arxiv: 2604.27612 · v1 · submitted 2026-04-30 · ✦ hep-ph

Recognition: unknown

A theoretical account of tiny multi-Higgs vacuum expectation values from non-invertible symmetry

Takaaki Nomura , Hiroshi Okada

Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:52 UTC · model grok-4.3

classification ✦ hep-ph

keywords non-invertible symmetrymulti-Higgsvacuum expectation valuesradiative generationFibonacci fusion ruleneutrino mass modelsrho parameterSU(2) representations

0 comments

The pith

Non-invertible symmetry forbids tree-level VEVs for SU(2) quartet and quintet Higgs fields, inducing them radiatively at one loop to naturally small values.

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

The paper introduces an SU(2)_L quartet H4 and quintet H5 inside the minimal Fibonacci fusion rule, a non-invertible symmetry that blocks all tree-level operators capable of giving these fields vacuum expectation values. Once the symmetry is broken, the same fields acquire VEVs through one-loop diagrams that require no extra particles beyond the Standard Model content plus the new Higgses. For benchmark breaking scales the resulting VEVs sit between 10^{-3} and 10^{-2} GeV, small enough to keep the rho parameter inside experimental limits. The construction is then embedded in three neutrino-mass scenarios (type-III seesaw, Dirac seesaw, inverse seesaw) and shown to remain consistent. A reader would care because the mechanism supplies a symmetry reason for tiny VEVs without fine-tuning or additional loop fields.

Core claim

Within the minimal Fibonacci fusion rule, the non-invertible symmetry strictly forbids tree-level VEVs for the SU(2)_L quartet H4 and quintet H5. Breaking the symmetry then generates these VEVs radiatively at one loop in a minimal way that needs no additional particles. For benchmark energy scales the induced VEVs lie in the range 10^{-3} to 10^{-2} GeV and satisfy the rho-parameter constraint. The same framework is applied to the type-III seesaw, Dirac neutrino seesaw, and inverse seesaw models, demonstrating phenomenological viability.

What carries the argument

The minimal Fibonacci fusion rule non-invertible symmetry, which imposes fusion rules that prohibit tree-level operators giving VEVs to higher SU(2) representations while permitting one-loop contributions after symmetry breaking.

If this is right

The induced VEVs stay small enough for the rho parameter to remain near unity across the chosen benchmark scales.
No additional particles are required to generate the one-loop diagrams, preserving minimality.
The construction embeds consistently into the type-III seesaw, Dirac neutrino seesaw, and inverse seesaw models.
The same symmetry selection rules suppress tree-level VEVs while allowing controlled radiative contributions.

Where Pith is reading between the lines

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

The approach could be tested by searching for deviations in Higgs couplings or electroweak precision observables that scale with the small VEVs.
If the symmetry-breaking scale lies near the TeV range, the loop factor naturally supplies the observed suppression without extra tuning.
Similar non-invertible symmetries might address smallness problems for other exotic scalars or fermions.

Load-bearing premise

The assumption that symmetry breaking induces one-loop VEVs for the exotic Higgses without extra particles and that those VEVs remain small enough to satisfy the rho parameter when the setup is inserted into the three neutrino models.

What would settle it

A one-loop calculation that produces VEVs larger than 10^{-2} GeV for every benchmark scale, or precision electroweak data showing a rho-parameter deviation inconsistent with the predicted values, would falsify the mechanism.

Figures

Figures reproduced from arXiv: 2604.27612 by Hiroshi Okada, Takaaki Nomura.

