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This paper claims the observed ninths/sixths fermion pattern forces a discrete gauge symmetry Z18, and that identifying the flavor flavon with the Peccei–Quinn field makes axion quality automatic and fixes a 7–12 μeV haloscope target.

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2026-08-02 19:04 UTC pith:6OYHQLBL

load-bearing objection The flavor fit is real, but the Z18 axion-origin story breaks against its own equations: rational exponents cannot come from a discrete gauge chain, and E/N and N_DW are internally inconsistent. the 5 major comments →

arxiv 2603.03631 v5 pith:6OYHQLBL submitted 2026-03-04 hep-ph hep-ex

Flavor in Ninths and a Discrete Gauge Origin of the QCD Axion

Vernon Barger This is my paper
classification hep-ph hep-ex
keywords flavor hierarchiesQCD axionaxion qualityPeccei–Quinn symmetryFroggatt–Nielsen mechanismdiscrete gauge symmetryZ18axion dark matter
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that the fermion mass and mixing data, recast as powers of one small parameter, fall into rational exponents with denominators 9 for quarks and 6 for charged leptons. If those denominators are real, the minimal discrete gauge group that can enforce the pattern is the least common multiple, Z18. Because the scalar flavon generating the flavor hierarchy carries the minimal Z18 charge, the first gauge-invariant operator that violates the associated Peccei–Quinn symmetry is Φ^18, which pushes the residual strong-CP shift to an unobservable level. The same structure yields N_DW=1, E/N=8/3, and an axion mass around 7–12 μeV, placing the proposed dark-matter axion within reach of resonant-cavity haloscope searches. The point is that the symmetry protecting the axion is forced by flavor arithmetic rather than chosen to solve the axion problem.

Core claim

On the paper's own terms, one discrete gauge symmetry, Z18, is the common origin of the 'flavor in ninths' pattern and of the QCD axion's stability. Quark hierarchies are fit by exponents in (1/9)Z with epsilon=14/75 and lepton ratios by exponents in (1/6)Z; the lcm of 9 and 6 forces Z18. Identifying the Froggatt–Nielsen flavon with the Peccei–Quinn field makes Φ^18 the leading Planck-suppressed PQ-violating operator, so the axion vacuum shift stays below 10^-39 in the body derivation (the abstract quotes 3×10^-27). The framework then predicts a Type-II DFSZ-like axion with E/N=8/3 (2 with light higgsinos), a dark-matter-compatible mass window m_a≈7–12 μeV, and a coupling C_aγ≈0.6–1.0; the b

What carries the argument

The central object is the discrete gauge symmetry Z18, realized as Z9×Z2, with its Z9 subgroup fixing all Froggatt–Nielsen charges to (1/9)Z and its lepton sector on the 1/6 sublattice. The Z18 charges are used to show that the flavon carrying minimal charge 1/18 is the Peccei–Quinn field, so the first allowed pure-flavon monomial is Φ^18. The rational exponents are generated by a four-site chain of vectorlike messenger pairs with nearest-neighbor hop charges (1,2,4) mod 9, giving every suppression as a sum of ninths; the same chain enters the anomaly sums that must vanish mod 18.

Load-bearing premise

The framework stands or falls on whether the denominators 9 and 6 are data-driven rather than chosen—if the rational exponents are a parametrization choice, then Z18 is imposed and the automatic axion quality is not a prediction—and it also depends on a full anomaly-free Z18 charge assignment for the messenger sector, whose explicit verification is deferred to a companion paper.

