REVIEW
A Catalog of Lepton Mass Textures from Koide and Alikhanov Relations
High Energy Physics - Phenomenology · Submitted 20 Oct 2026 · 22 references
Pith's verdict: A careful, internally consistent catalog of lepton textures that deserves refereeing, but the headline 402/400 family count rests on an enumeration the reader cannot independently audit from the paper alone.
T0 · AI review
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Referee report
The paper
The paper claims
This paper argues that the Koide and Alikhanov mass relations coexist with a large landscape of lepton mass-matrix textures: 400 representative families reproduce the measured mixing angles, and the full-rank branch fixes the lightest…
The reading
What carries the argument
The load-bearing objects are three. (i) The one-parameter Hermitian family $K(\theta) = I + \rho(\cos\theta\, T + \sin\theta\, S)$ with orthonormal traceless generators $T, S$; for $\rho = \sqrt{3}$ it satisfies the Koide trace condition $\mathrm{tr}(K^2)/(\mathrm{tr}K)^2 = 2/3$ identically, so the charged-lepton spectrum is controlled by a single shape angle. (ii) The $\zeta$-classification, which counts how many mutually orthogonal eigenvector directions remain fixed as $\theta$ varies; only $\zeta = 3, 1, 0$ are possible, realized by three explicit generator pairs called $K_1, K_2, K_3$. (iii) The Alikhanov ratio $q = 17 - 12\sqrt{2} \simeq 0.02943725$, imposed exactly on the neutrino textures, which for rank-two matrices reduces to the fixed mass ratio $m_2/m_3 = 3 - 2\sqrt{2}$ and for full-rank matrices fixes all singular-value ratios of the integer texture. Neutrino masses are read as the singular values of the real symmetric integer matrix $A$ with Takagi phases retained for Majorana observables, and the PMNS matrix is assembled as $U = U_e^\dagger \Phi W_\nu$, where $U_e$ diagonalizes the charged sector, $W_\nu$ is the neutrino Takagi diagonalizer, and $\Phi$ carries the relative alignment phases. The two exact relations do the heavy lifting by removing parameter freedom: spectra are fixed by the relations, and the search only tests relative orientations against the mixing data.
Extended reading notes
Core claim
The central claim is that the Koide and Alikhanov relations can be carried by the structure of the mass matrices themselves, not just by tuned spectra. For charged leptons the paper writes $M_e = m_* K^2$ with $K(\theta) = I + \rho(\cos\theta\, T + \sin\theta\, S)$ and $\rho = \sqrt{3}$, which makes the Koide ratio $\mathrm{tr}(K^2)/(\mathrm{tr}K)^2 = 2/3$ hold identically for every shape angle $\theta$, leaving one parameter to fit the charged masses; the possible families are classified by $\zeta$, the number of eigenvector directions common to the fixed generators $T$ and $S$, which can only be three, one, or none. For neutrinos, imposing the Alikhanov relation in the exact form $q \equiv \Delta m^2_{21}/\Delta m^2_{31} = 17 - 12\sqrt{2}$ on the singular-value spectrum of every real symmetric integer matrix with entries between $-5$ and $5$ leaves six signed-permutation classes with $m_1 = 0$ and six full-rank classes with $m_1 > 0$. Combining the twelve neutrino classes with the three charged-lepton families, over bounded relative orientations and alignment phases, yields 402 distinct prediction families, reducible to 400 representatives, all consistent with the adopted $3\sigma$ intervals for $\sin^2\theta_{12}$, $\sin^2\theta_{23}$, and $\sin^2\theta_{13}$. The sharpest output is that every retained full-rank texture shares a single ordered singular-value spectrum, which forces $m_1$ into $16.77$–$17.22\,\mathrm{meV}$ rather than leaving it free.