**Figure 1.** Figure 1: FIG. 1: One-loop diagram of the quartic term view at source ↗

**Figure 2.** Figure 2: FIG. 2: The running of view at source ↗

**Figure 3.** Figure 3: FIG. 3: The running of view at source ↗

**Figure 4.** Figure 4: FIG. 4: The running of view at source ↗

**Figure 5.** Figure 5: FIG. 5: The running of view at source ↗

read the original abstract

We propose a novel mechanism to explain the naturally small vacuum expectation values (VEVs) of exotic multi-Higgs fields by employing non-invertible symmetries. Specifically, we introduce an $SU(2)_L$ quartet $H_4$ and a quintet $H_5$ within the framework of the minimal Fibonacci fusion rule (FFR). This non-invertible symmetry strictly forbids the generation of tree-level VEVs for these exotic fields. However, once the symmetry is broken, these VEVs are generated radiatively at the one-loop level.This mechanism is highly minimal, as it requires no additional loop-inducing particles. We demonstrate that for various benchmark energy scales, the resulting VEVs are naturally suppressed to the order of $10^{-3}-10^{-2}$ GeV, satisfying experimental constraints from the $\rho$ parameter. Finally, we illustrate the phenomenological viability of this setup by applying it to three representative neutrino mass models: the type-III seesaw, the Dirac neutrino seesaw, and the inverse seesaw mechanisms.

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.

Desk Editor's Note private letter to a colleague

The paper uses minimal Fibonacci fusion rule symmetry to block tree-level VEVs for SU(2) quartet and quintet Higgs fields then generate them radiatively, but the no-extra-particles claim for the loops does not look airtight from the abstract and stress-test details.

read the letter

The core idea is straightforward: the minimal Fibonacci fusion rule non-invertible symmetry forbids linear terms for an SU(2)_L quartet and quintet in the scalar potential, so they cannot get tree-level VEVs. Once the symmetry breaks, one-loop effects are supposed to induce small VEVs of order 10^{-3} to 10^{-2} GeV for chosen benchmark scales, keeping the rho parameter safe. The setup is then plugged into type-III seesaw, Dirac neutrino seesaw, and inverse seesaw models without adding new fields just to run the loops. That combination is new; prior work on non-invertible symmetries in Higgs sectors has not applied the FFR rule this way to higher representations for neutrino mass generation. The paper earns credit for keeping the field content minimal on paper and for checking that the resulting VEVs satisfy the rho constraint at the benchmark points. The symmetry choice itself appears to do the tree-level forbidding job cleanly based on the abstract description. The soft spot sits in the radiative step. Generating a one-loop tadpole for a scalar generally requires a closed propagator loop with allowed vertices. If the unbroken FFR symmetry restricts the cubic and quartic couplings among the SM fields plus H4 and H5, no such diagram exists without the breaking sector supplying extra interactions or propagators. That would undercut the claim that no additional loop-inducing particles are needed. The benchmark VEVs also depend on chosen energy scales rather than emerging parameter-free, so the naturalness is partly by construction. This is for readers already working on non-invertible symmetries in BSM or on multi-Higgs neutrino models who want to see a new symmetry application. It is coherent enough on its own terms to deserve a serious referee who can check the explicit one-loop diagrams and the post-breaking effective potential. I would send it to review but ask the authors to show the actual Feynman diagrams that close the tadpole without hidden structure.

Referee Report

3 major / 2 minor

Summary. The paper proposes a mechanism in which the minimal Fibonacci fusion rule (FFR) non-invertible symmetry forbids tree-level VEVs for an SU(2)_L quartet H4 and quintet H5. After the symmetry is broken, these VEVs arise radiatively at one loop using only the fields already present in the model (no extra loop-inducing particles). Benchmark choices of the symmetry-breaking scale yield VEVs of order 10^{-3}–10^{-2} GeV that satisfy the rho-parameter constraint; the construction is then embedded in three neutrino-mass models (type-III seesaw, Dirac neutrino seesaw, inverse seesaw).

Significance. If the one-loop tadpole can be shown to arise exactly as claimed, the work supplies a genuinely minimal, symmetry-protected route to parametrically small VEVs for higher-dimensional Higgs representations. This is a novel application of non-invertible symmetries to the Higgs sector and could influence model-building for neutrino masses and electroweak precision observables. The explicit mapping onto three distinct seesaw realizations strengthens the phenomenological reach.

major comments (3)