What would settle it

Check whether the complete Z18 charge assignment (all SM fermions, the two Higgs doublets, the flavon, and the four vectorlike messenger pairs) satisfies the discrete anomaly sums AG=Σ Xi ℓ(ri) ≡ 0 mod 18 for SU(3), SU(2), U(1)_Y, and gravity; if no such assignment exists, the discrete-gauge origin fails. Alternatively, refit the seven observables of Table I with integer Froggatt–Nielsen exponents at epsilon≈0.22; if an integer-exponent fit works within the stated tolerances, the ninths denominators are not forced by the data.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Axion quality is automatic: no PQ-violating operator appears below dimension 18, so |Δθ̄| stays far below the experimental bound without any extra symmetry being imposed.
  • The axion mass is pinned to 7–12 μeV with the axion–photon coupling on the DFSZ-II line, making this a concrete target for existing resonant-cavity haloscope sensitivity.
  • With N_DW=1 the string–wall network has no stable domain walls, and the axion relic density can match the observed dark matter for f_a~(5–8)×10^11 GeV.
  • The same expansion parameter epsilon=14/75 and ninths exponents reproduce seven independent mass and mixing observables at roughly the percent level, with O(1) coefficients.
  • The neutrino sector inherits the lattice: normal mass ordering, m_2/m_3~m_1/m_2~epsilon, θ13~8–9°, δ_CP in 250–310°, and m_ββ~2 meV.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the Z18 identification is right, then a future axion detection well away from 7–12 μeV, or on the KSVZ rather than the DFSZ coupling line, would sever the flavon–PQ identification even if the ninths flavor fit survives.
  • The axion-quality calculation is brittle to small shifts in the measured exponents: if a refined fit moved one denominator or exponent by a unit of 1/9, the least common multiple of the denominators—and hence the minimal discrete gauge order and the allowed PQ-breaking operator—would change.
  • The manuscript itself leaves two checks open: the explicit Z18 charge assignments and numerical anomaly verification for the messenger sector are deferred to a companion paper, and the body's N_DW=1 differs from the abstract's N_DW=6; until these are reconciled, the 'prediction' framing should be read as conditional.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 3 minor

Summary. The paper proposes that quark and lepton mass hierarchies and CKM elements are organized by rational powers of a single small parameter ε = 14/75, with exponents quantized in units of 1/9 for quarks and 1/6 for leptons. It argues that this 'flavor in ninths' structure points to a gauged Z18 discrete symmetry, whose Z9 subgroup controls flavor. Identifying the Froggatt–Nielsen flavon with the QCD axion field, the paper claims that the first allowed pure-flavon PQ-violating operator is Φ^18, giving automatic axion quality, N_DW = 1 (in the body; abstract says 6), E/N = 8/3, and a dark-matter axion mass around 7–12 μeV. A vectorlike messenger chain is introduced as the UV completion that allegedly generates the rational exponents. The paper's central claim is that the Z18 order is fixed by flavor data rather than chosen to protect the axion.

Significance. If the construction were correct, it would unify the flavor hierarchy and the axion-quality problem in a single discrete gauge symmetry and produce a sharp, experimentally testable axion phenomenology. The numerical fits in Table I are legitimate and economical, and the predicted haloscope window m_a ≈ 7–12 μeV with C_{aγ} ≈ 0.6–1.0 is a genuine, falsifiable target. However, the paper's central derivations contain load-bearing errors: the E/N calculation is arithmetically inconsistent; the discrete gauge messenger chain cannot generate the advertised rational exponents from whole flavon insertions; and the axion-quality argument considers only pure-flavon monomials without proving the absence of lower-dimension mixed operators. These problems undermine the claimed prediction that the Z18 order and hence the axion quality are fixed by flavor data.

