservation in the leptonic sector with Majorana neutrinos, or whether four are required. The authors first show that the traditional set {I1,I2,I3} from Eqs. (2)-(4) is not sufficient: for the best-fit mixing parameters and m1=0.03 eV, the conditions I1=I2=I3=0 admit the nontrivial Majorana-phase solution ρ≈38.551°, σ≈173.146°, so CP is violated. They then propose two alternative three-invariant sets, {I1,I2,Î2} and {I1,Î2,Î3}, and argue that these are sufficient and necessary for CP conservation within the experimentally allowed parameter region, with critical lightest-neutrino masses m′*≈0.0557 eV and m″*≈0.142 eV respectively. Finally, they show that a four-invariant set, e.g. {I1,I2,Î2,Î3}, is sufficient and necessary independently of m1, using the determinant formula in Eq. (23). The paper concludes that the answer to the title question depends on the allowed mass range and on the choice of invariants.

Significance. If the claims are fully established, the paper makes a useful conceptual point: the number of necessary and sufficient CP-conservation conditions need not equal the number of independent CP-violating phases, and it supplies an explicit counterexample to the previously used three-invariant set. The counterexample at m1=0.03 eV is well supported by the numerical solutions in Eq. (16) and by the intersecting-zero curves in Fig. 1, and the determinant formula in Eq. (23) is compact and potentially reusable. The paper also makes a threshold prediction: the old conditions fail for m1 above about 0.0265 eV (at the adopted inputs), a falsifiable statement with future neutrino-mass measurements. However, the positive statement that the new three-invariant sets are sufficient for all experimentally allowed parameters rests on a numerical root scan at fixed best-fit values; the rigor of that part is below the standard of a proof, so the significance is conditional on completing that analysis.

major comments (3)

[Sec. 3, Eqs. (18)-(21), Figs. 3-4] The central positive claim—that the sets {I1,I2,Î2} and {I1,Î2,Î3} are sufficient for CP conservation for all experimentally allowed parameters—is not established by the evidence presented. The critical masses m′*≈0.0557 eV and m″*≈0.142 eV are obtained by numerically solving Eqs. (11) and (18) or (20) at the fixed best-fit values θ12=33.82°, θ13=8.61°, θ23=48.3°, Δ21=7.39×10^-5 eV^2 and Δ31=2.523×10^-3 eV^2. The statement that for m1 below these values the nontrivial solutions have complex ρ and σ, and are therefore unphysical, does not exclude the possibility of real solutions that the solver missed, and no analytic elimination, resultant, or monotonicity argument is supplied. Since the critical masses depend on all mixing parameters and mass-squared differences, a scan over the full experimentally allowed ranges, or an analytic argument, is needed before one can claim sufficiency for all physical parameters.
[Sec. 2, Eq. (13)] The translation of the cosmological bound m1+m2+m3<0.12 eV into m1<0.04 eV is numerically incorrect for normal neutrino mass ordering. With m1=0.04 eV, the sum m1+m2+m3 is about 0.145 eV, which already exceeds the quoted 0.12 eV bound; the correct upper limit is approximately m1<0.03 eV. This error does not destroy the counterexample at m1=0.03 eV, which remains allowed, but it means that all statements about 'the whole physically allowed space' and the ranges scanned in Figs. 2-4 should be corrected and re-checked with the correct upper bound on m1.
[Sec. 3, Eqs. (22)-(23)] The proof that the four-invariant set {I1,I2,Î2,Î3} is sufficient for all values of m1 relies on Det(A)≠0, but the nonzero determinant is only asserted with 'one can verify' at the adopted best-fit values and is not demonstrated over the full experimentally allowed parameter ranges. Because Det(A) factorizes as h12²h13²h23² times strictly positive factors, the possible failure modes are zeros of h12, h13, or h23; the explicit expressions in Eq. (29) show that this is not an empty concern in principle. A short analytic or numerical demonstration covering the allowed ranges would complete the proof as presented.

minor comments (4)

[Appendix A] There is a typographical error in the sentence preceding Eq. (35): 'indpendent' should read 'independent'.
[Fig. 2] The vertical-axis labels Re(ρ), Re(σ), Im(ρ), and Im(σ) do not indicate whether the angles are measured in degrees or radians; the units should be stated explicitly in the caption.
[References, [7]] Reference [7] is cited as 'to appear soon'; if any part of the argument depends on that work, the reference should be updated or the dependence removed.
[Eq. (28)] The display of the matrix U in Eq. (28) has line breaks that make the entries of the second row difficult to parse; please reformat the matrix for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counterexample and the critical-mass scans are direct computations from the defined invariants and standard PMNS inputs, not fitted or self-referential results.

