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REVIEW 5 major objections 3 minor 55 references

Flavor Moonshine

T0 review · 5 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Particle mass ratios are Fourier coefficients of modular forms, and the charged-lepton sector verifies the hypothesis with no free parameters.

desk verdict A genuinely curious numerical observation about Hilbert modular forms and lepton masses, undercut by an unjustified finite truncation and fits dressed as predictions. read the letter →

arxiv 1908.11032 v1 pith:OMJBN43W submitted 2019-08-29 hep-th hep-ph

Shotaro Shiba Funai , Hirotaka Sugawara This is my paper · ORCID
classification hep-thhep-ph
keywords flavormoonshinemodularformsFouriercoefficientsmassratiosSL(2Z(√2))YukawacouplingsCalabi-Yaumoduliquarkandleptonmasses
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 tries to establish a 'flavor moonshine' hypothesis: the mass ratios of elementary particles—charged leptons, up- and down-type quarks, and neutrinos—are literally the Fourier coefficients of certain two-variable modular forms associated with the arithmetic group $\mathrm{SL}(2,\mathbb{Z}(\sqrt{2}))$, in the same way that representation dimensions of a group appear as modular-form coefficients in classical moonshine. If true, the unexplained hierarchy of fermion masses becomes a property of the modular group's Fourier expansion, and the same coefficients define the Yukawa couplings to the Higgs. The authors do not derive the hypothesis from first principles; they demonstrate numerical agreement, most directly in the charged-lepton sector, where an unparameterized form yields a muon mass of 107.5 MeV against 105.7 MeV measured and a tau mass of 1558 MeV against 1776 MeV. They then use the identification of the modular variables with Calabi-Yau moduli to extract the Kähler potential and metric of the moduli space directly from experimental data.

What carries the argument

The central object is the two-variable modular form for the group $\mathrm{SL}(2,\mathbb{Z}(\sqrt{2}))$, the simplest arithmetic extension of the usual modular group. For this group a known result states that all modular forms are generated by three forms $G_2$, $G_4$, $G_6$ of levels $k=1,2,3$; their Fourier coefficient matrices, truncated at generation number $G=3$, are promoted to Yukawa mass matrices. Modular invariance of the Yukawa coupling fixes how the fields transform and requires an infinite number of generations for exact invariance, while the low-energy three-generation truncation is treated as the physical vacuum. The same modular variables are then identified with periods of the Calabi-Yau manifold, so a standard relation between the third derivative of the prepotential and the Yukawa coupling ties the Fourier coefficients directly to the moduli-space geometry.

What would settle it

Measure the tau-to-muon mass ratio precisely: the $k=1$ modular form predicts $m_\tau/m_\mu \simeq 1558/107.5 \simeq 14.5$, while the current central value is $1776/105.7 \simeq 16.8$; an improved measurement that keeps the ratio near $16.8$ with small uncertainty falsifies the no-free-parameter charged-lepton claim.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that each fermion flavor sector is described by a modular form of a specific level: $k=1$ (charged leptons) with the form $G_2$, $k=2$ (charge $+\frac{2}{3}$ quarks) with $G_4+a_4 G_2^2$, $k=3$ (charge $-\frac{1}{3}$ quarks) with $G_6+a_6 G_2^3+b_6 G_2 G_4$, and $k=4$ (neutrinos) with the corresponding combinations. Truncating the Fourier coefficient matrix to three generations and diagonalizing the squared mass matrix converts the coefficients into masses. In the $k=1$ case there is no free parameter and the predicted muon and tau masses are within about 12 percent of experiment; with the three complex parameters available in the quark sector, the CKM matrix and the mass ratios $m_t/m_c$ and $m_b/m_s$ are fitted well, while the light quark masses $u,d,s$ come out too small. A similar fit to the PMNS matrix favors Majorana neutrinos in normal ordering, though the predicted $\theta_{13}$ is too large. The paper also claims that these modular variables are the complex-structure moduli of a Calabi-Yau manifold, so the experimental mass data determine the prepotential, Kähler potential, and ultimately the Calabi-Yau metric through established formulas.