major comments (5)
  1. [Appendix B, Eq. (B4)] The derivation of E/N = 8/3 is internally inconsistent. Equation (B4) gives E/N = [2X(Hu) − 2X(Hd)] / [−3(X(Hu)+X(Hd))]. For the stated DFSZ-II condition X(Hu)=X(Hd), this is 0, not 8/3. The lepton contribution E_ℓ = −3X(Hd) is already included in (B3), so the statement that the numerator 'vanishes at leading order and one must retain the full expression including the lepton contribution' is contradicted by the equation itself. The central axion–photon prediction (Eq. (23), Table III) is therefore not derived.
  2. [Sec. VII/Appendix C, Eqs. (C2)–(C4), Table V] The discrete Z18 messenger chain cannot generate the rational ninths exponents. Each coupling in L_chain (Eq. (C3)) and in the endpoint terms (Eq. (C2)) contains an integer number of flavon fields, so with ε=⟨Φ⟩/Λ each insertion contributes ε^1. The effective suppression is ε^{n_tot}, not ε^{Σ n_a q_a/9}. In the paper's own example for m_s/m_b, the endpoint dressings require A_2=18, B_2=3 plus three internal links, i.e. 24 powers of ε, whereas Table V quotes ε^{7/3}; the down (2,2) row gives n_tot=6 hops but would yield ε^6 ≈ 4×10^-5, not ε^{7/3} ≈ 0.020. This is not a UV detail: it removes the mechanism advertised for generating the ninths lattice, and with it the claim that the Z18 order and the axion-quality argument are consequences of flavor data.
  3. [Sec. IV, Eq. (15); Appendix A] The axion-quality argument only considers pure-flavon monomials Φ^n. The claim that the first Planck-suppressed PQ-violating operator has dimension 18 requires that all mixed operators involving Higgs or matter fields are forbidden below that dimension, but no proof of this is given. The full Z18 charge assignments, including the messenger sector, are deferred to the companion paper [30]; Appendix A merely states that messenger charges are 'chosen so that' anomalies cancel. Since the automatic axion-quality conclusion rests on the identity of the lowest-dimension PQ-violating operator, this missing proof is load-bearing.
  4. [Abstract vs. Sec. IV] The headline predictions are internally contradictory. The abstract states N_DW = 6 and |Δθ̄| ≲ 3×10^-27, while Sec. IV (Eq. (19)) and Table III give N_DW = 1, and Eq. (16) with f_a ∼ 10^12 GeV gives |Δθ̄| ≲ 10^-39. These discrepancies are not explained and make the paper's central predictions ambiguous.
  5. [Secs. II–III, VIII] The claim that the Z18 order is 'fixed by flavor data rather than chosen' is not established. The denominators 9 and 6 are properties of the per-observable exponents in Table I, and Z18 = lcm(9,6) is then defined from them. As the paper acknowledges in Sec. VIII, integer-charge FN models with ε_C ≈ 0.22 fit the same quark data. Thus the denominator structure is a parametrization choice unless the discrete gauge mechanism independently forces it; the failure described in the previous comments means that forcing is absent.
minor comments (3)
  1. [Table II and Sec. III.B] The charge assignments are taken from companion papers [17,19] without derivations; since these charges determine all exponents, a self-contained presentation would need at least a summary of how they are obtained.
  2. [Fig. 2 and Eq. (C4)] The text and figure should clarify that A_i and B_j count actual flavon insertions, whereas the hop charges q_a/9 are charge differences; the present notation conflates charge normalization with operator suppression.
  3. [Sec. IV, Eq. (16)] The numerical estimate should state the exact values of f_a and Λ_QCD used; as written, the body's 10^-39 and the abstract's 3×10^-27 differ by twelve orders of magnitude.

Circularity Check

4 steps flagged

The claimed Z18-determined axion quality is the denominators of the fitted exponents plus a chosen charge normalization, not a data-driven prediction.

specific steps
  1. self definitional [Abstract; Sec. II.A Eq. (2); Sec. III.A Eq. (10)]
    "because the order of Z18 is fixed by the flavor data rather than chosen to protect the axion, the resulting high axion quality ... is a prediction of the flavor structure rather than an imposed ingredient. ... |Vus|=ϵ^{8/9}, |Vcb|=ϵ^{17/9}."

    The exponents 8/9 and 17/9 are fitted definitions, not independent inputs: they are chosen so that ϵ^p matches Vus and Vcb. The remaining exponents are also selected per observable (Table I) to make ϵ=14/75 reproduce the data. The denominator 9 in these fitted exponents is then used to compute lcm(9,6)=18, and that lcm is declared 'fixed by the flavor data'. Since the same data are fit by integer-charge FN models (as the paper concedes in Sec. VIII), the denominators are a parametrization choice; Z18 is the least common denominator of the chosen fit, so the claim is true by construction.

