Pith. sign in

REVIEW 3 major objections 4 minor 75 references

GOOFy-compatible 3HDMs and beyond

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A GOOFy-compatible quadratic sector exists only if the symmetry group admits a sign character realised in $R^* \otimes R$; the 2HDM's $r_0$-stable manifold is an exceptional $\mathrm{SU}(2)$ effect.

desk verdict Solid sign-orbit technique and a clean SU(2) argument, but the abstract's two-condition selection rule is overgeneralized: the paper's own partial sign-flipping examples contradict it as stated. read the letter →

arxiv 2608.05304 v1 pith:PMLS2EHV submitted 2026-08-05 hep-ph hep-th

I. de Medeiros Varzielas , A. Kunčinas This is my paper · ORCID
classification hep-phhep-th PACS 12.60.Fr11.30.-j11.10.Hi
keywords GOOFytransformationsthree-Higgs-doubletmodelmulti-Higgsdoubletmodelsquadraticscalarsectorsign-orbitclassificationrepresentationmismatchrenormalisation-groupstabilitySU(2)symmetrictensor
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper classifies which "GOOFy" transformations—non-standard operations that rotate scalar fields and their conjugates by related but inequivalent unitary matrices, producing a kinetic sign anomaly—can be compatible with a non-vanishing quadratic (mass) sector in three-Higgs-doublet models (3HDMs) and beyond. It proves that such a quadratic sector exists exactly when two conditions hold: the symmetry group must admit a non-trivial $\pm 1$ character, and that character must appear in the tensor product of the scalar representation with its conjugate. Applying this selection rule with a new sign-orbit technique, the authors enumerate the admissible quadratic potentials and show that the famous renormalisation-group stability of the 2HDM $r_0 \to -r_0$ parameter manifold does not generalise to $\mathrm{SU}(N\ge 3)$: the symmetric tensor $d_{abc}$, absent for $N=2$, opens radiative channels that regenerate the forbidden couplings. Overall sign-flipping GOOFy models are generically RG-stable, while partial sign-flipping models are generically radiatively unstable. The classification matters because it tells model-builders which non-standard symmetries can actually stabilise a multi-Higgs potential without introducing massless or tachyonic vacua.

What carries the argument

The main tool is the sign-orbit technique for cyclic groups $\mathbb{Z}_{2m}$: multiplication by the sign character $\chi = \rho_m$ partitions irreducible representations into orbits $O_k$ pairing $\rho_k$ with $\rho_{k+m}$, and a bilinear term is allowed only if the two fields' representations differ by $\chi$, so orbit multiplicities determine which $V_2$ structures survive. The action is packaged in the extended operator $W(U,X,\eta)$ on the doubled field space $H = h \oplus h^*$, where $U$ is the flavour rotation and $X$ is a unitary Hermitian anomaly matrix with eigenvalues $\pm 1$ encoding the mismatch between fields and conjugates; solving $W$-invariance of $Y = \mu^2$ reduces to the character condition above. A second load-bearing identity is $d_{abc}d^{abc} = (N^2-1)(N^2-4)/N$, whose vanishing only for $N=2$ removes the symmetric adjoint contraction that otherwise destabilises isolated $r_0$-type parities under renormalisation.

What would settle it

Compute the full two-loop $\beta$ functions, including gauge contributions, for a partial sign-flipping GOOFy 3HDM such as $W(I,\mathrm{diag}(-1,-1,1),0)$ with $V_2 = \mu_{33}^2 h_{33}$, and check whether the truncated maximal-torus couplings ($\mu_{11}^2$, $\mu_{22}^2$, $\mu_{12}^2$) are regenerated along every RG trajectory; a concrete counterexample with a non-trivial trajectory that keeps those couplings exactly zero through two loops would falsify the claimed generic instability.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a representation-theoretic selection rule: for a GOOFy transformation acting with a character $\chi$ on the conjugate fields, invariance of the quadratic potential requires $R(g)^\dagger Y R(g) = \chi(g) Y$ for all $g$, so $Y$ lives in the $\chi$-component of $R^* \otimes R$. A non-zero $Y$ exists precisely when the group has a non-trivial $\mathbb{Z}_2$-valued character (a $\mathbb{Z}_2$ quotient of its Abelianisation) and that $\chi$ occurs in $R^* \otimes R$. The paper also establishes that the 2HDM's $r_0 \to -r_0$ RG-invariant manifold is not a generic NHDM phenomenon: because $d_{abc}d^{abc} = (N^2-1)(N^2-4)/N$ vanishes only for $N=2$, $\mathrm{SU}(N\ge 3)$ algebras admit symmetric invariant contractions that generate $r_0$-odd couplings already at one loop. For the 3HDM, the sign-orbit construction classifies all realisable cyclic embeddings and gives the complete list of GOOFy-compatible quadratic structures with the quartic potentials attached to them; overall sign flips are radiatively stable, while partial sign flips are generically not.

