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A systematic catalogue of realisable symmetries in three-Higgs-doublet models, including newly studied GOOFy transformations, is presented as a practical reference for model builders.

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2026-08-03 17:51 UTC pith:ACGDSAML

load-bearing objection Careful, honest 3HDM symmetry catalogue that fixes the U(1)×D4 label and adds new GOOFy/T-GOOFy sections; referee it, but push back on the scan-based 'most complete' phrasing. the 4 major comments →

arxiv 2512.07657 v3 pith:ACGDSAML submitted 2025-12-08 hep-ph hep-th

Systematic analysis of 3HDM symmetries

A. Kun\v{c}inas , P. Osland , M. N. Rebelo This is my paper
classification hep-ph hep-th
keywords 3HDMHiggs family symmetriesgeneral CP transformationsGOOFy symmetriesT-GOOFy transformationsscalar potential classificationbilinear formalismdiscrete symmetries
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper aims to provide the most complete classification to date of realisable symmetries in three-Higgs-doublet models (3HDMs). It revisits conventional Higgs-family and general-CP transformations, identifies limitations in previous approaches, and compiles tables of realisable groups with their coupling counts and bilinear eigenvalue patterns. The authors then extend the analysis to GOOFy transformations, which act differently on Higgs doublets and their conjugates and can flip kinetic-term signs, providing the first systematic study of GOOFy- and T-GOOFy-invariant scalar potentials (restricted to quartic terms). If correct, model builders gain a practical diagnostic reference for identifying the symmetry of any 3HDM scalar potential, including previously overlooked structures such as U(1)∘V4.

Core claim

The central claim is that, by systematically imposing generator families of finite subgroups of U(3) (and continuous limits) on the most general 3HDM scalar potential, one obtains a complete practical catalogue of realisable symmetry groups, summarised in Tables 1–3. The paper also establishes that GOOFy HF-like transformations admit only two unique generator types, Z2 and Z4, and that GCP-like GOOFy transformations likewise reduce to Z2 and Z4, while T-GOOFy transformations (with independent unitary rotations of doublets and conjugates) yield additional distinct quartic potentials that can be organised by bilinear eigenvalue patterns.

What carries the argument

The central machinery is the extended 2N-dimensional Higgs space H = (h_i, h_i*) and the SU(2) bilinear singlets h_ij = h_i† h_j, together with the brute-force procedure of imposing symmetry generators iteratively on the most general 3HDM potential. Bilinear eigenvalue patterns (e.g., 8^1, 2^2 4^1) serve as basis-invariant diagnostics for identifying symmetry classes. For GOOFy/T-GOOFy transformations, the key structural device is the pair of independent unitary blocks U^(1), U^(2) acting on doublets and conjugates, with the requirement that the product of a transformation and its Hermitian conjugate be diagonal with ±1 eigenvalues.

Load-bearing premise

The catalogue's completeness rests on the assumption that the generator families of finite subgroups of U(3) of order below 2000, together with their continuous limits, exhaust all physically realisable 3HDM symmetries — an assumption the authors explicitly state cannot be claimed in a strict mathematical sense.

What would settle it

Finding a 3HDM scalar potential whose symmetry group is not listed in Tables 1–2 and whose bilinear eigenvalue pattern is not among those tabulated would falsify the completeness claim. Concretely, a discrete subgroup of U(3) with order above 36 that yields a new quartic potential, or a GOOFy transformation not expressible in the (U(1), U(2)) block form producing a distinct quartic potential, would invalidate the classification.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Tables 1–3 give model builders a direct way to identify the symmetry of a 3HDM scalar potential by comparing coupling counts and bilinear eigenvalue patterns.
  • The paper consolidates previously scattered results and corrects the earlier U(1)×D4 identification to the central-product structure U(1)∘V4, clarifying when different finite lifts (D4, Q8, P1) give the same potential.
  • GOOFy symmetries can eliminate bilinear terms while leaving quartic couplings intact, yielding Gildener–Weinberg-like scale-invariant scalar sectors and potentials that decouple a Higgs doublet at tree level.
  • The classification reveals that eigenvalue patterns alone do not uniquely identify a symmetry, motivating additional basis-invariant checks.
  • The GOOFy framework points toward new model-building directions, including the possibility of auxiliary scalar fields and RG-stable parameter relations without a conventional symmetry.