  2. fitted input called prediction [Sec. II.A (after Eq. 3); Table I; Sec. II.D; Table IV]
    "The system is overdetermined: once B is fixed by |Vus| and |Vcb|, the remaining CKM element |Vub| and the quark and lepton mass ratios become predictions."

    Table I assigns each observable an exponent p (10/3, 7/3, 5/3, 29/9, ...) that is the nearest ninth to log(data)/log(ϵ), with ϵ=14/75. These exponents are then used to compute charges in Table II, and Table IV's 'predictions' are just ϵ^p for those same assigned p. The 'predictions' are the fit values re-expressed, not independent outcomes. The good agreement only shows that the chosen ninths denominators can absorb the data; it does not show that the data force denominator 9.

  3. self definitional [Sec. IV, Eq. (15); Sec. III.A]
    "We now identify the FN flavon with the PQ field, so that Φ carries the minimal Z18 charge qΦ = 1/18. Any gauge-invariant singlet monomial must satisfy n qΦ ∈ Z, hence n = 18 is the first allowed power. Therefore the leading Planck-suppressed PQ-violating operator is ... Φ^18."

    The 'automatic' axion quality is not derived from the flavor structure; it is put in by the charge normalization qΦ=1/18. The ninths lattice itself is built from hops carrying charge qa/9 (Sec. III.A), i.e. Φ has Z9 charge 1, whose natural Z18 embedding is charge 2/18 = 1/9, which would allow Φ^9. The author's Eq. (15) instead chooses the minimal Z18 charge 1/18. With that choice, n=18 is of course the first invariant. Thus |Δθ| ≲ 10^-39 is equivalent to the chosen input qΦ=1/18, not to the flavor denominators.

  4. self citation load bearing [Sec. III.C; Appendix A]
    "The Z18 charges of the messenger pairs are chosen so that A_SM + A_VLQ ≡ 0 (mod 18). ... The explicit Z18 charge assignments for the full field content ... and the numerical verification of all anomaly conditions are given in the companion unified flavor paper [30]."

    The claim that a complete anomaly-free Z18 charge assignment exists is load-bearing for the discrete-gauge origin, but the present paper does not exhibit it: the messenger charges are 'chosen so that' the anomalies cancel, and the explicit verification is deferred to [30], an unpublished companion paper by the same author. This is a self-citation chain rather than an independent mathematical check or externally tested result, so it cannot independently support the central 'fixed by flavor data' assertion.

full rationale

The paper's central claim is that the Z18 order — and hence the axion-quality bound — is 'a prediction of the flavor structure rather than an imposed ingredient.' Walking the derivation chain, the exponent denominators that produce 18 are themselves fitted quantities: each p in Table I is chosen so that ϵ^p matches the measured value, with ϵ=14/75 fixed from the assumed 8/9 and 17/9 scalings. The arithmetic step lcm(9,6)=18 then simply computes the common denominator of the fitted exponents. The axion-quality step goes further and chooses qΦ=1/18, an extra normalization not forced by the ninths lattice (whose hops carry charge 1/9); this choice, not the flavor data, makes Φ^18 the first gauge-invariant monomial. The anomaly cancellation and UV messenger charges, needed to make the Z18 gauge symmetry real, are deferred to same-author companion papers and stated only as 'chosen so that' conditions. Thus the headline prediction reduces by construction to (i) the denominators of a fit and (ii) a chosen PQ charge. The model-building fits themselves are not circular, but the paper's framing of the discrete-gauge origin and axion quality as data-driven is. Score 7 reflects this partial but central circularity; the manuscript does contain genuinely independent standard results (e.g., DFSZ E/N=8/3, relic-density numerics) that are not circular.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 3 invented entities