Load-bearing premise

The argument that partial sign-flipping GOOFy models are radiatively unstable rests on the unproven premise that the maximal-torus $U(1)^{N-1}$-invariant sector is closed under the renormalisation-group flow.

Editorial extensions

If this is right

  • In a 3HDM, GOOFy quadratic sectors can only be built on groups such as $\mathbb{Z}_2$, $\mathbb{Z}_4$, $S_3$, $D_4$, $O(2)$ and $O(2)\times U(1)$; groups like $A_4$, $S_4$, odd cyclic groups and continuous $U(1)$ force $V_2 = 0$ entirely.
  • The 2HDM result that $m_{11}^2 + m_{22}^2 = 0$, $\lambda_1 - \lambda_2 = 0$ and $\lambda_6 + \lambda_7 = 0$ are RG-stable does not extend to three or more doublets, because the symmetric tensor $d_{abc}$ regenerates the forbidden couplings already at one loop.
  • Overall sign-flipping GOOFy transformations ($X = -I$) are generically RG-stable and include the scale-invariant $V_2 = 0$ models, while partial sign-flipping transformations (for example $X = \mathrm{diag}(-1,-1,1)$) are generically radiatively unstable because they truncate maximal-torus-allowed terms.
  • Most admissible GOOFy potentials have vacuum problems: stationary points yield massless scalars, tachyonic saddles, or degenerate spectra, so algebraic compatibility alone does not guarantee phenomenological viability.
  • The sign-orbit classification extends to 4HDMs and 5HDMs, where only the $X = -I$ eigenvalue patterns are RG-stable; the selection rule provides a template for classifying larger NHDMs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the two-condition rule is assumed correct, any future GOOFy candidate in an NHDM can be pre-screened group-theoretically: list the $\mathbb{Z}_2$ quotients of $G/[G,G]$, then check $R^* \otimes R$ for the corresponding character, before doing any Lagrangian computation.
  • The RG-instability argument for partial sign flips implies that a phenomenologically viable GOOFy model probably needs an enlarged UV field content to protect its parameter relations, since the tree-level truncation itself cannot survive quantum corrections.
  • The eigenvalue pairing forced by $X = -I$ (non-zero masses come in $\pm$ pairs) means any odd-$N$ GOOFy sector with an overall sign flip must contain at least one exactly massless scalar, a sharp no-go for building such models with an odd number of doublets.
  • The maximal-torus closure conjecture, if proven, would generalise the paper's instability result: any symmetry relation that sets to zero a maximal-torus-invariant parameter is radiatively unstable, which could be tested systematically with Hilbert-series or syzygy methods.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a group-theoretic 'sign-orbit' formalism for classifying quadratic (bilinear) sectors of multi-Higgs-doublet potentials that are compatible with GOOFy transformations, in which scalar fields and their conjugates transform inequivalently. The central claims are (i) a non-vanishing GOOFy-compatible quadratic sector exists only if the symmetry group admits a non-trivial sign character and that character appears in the tensor product of the scalar representation with its conjugate, and (ii) the RG stability of the r0 -> -r0 manifold in the 2HDM is an accident of SU(2), traced to the vanishing of the symmetric invariant d_abc. The paper applies the formalism to 3HDMs, lists overall and partial sign-flipping realisations, studies vacuum spectra, and uses PyR@TE and RGBeta to investigate two-loop RG stability, concluding that only the overall sign-flipping X=-I cases are RG stable. The group-theoretic derivation for X=-I is presented cleanly, but the abstract and conclusions overstate the selection rule by omitting the X=-I assumption, and the RG-stability verdicts rely on an explicitly conjectural closure property of the maximal-torus sector.

