pith. sign in

arxiv: 2510.03403 · v2 · pith:C7A4F6ERnew · submitted 2025-10-03 · ✦ hep-ph

Flavor hierarchies with nonminimal irreducible representations

Hannah Banks , Graeme Crawford , Matthew McCullough , Dave Sutherland This is my paper

Pith reviewed 2026-05-21 20:47 UTC · model grok-4.3

classification ✦ hep-ph
keywords flavor hierarchiesnonminimal representationsspurionnon-Abelian symmetryfermion massesflavor violating operatorsStandard Model extensions
0
0 comments X

The pith

A non-minimal spurion representation generates fermion mass hierarchies from untuned order-one parameters in a non-Abelian symmetry.

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

The paper proposes a new class of flavour models in which the spurion breaking Standard Model flavour symmetries transforms in a non-minimal irreducible representation. Hierarchies in fermion masses then arise from multiple insertions of this spurion and can be generated accidentally in a technically natural way from only a handful of untuned O(1) elements in the ultraviolet. The mechanism relies on the non-Abelian nature of the symmetry and therefore differs from standard Froggatt-Nielsen constructions. As a direct result the pattern of dimension-six flavour-violating operators can differ radically from earlier scenarios, which in turn requires a broader experimental flavour programme that spans all generations.

Core claim

We propose a new class of flavour models in which the spurion which breaks Standard Model flavour symmetries transforms in a non-minimal representation. Hierarchies in fermion masses, which arise from multiple insertions of this spurion, may be generated in a technically natural, accidental manner, from a handful of untuned O(1) elements in the UV. This relies explicitly on the non-Abelian nature of the symmetry, distinguishing it from standard Froggatt-Nielsen-like scenarios. The pattern of flavour violating operators at dimension-6 can radically differ from previously considered scenarios, and emphasises the need for a broad flavour programme across all generations.

What carries the argument

The non-minimal irreducible representation of the flavour symmetry group assigned to the spurion, which permits accidental hierarchies through repeated insertions while remaining technically natural.

Load-bearing premise

A spurion transforming in a non-minimal irreducible representation can be consistently embedded in an ultraviolet completion without introducing new tunings or inconsistencies with the non-Abelian symmetry structure.

What would settle it

Precision measurements that reveal or rule out the specific pattern of dimension-six flavour-violating operators predicted when multiple insertions of a non-minimal-representation spurion generate the observed mass hierarchies.

read the original abstract

We propose a new class of flavour models in which the spurion which breaks Standard Model flavour symmetries transforms in a non-minimal representation. Hierarchies in fermion masses, which arise from multiple insertions of this spurion, may be generated in a technically natural, accidental manner, from a handful of untuned $\mathcal{O}(1)$ elements in the UV. This relies explicitly on the non-Abelian nature of the symmetry, distinguishing it from standard Froggatt-Nielsen-like scenarios. The pattern of flavour violating operators at dimension-6 can radically differ from previously considered scenarios, and emphasises the need for a broad flavour programme across all generations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a new class of flavour models in which the spurion breaking Standard Model flavour symmetries transforms in a non-minimal irreducible representation. It claims that fermion mass hierarchies arise accidentally from multiple insertions of this spurion, generated in a technically natural manner from a handful of untuned O(1) parameters in the UV. This relies on the non-Abelian nature of the symmetry to distinguish it from Froggatt-Nielsen scenarios, and predicts that the pattern of dimension-6 flavour-violating operators can differ substantially from previous models, motivating a broad experimental flavour programme across all generations.