Where Pith is reading between the lines

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

  • If the catalogue is taken as complete, it suggests that the space of realisable 3HDM symmetries is closed under the operations considered — a useful constraint for future searches for exotic symmetries.
  • The GOOFy analysis hints that the distinction between 'realisable symmetry' and 'accidental parameter relation' may be more fluid than previously assumed, since GOOFy transformations enforce tree-level relations that are not symmetries of the full Lagrangian in the usual sense.
  • The restriction to quartic terms in the GOOFy study leaves open the question of whether bilinear-sector constraints could be classified systematically for arbitrary U^(1), U^(2) pairs; a general mapping from hij transformations to the block form is not provided.
  • The observation that U(1)∘V4 admits multiple finite lifts selected by the Yukawa sector suggests a general principle: scalar-sector symmetries may define equivalence classes of group extensions, with fermion embeddings choosing a representative.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript presents a systematic, scan-based classification of symmetry groups realisable by the scalar potential of three-Higgs-doublet models. For conventional Higgs-family (HF) and general CP (GCP) transformations, the authors construct invariant potentials by imposing generator families of finite subgroups of U(3) taken from Ref. [52] (order below 2000), and organise the results into Tables 1–3 by numbers of couplings and bilinear eigenvalue patterns. They also analyse GOOFy and T-GOOFy transformations of the block forms (6.11)–(6.12), providing invariant quartic potentials in Sections 6–7 and Appendices D–E, and they correct the earlier U(1)×D4 label by identifying the underlying central-product structure and using cohomology (Appendix B). The authors state explicitly that strict mathematical completeness is not claimed, and that the GOOFy analysis is restricted to the families of Ref. [68].

Significance. The paper has substantial reference value. The explicit scalar potentials and the bilinear eigenvalue-pattern tables give model builders a concrete diagnostic tool, the basis-invariant counting in Appendix A is useful, and the Appendix B analysis of U(1)◦V4 is a genuine improvement over the earlier U(1)×D4 label. The GOOFy sections are, to my knowledge, the first systematic survey of 3HDM scalar potentials invariant under the Ref. [68] block-form transformations. However, the central HF/GCP catalogue is established by an empirical scan rather than by a proof of completeness, and the GOOFy survey is restricted by construction. These caveats should be made explicit in the abstract and conclusions.