Significance. If the claims are properly qualified, the paper makes a useful contribution to the classification of non-standard scalar-sector symmetries. The derivation of eq. (4.9) and the SU(2) vs SU(N>=3) argument in Section 3.2 are transparent and correct, and the explicit use of independent beta-function packages (PyR@TE, RGBeta) is a strength. The sign-orbit classification of cyclic embeddings is a practical bookkeeping device that can be extended to larger NHDMs. However, the main classification statement as written is not valid for the partial sign-flipping models that the paper itself treats, and the RG-stability claims are conditional on an unproven conjecture; these issues affect the paper's headline results and therefore require correction before publication.

major comments (3)
  1. [Abstract, §4.3, §5.2] The two-condition selection rule stated in the abstract and in Section 4.3 is overgeneralized. The derivation of eq. (4.9), R(g)^dagger Y R(g)=chi(g) Y, is performed under the explicit assumption X=-I, as the section itself notes with 'The preceding discussion assumes X=-I.' However, Section 5.2 studies partial sign-flipping transformations with X'=diag(-1,-1,1). For the case Z1,± with U=I, the flavour group is trivial and admits no non-trivial homomorphism to Z2, yet eq. (5.16) gives V2=mu^2_33 h33 with a non-zero quadratic coupling; similarly, eq. (5.19) for Z2,± gives V2=mu^2_12 h12 + h.c. + mu^2_33 h33. These examples contradict the claim that a non-trivial sign character of the underlying group is necessary. The abstract and conclusions must either restrict the selection rule to overall sign-flipping transformations with X=-I or be reformulated to accommodate partial sign-flipping cases as a separate class with a generalized criterion.
  2. [§5.3, Conclusions] The RG-instability verdict for partial sign-flipping GOOFy models rests on the unproven premise that the maximal-torus U(1)^{N-1}-invariant sector is closed under the renormalization group flow. The paper states this explicitly: 'While this observation is based on explicit multi-loop calculations rather than a general proof, it motivates the conjecture...'. The subsequent conclusion that 'these radiative effects inevitably destabilise partial sign-flipping configurations' and the unqualified statement 'only the cases with X=-I are RG stable' therefore go beyond what is demonstrated. To make the claim load-bearing, the paper should either prove the closure property for the maximal-torus sector or present explicit beta-function computations showing regeneration of the forbidden couplings for at least one representative partial sign-flipping model, and otherwise present the RG-stability statement as a conjecture.
  3. [§5.1, Table 1] The claim in Section 5.1 that 'in none of these cases are the underlying conditions found to be RG-stable' is not supported by any displayed beta-function result for the overall sign-flipping cases with V2 != 0. The sentence appears in Table 1's caption area, but no explicit two-loop beta-function expressions or numerical checks are shown. Since this is one of the paper's key physical conclusions, at least one illustrative calculation (e.g., beta(mu^2_11+mu^2_22) for a representative case) should be provided or the statement should cite a specific reproducible output file.
minor comments (4)
  1. [§4.4] The orbit notation O(n,m) is used in eq. (4.20) and in the surrounding text, but the reader must infer the full definition from the discussion; a one-sentence formal definition before eq. (4.20) would improve clarity.
  2. [§5.3, eq. (5.33)] The maximal-torus potential V_{U(1)^{N-1}} in eq. (5.33) is written with a sum over i<j for both lambda_iijj and lambda_ijji, but the hermiticity conditions for the latter are not spelled out; specifying the reality/phase conventions for these couplings would remove ambiguity.
  3. [§4.1, Table after eq. (4.5)] The table of parities under K_S, K_O, K_P is not explicitly referenced in the text that follows; adding a sentence explaining how the Z2 x Z2 grading is used in later sections would make the table easier to interpret.
  4. [§3.2, eq. (3.13)] The normalization of the generators t_a is not stated before eq. (3.13); since d_abc depends on the normalization convention, a brief statement (e.g., Tr(t_a t_b)=1/2 delta_ab) would make the identity d_abc d^abc=(N^2-1)(N^2-4)/N unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations are self-contained, and the main caveats are overgeneralization and an unproven conjecture, not circular inputs.

