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arxiv: 2606.07799 · v2 · pith:MIV6BOWLnew · submitted 2026-06-05 · ✦ hep-ph

Maximal Abelian Flavor Symmetries

Juanca Carrasco-Martinez , Lawrence J. Hall This is my paper

Pith reviewed 2026-06-29 05:42 UTC · model grok-4.3

classification ✦ hep-ph
keywords flavor hierarchiesquark masseslepton massesneutrino mixingSU(5) unificationSO(10) unificationleptogenesisMaximal Abelian Flavor Symmetries
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The pith

Assigning one small suppression factor to each fermion multiplet describes fifteen observed mass ratios and mixing angles at the factor-of-two level in SU(5) unified theories.

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

The paper introduces the MAFS framework, which assigns a small parameter ε_a to each fermion multiplet to generate the observed hierarchies in quark and lepton masses and mixings. Unlike mechanisms that require choosing charges for fermions, MAFS needs no such assignments and grows more predictive as the number of multiplets shrinks. In SU(5) theories this accounts for fifteen quantities with five parameters, and the resulting structure forces large neutrino mixing angles together with small neutrino mass hierarchies. In SO(10) theories the same fifteen hierarchies are captured by only three parameters with values 0.01, 0.02 and 0.002. Leptogenesis then yields the observed baryon asymmetry in SU(5) without any further small parameter.

Core claim

MAFS provides an approximate description of the hierarchies of quark and lepton masses and mixing angles in terms of a set of small parameters, ε_a, one for each fermion multiplet. In SU(5) unified theories, 15 observed mass ratios and mixing angles are described, at the factor of two level, by five small ε_a parameters. Even though quarks and leptons are unified, the observed hierarchical pattern of quark masses and mixings requires large neutrino mixing angles and small neutrino mass hierarchies. In an SO(10) unified theory, MAFS successfully describes the 15 observed flavor hierarchies with just three small ε_a, taking values of 0.01, 0.02 and 0.002. The observed cosmological baryon asymm

What carries the argument

The assignment of independent small suppression factors ε_a to each fermion multiplet, which multiplicatively suppress the corresponding Yukawa couplings to generate the mass and mixing hierarchies.

If this is right

  • In SU(5), the hierarchical pattern of quark masses and mixings necessarily produces large neutrino mixing angles and small neutrino mass hierarchies.
  • Leptogenesis accounts for the observed baryon asymmetry in SU(5) without introducing any extra small parameter beyond the five ε_a.
  • In SO(10), the same fifteen hierarchies are reproduced with only three ε_a parameters.
  • An additional parameter of order 0.2 is required in SO(10) to obtain the correct baryon asymmetry via leptogenesis.

Where Pith is reading between the lines

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

  • The approach could be applied directly to other grand-unified or non-unified models by counting their fermion multiplets and fitting the resulting ε_a set.
  • Future precision data on neutrino mass ratios could be compared against the small-hierarchy prediction that follows from the quark-lepton unification pattern.
  • Model builders might use the fitted ε_a values as targets when constructing explicit ultraviolet completions that generate the suppressions dynamically.

Load-bearing premise

The observed flavor hierarchies arise from and can be captured by assigning independent small suppression factors to each fermion multiplet without additional structure or charges.

What would settle it

A precise measurement establishing that the neutrino mass hierarchy is not small, or that the neutrino mixing angles deviate by more than a factor of two from the values implied by the five ε_a parameters fitted to the quark and charged-lepton data, would falsify the SU(5) description.

read the original abstract

A framework, MAFS, is introduced that provides an approximate description of the hierarchies of quark and lepton masses and mixing angles in terms of a set of small parameters, $\epsilon_a$, one for each fermion multiplet. MAFS is an alternative to the Froggatt-Nielsen mechanism and has a unique application in any theory, as there are no fermion charges to choose. It becomes more powerful as the number of multiplets is reduced. In $SU(5)$ unified theories, 15 observed mass ratios and mixing angles are described, at the factor of two level, by five small $\epsilon_a$ parameters. Even though quarks and leptons are unified, the observed hierarchical pattern of quark masses and mixings {\it requires} large neutrino mixing angles and small neutrino mass hierarchies. In an $SO(10)$ unified theory, MAFS successfully describes the 15 observed flavor hierarchies with just three small $\epsilon_a$, taking values of $0.01, 0.02$ and $0.002$. The observed cosmological baryon asymmetry results approximately from leptogenesis using MAFS in $SU(5)$, without the need for any additional small parameter; while in $SO(10)$, a further small parameter of about 0.2 appears necessary.