Significance. If the mechanism can be shown to work in a consistent UV embedding without new tunings, it would provide a representation-theory-based alternative to standard flavour models that generates hierarchies accidentally while preserving technical naturalness. The distinct predictions for higher-dimensional operators could alter expectations for flavour violation and encourage broader experimental searches beyond the usual third-generation focus.

major comments (2)
  1. [Abstract] Abstract: The central claim that hierarchies 'may be generated in a technically natural, accidental manner, from a handful of untuned O(1) elements in the UV' and rely explicitly on the non-Abelian nature is presented without an explicit choice of flavour group, non-minimal irrep, or derivation showing how multiple insertions produce the observed pattern while forbidding lower-order invariants. This leaves the technical naturalness unverified and vulnerable to the stress-test concern that UV consistency may require additional tuning to preserve the non-Abelian structure.
  2. [Model setup] The manuscript does not demonstrate that the chosen non-minimal representation avoids invariant contractions at leading order that would force unsuppressed entries in the mass matrices unless the spurion vev alignment or UV coefficients are tuned by hand. An explicit example with the representation theory and effective operator analysis is needed to secure the claim that the pattern differs radically from Froggatt-Nielsen-like scenarios.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief statement of the specific flavour symmetry group (e.g., a subgroup of U(3)^3) and the dimension of the non-minimal irrep to make the proposal concrete for readers.
  2. Notation for the spurion and its insertions should be defined clearly at first use, including how the effective Yukawa matrices are constructed from the multiple insertions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major points below and will revise the manuscript to include explicit examples that strengthen the claims.

read point-by-point responses

  1. Referee: [Abstract] The central claim that hierarchies 'may be generated in a technically natural, accidental manner, from a handful of untuned O(1) elements in the UV' and rely explicitly on the non-Abelian nature is presented without an explicit choice of flavour group, non-minimal irrep, or derivation showing how multiple insertions produce the observed pattern while forbidding lower-order invariants. This leaves the technical naturalness unverified and vulnerable to the stress-test concern that UV consistency may require additional tuning to preserve the non-Abelian structure.

    Authors: We agree that an explicit example would make the central claim more concrete. In the revised version we will add a specific choice of non-Abelian flavour group (e.g., SU(3) factors) together with a non-minimal irrep such as the 6 or 10, and derive the leading operators to show how representation theory forbids lower-order invariants. This will explicitly demonstrate the accidental hierarchy generation from O(1) UV parameters while preserving technical naturalness, directly addressing the concern about unverified claims. revision: yes

  2. Referee: [Model setup] The manuscript does not demonstrate that the chosen non-minimal representation avoids invariant contractions at leading order that would force unsuppressed entries in the mass matrices unless the spurion vev alignment or UV coefficients are tuned by hand. An explicit example with the representation theory and effective operator analysis is needed to secure the claim that the pattern differs radically from Froggatt-Nielsen-like scenarios.

    Authors: We acknowledge the need for an explicit demonstration. The revised manuscript will include a concrete representation-theoretic analysis for the chosen non-minimal irrep, showing that certain invariant contractions are forbidden at leading order by the non-Abelian symmetry without requiring hand-tuned vev alignments or coefficients. This will illustrate the radical difference from Froggatt-Nielsen scenarios, where Abelian charges permit sequential suppressions, and confirm that the mass-matrix pattern arises accidentally from multiple insertions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent symmetry and representation theory

full rationale

The paper proposes a new class of flavor models where a spurion in a non-minimal irreducible representation breaks flavor symmetries. Hierarchies arise accidentally from multiple insertions of O(1) UV parameters in a technically natural way, distinguished by non-Abelian structure from Froggatt-Nielsen scenarios. No load-bearing step reduces by construction to fitted inputs, self-definitions, or self-citation chains. The central mechanism is grounded in representation theory and effective operator analysis, which are externally checkable and do not presuppose the target mass hierarchies. This is the most common honest finding for model-building papers with untuned inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on introducing a non-minimal spurion representation as the key new element, with a small number of O(1) UV parameters treated as untuned inputs and the non-Abelian character of the flavor symmetry as a background assumption.