full rationale

The paper's two central claims are derived from independent mathematical and computational inputs rather than from their conclusions. The quadratic-sector selection rule in Section 4.3 follows from the invariance equation R(g)^\dagger Y R(g) = \chi(g)Y, which is a direct application of standard representation theory to the GOOFy transformation law h^* -> \chi(g)R(g)^* h^*; the 'two conditions' are the existence of the character and existence of a Y in the appropriate R^* \otimes R component, not fitted parameters relabeled as predictions. The RG-stability claim for the r0 -> -r0 manifold is supported by explicit two-loop beta functions computed with PyR@TE and RGBeta and by the identity d_abc d_abc = (N^2-1)(N^2-4)/N, which vanishes iff N=2; no fitted quantity is renamed as a prediction. The reliance on Ref. [31], co-authored by one of the present authors, for the quartic-sector classification is not load-bearing: the relevant potentials are re-derived in Section 5 via the W(U,X,\eta) construction, and the paper explicitly identifies and corrects the G4 error in Ref. [31]; moreover that classification is externally checkable through GAP/SmallGrp. Two caveats are present but are not circularity. First, Section 4.3 explicitly states 'The preceding discussion assumes X=-I', yet the abstract and conclusions state the two-condition rule without this qualifier, and the Section 5.2 partial sign-flipping examples (e.g., V2 = \mu^2_{33} h_{33} for U=I and X'=diag(-1,-1,1)) show the rule is not necessary in the partial-X class; this is an overgeneralization and internal-consistency issue rather than a circular reduction. Second, Section 5.3's conclusion that partial sign-flipping models are radiatively unstable relies on the unproven maximal-torus closure conjecture, which the authors explicitly flag as a conjecture; this is an evidence-limitation, not a circular step. No part of the derivation reduces by construction to its own inputs or to a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The load-bearing content is carried by standard group theory plus three domain assumptions: the X=+/-1 restriction, the acceptance of Ref [31]'s classification, and the maximal-torus closure conjecture. The last is the only genuinely ad hoc premise.

assumptions (5)
  • domain assumption X (the kinetic anomaly matrix) is restricted to unitary Hermitian matrices with X^2=I, so eigenvalues are +1 or -1.
    Section 3.4 argues this is required to avoid ghosts and non-Hermitian kinetic terms; the entire classification is built on this restriction.
  • domain assumption The scalar potential is a Hermitian polynomial in h_ij = h_i^dagger h_j with the form of eq (2.2).
    Standard NHDM setup used throughout without derivation.
  • domain assumption The list of realisable HF/GCP symmetry groups and quartic structures in the 3HDM from Ref [31] and the finite subgroup data of Ref [46] are accepted without re-derivation.
    The paper extends Ref [31] and relies on its completeness; it even corrects one of its entries (G4).
  • ad hoc to paper The maximal-torus U(1)^{N-1}-invariant sector is closed under RG flow.
    Explicitly flagged as a conjecture in Section 5.3, yet used as a premise for the radiative instability of partial sign-flipping models.
  • domain assumption Two-loop beta functions without fermions, computed with PyR@TE and RGBeta, capture the relevant RG structure.
    Section 5.3 restricts the analysis to two loops excluding fermions; the stability verdicts depend on these computations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of GOOFy-compatible 3HDMs and beyond." pith.science (2026). https://pith.science/paper/PMLS2EHV

@misc{pith2026260805304,
  author       = {Pith},
  title        = {Pith review of: GOOFy-compatible 3HDMs and beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMLS2EHV}},
  note         = {Machine review of arXiv:2608.05304}
}
abstract

Beyond conventional Higgs-family and general CP transformations, one may also consider a broader class of non-standard "GOOFy" transformations, in which the scalar fields and their conjugates are assigned related but inequivalent transformations. Although these transformations are not symmetries of a full Lagrangian in the conventional sense, some nevertheless stabilise the scalar potential. Their impact on the quadratic sector has not yet been systematically classified. We develop a sign-orbit technique to determine the bilinear structures compatible with these transformations. Applied to three-Higgs-doublet models, and formulated so as to extend to larger multi-Higgs sectors, the method shows that a non-vanishing GOOFy-compatible quadratic sector exists only when two independent conditions are met: the underlying group admits a non-trivial sign character, and that character is realised within the tensor product of the scalar representation and its conjugate. As a key consequence, we demonstrate that the renormalisation-group stability of the $r_0 \to -r_0$ manifold in isolation is not a generic feature of multi-Higgs potentials, but rather a direct consequence of special algebraic properties of $SU(2)$. These results provide a classification of the admissible GOOFy quadratic structures and their invariant bilinear subspaces, highlighting the algebraic obstructions to their renormalisation-group stability.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