Load-bearing premise
The catalog rests on treating the Alikhanov ratio $q = \Delta m^2_{21}/\Delta m^2_{31} = 17 - 12\sqrt{2}$ as an exact selection condition even though it is a very recent empirical coincidence, agreeing with current central values at roughly the $0.3\%$ level and supported only by present oscillation data and the cited Alikhanov preprints; if future precision measurements, for example from JUNO, move the two mass-squared differences off this line, the entire texture catalog, the $m_1$ window, and the $m_{\beta\beta}$ bands cease to apply.
Editorial extensions
If this is right
- If the catalog is right, the Koide and Alikhanov relations are compatible with a large landscape of textures rather than the two explicit examples in Ref. [14]: 400 representative families reproduce the complete set of allowed predictions.
- The full-rank branch predicts the lightest neutrino mass $m_1 \simeq 16.77$–$17.22\,\mathrm{meV}$, a narrow enough window for direct kinematics and cosmological bounds to confirm or exclude.
- The two branches give distinct neutrinoless double-beta targets — $m_{\beta\beta} \le 4.041\,\mathrm{meV}$ for $m_1 = 0$ and $m_{\beta\beta} \le 18.775\,\mathrm{meV}$ for $m_1 > 0$ — while all retained solutions satisfy $\sum m_\nu < 0.11\,\mathrm{eV}$, the KATRIN bound, and the KamLAND-Zen limit.
- A stronger cosmological bound of $\sum m_\nu < 0.072\,\mathrm{eV}$, as obtained in analyses incorporating DESI data, would exclude the full-rank branch, giving cosmology a way to pick between the two spectral branches.
- The paper shows the Alikhanov line is consistent with the 207.2-live-day JUNO result, so continued JUNO exposure provides the direct test of the relation on which the whole catalog rests.
Reading between the lines
- The same $\zeta = 3, 1, 0$ classification of fixed-eigenvector families is a general matrix-theory construction, so a natural extension the paper leaves implicit is to apply it to quark mass matrices or to Koide-like relations for running masses and see whether the three structural categories reappear.
- Because the Alikhanov ratio correlates the two splittings, the catalog's sharp predictions would change if the adopted marginal $3\sigma$ intervals were replaced by a full correlated fit of oscillation parameters; rerunning the enumeration with updated inputs is a cheap, direct update of every claim.
- If JUNO's final precision places the split ratio measurably off $q = 17 - 12\sqrt{2}$, the viable texture space would probably collapse to a much smaller set, which means the size of the catalog itself is a diagnostic of how seriously the relation should be taken.
- The massless branch's $m_{\beta\beta}$ ceiling near $4\,\mathrm{meV}$ sits just below the projected sensitivity of some next-generation double-beta experiments, so a positive signal above that value would select the full-rank branch, while a null result would leave the two branches unresolved.
Original abstract
The empirical Koide relation for charged-lepton masses and the recently proposed Alikhanov relation for neutrino mass-squared differences offer intriguing hints toward understanding the structure of lepton mass matrices. Motivated by these observations, we systematically explore charged-lepton and neutrino mass-matrix textures that satisfy these relations and confront their predictions for lepton mixing with experimental data. For the charged-lepton sector, we investigate representative positive Hermitian square-root matrices classified by the number of common eigenvector directions of their generators: three, one, or none. For the neutrino sector, an exhaustive enumeration of real symmetric integer matrices with entries between $-5$ and $5$ identifies six signed-permutation classes with $m_1=0$ and six with $m_1>0$ that satisfy the Alikhanov relation. Combining these neutrino textures with the charged-lepton constructions, allowing for different relative flavor orientations and phase alignments, yields $400$ viable structural pairs consistent with the measured lepton mixing parameters. All retained solutions satisfy the adopted cosmological bound on the sum of neutrino masses and existing laboratory constraints. In particular, the full-rank neutrino textures predict $m_1\simeq16.77$-$17.22\,\mathrm{meV}$. Representative viable solutions yield effective Majorana masses $m_{\beta\beta}\simeq1.29$-$4.03\,\mathrm{meV}$ and $4.07$-$18.67\,\mathrm{meV}$ for the $m_1=0$ and $m_1>0$ branches, respectively, providing potential targets for future high-sensitivity searches.