free parameters (1)
  • handful of untuned O(1) elements in the UV
    These are the source parameters from which mass hierarchies emerge accidentally via multiple spurion insertions; they are described as untuned but not derived from first principles.
axioms (1)
  • domain assumption The flavor symmetry is non-Abelian
    Invoked explicitly to enable the accidental hierarchy generation and to distinguish the scenario from Abelian Froggatt-Nielsen-like models.
invented entities (1)
  • Spurion transforming in a non-minimal irreducible representation no independent evidence
    purpose: Breaks Standard Model flavour symmetries to generate fermion mass hierarchies through multiple insertions in a technically natural way
    This is the core new postulated transformation property introduced by the model class; no independent evidence outside the proposal is provided in the abstract.

pith-pipeline@v0.9.0 · 5626 in / 1616 out tokens · 54378 ms · 2026-05-21T20:47:23.416591+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Hierarchies in fermion masses, which arise from multiple insertions of this spurion, may be generated in a technically natural, accidental manner, from a handful of untuned O(1) elements in the UV. This relies explicitly on the non-Abelian nature of the symmetry

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Hierarchies from Higher Flavor Spin

    hep-ph 2026-05 unverdicted novelty 7.0

    Yukawa hierarchies are generated from powers of fully anarchic spurions in higher SU(2) and SU(3) flavor representations via progressive rank lifting through composite outer products, with explicit models and predicti...

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · cited by 1 Pith paper · 19 internal anchors

  1. [1]

    A notable example are the light quark masses, which explicitly break chiral symmetry at the QCD scale, but arise from spontaneous breaking of the electroweak symmetry in the UV

    SU(3) diag(0,0, v) SU(2)×SU(2)×U(1) Q8+q8 itly breaks a global symmetry in the IR, is in fact the expectation of a scalar field which spontaneously breaks a gauge symmetry in the UV, with the UV dynamics integrated out. A notable example are the light quark masses, which explicitly break chiral symmetry at the QCD scale, but arise from spontaneous breakin...

    work page

  2. [2]

    R. S. Chivukula and H. Georgi, Composite Technicolor Standard Model, Phys. Lett. B188, 99 (1987)

    work page 1987

  3. [3]

    L. J. Hall and L. Randall, Weak scale effective supersym- metry, Phys. Rev. Lett.65, 2939 (1990)

    work page 1990

  4. [4]

    D’Ambrosio, G

    G. D’Ambrosio, G. F. Giudice, G. Isidori, and A. Stru- mia, Minimal flavor violation: An Effective field theory approach, Nucl. Phys. B645, 155 (2002), arXiv:hep- ph/0207036

    work page arXiv 2002

  5. [5]

    A. J. Buras, P. Gambino, M. Gorbahn, S. Jager, and L. Silvestrini, Universal unitarity triangle and physics be- yond the standard model, Phys. Lett. B500, 161 (2001), arXiv:hep-ph/0007085

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  6. [6]

    Minimal Flavor Violation in the Lepton Sector

    V. Cirigliano, B. Grinstein, G. Isidori, and M. B. Wise, Minimal flavor violation in the lepton sector, Nucl. Phys. B728, 121 (2005), arXiv:hep-ph/0507001

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  7. [7]

    Minimal Flavour Violation and Beyond

    G. Isidori and D. M. Straub, Minimal Flavour Viola- tion and Beyond, Eur. Phys. J. C72, 2103 (2012), arXiv:1202.0464 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  8. [8]

    Horizontal Symmetries for the Supersymmetric Flavor Problem

    A. Pomarol and D. Tommasini, Horizontal symmetries for the supersymmetric flavor problem, Nucl. Phys. B 466, 3 (1996), arXiv:hep-ph/9507462

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  9. [9]

    Predictions From A U(2) Flavour Symmetry In Supersymmetric Theories

    R. Barbieri, G. R. Dvali, and L. J. Hall, Predictions from a U(2) flavor symmetry in supersymmetric theories, Phys. Lett. B377, 76 (1996), arXiv:hep-ph/9512388

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  10. [10]