full rationale

The paper's central counterexample is self-contained. It defines the weak-basis invariants I1, I2, I3 in Eqs. (2)-(4), computes their explicit forms in the standard PMNS parametrization, and then solves Eqs. (11)-(12) for the Majorana phases. The nontrivial solution at m1 = 0.03 eV is an output of those equations, not an input, and no parameter is fitted to the target conclusion. The critical mass m* in Sec. 2 is likewise computed from the same equations, not imposed. Section 3 follows the same pattern: for the new invariant sets {I1, I2, Ihat2} and {I1, Ihat2, Ihat3}, the critical values m'_* and m''_* are obtained by solving the corresponding equations (11), (18), and (20). These critical values are outputs of the calculation, and the sufficiency claims are conditional on them being above the cosmological bound. The numerical scan is restricted to best-fit values and could be criticized as under-powered — complex roots returned by a solver do not rigorously exclude real roots, and parameter uncertainties could shift the critical masses — but that is a correctness/robustness concern, not circularity. No invariant is defined in terms of the conclusion, no fitted parameter is relabeled as a prediction, and no load-bearing claim is justified by a self-citation. The only self-citation is Ref. [7], a forthcoming paper on degenerate neutrino masses, which is mentioned as future work and as a caveat in the introduction; it is not used to establish any of the paper's main results. The four-invariant sufficiency proof is credited to the external Ref. [4], and the paper's determinant check in Eq. (23) is an independent algebraic verification. The derivation chain is therefore self-contained and non-circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted constants. The central claims rest on standard PMNS parametrization, the non-degeneracy of masses, the cosmological bound on neutrino masses, and the completeness of numerical root-finding; the last is the main unproven input.

free parameters (1)

m1 (lightest neutrino mass) = 0.03 eV for counterexample; scanned in [0, 0.04] eV
The critical thresholds m*, m'*, m''* and the existence of CP-violating solutions depend on m1. The paper uses a specific example and parameter scans, so m1 is a scanned physical parameter rather than a fitted constant.

assumptions (4)

domain assumption Charged-lepton and neutrino masses are non-degenerate
Invoked in Sec. 1 following Ref. [2]; degeneracy would change the number of conditions (Refs. [4-6]).
standard math The three phases delta, rho, sigma in the standard parametrization (9) are the only physical CP-violating phases for three Majorana neutrinos
Used throughout to express the WB invariants as trigonometric equations in these phases.
domain assumption The Planck 2018 cosmological bound m1 + m2 + m3 < 0.12 eV restricts m1 < 0.04 eV for normal ordering
Defines the 'experimentally allowed range' within which the new three-invariant sets are claimed to be sufficient; from Ref. [11].
ad hoc to paper Numerical root-finding of Eqs. (11) and (12) correctly identifies all real solutions for rho and sigma
The existence of nontrivial solutions above the critical masses and their absence below are inferred from numerical solving and plotting, not from an analytic enumeration.

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Cite this review

Pith. "Pith review of The number of sufficient and necessary conditions for CP conservation with Majorana neutrinos: three or four?." pith.science (2026). https://pith.science/paper/PJGAZCSC

@misc{pith2026190809306,
  author       = {Pith},
  title        = {Pith review of: The number of sufficient and necessary conditions for CP conservation with Majorana neutrinos: three or four?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJGAZCSC}},
  note         = {Machine review of arXiv:1908.09306}
}

read the original abstract

As is well-known, there exist totally three CP-violating phases in the leptonic sector if three ordinary neutrinos are massive Majorana particles. In this short note, we raise the question whether the number of sufficient and necessary conditions for CP conservation in the leptonic sector with massive Majorana neutrinos is three or four. An intuitive answer to this question would be three, which is also the total number of independent CP-violating phases. However, we give a counter example, in which three conditions are in general not sufficient for CP conservation. Only for all the lepton masses and mixing angles within their experimentally allowed ranges can we demonstrate that it is possible to find out three weak-basis invariants, which should be vanishing to guarantee leptonic CP conservation.

Figures

Figures reproduced from arXiv: 1908.09306 by the authors.

**Figure 2.** Illustration for the real parts (the upper panel) and the imaginary parts (the lower [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Illustration for the real parts (the upper panel) and the imaginary parts (the lower [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Illustration for the real parts (the upper panel) and the imaginary parts (the lower [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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

[1] [1]

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Z. z. Xing and S. Zhou, “Neutrinos in particle physics, astronomy and cosmology,” Springer- Verlag, Berlin Heidelberg (2011)

work page 2011

[2] [2]

Supersymmetric Jarlskog invari- ants: The Neutrino sector,

H. K. Dreiner, J. S. Kim, O. Lebedev and M. Thormeier, “Supersymmetric Jarlskog invari- ants: The Neutrino sector,” Phys. Rev. D 76, 015006 (2007) [hep-ph/0703074]

arXiv 2007

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M. Tanabashi et al. [ParticleDataGroup], “Review of Particle Physics,” Phys. Rev. D 98, no. 3, 030001 (2018)

work page 2018

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G. C. Branco, L. Lavoura and M. N. Rebelo, “Majorana Neutrinos and CP Violation in the Leptonic Sector,” Phys. Lett. B 180, 264 (1986)

work page 1986

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Degenerate and Quasi Degenerate Majorana Neutrinos

G. C. Branco, M. N. Rebelo and J. I. Silva-Marcos, “Degenerate and quasidegenerate Majo- rana neutrinos,” Phys. Rev. Lett. 82, 683 (1999) [hep-ph/9810328]

work page Pith review arXiv 1999

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J. w. Mei and Z. z. Xing, “Impact of fermion mass degeneracy on ﬂavor mixing,” J. Phys. G 30, 1243 (2004) [hep-ph/0312382]

work page Pith review arXiv 2004

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