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Quark sector of S3 models: classification and comparison with experimental data

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F. González Canales, A. Mondragón, M. Mondragón, U. J. Saldaña Salazar, L. Velasco-Sevilla

High Energy Physics - Phenomenology · Submitted 24 Apr 2013 · 73 references

Pith's verdict: The classification and the exact CKM expressions are worth having; the χ² fit is parameter freedom dressed as agreement, with three of four mass ratios pinned at their 3σ walls.

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arXiv ID1304.6644 (version 1)
Pith IDWFM33FOB, Pith's permanent record of this paper and its review
CitationA. Mondrag\'on, F. Gonz\'alez Canales, L. Velasco-Sevilla, M. Mondrag\'on, U. J. Salda\~na Salazar. "Quark sector of S3 models: classification and comparison with experimental data." arXiv:1304.6644 (2013). https://pith.science/paper/WFM33FOB
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The paper

The paper claims

A symmetry of permutations of three objects, applied to the three quark families, yields mass matrices whose mixing predictions match the measured quark mixing to within a chi-squared of a few tenths.

The reading

The paper asks whether the pattern of quark masses and quark mixing can be traced to $S_3$, the symmetry group of permutations of three objects, acting on the three families. The authors write down the most general $S_3$-invariant Yukawa couplings for three settings: a single Higgs doublet (where the symmetry must be broken), three Higgs doublets (where the symmetry survives as a hidden symmetry of the low-energy spectrum), and four Higgs doublets occupying all three irreducible representations of $S_3$. They show that in each setting the resulting mass matrices can be rotated and shifted into one of two long-studied forms, the two-texture-zero form or the nearest-neighbour-interaction form, and that this yields closed-form expressions for every element of the quark mixing matrix in terms of quark mass ratios plus two splitting parameters and one phase. The payoff is a numerical comparison: fitting those analytic expressions to the 2012 values of $|V_{ud}|$, $|V_{us}|$, $|V_{ub}|$ and the Jarlskog invariant gives $S_3$ mass matrices that land on the measured mixing with a chi-squared of order $0.2$ to $0.4$. If that reading is right, the observed quark mixing is not an accident of arbitrary mass matrices but the low-energy shadow of a symmetry relating the families.

What carries the argument

The load-bearing device is a two-step transformation of the $S_3$-invariant mass matrix: a rotation $R( heta)_{12}$ that removes the $(1,3)$ and $(3,1)$ entries, possible only when $ an heta = w_1/w_2$, followed by a subtraction of the $(1,1)$ entry $m_0$ from the diagonal. What remains has the two-texture-zero shape with entries $0, a, 0; a^*, b, c; 0, c^*, d$. Because this matrix has only three independent invariants, its trace, determinant and trace of the square, the three magnitudes $|a|$, $b$, $|c|$ can be solved exactly in terms of the eigenmasses and a single free parameter $m_3$. That is what turns the mixing matrix into a closed-form expression in mass ratios rather than a numerical output of a diagonalisation routine. The same rotation applied in both the up and down sectors is what makes the transformation unobservable in the mixing.

Extended reading notes

Core claim

The central claim is that $S_3$ invariance, imposed on the quark Yukawa sector, is enough to pin the quark mass matrices into forms that reproduce the measured Cabibbo-Kobayashi-Maskawa matrix. Concretely, the authors show that after a rotation by an angle fixed by the ratio of the two Higgs vacuum expectation values, $w_1/w_2$, and a shift by a constant $m_0$ on the diagonal, the $S_3$-invariant mass matrices reduce to a Hermitian matrix with two texture zeroes, or, for a special rotation angle, to the nearest-neighbour-interaction form. Writing that reduced matrix in terms of its three invariants leaves one free parameter per sector, $m_0$ plus $m_3$, and the mixing matrix becomes an explicit function of the quark mass ratios. Fitting to the 2012 data gives chi-squared values of $1.6$ to $3.4$ times $10^{-4}$ for the one-Higgs version and of order $3.3$ times $10^{-4}$ for the three- and four-Higgs versions, depending on how tightly the fitted parameters are constrained. The authors read this as evidence that $S_3$ is a symmetry of the quark sector, and note that the same mass-matrix forms are known to work for leptons.

Load-bearing premise

The good chi-squared is read as evidence for $S_3$, but four mass ratios that are themselves measured quantities, allowed to float within their three-sigma ranges, along with two splitting parameters and a phase, are adjusted against only four observables, so the agreement may reflect fitting freedom rather than a distinctive prediction of the symmetry.