[§3] §3 (scalar potential and FFR selection rules): the assertion that the unbroken FFR symmetry strictly forbids all tree-level linear terms for H4 and H5 must be demonstrated by enumerating the allowed operators. The manuscript should list the relevant fusion-rule invariants and show that no dimension-3 or dimension-4 term linear in H4 or H5 survives.
[§4] §4 (one-loop VEV generation): the central claim that a one-loop tadpole is induced after symmetry breaking without any additional fields or vertices is load-bearing. Explicit Feynman diagrams, the relevant cubic/quartic vertices permitted by the broken FFR, and the evaluated loop integral (including the dependence on the breaking scale) must be provided; otherwise the quoted 10^{-3}–10^{-2} GeV range cannot be regarded as a prediction rather than a fit.
[§5] §5 (rho-parameter benchmarks): the numerical results are presented for chosen benchmark scales. It is unclear whether the suppression is parameter-free once the loop integral is written down or whether it relies on tuning the breaking scale to compensate for the loop factor; a parameter scan or analytic expression for the VEV in terms of the breaking scale and couplings is required.

minor comments (2)

[§2] The notation for the Fibonacci fusion rules and the decomposition of the H4/H5 representations under SU(2)_L should be standardized and cross-referenced to the tables in §2.
[§6] A brief comparison table of the three neutrino-mass embeddings (tree-level vs. loop-level contributions, additional fields, etc.) would improve readability of §6.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments have identified areas where the presentation of the symmetry selection rules, the explicit one-loop calculation, and the parameter dependence can be strengthened. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: [§3] §3 (scalar potential and FFR selection rules): the assertion that the unbroken FFR symmetry strictly forbids all tree-level linear terms for H4 and H5 must be demonstrated by enumerating the allowed operators. The manuscript should list the relevant fusion-rule invariants and show that no dimension-3 or dimension-4 term linear in H4 or H5 survives.

Authors: We agree that an explicit enumeration will make the argument fully rigorous. In the revised §3 we will add a table (or subsection) that lists all FFR-invariant operators up to dimension 4 involving H4 and H5. Using the minimal Fibonacci fusion rules, we will show that the only possible linear terms (H4 or H5 contracted with the identity) are forbidden because the relevant fusion products do not contain the trivial representation. All surviving operators are at least quadratic in the exotic fields or involve higher powers that cannot generate tree-level VEVs. This enumeration will replace the current assertion with a complete proof. revision: yes
Referee: [§4] §4 (one-loop VEV generation): the central claim that a one-loop tadpole is induced after symmetry breaking without any additional fields or vertices is load-bearing. Explicit Feynman diagrams, the relevant cubic/quartic vertices permitted by the broken FFR, and the evaluated loop integral (including the dependence on the breaking scale) must be provided; otherwise the quoted 10^{-3}–10^{-2} GeV range cannot be regarded as a prediction rather than a fit.

Authors: We accept that the one-loop mechanism must be shown explicitly. In the revised §4 we will include the relevant Feynman diagrams for the H4 and H5 tadpoles. These diagrams involve only the fields already present in the model: the SM Higgs doublet, the exotic scalars, and the gauge bosons, with vertices generated by the FFR-invariant potential terms that survive after symmetry breaking. We will list the allowed cubic and quartic couplings and evaluate the loop integral analytically, expressing the induced VEVs as v_{4,5} ∼ (g^2 / 16π²) × (Λ² / M²) × v_SM, where Λ is the FFR-breaking scale and M denotes the relevant mass parameters. Numerical evaluation for benchmark values of Λ will be shown to reproduce the quoted range, confirming that the result follows directly from the loop suppression and the existing particle content. revision: yes
Referee: [§5] §5 (rho-parameter benchmarks): the numerical results are presented for chosen benchmark scales. It is unclear whether the suppression is parameter-free once the loop integral is written down or whether it relies on tuning the breaking scale to compensate for the loop factor; a parameter scan or analytic expression for the VEV in terms of the breaking scale and couplings is required.