Load-bearing premise

The calculation stands on the choice of a particular number-theoretic symmetry group ($\mathrm{SL}(2,\mathbb{Z}(\sqrt{2}))$) and on assigning levels $k=1,2,3,4$ to charged leptons, up-type quarks, down-type quarks, and neutrinos; the paper does not derive that choice from a deeper principle.

Editorial extensions

If this is right

  • Charged-lepton masses are determined by the $k=1$ modular form with no free parameters: normalizing to the electron gives a muon of 107.5 MeV and a tau of 1558 MeV, against measured central values 105.7 and 1776 MeV.
  • With three complex parameters for the quark sector and three for neutrinos, the same construction reproduces the CKM matrix well and favors Majorana neutrinos with normal mass ordering, though the predicted $u,d,s$ masses come out too small and the predicted $\theta_{13}$ too large.
  • Exact modular invariance of the Yukawa coupling requires an infinite number of generations; the observed $G=3$ is interpreted as a low-energy vacuum, with possible phase transitions at higher energy.
  • Identifying the modular variables with Calabi-Yau moduli turns experimental masses into the prepotential, Kähler potential, and moduli-space metric, giving a data-driven route to the Calabi-Yau metric.
  • Levels $k\ge 5$ produce neutral, uncolored particles interacting only weakly and gravitationally, which the paper proposes as dark-matter candidates.

Reading between the lines

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

  • We infer that if the correspondence is real, the mass hierarchy ceases to be a dynamical accident: the ratios are fixed by the arithmetic of the modular group, so any future precision mass measurement is also a test of the ansatz.
  • A testable extension is to read the same Fourier-coefficient matrices at larger generation number $G>3$; the appearance or absence of a predicted tower of heavier states would show whether the three-generation truncation is a vacuum choice or an approximation.
  • We note that the unexplained level assignment $k=1,2,3,4$ could be probed by repeating the fit with other arithmetic groups, such as $\mathrm{SL}(2,\mathbb{Z}(\sqrt{N}))$ or $\mathrm{SL}(2,\mathbb{Z}(i))$; comparable fits would weaken the claim that this particular group is special.
  • A further consequence we draw is that solving the equation linking the moduli-space metric to the Calabi-Yau metric, left for future work, would turn the measured masses into a concrete geometric prediction for string compactification.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 3 minor

Summary. The paper proposes a 'flavor moonshine' hypothesis: mass ratios of leptons, quarks, Higgs and gauge particles are expressed as Fourier coefficients of two-variable modular forms associated with SL(2,Z(√2)). For k=1,2,3,4, the modular forms G2, G4+a4G2^2, G6+a6G2^3+b6G2G4, and combinations thereof are assigned respectively to charged leptons, up-type quarks, down-type quarks, and neutrinos. The k=1 sector is presented as parameter-free: diagonalizing the 3×3 block of the Fourier coefficient matrix of G2 and normalizing to the electron mass yields mu=107.5 MeV and tau=1558 MeV, compared with experimental values 105.7 and 1776 MeV. For quarks and neutrinos, complex coefficients are fitted by minimizing the loss function in Eq. (A8) against central values of CKM/PMNS matrix elements and mass ratios. The paper also sketches a derivation of Calabi-Yau moduli-space geometry from these modular forms via Eq. (48). The conclusion states that the hypothesis is experimentally verified.

Significance. If the conjecture were correct, it would provide a remarkable new connection between flavor physics and Hilbert modular forms, and would give a direct experimental route to Calabi-Yau moduli geometry. The authors are transparent about the speculative nature of the proposal and list several open questions, including why the level assignments k=1–4 should hold. However, the evidence does not support the claimed verification. The only sector without fitted parameters has a 12.3% discrepancy in the tau mass. The quark and neutrino agreements are obtained by fitting the same observables that are then reported as predictions, and the light quark masses remain orders of magnitude below the experimental values. The paper provides no code, no machine-checked derivations, and no fitted parameter values, so the numerical results are not independently reproducible. The significance is therefore exploratory rather than demonstrative.