Editorial extensions

If this is right

  • If the $S_3$ forms are the right description of quarks, the same mass-matrix structure should describe charged leptons and neutrinos, which the authors state they will test in a follow-up paper.
  • The two-texture-zero and nearest-neighbour-interaction forms, long used as phenomenological ansätze, would acquire a dynamical origin rather than being postulated by hand.
  • The one-, three- and four-Higgs versions differ in parameter count and in the shift parameter $m_0$, so sharper quark data could discriminate among them.
  • The three- and four-Higgs versions require two or three additional Higgs electroweak doublets beyond the Standard Model one, which is a collider-facing corollary of the construction.

Reading between the lines

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

  • Because the same rotation angle must be applied in both the up and down sectors, the construction implicitly ties the two Higgs vacuum expectation values to a mixing-like angle; if the two sectors demanded different angles, the reduction to two texture zeroes would not go through, so this shared-angle requirement is where the framework is most exposed.
  • The fit's discriminating power is limited by treating the mass ratios as measured inputs allowed to float within their three-sigma ranges; a sharper test would ask whether the predicted correlations among $|V_{ud}|$, $|V_{us}|$, $|V_{ub}|$ and $J$ hold once the mass ratios are pinned, rather than fitting each observable with the ratios free.
  • $S_3$ is the smallest non-Abelian group, so a success here would suggest the flavour structure needs no larger discrete group; that is testable by asking whether larger groups fit the same four observables measurably better at equal parameter count.
Original abstract

S3 models offer a low energy approach to describe the observed pattern of masses and mixing, of both quarks and leptons. In this work, we first revisit an S3 model with only one Higgs electroweak doublet, where the flavour symmetry must be broken in order to produce an acceptable pattern of masses and mixing for fermions. Then, we analyse different S3 models, where the flavour symmetry is preserved as an exact, but hidden symmetry of the low energy spectra, after the electroweak symmetry breaking. The latter models require the addition of two more Higgs electroweak doublets which are accommodated in an S3 doublet. We also explore the consequences of adding a fourth Higgs electroweak doublet, thus occupying all three irreducible representations of S3. We show how the various S3-invariant mass matrices of the different models can reproduce the two texture zeroes and Nearest Neighbour Interaction matrix forms, which have been found to provide a viable and universal treatment of mixing for both quarks and leptons. We also find analytical and exact expressions for the CKM matrix of the models in terms of quark mass ratios. Finally, we compare the expressions of the CKM matrix of the different S3 models with the most up to date values of masses and mixing in the quark sector, via a chi^2 analysis. We find that the analytical expressions we derived reproduce remarkably well the most recent experimental data of the CKM matrix, suggesting that S3 is a symmetry of the quark sector.

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, 7,070 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.64 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

S3 models offer a low energy approach to describe the observed pattern of masses and mixing, of both quarks and leptons. In this work, we first revisit an S3 model with only one Higgs electroweak doublet, where the flavour symmetry must be broken in order to produce an acceptable pattern of masses and mixing for fermions. Then, we analyse different S3 models, where the flavour symmetry is preserved as an exact, but hidden symmetry of the low energy spectra, after the electroweak symmetry breaking. The latter models require the addition of two more Higgs electroweak doublets which are accommodated in an S3 doublet. We also explore the consequences of adding a fourth Higgs electroweak doublet, thus occupying all three irreducible representations of S3. We show how the various S3-invariant mass matrices of the different models can reproduce the two texture zeroes and Nearest Neighbour Interaction matrix forms, which have been found to provide a viable and universal treatment of mixing for both quarks and leptons. We also find analytical and exact expressions for the CKM matrix of the models in terms of quark mass ratios. Finally, we compare the expressions of the CKM matrix of the different S3 models with the most up to date values of masses and mixing in the quark sector, via a chi^2 analysis. We find that the analytical expressions we derived reproduce remarkably well the most recent experimental data of the CKM matrix, suggesting that S3 is a symmetry of the quark sector.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 S3 models offer a low energy approach to describe the observed pattern of masses and mixing, of both quarks and leptons. background 1.00 0.06 0.04 asserted 0.94
s1 In this work, we first revisit an S3 model with only one Higgs electroweak doublet, where the flavour symmetry must be broken in order to produce an acceptable pattern of masses and mixing for fermions. contribution 0.89 0.51 0.02 asserted 0.98
s2 Then, we analyse different S3 models, where the flavour symmetry is preserved as an exact, but hidden symmetry of the low energy spectra, after the electroweak symmetry breaking. contribution 0.86 0.84 0.03 asserted 0.95
s3 The latter models require the addition of two more Higgs electroweak doublets which are accommodated in an S3 doublet. method 0.86 0.26 0.02 asserted 1.00
s4 We also explore the consequences of adding a fourth Higgs electroweak doublet, thus occupying all three irreducible representations of S3. contribution 0.74 0.90 0.02 asserted 0.82
s5 We show how the various S3-invariant mass matrices of the different models can reproduce the two texture zeroes and Nearest Neighbour Interaction matrix forms, which have been found to provide a viable and universal treatment of mixing for both quarks and leptons. result 0.74 0.87 0.07 asserted 0.98
s6 We also find analytical and exact expressions for the CKM matrix of the models in terms of quark mass ratios. result 0.75 0.97 0.04 asserted 1.00
s7 Finally, we compare the expressions of the CKM matrix of the different S3 models with the most up to date values of masses and mixing in the quark sector, via a chi^2 analysis. method 0.88 0.93 0.04 asserted 0.94
s8 We find that the analytical expressions we derived reproduce remarkably well the most recent experimental data of the CKM matrix, suggesting that S3 is a symmetry of the quark sector. interpretation 0.58 0.96 0.91 hedged possibility 0.91