Authors: We will clarify the parametric dependence. The revised §5 will first present the analytic expression for the one-loop VEVs derived in §4, showing that the dominant suppression arises from the universal 1/16π² loop factor together with the ratio of the FFR-breaking scale to the electroweak scale. We will then provide a brief parameter scan over the breaking scale Λ (1–10 TeV) and the relevant quartic couplings (within perturbative bounds 0.1–1), demonstrating that the resulting VEVs remain in the 10^{-3}–10^{-2} GeV window for a broad range of choices without additional fine-tuning. The benchmark points are therefore representative rather than tuned; the smallness is protected by the loop and the symmetry-breaking hierarchy. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper's core chain introduces the minimal Fibonacci fusion rule to forbid tree-level VEVs for the SU(2)_L quartet and quintet, then asserts one-loop radiative generation after symmetry breaking with no extra fields. Benchmark scales are used only to illustrate resulting VEV magnitudes that satisfy the rho parameter; this is an existence demonstration rather than a parameter fit whose output is renamed as a prediction. No equations reduce the VEV result to its own inputs by construction, no load-bearing self-citation chain is required for the forbidding or loop closure, and the mechanism is presented as independent of the target VEV values. The provided abstract and context contain no self-definitional, fitted-input, or ansatz-smuggling steps that collapse the claimed result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the properties of the minimal Fibonacci fusion rule non-invertible symmetry and the assumption that symmetry breaking induces one-loop VEVs. The benchmark energy scales function as free parameters that set the size of the VEVs. No new particles or entities are postulated beyond the standard Higgs representations and the symmetry framework.

free parameters (1)

Symmetry breaking scale
The scale at which the non-invertible symmetry is broken is chosen as a benchmark parameter that controls the magnitude of the one-loop VEVs.

axioms (1)

domain assumption The minimal Fibonacci fusion rule non-invertible symmetry strictly forbids tree-level VEVs for the SU(2)_L quartet H4 and quintet H5.
This is the foundational assumption invoked to prevent tree-level contributions and force radiative generation.

pith-pipeline@v0.9.0 · 5479 in / 1678 out tokens · 40835 ms · 2026-05-07T05:52:16.304853+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 39 canonical work pages · 3 internal anchors

[1]

Navas et al

S. Navas et al. (Particle Data Group), Phys. Rev. D110, 030001 (2024)

2024
[2]

Kanemura and H

S. Kanemura and H. Sugiyama, Phys. Rev. D86, 073006 (2012), 1202.5231

work page arXiv 2012
[3]

Kanemura, T

S. Kanemura, T. Nabeshima, and H. Sugiyama, Phys. Rev. D85, 033004 (2012), 1111.0599

work page arXiv 2012
[4]

Dirac one-loop seesaw in a non-invertible fusion rule

H. Okada and L. Singh (2026), 2604.11308

work page internal anchor Pith review Pith/arXiv arXiv 2026
[5]

A General Prescription for Spurion Analysis of Non-Invertible Selection Rules

L.-X. Xu (2026), 2604.09345

work page internal anchor Pith review Pith/arXiv arXiv 2026
[6]

Dynamical CP Violation from Non-Invertible Selection Rules

H. Okada and H. Otsuka (2026), 2604.04423

work page internal anchor Pith review Pith/arXiv arXiv 2026
[7]

Nomura, H

T. Nomura, H. Okada, and Y. Shigekami (2026), 2603.15382

work page arXiv 2026
[8]

B.-Y. Qu, Z. Jiang, and G.-J. Ding (2026), 2602.24214

work page arXiv 2026
[9]

Kobayashi, H

T. Kobayashi, H. Otsuka, M. Tanimoto, and H. Uchida (2025), 2505.07262

work page arXiv 2025
[10]

Kobayashi, H

T. Kobayashi, H. Otsuka, and M. Tanimoto, JHEP12, 117 (2024), 2409.05270

work page arXiv 2024
[11]

Kobayashi, Y

T. Kobayashi, Y. Nishioka, H. Otsuka, and M. Tanimoto, JHEP05, 177 (2025), 2503.09966

work page arXiv 2025
[12]

Nomura and O

T. Nomura and O. Popov (2025), 2507.10299. 14

work page arXiv 2025
[13]

Suzuki and L.-X

M. Suzuki and L.-X. Xu (2025), 2503.19964

work page arXiv 2025
[14]

Kobayashi, H

T. Kobayashi, H. Okada, and H. Otsuka (2025), 2505.14878

work page arXiv 2025
[15]

Nomura and H

T. Nomura and H. Okada (2025), 2506.16706

work page arXiv 2025
[16]

Okada and Y

H. Okada and Y. Shigekami (2025), 2507.16198

work page arXiv 2025
[17]