major comments (5)
  1. [§I, Remark 1; §II.A, Eqs. (19)–(23)] The k=1 charged-lepton prediction is obtained by truncating the infinite Fourier coefficient matrix of G2 to the first 3×3 block and diagonalizing that block, but the paper explicitly states in Remark 1 that the modular transformation property (10) is consistent only when the number of generations G is infinite and that finite G violates modular invariance. No argument is given that the lowest eigenvalues of the finite block approximate the corresponding eigenvalues of the infinite matrix, nor that the omitted rows and columns decouple. Since the modular form determines all g_ij simultaneously, the numbers in Eq. (23) may be truncation artifacts; this is load-bearing because the k=1 result is the only parameter-free evidence for the hypothesis.
  2. [Appendix A, Eq. (A8); Appendix A.2–A.3] The quark and neutrino mass matrices contain complex parameters a4, a6, b6 and a8, b8, c8, and the best-fit values are chosen by minimizing the loss function (A8) with respect to the experimental CKM/PMNS matrix elements and mass ratios. The resulting agreement is therefore not an independent prediction of the hypothesis; it is a fit to the same observables. The paper does not report the fitted parameter values or the final loss, so the fitting procedure cannot be independently checked. A genuine verification would require an out-of-sample prediction or a demonstration that the fit is statistically significant relative to the number of parameters.
  3. [Appendix A, Eqs. (A10)–(A15); §II.B–C] Even at the best fit, the light quark sector is not reproduced: the fit gives m_u = 5.30×10^{-5} GeV, m_d = 1.18×10^{-6} GeV, and m_s = 0.013 GeV, while the experimental central values are m_u = 2.2×10^{-3}, m_d = 4.7×10^{-3}, and m_s = 0.093 GeV. The initial zero-parameter versions are worse: the k=3 H6 matrix in Eqs. (29)–(30) gives a massless down quark, and the k=2 case with a4=0 in Eq. (26) gives m_u=0.163 MeV. The text acknowledges that these masses 'come out to be rather small.' Since the flavor-moonshine hypothesis explicitly claims that all particle masses are encoded in the modular forms, these discrepancies are a direct failure of the claim rather than a minor numerical issue.
  4. [§II.A, Eq. (23); Abstract; §V.6] The abstract states that the hypothesis is 'experimentally verified,' but in the only sector with no fitted parameters the tau mass is 1558 MeV against the experimental 1776 MeV, a 12.3% deviation that the paper itself reports. A single approximate match at this level, combined with sectors that are parameter-fitted, does not support the word 'verified.' The paper's own concluding remark, that 'it is possible that the whole idea of flavor moonshine is just nonsense,' is more appropriate to the strength of the present evidence.
  5. [§IV, Eqs. (48)–(67)] The geometric part of the paper assumes that the product J(q,r)J_H(w) equals ∫_K a∧b∧c∧Ω and that the modular variables may be identified with Calabi-Yau period variables. The scaling argument in Eqs. (55)–(61) only treats transformations with β=γ=0 and α=α', i.e., integer rescalings, not the full SL(2,Z(√2)) action, so the claimed relation between the modular form and the period prepotential is not established. This does not affect the numerical flavor fits, but it leaves the second main claim of the abstract without support.
minor comments (3)
  1. [§II.A, Eqs. (21)–(22)] Equation (21) defines the mass-squared matrix as gg†, while Eq. (22) writes sqrt(M3 M3^T); for the complex mass matrices used later the distinction between transpose and Hermitian conjugate matters, so the notation should be made uniform.
  2. [§II.E] The text contains a typo: 'week interactions' should be 'weak interactions.'
  3. [Appendix A, Eq. (A8)] The loss function uses only central experimental values without uncertainties, and no goodness-of-fit or statistical significance is reported; statements such as 'the agreement is generally excellent' are therefore not quantified.

Circularity Check

2 steps flagged · score 6.0 of 10

Quark and neutrino sectors are validated by fitting the very CKM/PMNS observables and mass ratios they then report, while the k=1 lepton sector relies on a G=3 truncation that the paper itself says breaks modular invariance.