major comments (4)
  1. [Sec. 3; Tables 1–2] The central HF/GCP catalogue is presented as the 'most complete overview to date', but its completeness is not proven. The scan starts from generator families of finite subgroups of U(3) of order below 2000 listed in Ref. [52], a reference that itself contains 'special exceptions', and the saturation criterion is the empirical observation that no new potentials appear beyond order 36 (Sec. 3). This does not logically exclude a realisable group of order 37–2000, nor a missing subgroup below 2000. Because Tables 1–2 are offered as a practical diagnostic, an omitted realisable symmetry would break the headline claim. Please either provide a rigorous completeness argument, or explicitly and consistently downgrade the claims to a scan-based catalogue with completeness left open; the abstract and Sec. 8 currently overstate the status.
  2. [Sec. 6.2; Sections 6–7] The GOOFy/T-GOOFy analysis is restricted to transformations of the block forms (6.11)–(6.12) from Ref. [68]. The paper admits in Sec. 6.2 that no general mapping from arbitrary h_ij transformations to these forms is provided, and the T-GOOFy survey is limited to quartic terms. The abstract and Sec. 8 describe this as an expansion of the set of symmetries and a 'first systematic study'; this is defensible only within the stated block-form restriction. Please state the restriction prominently in the abstract and treat the GOOFy part as a survey of the Ref. [68] family, not of generalised symmetries beyond that family.
  3. [Sec. 5; Tables 1–3] There are internal discrepancies in the summary tables. Table 3 lists S3×Z*2 at N≤10 and Z3⋊Z*2 at N≤15 in the GCP column, but neither appears in the GCP column of Table 1 or in the eigenvalue-pattern list of Table 2. The preamble to Table 1 explains some exclusions (real-parameter restrictions), but the Table 3 entries are not footnoted or cross-referenced. Since these tables are meant to be a practical guide, the authors should either list these groups in Tables 1–2 with the relevant eigenvalue patterns, or remove them from Table 3 with an explanation.
  4. [Sec. 6.1] The physical motivation for GOOFy symmetries rests on all-order RG stability, but the paper states that a full proof is expected in a future publication. The present manuscript therefore classifies potentials invariant under transformations that are not symmetries of the kinetic term, with the dynamical stability claim unproven. Please separate the classification (which is well defined) from the RG-stability claim, and either prove or clearly mark the stability assertions as conjectures based on Ref. [16].
minor comments (5)
  1. [Sec. 7.1, after Eq. (7.11)] The claim that V4_G1, V4_G3 and V4_G4 are unitarily inequivalent is asserted without proof. Please give a short argument or cite a computation.
  2. [Sec. 4.1, Eq. (4.10)] The two generators in Eq. (4.10) use phases θ1, θ2 without stating their ranges or independence. A brief specification would let the reader verify that the generated group is indeed O(2)×U(1).
  3. [Sec. 7.1, Eqs. (7.3)–(7.13)] The notation G1, G2, ... is used for both individual generators and generated symmetry groups. Consider using calligraphic labels for groups, or add a sentence disambiguating the notation.
  4. [Sec. 6.3, Table 4] In several rows the 'GHF' sub-column lists the identity matrix, which is confusing. Leaving those cells empty or marking them as 'trivial' would improve readability.
  5. [Sec. 2 and Appendix A] The scan-based 'discarding cases that reproduce previously obtained potentials' step is not described in enough detail to be independently reproduced. Publishing the GAP scripts or a pseudo-code description of the potential-comparison algorithm would strengthen confidence in the catalogue.

Circularity Check

0 steps flagged

No significant circularity: potentials and eigenvalue patterns are computed from imposed generators, with completeness explicitly caveated rather than assumed by construction.

full rationale

The paper's derivation chain is constructive: starting from the general 3HDM potential, invariance under given generators is imposed through eqs. (2.8)-(2.9), producing linear constraints on couplings; the listed potentials and bilinear eigenvalue patterns are outputs of this computation, not inputs. No fitted parameter is later renamed as a prediction, and no group or target potential is defined in terms of the quantity it is supposed to determine. The strongest claim—completeness of Tables 1-2—rests on the external finite-subgroup scan of Ref. [52] plus a saturation observation, but the authors explicitly disclaim strict mathematical completeness ('mathematical completeness cannot be claimed in a strict sense', Sec. 3) and state that no new potentials appear above order 36; this is an acknowledged empirical limitation, not a circular reduction. The GOOFy analysis is similarly restricted by an explicitly stated scope choice: 'we restrict our attention to the GOOFy symmetry transformations explicitly discussed in Ref. [68]' (Sec. 6.2), which narrows the claimed classification rather than smuggling in the conclusion. Self-citations (Refs. [10,12,14,16]) are contextual: Ref. [14] is revisited and corrected by an independent cohomological argument in Appendix B, and Ref. [16] supplies the definition of GOOFy, while the present potentials are newly derived by invariance constraints. No uniqueness theorem from the authors' prior work is invoked to forbid alternatives. Thus there is no exhibited reduction of the results to their inputs, and the paper should be assessed for completeness/correctness risk rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The classification rests on the group-theory infrastructure of Refs [2,9,52] and on the GOOFy framework of Refs [16,68]. No free parameters are fitted to data. The principal added assumptions are (i) that the U(3) subgroup scan up to order 2000 is effectively complete for 3HDM purposes, and (ii) that the unproved RG-stability claim makes GOOFy symmetries physically relevant. Both are flagged in the text.