75 extracted references · 24 canonical work pages

  1. [31]

    Kunˇ cinas, P

    A. Kunˇ cinas, P. Osland and M. N. Rebelo,Systematic analysis of 3HDM symmetries,Phys. Rev. D113(2026) 115054, [2512.07657]

    arXiv 2026

  2. [3]

    Olaussen, P

    K. Olaussen, P. Osland and M. A. Solberg,Symmetry and Mass Degeneration in Multi-Higgs-Doublet Models,JHEP07(2011) 020, [1007.1424]

    arXiv 2011

  3. [4]

    I. P. Ivanov,Two-Higgs-doublet model from the group-theoretic perspective,Phys. Lett. B632(2006) 360–365, [hep-ph/0507132]

    arXiv 2006

  4. [5]

    I. P. Ivanov,Minkowski space structure of the Higgs potential in 2HDM,Phys. Rev. D75(2007) 035001, [hep-ph/0609018]

    arXiv 2007

  5. [6]

    J. M. Gerard and M. Herquet,A Twisted custodial symmetry in the two-Higgs-doublet model,Phys. Rev. Lett.98(2007) 251802, [hep-ph/0703051]

    arXiv 2007

  6. [7]

    I. P. Ivanov,Minkowski space structure of the Higgs potential in 2HDM. II. Minima, symmetries, and topology,Phys. Rev. D77(2008) 015017, [0710.3490]

    arXiv 2008

  7. [8]

    P. M. Ferreira, H. E. Haber and J. P. Silva,Generalized CP symmetries and special regions of parameter space in the two-Higgs-doublet model,Phys. Rev. D79(2009) 116004, [0902.1537]

    arXiv 2009

  8. [9]

    P. M. Ferreira, H. E. Haber, M. Maniatis, O. Nachtmann and J. P. Silva,Geometric picture of generalized-CP and Higgs-family transformations in the two-Higgs-doublet model,Int. J. Mod. Phys. A26(2011) 769–808, [1010.0935]

    arXiv 2011

Show all 75 references
  1. [10]

    R. A. Battye, G. D. Brawn and A. Pilaftsis,Vacuum Topology of the Two Higgs Doublet Model,JHEP08(2011) 020, [1106.3482]. 37

    2011 arXiv

  2. [11]

    Pilaftsis,On the Classification of Accidental Symmetries of the Two Higgs Doublet Model Potential,Phys

    A. Pilaftsis,On the Classification of Accidental Symmetries of the Two Higgs Doublet Model Potential,Phys. Lett. B706(2012) 465–469, [1109.3787]

    2012 arXiv

  3. [12]

    H. E. Haber, O. M. Ogreid, P. Osland and M. N. Rebelo,Symmetries and Mass Degeneracies in the Scalar Sector,JHEP01(2019) 042, [1808.08629]

    2019 arXiv

  4. [13]

    M. P. Bento, R. Boto, J. P. Silva and A. Trautner,A fully basis invariant Symmetry Map of the 2HDM,JHEP02(2021) 220, [2009.01264]

    2021 arXiv

  5. [14]

    P. M. Ferreira, B. Grzadkowski, O. M. Ogreid and P. Osland,Symmetries of the 2HDM: an invariant formulation and consequences,JHEP02(2021) 196, [2010.13698]

    2021 arXiv

  6. [15]

    P. M. Ferreira, B. Grzadkowski, O. M. Ogreid and P. Osland,Softly broken symmetries in the 2HDM – an invariant formulation,JHEP01(2023) 143, [2209.00152]

    2023 arXiv

  7. [16]

    M. A. Solberg,Lie symmetry analysis of the two-Higgs-doublet model field equations,Phys. Scripta101(2026) 035209, [2510.09542]

    2026

  8. [17]

    P. M. Ferreira and J. P. Silva,Discrete and continuous symmetries in multi-Higgs-doublet models,Phys. Rev. D78(2008) 116007, [0809.2788]