The construction rests on the standard FN and discrete-gauge technologies plus the author's choices: 6 fitted/selected parameters of which the per-observable flavor exponents are the most consequential (they fix the denominators whose lcm is Z18). The three invented structures (Z18, the VLQ chain, the flavon-as-PQ identification) all carry falsifiable handles. The two most fragile axioms — anomaly-free charge assignment and the pure-flavon-only violation claim — are asserted and deferred to companion [30]. Net contribution of the text itself: a parametrization plus a conditional axion-quality estimate; the verification work is external.

free parameters (6)
  • ε (expansion parameter) = 14/75 ≈ 0.1867
    Determined from |Vus| and |Vcb| via the chosen exponents 8/9 and 17/9 (Sec II.A); rational approximation of 5.357. All hierarchies are powers of this single number.
  • Rational flavor exponents = 8/9, 17/9, 10/3, 7/3, 29/9, 5/3
    One exponent per observable in Table I; each is chosen so that ε^p matches the measured ratio. me/mμ = ε^{29/9} is especially unconstrained. These exponents are the 'flavor data' later claimed to fix the Z18 order (circularity).
  • O(1) prefactors c_ij = of order unity (some as large as ~2-3)
    Invoked to absorb residual mismatches; the paper acknowledges mu/mc requires 'a partial cancellation among O(1) prefactors' (Sec II.C), an ad hoc adjustment.
  • Higgs charges X(Hu), X(Hd) = claimed equal (DFSZ-II)
    The E/N 'derivation' in Appendix B assumes X(Hu) = X(Hd); with that assumption Eq. (B4) evaluates to 0, not 8/3, so the actual charge values are unspecified and the derivation is self-contradictory.
  • Axion decay constant fa = (5-8)×10^11 GeV
    Chosen so the post-inflation relic density formula (Eq. 26) matches Ωh² ≈ 0.12; sets the 7-12 μeV mass window.
  • Z9 site/hop charges (0,8,6,2)/(1,2,4) and messenger masses = Λ = fa/ε ~ (3-4)×10^12 GeV
    Model-building inputs for the messenger chains; the hop charges are chosen so every residue is reachable with ≤3 hops (Sec III.A).
axioms (7)
  • domain assumption Froggatt-Nielsen operator expansion: Yukawas = c_ij (⟨Φ⟩/Λ)^p
    Standard FN mechanism (Ref. [13]); the entire exponent lattice rests on it (Eq. 1).
  • domain assumption Discrete gauge symmetries are exactly respected by quantum gravity
    Krauss-Wilczek [9], Babu et al. [10]; needed for the axion-quality protection to be truly automatic.
  • standard math Z18 discrete anomaly conditions A_G = Σ X_i ℓ(r_i) ≡ 0 (mod 18)
    Ibanez-Ross [11], Banks-Dine [12]; invoked in Eq. (14), verification deferred.
  • ad hoc to paper An anomaly-free full Z18 charge assignment exists (SM + 2 Higgses + 4 VLQ pairs + flavon)
    Appendix A states messenger charges are 'chosen so that' anomalies cancel; explicit assignment and verification are deferred to companion [30] ('submitted'). Load-bearing for the whole discrete-gauge edifice.
  • ad hoc to paper The leading PQ-violating operator below dimension 18 involves only flavons; mixed operators with SM fields are forbidden
    Sec IV only demonstrates Φ^18 is the first pure-flavon invariant; 'forbids every Planck-suppressed PQ-violating operator below dimension eighteen' is asserted without enumerating mixed operators.
  • ad hoc to paper Messenger-chain inversion theorem (integrating out 4 VLQ pairs yields effective Yukawas with hop-count exponents)
    Eq. (C4) in Appendix C is attributed to companion [30]; the theorem's proof is not included here.
  • domain assumption Post-inflationary PQ breaking with N_DW = 1 so the string-wall network collapses
    Used for the relic-density target Eq. (26); depends on the body's N_DW = 1 claim, contradicted by the abstract's N_DW = 6.
invented entities (3)
  • Gauged Z18 discrete symmetry (with Z9 flavor subgroup) independent evidence
    purpose: Quantize FN charges in ninths/sixths and forbid PQ-violating operators below dimension 18
    Testable through the claimed flavor lattice and axion mass/coupling predictions; the order is claimed to be data-fixed, but is in fact inherited from the chosen fit exponents.
  • Vectorlike messenger chain: 4 VLQ pairs D_a + D̄_a with Z9 site charges (0,8,6,2) independent evidence
    purpose: UV completion generating the rational FN exponents via nearest-neighbor flavon hops
    Has a falsifiable collider handle: D → bZ, bH, tW with M_D ≳ 1.5 TeV from LHC searches (Sec VII).
  • Identification of the FN flavon Φ with the PQ axion field independent evidence
    purpose: The same scalar's VEV sets ε and its phase is the axion, unifying flavor and strong CP
    Falsifiable through the predicted axion mass window and DFSZ-II coupling line; this identification is the paper's central constructive move.

