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arxiv: 2606.07799 · v1 · pith:MIV6BOWLnew · submitted 2026-06-05 · ✦ hep-ph

Maximal Abelian Flavor Symmetries

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

Pith reviewed 2026-06-27 21:21 UTC · model grok-4.3

classification ✦ hep-ph
keywords flavor hierarchiesunified theoriesmass ratiosneutrino mixingleptogenesisSU(5)SO(10)
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The pith

In SU(5) and SO(10) theories a handful of small parameters per fermion multiplet reproduce all observed quark and lepton mass ratios and mixing angles to within a factor of two.

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

The paper introduces MAFS, a framework that assigns one small parameter ε_a to each fermion multiplet and builds all mass and mixing hierarchies from products of these parameters. In SU(5) unified models the approach accounts for fifteen measured quantities using five such parameters. In SO(10) the same fifteen quantities are accounted for with only three parameters whose values are 0.01, 0.02 and 0.002. The framework forces the observed quark pattern to imply large neutrino mixing angles together with a small hierarchy among neutrino masses. It also shows that the cosmological baryon asymmetry arises from leptogenesis in the SU(5) case with no extra small parameter required.

Core claim

MAFS describes the hierarchies of quark and lepton masses and mixing angles by a product of independent small parameters ε_a, one assigned to each fermion multiplet. In an SU(5) theory five such parameters suffice to match the fifteen observed quantities at the factor-of-two level; in an SO(10) theory three parameters of size 0.01, 0.02 and 0.002 achieve the same description. The resulting pattern automatically requires large neutrino mixing angles and only a mild neutrino mass hierarchy, while leptogenesis in SU(5) yields the observed baryon asymmetry without any further small parameter.

What carries the argument

The set of small parameters ε_a, one per fermion multiplet, whose products generate all mass ratios and mixing angles.

If this is right

  • The pattern of quark masses and mixings forces neutrino mixing angles to be large and the neutrino mass hierarchy to be small.
  • Leptogenesis in the SU(5) case produces the observed baryon asymmetry with no additional small parameter.
  • In SO(10) an extra parameter of order 0.2 is needed to match the baryon asymmetry via leptogenesis.
  • Reducing the number of multiplets makes the description more predictive because fewer ε_a parameters are available.

Where Pith is reading between the lines

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

  • The same ε_a values that fit the charged fermions also fix the scale of the lightest neutrino mass once the overall seesaw scale is chosen.
  • Any high-scale completion that generates the ε_a must simultaneously suppress all other possible operators that would spoil the observed hierarchies.
  • Extending the counting to include the three right-handed neutrinos would require checking whether their ε parameters can be chosen consistently with the existing three-parameter fit.

Load-bearing premise

The observed hierarchies can be reproduced to within a factor of two by multiplying independent small parameters assigned separately to each fermion multiplet, without any further charge assignments or symmetry structure.

What would settle it

A precise measurement showing that the ratio of the two heaviest neutrino masses lies outside the narrow range predicted once the three ε_a values are fixed by the quark and charged-lepton data.

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 / 2 minor

Summary. The manuscript introduces the Maximal Abelian Flavor Symmetries (MAFS) framework, which assigns one small parameter ε_a to each fermion multiplet to approximately describe the hierarchies of 15 quark and lepton mass ratios and mixing angles. In SU(5) GUTs five such parameters suffice at the factor-of-two level; the observed quark pattern is stated to require large neutrino mixing angles and small neutrino mass hierarchies. In SO(10) the same 15 hierarchies are described by three parameters with values 0.01, 0.02 and 0.002. Leptogenesis is claimed to reproduce the baryon asymmetry approximately in SU(5) without extra parameters.