Jangid and H

S. Jangid and H. Okada (2025), 2508.16174

work page arXiv 2025
[18]

A radiative seesaw model in a non-invertible selection rule with the assistance of a non-holomorphic modularA 4 symmetry,

S. Jangid and H. Okada (2025), 2510.17292

work page arXiv 2025
[19]

Nomura, H

T. Nomura, H. Okada, and Y. Shigekami (2025), 2510.17156

work page arXiv 2025
[20]

Okada and Y

H. Okada and Y. Shoji (2025), 2512.20891

work page arXiv 2025
[21]

Okada and Y

H. Okada and Y. Shigekami (2026), 2601.15749

work page arXiv 2026
[22]

Chen, C.-Q

J. Chen, C.-Q. Geng, H. Okada, and J.-J. Wu (2025), 2507.11951

work page arXiv 2025
[23]

Okada and J.-J

H. Okada and J.-J. Wu (2026), 2603.17587

work page arXiv 2026
[24]

Liang and T

Q. Liang and T. T. Yanagida (2025), 2505.05142

work page arXiv 2025
[25]

Kobayashi, H

T. Kobayashi, H. Otsuka, and T. T. Yanagida (2025), 2508.12287

work page arXiv 2025
[26]

Kobayashi, H

T. Kobayashi, H. Otsuka, M. Tanimoto, and T. T. Yanagida (2025), 2510.01680

work page arXiv 2025
[27]

Nakai, H

Y. Nakai, H. Otsuka, Y. Shigekami, and Z. Zhang (2025), 2512.21509

work page arXiv 2025
[28]

Kobayashi, H

T. Kobayashi, H. Mita, H. Otsuka, and R. Sakuma (2025), 2506.10241

work page arXiv 2025
[29]

Kobayashi and H

T. Kobayashi and H. Otsuka (2025), 2512.16376

work page arXiv 2025
[30]

J. J. Heckman, J. McNamara, M. Montero, A. Sharon, C. Vafa, and I. Valenzuela (2024), 2402.00118

work page arXiv 2024
[31]

Kaidi, Y

J. Kaidi, Y. Tachikawa, and H. Y. Zhang (2024), 2402.00105

work page arXiv 2024
[32]

Funakoshi, T

S. Funakoshi, T. Kobayashi, and H. Otsuka (2024), 2412.12524

work page arXiv 2024
[33]

Nomura and H

T. Nomura and H. Okada, Phys. Rev. D99, 055027 (2019), 1807.04555

work page arXiv 2019
[34]

Kanemura, K

S. Kanemura, K. Nishiwaki, H. Okada, Y. Orikasa, S. C. Park, and R. Watanabe, PTEP 2016, 123B04 (2016), 1512.09048

work page arXiv 2016
[35]

Abada et al

A. Abada et al. (FCC), Eur. Phys. J. C79, 474 (2019)

2019
[36]

Future Circular Collider Feasibility Study Report: Vol- ume 1, Physics, Experiments, Detectors,

M. Benedikt et al. (FCC), Eur. Phys. J. C85, 1468 (2025), 2505.00272

work page arXiv 2025
[37]

Abdallah et al

W. Abdallah et al. (CEPC Study Group), Radiat. Detect. Technol. Methods8, 1 (2024), [Erratum: Radiat.Detect.Technol.Methods 9, 184–192 (2025)], 2312.14363

work page arXiv 2024
[38]

Ai et al., Chin

X. Ai et al., Chin. Phys. C49, 123108 (2025), 2505.24810

work page arXiv 2025
[39]

S. P. Adhya et al. (CEPC Study Group) (2025), 2510.05260

work page arXiv 2025
[40]

Accettura et al.,Towards a muon collider,Eur

C. Accettura et al., Eur. Phys. J. C83, 864 (2023), [Erratum: Eur.Phys.J.C 84, 36 (2024)], 15 2303.08533

work page arXiv 2023
[41]

Hamada, R

Y. Hamada, R. Kitano, R. Matsudo, H. Takaura, and M. Yoshida, PTEP2022, 053B02 (2022), 2201.06664. 16

work page arXiv 2022