axioms (6)
  • domain assumption Finite subgroups of U(3) of order <2000 from Ref [52] are a sufficient generator source for realisable 3HDM symmetries
    Used in Section 3 for the HF scan; completeness beyond order 2000 is not proven and the authors concede this.
  • domain assumption The scalar potential is fully determined by the SU(2) bilinear singlets hij, so symmetry constraints can be imposed and read off at the level of hij
    Section 2, eqs. (2.5)–(2.9); standard NHDM formalism for the potential and its transformations.
  • standard math Finite Abelian realisable groups satisfy |G| ≤ 2^{N-1} (Ref [2]); Z4 is the maximal Abelian group for 3HDM
    Invoked in Section 3 to justify Z4 maximality and in Appendix B for U(1)◦V4 reasoning.
  • domain assumption The eigenvalue multiplicity pattern of the bilinear Λ matrix is a reliable basis-invariant symmetry diagnostic (Ref [9])
    Used throughout Sections 4–7 and Table 2 to classify potentials; the authors note it is not a full identifier by itself.
  • standard math Cohomology classification of central extensions (H^2(V4, Z_n)) correctly describes the U(1)◦V4 structure
    Appendix B, eqs. (B.11)–(B.14), used to argue the central-product structure of the previously mislabeled U(1)×D4 symmetry.
  • domain assumption Global-sign-flipping GOOFy transformations yield all-order RG-stable parameter relations
    Section 6.1 states this with proof deferred to a future publication; load-bearing for the physical relevance of GOOFy symmetries, not for the potential classification itself.

pith-pipeline@v1.3.0-alltime-deepseek · 53975 in / 12453 out tokens · 101161 ms · 2026-08-03T17:51:11.921850+00:00 · methodology

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read the original abstract

Symmetries play a crucial role in shaping the structure and predictions of multi-Higgs-doublet models. In three-Higgs-doublet models considerable effort has been put into classifying possible symmetry groups and the conditions for their realisation, yet the completeness of existing classifications remains an open question. In this work, we revisit the problem of identifying realisable symmetries by re-examining conventional Higgs family and general CP transformations from an alternative perspective. Our analysis identifies certain limitations in previous approaches and introduces a clearer, more systematic framework for model builders. We expand our classification by investigating more generalised symmetry structures - the recently identified GOOFy transformations, which act non-trivially on the Higgs doublets and their conjugates. Our analysis consolidates known results, uncovers previously overlooked structures, and expands the set of symmetries in three-Higgs-doublet models, offering both a clearer theoretical foundation and a practical reference for symmetry-based model building.

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Forward citations

Cited by 5 Pith papers

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

  1. Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries

    hep-th 2026-05 unverdicted novelty 6.0

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  2. Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries

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    A spurion-field approach using scale-invariant directions and non-overlapping symmetries constructs all-loop RGIs for bilinear operators in multi-scalar potentials.

  3. The Hilbert Series and the Flavor Invariants of the 3HDM

    hep-th 2026-04 conditional novelty 6.0

    The full multigraded Hilbert series of the 3HDM is computed in closed form, and a basis of SU(3)-invariant operators is constructed up to cubic order in the quartic couplings.

  4. Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries

    hep-th 2026-05 unverdicted novelty 5.0

    Scaling and non-overlapping global symmetries produce RGIs for bilinear operators to all loops via scale-invariant field directions, demonstrated in non-SUSY models including the 2HDM.

  5. The Hilbert Series and the Flavor Invariants of the 3HDM

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    Hilbert series for 3HDM global symmetries is computed with explicit invariants constructed up to cubic order.

Reference graph

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