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read the original abstract

Quark and lepton hierarchies are organized by rational powers of a single parameter (the quark masses and mixings in ninths, the charged-lepton mass ratios in sixths). We argue that this structure points to a discrete $\mathbb{Z}_{18}$ origin of Froggatt--Nielsen symmetry, fixed as the least common denominator $\mathrm{lcm}(9,6)=18$, with its $\mathbb{Z}_9$ subgroup governing the quark lattice and the full $\mathbb{Z}_{18}\cong\mathbb{Z}_9\times\mathbb{Z}_2$ required to accommodate the half-integer lepton charges. The flavon is the unique $\mathbb{Z}_{18}$-charged scalar with a large VEV, so its phase is the Peccei--Quinn axion; the \emph{same} symmetry then solves the strong $CP$ problem and forbids every Planck-suppressed PQ-violating operator below dimension eighteen. Our central result is that, because the order of $\mathbb{Z}_{18}$ is fixed by the flavor data rather than chosen to protect the axion, the resulting high axion quality (the residual shift of the axion vacuum from $\bar\theta=0$ being only $|\Delta\bar\theta|\lesssim 3\times10^{-27}$) is a prediction of the flavor structure rather than an imposed ingredient. The axion is a generation-dependent Type-II DFSZ axion with domain-wall number $N_{\rm DW}=6$. The decisive phenomenological consequence is that the same non-universal charges that reproduce the fermion masses enhance the axion--photon coupling to an observable level, $C_{a\gamma}\simeq0.6$--$1.0$ (well above the higgsino-suppressed MSSM value), placing the dark-matter axion at $m_a\simeq 7$--$12~\mu$eV within reach of upcoming haloscope searches.

Figures

Figures reproduced from arXiv: 2603.03631 by Vernon Barger.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Chain topology for the down-type messenger sector. Circles denote the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

82 extracted references · 22 linked inside Pith · cited by 2 Pith papers

  1. [1]

    = 3 + 10 9 = 37 9 ,(13) matching the(1,1)entry ofp d ij in Eq. (6). The un- suppressed(3,3)entries follow fromQ(Q 3) =Q(u c

  2. [2]

    To connect this to the chain diagram in Fig

    Hops rule The Suppression Rule gives the exponentpij from the FN charges. To connect this to the chain diagram in Fig. 2, each charge must be decomposed into flavon in- sertions at the chain endpoints. Since the flavon carries unitZ 9 charge, each insertion contributes1/9to the ex- ponent and falls into one of three hop types with charges (1,2,4) mod 9: T...

  3. [3]

    Concrete charge assignments and full tex- ture fits are presented in the companion papers [17, 18]

    = 0; the stepwise suppressions in ninths capture the observed hi- erarchies and are stable under diagonalization (not a ba- sis artifact). Concrete charge assignments and full tex- ture fits are presented in the companion papers [17, 18]. Worked examples showing how specific mass ratios and CKM elements emerge from the hop framework are col- lected in App...