    Unified Theories with U(2) Flavor Symmetry

    R. Barbieri, L. J. Hall, S. Raby, and A. Romanino, Uni- fied theories with U(2) flavor symmetry, Nucl. Phys. B 493, 3 (1997), arXiv:hep-ph/9610449

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  11. [11]

    Consequences of a U(2) Flavour Symmetry

    R. Barbieri, L. J. Hall, and A. Romanino, Consequences of a U(2) flavor symmetry, Phys. Lett. B401, 47 (1997), arXiv:hep-ph/9702315

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  12. [12]

    Large Top Mass and Non-Linear Representation of Flavour Symmetry

    T. Feldmann and T. Mannel, Large Top Mass and Non- Linear Representation of Flavour Symmetry, Phys. Rev. Lett.100, 171601 (2008), arXiv:0801.1802 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  13. [13]

    A. L. Kagan, G. Perez, T. Volansky, and J. Zupan, Gen- eral Minimal Flavor Violation, Phys. Rev. D80, 076002 (2009), arXiv:0903.1794 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  14. [14]

    Flavour physics from an approximate U(2)^3 symmetry

    R. Barbieri, D. Buttazzo, F. Sala, and D. M. Straub, Flavour physics from an approximateU(2) 3 symmetry, JHEP07, 181, arXiv:1203.4218 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    U(2) and Minimal Flavour Violation in Supersymmetry

    R. Barbieri, G. Isidori, J. Jones-Perez, P. Lodone, and D. M. Straub,U(2) and Minimal Flavour Violation in Supersymmetry, Eur. Phys. J. C71, 1725 (2011), arXiv:1105.2296 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  16. [16]

    Fuentes-Mart´ ın, G

    J. Fuentes-Mart´ ın, G. Isidori, J. Pag` es, and K. Ya- mamoto, With or without U(2)? Probing non-standard flavor and helicity structures in semileptonic B decays, Phys. Lett. B800, 135080 (2020), arXiv:1909.02519 [hep- ph]

    work page arXiv 2020

  17. [17]

    Greljo and A

    A. Greljo and A. E. Thomsen, Rising through the ranks: flavor hierarchies from a gauged SU(2) symmetry, Eur. Phys. J. C84, 213 (2024), arXiv:2309.11547 [hep-ph]

    work page arXiv 2024

  18. [18]

    Greljo, A

    A. Greljo, A. E. Thomsen, and H. Tiblom, Flavor hier- archies from SU(2) flavor and quark-lepton unification, JHEP08, 143, arXiv:2406.02687 [hep-ph]

    work page arXiv

  19. [19]

    Next to Minimal Flavor Violation

    K. Agashe, M. Papucci, G. Perez, and D. Pirjol, Next to minimal flavor violation, (2005), arXiv:hep-ph/0509117

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  20. [20]

    Aligned and Spontaneous Flavor Violation

    D. Egana-Ugrinovic, S. Homiller, and P. Meade, Aligned and Spontaneous Flavor Violation, Phys. Rev. Lett.123, 031802 (2019), arXiv:1811.00017 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  21. [21]

    C. D. Froggatt and H. B. Nielsen, Hierarchy of Quark Masses, Cabibbo Angles and CP Violation, Nucl. Phys. B147, 277 (1979)

    work page 1979

  22. [22]

    Mass Matrix Models

    M. Leurer, Y. Nir, and N. Seiberg, Mass matrix models, Nucl. Phys. B398, 319 (1993), arXiv:hep-ph/9212278

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  23. [23]

    Leurer, Y

    M. Leurer, Y. Nir, and N. Seiberg, Mass matrix models: The Sequel, Nucl. Phys. B420, 468 (1994), arXiv:hep- ph/9310320

    work page arXiv 1994

  24. [24]

    Fedele, A

    M. Fedele, A. Mastroddi, and M. Valli, Mini- mal Froggatt-Nielsen textures, JHEP03, 135, arXiv:2009.05587 [hep-ph]

    work page arXiv 2009

  25. [25]