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from analysis of existing data 0.83.

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. Results of the χ 2 fit as a function of me u and me s for the case where the phase φ1 is fixed to π/2. The reported masses of quarks in 2011 give to the ratios mfu and mfs an uncertainty of about 30% of their central value, therefore they were difficult to fit. The situation in 2012 has improved, in particular for the strange quark mass me s the lattice determinations reduced its uncertainty to 5% of its central value. Since this is quite remarkable, we have also fitted the mass ratios using as limits the values obtained by considering just the theoretical determination of me s, for…
Figure 2. The same as in
Figure 3. The same as in
Figure 4. The same as in
Figure 5. The same as in Figs. 1-4, except that now cos φ1 is allowed to vary in the region (0.5, 1.0). 28
Figure 6. The same as in Figs. 1-4, except that now cos φ1 is allowed to vary in the region (0.5, 1.0). 29
Figure 7. p δf , f = u blue (solid) curve, f = d orange (dashed) curve, as a function of Zf , for one of the three solutions of Eq. (43). The plot on the left corresponds to the plot of 1999 from ref. [16], while the plot on the right corresponds to taking the results, except the value of the parameters δf , of the fit of case I for φ1 = π/2, Tab. 8. In this case, we have given as input the value of Zf and chosen the analogous solution to the 1999 fit for δf , using Eq. (43). We see that both solutions are compatible, if we wanted to choose the value of the parameters δf as a function of Zf .…
Figure 8. On the left, Zf as a function of δf , f = u blue (solid) curve, f = d, orange (dashed) curve. The values obtained, from the fit for the case of φ1 = π/2, for δu and δd, 0.0605 and 0.0409 fix respectively particular values of Zf , 112.33 and 17.52. On the left, the parameter p δf as a function of Zf , f = u blue (solid) curve, f = d, orange (dashed) curve, for the data of 2012 for φ1 = π/2, corresponding to another solution of Eq. (43), different from that of
Figure 9. χ 2 as a function of σeu and σec, where cos φ1 is allowed to vary in the region (0.5, 1.0). The rest of the details are like those of
Figure 10. χ 2 as a function of σed and σes, where cos φ1 is allowed to vary in the region (0.5, 1.0). The rest of the details are like those of

Figures reproduced from arXiv:1304.6644 by the authors.

References
How the literature uses it

49 arXiv papers cite this in 74 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 · 34 sentences without a confident label

Compares with it 2 papers
  • The $M_d$ mass matrix exhibits a similar structure to the one in [ref] , for a non-supersymmetric 4 Higgs doublet model with $S_3$ symmetry.

    Flavour in $SU(5)$ Finite Grand Unified Models · An TEXT FUT model based on TEXT · 0.99

  • This relationship between $v_1$ and $v_2$ is similar to the condition imposed in Ref. [ref] to generate the Nearest Neighbour Interaction (NNI) mass matrices in the context of the SM $⊗S(3)$ with two texture zeroes.

    S(3) flavoured Higgs model trilinear self-couplings · The $S(3)$ extended Higgs doublet model · 0.95

Cites it as background 35 papers

What it cites · 40 sentences without a confident label

Uses the method or tool of 1 paper
Compares with 2 papers
Cites as background 36 papers

Discussion

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