    2008 arXiv

  9. [18]

    I. P. Ivanov, V. Keus and E. Vdovin,Abelian symmetries in multi-Higgs-doublet models,J. Phys. A45(2012) 215201, [1112.1660]

    2012 arXiv

  10. [19]

    I. P. Ivanov and E. Vdovin,Discrete symmetries in the three-Higgs-doublet model, Phys. Rev. D86(2012) 095030, [1206.7108]

    2012 arXiv

  11. [20]

    I. P. Ivanov and E. Vdovin,Classification of finite reparametrization symmetry groups in the three-Higgs-doublet model,Eur. Phys. J. C73(2013) 2309, [1210.6553]

    2013 arXiv

  12. [21]

    V. Keus, S. F. King and S. Moretti,Three-Higgs-doublet models: symmetries, potentials and Higgs boson masses,JHEP01(2014) 052, [1310.8253]

    2014 arXiv

  13. [22]

    I. P. Ivanov and C. C. Nishi,Symmetry breaking patterns in 3HDM,JHEP01 (2015) 021, [1410.6139]

    2015 arXiv

  14. [23]

    I. P. Ivanov and J. P. Silva,CP-conserving multi-Higgs model with irremovable complex coefficients,Phys. Rev. D93(2016) 095014, [1512.09276]

    2016 arXiv

  15. [24]

    Pilaftsis,Symmetries for standard model alignment in multi-Higgs doublet models,Phys

    A. Pilaftsis,Symmetries for standard model alignment in multi-Higgs doublet models,Phys. Rev. D93(2016) 075012, [1602.02017]

    2016 arXiv

  16. [25]

    de Medeiros Varzielas and I

    I. de Medeiros Varzielas and I. P. Ivanov,Recognizing symmetries in a 3HDM in a basis-independent way,Phys. Rev. D100(2019) 015008, [1903.11110]

    2019 arXiv

  17. [26]

    Kunˇ cinas,Properties ofS 3-symmetric three-higgs-doublet models, Master’s thesis

    A. Kunˇ cinas,Properties ofS 3-symmetric three-higgs-doublet models, Master’s thesis. http://bora.uib.no/handle/1956/20467

    1956

  18. [27]

    Darvishi and A

    N. Darvishi and A. Pilaftsis,Classifying Accidental Symmetries in Multi-Higgs Doublet Models,Phys. Rev. D101(2020) 095008, [1912.00887]. 38

    2020 arXiv

  19. [28]

    Kunˇ cinas, O

    A. Kunˇ cinas, O. Ogreid, P. Osland and M. Rebelo,S 3-inspired three-Higgs-doublet models: A class with a complex vacuum,Phys. Rev. D101(2020) 075052, [2001.01994]

    2020 arXiv

  20. [29]

    Kunˇ cinas, P

    A. Kunˇ cinas, P. Osland and M. N. Rebelo,U(1)-charged Dark Matter in three-Higgs-doublet models,JHEP11(2024) 086, [2408.02728]

    2024 arXiv

  21. [30]

    D¨ oring and A

    C. D¨ oring and A. Trautner,Symmetries from outer automorphisms and unorthodox group extensions,J. Phys. A58(2025) 475401, [2410.11052]

    2025

  22. [32]

    Shao and I

    J. Shao and I. P. Ivanov,Symmetries for the 4HDM: extensions of cyclic groups, JHEP10(2023) 070, [2305.05207]

    2023 arXiv

  23. [33]

    J. Shao, I. P. Ivanov and M. Korhonen,Symmetries for the 4HDM: II. Extensions by rephasing groups,J. Phys. A57(2024) 385401, [2404.10349]

    2024 arXiv

  24. [34]

    P. M. Ferreira, B. Grzadkowski, O. M. Ogreid and P. Osland,New symmetries of the two-Higgs-doublet model,Eur. Phys. J. C84(2024) 234, [2306.02410]

    2024 arXiv

  25. [35]

    H. E. Haber and P. M. Ferreira,RG-stable parameter relations of a scalar field theory in absence of a symmetry,Eur. Phys. J. C85(2025) 541, [2502.11011]

    2025

  26. [36]

    Trautner,Goofy is the new Normal,JHEP10(2025) 051, [2505.00099]