    Smolkoviˇ c, M

    A. Smolkoviˇ c, M. Tammaro, and J. Zupan, Anomaly free Froggatt-Nielsen models of flavor, JHEP10, 188, [Erra- tum: JHEP 02, 033 (2022)], arXiv:1907.10063 [hep-ph]

    work page arXiv 2022

  26. [26]

    Altmannshofer and A

    W. Altmannshofer and A. Greljo, Recent Progress in Fla- vor Model Building 10.1146/annurev-nucl-121423-100950 (2024), arXiv:2412.04549 [hep-ph]

    work page doi:10.1146/annurev-nucl-121423-100950 2024

  27. [27]

    Altmannshofer and P

    W. Altmannshofer and P. Stangl, Flavour Physics Be- yond the Standard Model, (2025), arXiv:2508.03950 [hep-ph]

    work page arXiv 2025

  28. [28]

    Beyond MFV in family symmetry theories of fermion masses

    Z. Lalak, S. Pokorski, and G. G. Ross, Beyond MFV in 9 family symmetry theories of fermion masses, JHEP08, 129, arXiv:1006.2375 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    Antusch, A

    S. Antusch, A. Greljo, B. A. Stefanek, and A. E. Thom- sen, U(2) Is Right for Leptons and Left for Quarks, Phys. Rev. Lett.132, 151802 (2024), arXiv:2311.09288 [hep- ph]

    work page arXiv 2024

  30. [30]

    Phenomenology of Minimal Lepton Flavor Violation

    V. Cirigliano and B. Grinstein, Phenomenology of mini- mal lepton flavor violation, Nucl. Phys. B752, 18 (2006), arXiv:hep-ph/0601111

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  31. [31]

    Silvestrini and M

    L. Silvestrini and M. Valli, Model-independent Bounds on the Standard Model Effective Theory from Flavour Physics, Phys. Lett. B799, 135062 (2019), arXiv:1812.10913 [hep-ph]

    work page arXiv 2019

  32. [32]

    Chau and W.-Y

    L.-L. Chau and W.-Y. Keung, Comments on the Parametrization of the Kobayashi-Maskawa Matrix, Phys. Rev. Lett.53, 1802 (1984)

    work page 1984

  33. [33]

    Cornella, D

    C. Cornella, D. Curtin, E. T. Neil, and J. O. Thompson, Mapping and probing Froggatt-Nielsen solutions to the quark flavor puzzle, Phys. Rev. D111, 015042 (2025), arXiv:2306.08026 [hep-ph]

    work page arXiv 2025

  34. [34]

    Neutrino Mixing and Masses from a Minimum Principle

    R. Alonso, M. B. Gavela, G. Isidori, and L. Maiani, Neutrino Mixing and Masses from a Minimum Principle, JHEP11, 187, arXiv:1306.5927 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  35. [35]

    Naturally large Yukawa hierarchies

    E. Nardi, Naturally large Yukawa hierarchies, Phys. Rev. D84, 036008 (2011), arXiv:1105.1770 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  36. [36]

    Michel, MINIMA OF HIGGS-LANDAU POLYNO- MIALS, inColloquium on Fundamental Interactions, in honor of A

    L. Michel, MINIMA OF HIGGS-LANDAU POLYNO- MIALS, inColloquium on Fundamental Interactions, in honor of A. Visconti(1979)

    work page 1979

  37. [37]

    J. R. Espinosa, C. S. Fong, and E. Nardi, Yukawa hierar- chies from spontaneous breaking of theSU(3) L ×SU(3) R flavour symmetry?, JHEP02, 137, arXiv:1211.6428 [hep- ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [38]

    Slansky, Group Theory for Unified Model Building, Phys

    R. Slansky, Group Theory for Unified Model Building, Phys. Rept.79, 1 (1981)

    work page 1981