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arxiv: 2606.25027 · v1 · pith:BIWR7WF4new · submitted 2026-06-23 · ✦ hep-ph · hep-th

Fermion mass relations in one-parameter modular models

Salvador Centelles Chuli\'a , Xueqi Li , Xiang-Gan Liu , Omar Medina , J. T. Penedo This is my paper

Pith reviewed 2026-06-25 22:57 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords modular symmetryfermion mass relationsflavour modelsone-parameter modelscharged fermionsrenormalization group evolutionsupersymmetric thresholds
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The pith

In the one-parameter modular limit, a shared modulus fixes exact high-scale mass relations between down quarks and charged leptons.

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

The paper examines the extreme one-parameter modular limit in which each charged-fermion mass matrix is fixed by a single modular invariant contraction controlled by one complex modulus. In a concrete realization, the charged-lepton and down-quark sectors share this modulus and therefore obey three exact relations at the flavour scale: m_s^5 equals 2 times the square root of 2 times m_d cubed times m_b squared, m_mu cubed equals the square root of 2 times m_e times m_tau squared, and m_s squared times m_tau equals the square root of 2 times m_e times m_b squared. These high-scale relations become compatible with measured low-energy masses once renormalization-group evolution and selected supersymmetric threshold corrections are included. The construction demonstrates that the one-parameter requirement already imposes strong constraints on possible fermion hierarchies.

Core claim

In the extreme OPM limit each charged-fermion mass matrix arises from exactly one modular invariant contraction. When the charged-lepton and down-quark sectors are controlled by the same modulus, the resulting mass matrices at the flavour scale satisfy the three exact relations m_s^5 = 2√2 m_d^3 m_b^2, m_μ^3 = √2 m_e m_τ^2 and m_s^2 m_τ = √2 m_e m_b^2. These relations remain consistent with low-energy data after renormalization-group running and selective supersymmetric threshold effects are taken into account.

What carries the argument

The one-parameter modular (OPM) limit, in which each charged-fermion mass matrix is fixed by a single modular invariant contraction of one complex modulus.

If this is right

  • The three mass relations hold exactly at the flavour scale before any running occurs.
  • Renormalization-group evolution from the flavour scale down to the electroweak scale must be included to compare predictions with data.
  • Selective supersymmetric threshold corrections can adjust the running to restore compatibility with observed masses.
  • The up-quark sector is not constrained by the same common modulus and may be treated separately.

Where Pith is reading between the lines

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

  • The relations link quark and lepton masses without extra free parameters beyond the single modulus.
  • Precision flavour measurements at future facilities could test whether the high-scale values satisfy the predicted ratios after accounting for running.
  • Similar one-parameter structures might be applied to the neutrino sector to generate additional testable predictions.

Load-bearing premise

The model must realize the extreme limit in which every charged-fermion mass matrix is fixed by exactly one modular invariant contraction while keeping modular invariance across sectors.

What would settle it

A calculation that evolves the measured low-energy masses upward, including all relevant thresholds, and finds that the resulting high-scale values violate any of the three stated relations outside combined experimental and theoretical uncertainties.

Figures

Figures reproduced from arXiv: 2606.25027 by J. T. Penedo, Omar Medina, Salvador Centelles Chuli\'a, Xiang-Gan Liu, Xueqi Li.

Figure 1
Figure 1. Figure 1: FIG. 1: RG evolution of the experimentally extracted Yukawa ratios from [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Running of the mass relations in eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Low-energy predictions of the OPM before finite SUSY threshold corrections. The curves show the [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: RG evolution of the rescaled Yukawa ratios from [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: High-scale hierarchy plane illustrating the non-triviality of the one-parameter predictions and the [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

Modular flavour symmetries provide a possible organizing principle for the Standard Model Yukawa sector, by replacing generic couplings with a potentially small number of modular forms controlled by a single complex modulus. We study the extreme limit of this idea: \acp{OPM}, in which each charged-fermion mass matrix is fixed by a single modular invariant contraction. We develop a systematic method to construct such models, showing that the \ac{OPM} requirement is already highly constraining at the level of possible fermion hierarchies. In a concrete realization, the charged-lepton and down-quark sectors are controlled by the common modulus, leading to exact mass relations at the flavour scale, \[ m_s^5 = 2\sqrt{2}\,m_d^3m_b^2, \qquad m_\mu^3 = \sqrt{2}\,m_e m_\tau^2, \qquad m_s^2m_\tau = \sqrt{2}\,m_e m_b^2. \] We show that, once renormalization-group evolution and selective supersymmetric threshold effects are included, these high-scale relations can be made compatible with low-energy charged-fermion data. Our results provide a working proof of principle for \acp{OPM} and point towards a possible route to the flavour puzzle through highly

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 / 1 minor

Summary. The manuscript develops a systematic approach to one-parameter modular (OPM) models in which each charged-fermion mass matrix arises from exactly one modular invariant contraction. In a concrete realization with a shared modulus for the down-quark and charged-lepton sectors, this yields exact high-scale mass relations m_s^5 = 2√2 m_d^3 m_b^2, m_μ^3 = √2 m_e m_τ^2 and m_s^2 m_τ = √2 m_e m_b^2. The authors show that renormalization-group evolution together with selective supersymmetric threshold corrections can bring these relations into agreement with low-energy data.