    A. Trautner,Goofy is the new Normal,JHEP10(2025) 051, [2505.00099]

    2025 arXiv

  27. [37]

    P. M. Ferreira, B. Grzadkowski and O. M. Ogreid,Imaginary scaling invariance of the one-loop effective potential,Phys. Rev. D113(2026) 075036, [2506.21145]

    2026 arXiv

  28. [38]

    de Boer, F

    T. de Boer, F. Goertz and A. Incrocci,The goofy-symmetric Standard Model and the Hierarchy Problem,2507.22111

    arXiv

  29. [39]

    Trautner,Goofy transformations and the hierarchy problem,Phys

    A. Trautner,Goofy transformations and the hierarchy problem,Phys. Lett. B873 (2026) 140190, [2508.02646]

    2026

  30. [40]

    Grzadkowski and O

    B. Grzadkowski and O. M. Ogreid,GOOFy — a systematic approach,JHEP07 (2026) 038, [2602.20849]

    2026 arXiv

  31. [41]

    P. M. Ferreira,GOOFy fermions,2604.07458

    arXiv

  32. [42]

    T. D. Lee and G. C. Wick,Space Inversion, Time Reversal, and Other Discrete Symmetries in Local Field Theories,Phys. Rev.148(1966) 1385–1404

    1966

  33. [43]

    Ecker, W

    G. Ecker, W. Grimus and W. Konetschny,Quark Mass Matrices in Left-right Symmetric Gauge Theories,Nucl. Phys. B191(1981) 465–492

    1981

  34. [44]

    Ecker, W

    G. Ecker, W. Grimus and H. Neufeld,A Standard Form for Generalized CP Transformations,J. Phys. A20(1987) L807

    1987

  35. [45]

    Neufeld, W

    H. Neufeld, W. Grimus and G. Ecker,Generalized CP Invariance, Neutral Flavor Conservation and the Structure of the Mixing Matrix,Int. J. Mod. Phys. A03 (1988) 603–616. 39

    1988

  36. [46]

    Jurciukonis and L

    D. Jurciukonis and L. Lavoura,GAP listing of the finite subgroups of U(3) of order smaller than 2000,PTEP2017(2017) 053A03, [1702.00005]

    2017 arXiv

  37. [47]

    SmallGrp, the gap small groups library, Version 1.5.4

    H. U. Besche, B. Eick, E. O’Brien and M. Horn, “SmallGrp, the gap small groups library, Version 1.5.4.”https://gap-packages.github.io/smallgrp/, Jul, 2024. [48]GAP – Groups, Algorithms, and Programming, Version 4.13.1, https: // www. gap-system. org, 2024

    2024

  38. [49]

    W. A. Bardeen,On naturalness in the standard model, inOntake Summer Institute on Particle Physics, 8, 1995

    1995

  39. [50]

    S. R. Coleman and E. J. Weinberg,Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,Phys. Rev. D7(1973) 1888–1910

    1973

  40. [51]

    Gildener and S

    E. Gildener and S. Weinberg,Symmetry Breaking and Scalar Bosons,Phys. Rev. D 13(1976) 3333

    1976

  41. [52]

    J. S. Lee and A. Pilaftsis,Radiative Corrections to Scalar Masses and Mixing in a Scale Invariant Two Higgs Doublet Model,Phys. Rev. D86(2012) 035004, [1201.4891]

    2012 arXiv

  42. [53]

    Hashino, S

    K. Hashino, S. Kanemura and Y. Orikasa,Discriminative phenomenological features of scale invariant models for electroweak symmetry breaking,Phys. Lett. B752 (2016) 217–220, [1508.03245]

    2016 arXiv

  43. [54]

    Lane and E

    K. Lane and E. Pilon,Phenomenology of the new light Higgs bosons in Gildener-Weinberg models,Phys. Rev. D101(2020) 055032, [1909.02111]

    2020 arXiv

  44. [55]

    Braathen, S

    J. Braathen, S. Kanemura and M. Shimoda,Two-loop analysis of classically scale-invariant models with extended Higgs sectors,JHEP03(2021) 297, [2011.07580]

    2021 arXiv

  45. [56]

    E. J. Eichten and K. Lane,Gildener-Weinberg two-Higgs-doublet model at two loops,Phys. Rev. D107(2023) 075038, [2209.06632]