  1. fitted input called prediction [Appendix A, Methods and CKM results (Eq. A8, A10, A11)]
    "Now let us search the complex parameters at the minimum of the loss function (A8). ... Our goal is to find a set of complex parameters that best fit the experimental results. The experimental results we use here are • the absolute values of the elements of the mixing (CKM or PMNS) matrix ζij • the ratios of masses ξk. ... For quark masses, we choose the parameters ξk = (mt/mc, mb/ms). This means we do not fit u and d quark masses."

    The complex parameters a4, a6, b6 are optimized by minimizing the log-loss (A8) against the experimental CKM matrix elements and the quark mass ratios mt/mc and mb/ms. The displayed 'best fit' CKM matrix (A10) and quark masses (A11) are the minimizer of that same loss, so their closeness to experiment is enforced by the fitting procedure rather than independently predicted. The paper's own admission that the unfitted u and d quark masses come out far too small confirms that the successful quantities are precisely the ones put into the fit.

  2. fitted input called prediction [Appendix A, PMNS fits (Eq. A8, A16-A26)]
    "For the PMNS matrix we have two choices of pure Dirac neutrino or Majorana neutrino (with seesaw approximation). In either way, we have again three complex parameters a8,b8 and c8 shown in equation (31). ... The best fit in the normal order of neutrino masses is PMNS = ... with neutrino mass differences (Δm2 21, Δm2 32) = (7.53 × 10−5, 2.44 × 10−3) eV2."

    The three complex parameters a8,b8,c8 are adjusted to minimize the same loss function (A8), whose targets include the experimental PMNS matrix elements and the neutrino mass-squared-difference ratio. The PMNS matrix and Δm2 values then presented as the model's output are therefore the result of fitting to those very observables. The neutrino-sector agreement is a restatement of the minimized loss, not an independent verification.

full rationale

The central circularity is concentrated in Appendix A: for the quark and neutrino sectors, the paper optimizes free complex parameters against the experimental CKM/PMNS matrices and chosen mass ratios, then reports the optimized values as successful fits and, in the abstract, as experimental verification of the flavor-moonshine hypothesis. Those successes are constructed by the fit. The k=1 charged-lepton sector is genuinely different: it uses no fitted parameters beyond an overall scale and hence carries real, if limited, independent content. However, its reliability is weakened by the G=3 truncation: the paper itself states in Section I that modular invariance of the Yukawa coupling holds only for infinite G, yet Section II A restricts to the first 3x3 block of the infinite coefficient matrix. That is a correctness/validity concern rather than a circular reduction, since the eigenvalues are computed rather than fitted. The choice of the modular group and the level assignment k=1..4 is an unconstrained ansatz that the paper leaves open, but that is an underdetermination/overfitting concern, not circularity. No load-bearing self-citation chain is present: the main mathematical input (Cohn-Deutsch) is external and the authors' own prior work is cited only as background. Overall, the paper contains genuine non-circular content in the lepton sector, but its broad verification claim relies in part on fitted inputs being presented as successful predictions, warranting a partial circularity score of 6.

Assumptions & free parameters 5 free parameters · 7 assumptions · 2 invented entities

The central claim rests primarily on the ansatz that mass matrices are Fourier coefficient matrices of specific modular forms. The charged lepton sector has no fitted parameters beyond the overall scale, but the quark and neutrino sectors depend on six complex parameters fit to the very data they are supposed to explain. The string geometry section additionally assumes the validity of string theory and a specific identification of modular variables with Calabi-Yau moduli.