  4. [4]

    three messenger chains

    = 0. Theexponentdifferences(whichcontrolmass ratios and mixing angles) are the physically meaningful quantities, and the charge assignments in Table II repro- duce all seven entries of Table I withO(1)coefficients close to unity [17]. C. Anomaly cancellation IfZ 18 is a discrete gauge symmetry, it is constrained by mixed anomaly conditions. A compact stat...

  5. [5]

    Calculation rule Every Yukawa suppression in the ninths framework is obtained by a single rule: Suppression Rule.For any fermion bilinear ¯ψiψc jH, the effective Yukawa coupling is Yij =c ijϵpij, p ij =Q(ψ i) +Q(ψc j),(E1) whereQ(ψ i)andQ(ψ c j)are the FN charges in Ta- ble II,ϵ= 14/75, andc ij is anO(1)complex co- efficient. Mass ratios and mixing angles...

  6. [6]

    0 1— ms/mb Q(Q2) +Q(dc

  7. [7]

    7/3 0.020 0.019 md/mb Q(Q1) +Q(dc

  8. [8]

    37/9 0.0010 0.0010 Up-type quarks mt Q(Q3) +Q(uc

  9. [9]

    0 1— mc/mt Q(Q2) +Q(uc

  10. [10]

    10/3 0.0037 0.0036 mu/mt Q(Q1) +Q(uc

  11. [11]

    64/9 6.6×10 −6 7.5×10 −6 Charged leptons mτ Q(L3) +Q(ec

  12. [12]

    0 1— mµ/mτ Q(L2) +Q(ec

  13. [13]

    5/3 0.061 0.060 me/mτ Q(L1) +Q(ec

  14. [14]

    29/6 3.0×10 −4 2.8×10 −4 CKM elements (FX-corrected) |Vus|FX(θ d 12,θu 12)8/9 0.225 0.225 |Vcb|FX(θ d 23,θu 23)17/9 0.042 0.042 |Vub|FX(θ d 13,θu 13)10/3 0.0037 0.0038 Jsin −1δ|V us|·|V cb|·|V ub|55/9 3.5×10 −5 3.1×10 −5 Hops Rule.To compute the suppression for entry (i,j)from the chain diagram (Fig. 2):

  15. [15]

    Look up the chargesQ(ψi)andQ(ψ c j)from Table II

  16. [16]

    Express9Qas a non-negative integer:A i = 9Q(ψi)(entrance),B j = 9Q(ψc j)(exit)

  17. [17]

    Decompose each into flavon insertions:N= n1·1 +n 2·2 +n 3·4, wherena≥0counts the number of type-ahops at that endpoint

  18. [18]

    Total” column is the full Yukawa suppression ϵ(Ai+7+Bj)/9 as read off from the chain diagram in Fig. 2; the “ϵp

    The full Yukawa suppression is the product of three factors: Yij∼ϵ Ai/9  entrance ×ϵ 7/9  internal ×ϵ Bj/9  exit =ϵ (Ai+7+Bj)/9. (E3) The internal factorϵ 7/9 ≃0.27is common to all entries and sets the overall Yukawa scale (absorbed intom b,m t,m τ). Mass ratios depend only on the endpoint dressings:m i/mj∼ϵ (Ai+Bi−Aj−Bj)/9. Table V shows the ...

  19. [19]

    Bottom quark: the unsuppressed entry The(3,3)entry of the down-type texture hasp d 33 = 0, since both the third-generation left-handed charge Q(Q3) = 0and the third-generation right-handed charge Q(dc

  20. [20]

    = 0vanish. In the chain language, the end- point couplings¯qL,3 ˜HD 4 and ¯D1dR,3 carry no addi- tional flavon suppression beyond the chain traversal, and the overall exponent is normalized to zero. The bottom Yukawa coupling is thereforeO(1), setting the reference scale for all other down-type masses

  21. [21]

    Strange-to-bottom mass ratio The second-generation diagonal entry ispd 22 = 21/9 = 7/3, decomposing via the additive rule as pd 22 =Q(Q 2) +Q(dc

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