Significance. If the OPM construction is internally consistent, the work supplies a concrete proof-of-principle that modular symmetries can produce parameter-free fermion mass relations controlled by a single complex modulus. The explicit demonstration that the high-scale relations survive RG running and threshold effects adds phenomenological relevance. The systematic construction method itself is a useful contribution to the modular-flavour literature.

major comments (2)
  1. [concrete realization (section containing the explicit model)] The central claim that the quoted mass relations follow directly from the single-contraction OPM requirement rests on the viability of assigning modular weights such that each of the down-quark and lepton mass matrices is fixed by precisely one invariant while sharing the same modulus τ. The manuscript must supply the explicit weight assignments and verify that no additional contractions are permitted by modular invariance; without this verification the exact numerical coefficients (including the factor 2√2) cannot be confirmed to arise solely from the OPM condition.
  2. [concrete realization] The up-quark sector is not included in the quoted relations. If the model is intended to be complete, the same single-contraction constraint must be satisfied simultaneously for the up sector with the identical modulus; otherwise the claim that the framework is realized with a common τ for all charged fermions is incomplete.
minor comments (1)
  1. [RG evolution discussion] The abstract refers to 'selective supersymmetric threshold effects' without specifying which thresholds are retained or suppressed; a brief table or paragraph listing the retained thresholds would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work's significance and for the constructive major comments. We address each point below, indicating the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [concrete realization (section containing the explicit model)] The central claim that the quoted mass relations follow directly from the single-contraction OPM requirement rests on the viability of assigning modular weights such that each of the down-quark and lepton mass matrices is fixed by precisely one invariant while sharing the same modulus τ. The manuscript must supply the explicit weight assignments and verify that no additional contractions are permitted by modular invariance; without this verification the exact numerical coefficients (including the factor 2√2) cannot be confirmed to arise solely from the OPM condition.

    Authors: We agree that explicit verification is essential. The concrete realization section of the manuscript already specifies the modular weights that enforce single contractions, but we will expand this in the revision by adding a dedicated table listing all field weights together with a short proof that modular invariance forbids additional independent contractions for each matrix entry. This will directly confirm the numerical prefactors in the mass relations. revision: yes

  2. Referee: [concrete realization] The up-quark sector is not included in the quoted relations. If the model is intended to be complete, the same single-contraction constraint must be satisfied simultaneously for the up sector with the identical modulus; otherwise the claim that the framework is realized with a common τ for all charged fermions is incomplete.

    Authors: The manuscript explicitly limits the concrete realization to the down-quark and charged-lepton sectors sharing a common modulus, as stated in the abstract. No claim is made that the up-quark sector must employ the identical τ. The up sector can be realized separately within the OPM framework (possibly with its own modulus). We will insert a clarifying sentence in the revised text to restate the intended scope and note that a fully unified three-sector construction lies outside the present proof-of-principle study. revision: partial

Circularity Check

0 steps flagged

No circularity: mass relations derived from modular invariance and OPM construction

full rationale

The paper constructs OPM models where each charged-fermion mass matrix is fixed by a single modular invariant contraction. In the concrete realization with shared modulus for down-quarks and leptons, the quoted mass relations follow directly as algebraic consequences of that single-contraction requirement and modular invariance. These are presented as exact high-scale outputs of the model, not as fits to data. Compatibility with low-energy observables is checked via standard RG evolution and threshold corrections, which are external to the model. No self-citations, fitted inputs renamed as predictions, or self-definitional steps appear in the derivation chain. The central claim therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The construction rests on the domain assumption of modular invariance for the Yukawa sector and the modeling choice that each mass matrix uses exactly one modular invariant contraction; no new particles or forces are introduced.

free parameters (1)
  • complex modulus tau
    Single complex parameter that sets the value of all modular forms in the model.
axioms (1)
  • domain assumption Yukawa couplings are modular forms of definite weight under a modular group
    Standard assumption of modular flavour models invoked to replace generic couplings.

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Reference graph

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