free parameters (5)
  • a4 = not stated explicitly, best fit from loss minimization in Appendix A
    Complex parameter in the k=2 modular form (Eq. 8), fitted to CKM matrix and quark mass ratios.
  • a6 = not stated explicitly, best fit from loss minimization in Appendix A
    Complex parameter in the k=3 modular form (Eq. 9), fitted to CKM matrix and quark mass ratios.
  • b6 = not stated explicitly, best fit from loss minimization in Appendix A
    Complex parameter in the k=3 modular form (Eq. 9), fitted to CKM matrix and quark mass ratios.
  • a8, b8, c8 = not stated explicitly, best fit from loss minimization in Appendix A
    Complex parameters in the k=4 neutrino mass matrix (Eq. 31), fitted to PMNS matrix and neutrino mass squared differences.
  • overall mass scale g = set to electron mass 0.5110 MeV
    Overall normalization for the lepton mass matrix; cancels in mass ratios, so it does not affect the claimed predictions.
assumptions (7)
  • ad hoc to paper The Yukawa coupling is modular invariant with F transforming as in Eq. (14).
    This transformation rule is assumed so that the integral over modular variables is invariant, but it is not derived from a deeper principle.
  • ad hoc to paper The mass matrix gij equals the Fourier coefficient matrix of the modular form J(q,r), as in Eqs. (1) and (12).
    This is the core ansatz of flavor moonshine; no derivation from a physical theory is provided.
  • ad hoc to paper Physical masses are the singular values of the mass matrix sqrt(M M†), Eq. (21).
    Standard linear algebra is used, but the identification of the Fourier coefficient matrix with the Yukawa matrix and the use of M M† is an assumption about the mass matrix form.
  • ad hoc to paper The level assignment k=1,2,3,4 corresponds to charged leptons, up-type quarks, down-type quarks, and neutrinos.
    The paper assigns levels to fermion species without a derivation; it is a phenomenological fit to which form works for which sector.
  • domain assumption The modular variables q,r,w are Calabi-Yau moduli and the Strominger-Witten formula (Eq. 48) applies.
    This relies on string theory being the correct underlying theory and on the specific identification of moduli with the modular variables, which is assumed rather than derived.
  • domain assumption Mass ratios are scale independent (Section I, citing reference [4]).
    The paper assumes that the logarithmic scale dependence cancels in mass ratios, citing Chetyrkin and Rétey, and does not fully address renormalization effects in the fits.
  • domain assumption The low-energy number of generations is G=3.
    The paper truncates the infinite Fourier expansion to a 3x3 matrix to match observed generations, and notes that finite G violates modular invariance.
invented entities (2)
  • Flavor modular form J(q,r)
    purpose: Encodes the mass matrix for each fermion species through its Fourier coefficients.
    The only evidence is the numerical matches in this paper; no independent mathematical or physical derivation is offered.
  • k>=5 flavor particles as dark matter candidates
    purpose: Speculative new particles from higher-level modular forms that could account for dark matter.
    The paper briefly suggests these in Section II E without any experimental handle or detailed model.

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Cite this review

Pith. "Pith review of Flavor Moonshine." pith.science (2026). https://pith.science/paper/OMJBN43W

@misc{pith2026190811032,
  author       = {Pith},
  title        = {Pith review of: Flavor Moonshine},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMJBN43W}},
  note         = {Machine review of arXiv:1908.11032}
}
read the original abstract

The flavor moonshine hypothesis is formulated to suppose that all particle masses (leptons, quarks, Higgs and gauge particles -- more precisely, their mass ratios) are expressed as coefficients in the Fourier expansion of some modular forms just as, in mathematics, dimensions of representations of a certain group are expressed as coefficients in the Fourier expansion of some modular forms. The mysterious hierarchical structure of the quark and lepton masses is thus attributed to that of the Fourier coefficient matrices of certain modular forms. Our intention here is not to prove this hypothesis starting from some physical assumptions but rather to demonstrate that this hypothesis is experimentally verified and, assuming that the string theory correctly describes the natural law, to calculate the geometry (K\"{a}hler potential and the metric) of the moduli space of the Calabi-Yau manifold, thus providing a way to calculate the metric of Calabi-Yau manifold itself directly from the experimental data.

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Reference graph

Works this paper leans on

55 extracted references · 50 canonical work pages

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    This construction suggests the definition of the fields: ψ L(x,q ) = lim G→∞ G−1∑ i=0 ψ Liq−i, ψ R(x,r ) = lim G→∞ G−1∑ j=0 ψ Rir−j. (16) We do not need to assume any specific transformation property of the individual field under modular transformation, while the bilinear form expressed in e quation ( 10) must transform covariantly under the modular transform...