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.
The abstract, sentence by sentence
The empirical Koide relation for charged-lepton masses and the recently proposed Alikhanov relation for neutrino mass-squared differences offer intriguing hints toward understanding the structure of lepton mass matrices. Motivated by these observations, we systematically explore charged-lepton and neutrino mass-matrix textures that satisfy these relations and confront their predictions for lepton mixing with experimental data. For the charged-lepton sector, we investigate representative positive Hermitian square-root matrices classified by the number of common eigenvector directions of their generators: three, one, or none. For the neutrino sector, an exhaustive enumeration of real symmetric integer matrices with entries between $-5$ and $5$ identifies six signed-permutation classes with $m_1=0$ and six with $m_1>0$ that satisfy the Alikhanov relation. Combining these neutrino textures with the charged-lepton constructions, allowing for different relative flavor orientations and phase alignments, yields $400$ viable structural pairs consistent with the measured lepton mixing parameters. All retained solutions satisfy the adopted cosmological bound on the sum of neutrino masses and existing laboratory constraints. In particular, the full-rank neutrino textures predict $m_1\simeq16.77$-$17.22\,\mathrm{meV}$. Representative viable solutions yield effective Majorana masses $m_{\beta\beta}\simeq1.29$-$4.03\,\mathrm{meV}$ and $4.07$-$18.67\,\mathrm{meV}$ for the $m_1=0$ and $m_1>0$ branches, respectively, providing potential targets for future high-sensitivity searches.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The empirical Koide relation for charged-lepton masses and the recently proposed Alikhanov relation for neutrino mass-squared differences offer intriguing hints toward understanding the structure of lepton mass matrices. | background 1.00 | 0.05 | 0.03 | hedged possibility 0.89 |
| s1 | Motivated by these observations, we systematically explore charged-lepton and neutrino mass-matrix textures that satisfy these relations and confront their predictions for lepton mixing with experimental data. | contribution 0.96 | 0.83 | 0.04 | asserted 0.95 |
| s2 | For the charged-lepton sector, we investigate representative positive Hermitian square-root matrices classified by the number of common eigenvector directions of their generators: three, one, or none. | method 0.89 | 0.92 | 0.03 | asserted 0.95 |
| s3 | For the neutrino sector, an exhaustive enumeration of real symmetric integer matrices with entries between $-5$ and $5$ identifies six signed-permutation classes with $m_1=0$ and six with $m_1>0$ that satisfy the Alikhanov relation. | method 0.46 | 0.95 | 0.07 | asserted 1.00 |
| s4 | Combining these neutrino textures with the charged-lepton constructions, allowing for different relative flavor orientations and phase alignments, yields $400$ viable structural pairs consistent with the measured lepton mixing parameters. | result 0.94 | 0.93 | 0.40 | asserted 0.99 |
| s5 | All retained solutions satisfy the adopted cosmological bound on the sum of neutrino masses and existing laboratory constraints. | result 0.98 | 0.74 | 0.10 | asserted 1.00 |
| s6 | In particular, the full-rank neutrino textures predict $m_1\simeq16.77$-$17.22\,\mathrm{meV}$. | result 1.00 | 0.90 | 0.16 | asserted 0.99 |
| s7 | Representative viable solutions yield effective Majorana masses $m_{\beta\beta}\simeq1.29$-$4.03\,\mathrm{meV}$ and $4.07$-$18.67\,\mathrm{meV}$ for the $m_1=0$ and $m_1>0$ branches, respectively, providing potential targets for future high-sensitivity searches. | result 0.95 | 0.92 | 0.24 | asserted 0.63 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
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- references and citing sentences
- relations to other papers
- P abstract
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Figures
Figures reproduced from arXiv:2610.28143 by the authors.
References
Discussion
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