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 / 0 minor

Summary. The paper introduces the Maximal Abelian Flavor Symmetries (MAFS) framework, an alternative to Froggatt-Nielsen, in which each fermion multiplet is assigned an independent small suppression factor ε_a so that effective mass-matrix entries are products of the relevant ε's. It claims that in SU(5) five such parameters describe 15 observed mass ratios and mixing angles at the factor-of-two level, that the resulting quark pattern forces large neutrino mixing angles and small neutrino-mass hierarchies, that in SO(10) the same 15 hierarchies are described by three ε_a values (0.01, 0.02, 0.002), and that leptogenesis in SU(5) reproduces the observed baryon asymmetry without any additional small parameter.

Significance. If the parameterization is accepted, MAFS supplies a minimal, charge-free description of flavor hierarchies that becomes more constraining as the number of multiplets decreases in unified theories, directly linking the quark and lepton sectors and yielding testable implications for neutrino mixing and leptogenesis from the same small set of ε_a.

major comments (2)
  1. [Abstract] The central results consist of selecting the five (SU(5)) or three (SO(10)) ε_a values to reproduce the observed hierarchies at the factor-of-two level; this reduces the claimed description to a post-hoc parameterization of the input data rather than an independent derivation, as noted in the abstract's statements on the number of parameters and the quoted numerical values.
  2. [Abstract] The assertion that the observed quark pattern 'requires' large neutrino mixing and small neutrino-mass hierarchies follows directly from the unified multiplet assignments once the ε_a are fixed by the quark data, but the loose (factor-of-two) matching tolerance for 15 observables with only five parameters leaves the strength of this requirement unclear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We respond to each major comment below and will revise the abstract accordingly.

read point-by-point responses

  1. Referee: [Abstract] The central results consist of selecting the five (SU(5)) or three (SO(10)) ε_a values to reproduce the observed hierarchies at the factor-of-two level; this reduces the claimed description to a post-hoc parameterization of the input data rather than an independent derivation, as noted in the abstract's statements on the number of parameters and the quoted numerical values.

    Authors: The ε_a values are fitted to the data, but the assignments are fixed by the GUT multiplet structure rather than chosen freely. This imposes relations between sectors that a generic fit would not. We will revise the abstract to clarify that the parameters are determined within this constrained assignment scheme. revision: yes

  2. Referee: [Abstract] The assertion that the observed quark pattern 'requires' large neutrino mixing and small neutrino-mass hierarchies follows directly from the unified multiplet assignments once the ε_a are fixed by the quark data, but the loose (factor-of-two) matching tolerance for 15 observables with only five parameters leaves the strength of this requirement unclear.

    Authors: The referee is correct that the factor-of-two tolerance makes the predictions approximate. We will replace 'requires' with 'leads to' in the abstract to better reflect the approximate nature of the matching. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents MAFS explicitly as a phenomenological parameterization that assigns one small ε_a per fermion multiplet and forms effective mass-matrix entries as products of the relevant ε's. It states that this describes (rather than derives or predicts) the 15 observed hierarchies at the factor-of-two level. The statement that the quark pattern 'requires' large neutrino mixing follows directly from the unified multiplet content once the ε_a are assigned; it is a logical consequence of the parameterization, not an independent first-principles result. No self-citations, uniqueness theorems, or ansatze imported from prior work appear in the provided text. The leptogenesis claim is presented as an approximate outcome using the same ε_a without extra parameters, but remains a consistency check rather than a reduction of the central claim to its inputs by construction. The framework is therefore self-contained as a descriptive tool.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the introduction of independent small ε_a parameters for each multiplet and the assumption that these suffice to organize all flavor data once the GUT multiplet structure is fixed.

free parameters (1)
  • ε_a (one per fermion multiplet)
    Small suppression factors whose specific values (e.g., 0.01, 0.02, 0.002 in SO(10)) are selected to reproduce the 15 observed mass ratios and mixings at factor-of-two accuracy.
axioms (1)
  • domain assumption SU(5) or SO(10) grand unification fixes the fermion multiplet content
    The reduction from 15 observables to 5 or 3 parameters is possible only because the GUT embedding relates quarks and leptons into shared multiplets.

pith-pipeline@v0.9.1-grok · 5750 in / 1479 out tokens · 30873 ms · 2026-06-29T05:42:12.211242+00:00 · methodology

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