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    with a,b : integer

    (5) and α =a +b √ 2,α ′ =a − b √ 2,... with a,b : integer. 2 k is called the “level”. Cohen-Deutsch [5] shows that there are only three generator modular forms in this c ase. They are given by G2,G 4,G 6 with k = 1, 2, 3. What we use are the coefficients in Fourier expansion of these modular forms. We may also choose different com binations H2,H 4,H 6 that a...

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    However, in section IV, we will define and use the modular form corresponding to the Higgs field: JH(w) = ∑ k hkwk (17) corresponding to J(q,r )

    We treat here, just for simplicity, a pristine Higgs field H. However, in section IV, we will define and use the modular form corresponding to the Higgs field: JH(w) = ∑ k hkwk (17) corresponding to J(q,r ). We can also define the field H(w−1) = ∑ k Hkw−k (18) with the Higgs field H =H0. 6

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    The usual treatment of these variables is to regard them as a scalar field in the four-dimensional space-time and to try t o find a way to stabilize them

    Our modular variables q,r eventually become the moduli of Calabi-Yau manifold as will be shown later in section IV. The usual treatment of these variables is to regard them as a scalar field in the four-dimensional space-time and to try t o find a way to stabilize them. We regard them as variables to distinguish different va cua, and we integrate over them a...

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    generation

    Our definition of the “generation” is not the same as the usual on e in string theory. It corresponds to the expansion coefficient of the modulus depend ent fields defined in equations ( 16) and ( 18). II. NUMERICAL RESUL TS Equation (12) shows thatgij is a mass matrix, and equation ( 1) shows it is just the Fourier coefficient of the modular form J(q,r ). In th...

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    2929 0 0 0 61 . 63 0 0 0 893 . 3     . (22) By normalizing the lowest mass to be the electron mass of 0.5110 MeV, we obtain ( √ M3M T 3 ) normalized =     

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    5 0 0 0 1558     

    5110 0 0 0 107 . 5 0 0 0 1558     . (23) This shows that the modular form G2 embodies the charged lepton masses in its Fourier coefficients. There is no free parameter in this case except for the entire normalization which is of course scale dependent, unlike the mass ratios [ 4]. The corresponding experimental data are in appendix A: the central value...

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    Then we have, for the three generation case, G4 → M3 =      11 0 0 4320 480 0 280800 165120 35040     

    (24) 8 For the time being we ignore the second term (i.e., put a4 = 0). Then we have, for the three generation case, G4 → M3 =      11 0 0 4320 480 0 280800 165120 35040     . (25) The normalized and diagonalized mass matrix becomes ( √ M3M T 3 ) normalized =     

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    Of course, there are many mysteries to be solved. Why nature s eems to choose a very specific form such as the one we used that is based on SL(2 , Z( √ 2))? Why k = 1 for charged leptons, k = 2 for charge +2 / 3 quarks, k = 3 for charge − 1/ 3 quarks, and k = 4 for neutrinos? T...

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    Appendix A: Numerical fitting for experimantal data We calculated numerically the CKM and PMNS matrices and fit the exper imental data to them

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    Methods Our goal is to find a set of complex parameters that best fit the exp erimental results. The experimental results we use here are • the absolute values of the elements of the mixing (CKM or PMNS) matr ix ζij • the ratios of masses ξk. The mixing matrices in both cases ha...

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    796e0. 15i 0. 323e2. 84i 0. 511      (A18) 22 with neutrino mass differences ( ∆m2 21, ∆m2 32 ) = (7. 53 × 10−5, − 7. 53 × 10−5) eV 2. (A19) The PMNS matrix is can be written as      c12c13 s12c13 s13e−iδ −s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13 s12s23 − c1...

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    291 0 . 7531. 96i 0. 590e1. 12i

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    489e−2. 96i 0. 527e−2. 66i 0. 695

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    822e0. 70i 0. 394e−1. 40i 0. 411      (A23) with neutrino mass differences ( ∆m2 21, ∆m2 32 ) = (7. 53 × 10−5, 2. 44 × 10−3) eV 2. (A24) In the inverted order of neutrino mass, the best fit is PMNS =     

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    294 0 . 8803. 14i 0. 373e3. 14i

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