Significance. If the mapping from the ε_a set to the Yukawa matrices is uniquely fixed by the symmetry and reproduces the data at the stated level without additional structure, the framework would constitute a compact, predictive alternative to Froggatt-Nielsen mechanisms, with the notable feature that the quark sector forces large PMNS mixing. The reduction to three parameters in SO(10) and the approximate leptogenesis result would be strengths.

major comments (2)
  1. [MAFS mass-matrix construction (section describing the Yukawa matrices in SU(5) and SO(10))] The explicit functional dependence that determines each entry of the Yukawa matrices from the ε_a parameters (e.g., whether every M_ij scales as ε_i ε_j, ε_i^2, or a selected subset of products) is not shown to be fixed uniquely by the maximal Abelian symmetry. This choice is load-bearing for the central claim that the quark hierarchies automatically require large neutrino mixing, because different assignments can alter the predicted PMNS angles while preserving the same ε values.
  2. [SU(5) and SO(10) numerical results (sections presenting the fits)] The factor-of-two agreement for all 15 observables must be demonstrated with an explicit table listing the fitted ε_a values, the resulting predictions for every mass ratio and mixing angle, and the experimental values, so that it can be verified that the agreement is not achieved by post-hoc selection of which entries receive which ε products.
minor comments (2)
  1. The abstract states that leptogenesis works 'approximately' in SU(5); a short paragraph or equation showing the relevant CP asymmetry and washout factors would make the claim easier to assess.
  2. Clarify whether the three numerical values quoted for the SO(10) ε_a are obtained from a global fit or chosen by hand to illustrate the framework.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [MAFS mass-matrix construction (section describing the Yukawa matrices in SU(5) and SO(10))] The explicit functional dependence that determines each entry of the Yukawa matrices from the ε_a parameters (e.g., whether every M_ij scales as ε_i ε_j, ε_i^2, or a selected subset of products) is not shown to be fixed uniquely by the maximal Abelian symmetry. This choice is load-bearing for the central claim that the quark hierarchies automatically require large neutrino mixing, because different assignments can alter the predicted PMNS angles while preserving the same ε values.

    Authors: In the MAFS construction the maximal Abelian symmetry is realized by assigning a single suppression parameter ε_a to each fermion multiplet; the resulting Yukawa matrix entries are then fixed to scale as the product of the two ε parameters belonging to the fields that enter the operator (M_ij ∝ ε_i ε_j). This product form follows directly from the Abelian character of the symmetry and the requirement that it be maximal (no additional U(1) charges or selective higher powers are introduced). The same rule applies uniformly to all sectors, which is why the observed quark hierarchies force large neutrino mixing angles. We recognize that the manuscript did not contain an explicit derivation of this functional dependence from the symmetry and will add a short dedicated paragraph in the revised version to make the mapping transparent. revision: yes

  2. Referee: [SU(5) and SO(10) numerical results (sections presenting the fits)] The factor-of-two agreement for all 15 observables must be demonstrated with an explicit table listing the fitted ε_a values, the resulting predictions for every mass ratio and mixing angle, and the experimental values, so that it can be verified that the agreement is not achieved by post-hoc selection of which entries receive which ε products.

    Authors: We agree that an explicit table is the clearest way to demonstrate the claimed level of agreement and to confirm that no post-hoc choices were made. The revised manuscript will contain a single table (or two side-by-side tables) that lists, for both SU(5) and SO(10), the fitted ε_a values, the predicted value of each of the 15 observables, and the corresponding experimental central values (with the factor-of-two tolerance indicated). revision: yes

Circularity Check

0 steps flagged

No significant circularity in MAFS framework

full rationale

The paper defines MAFS as a parametrization assigning one ε_a per fermion multiplet and constructing Yukawa matrices from these parameters to approximate observed hierarchies at the factor-of-two level. The ε_a are fitted to the 15 quark and lepton observables in SU(5) or SO(10), with the same construction applied uniformly across sectors due to unified multiplet assignments. The claim that the quark pattern requires large neutrino mixing follows directly from sharing the fitted ε_a values across quark and lepton matrices in the unified theory, rather than from any self-definitional loop, fitted-input renaming, or load-bearing self-citation. No equations or steps in the provided text reduce the central result to its inputs by construction; the framework is self-contained as an alternative parametrization to Froggatt-Nielsen with explicit parameter counting and no external uniqueness theorems invoked.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the introduction of one small ε_a per fermion multiplet whose values are chosen to reproduce the observed hierarchies; these parameters function as free parameters fitted to data.

free parameters (1)
  • ε_a (one per fermion multiplet)
    Small suppression factors introduced to generate the mass and mixing hierarchies; their specific values are adjusted to match the 15 observables.

pith-pipeline@v0.9.1-grok · 5750 in / 1209 out tokens · 16330 ms · 2026-06-27T21:21:46.666679+00:00 · methodology

discussion (0)

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Reference graph

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