    2023 arXiv

  46. [57]

    Slavich,Revisiting the Higgs-mass calculation in the scale-invariant THDM, Phys

    P. Slavich,Revisiting the Higgs-mass calculation in the scale-invariant THDM, Phys. Rev. D113(2026) 095022, [2602.17643]

    2026 arXiv

  47. [58]

    Velhinho, R

    J. Velhinho, R. Santos and A. Barroso,Tree level vacuum stability in two Higgs doublet models,Phys. Lett. B322(1994) 213–218

    1994

  48. [59]

    Nagel,New aspects of gauge-boson couplings and the Higgs sector

    F. Nagel,New aspects of gauge-boson couplings and the Higgs sector. PhD thesis, University of Heidelberg, 2004. [10.11588/heidok.00004803]

    2004 doi

  49. [60]

    Maniatis, A

    M. Maniatis, A. von Manteuffel, O. Nachtmann and F. Nagel,Stability and symmetry breaking in the general two-Higgs-doublet model,Eur. Phys. J. C48 (2006) 805–823, [hep-ph/0605184]

    2006 arXiv

  50. [61]

    Maniatis, A

    M. Maniatis, A. von Manteuffel and O. Nachtmann,Determining the global minimum of Higgs potentials via Groebner bases: Applied to the NMSSM,Eur. Phys. J. C49(2007) 1067–1076, [hep-ph/0608314]. 40

    2007 arXiv

  51. [62]

    Maniatis, A

    M. Maniatis, A. von Manteuffel and O. Nachtmann,CP violation in the general two-Higgs-doublet model: A Geometric view,Eur. Phys. J. C57(2008) 719–738, [0707.3344]

    2008 arXiv

  52. [63]

    C. C. Nishi,Physical parameters and basis transformations in the Two-Higgs-Doublet model,Phys. Rev. D77(2008) 055009, [0712.4260]

    2008 arXiv

  53. [64]

    A. V. Bednyakov,On three-loop RGE for the Higgs sector of 2HDM,JHEP11 (2018) 154, [1809.04527]

    2018 arXiv

  54. [65]

    A. V. Bednyakov,On the scalar sector of 2HDM: ring of basis invariants, syzygies, and six-loop renormalization-group equations,JHEP06(2025) 081, [2501.14087]

    2025 arXiv

  55. [66]

    Pilaftsis,Dirac algebra formalism for Two Higgs Doublet Models: The one-loop effective potential,Phys

    A. Pilaftsis,Dirac algebra formalism for Two Higgs Doublet Models: The one-loop effective potential,Phys. Lett. B860(2025) 139147, [2408.04511]

    2025 arXiv

  56. [67]

    Sartore and I

    L. Sartore and I. Schienbein,PyR@TE 3,Comput. Phys. Commun.261(2021) 107819, [2007.12700]

    2021 arXiv

  57. [68]

    A. E. Thomsen,Introducing RGBeta: a Mathematica package for the evaluation of renormalization groupβ-functions,Eur. Phys. J. C81(2021) 408, [2101.08265]

    2021 arXiv

  58. [69]

    Pilaftsis,Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries,2605.18341

    A. Pilaftsis,Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries,2605.18341

    arXiv

  59. [70]

    de Boer and A

    T. de Boer and A. Trautner,Outer automorphisms are sufficient conditions for RG fixed points,2603.12318

  60. [71]

    Benvenuti, B

    S. Benvenuti, B. Feng, A. Hanany and Y.-H. He,Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics,JHEP11(2007) 050, [hep-th/0608050]

    2007 arXiv

  61. [72]

    B. Feng, A. Hanany and Y.-H. He,Counting gauge invariants: The Plethystic program,JHEP03(2007) 090, [hep-th/0701063]

    2007 arXiv

  62. [73]

    E. E. Jenkins and A. V. Manohar,Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation,JHEP10(2009) 094, [0907.4763]

    2009 arXiv

  63. [74]

    Lehman and A

    L. Lehman and A. Martin,Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox,Phys. Rev. D91(2015) 105014, [1503.07537]

    2015 arXiv

  64. [75]

    Henning, X

    B. Henning, X. Lu, T. Melia and H. Murayama,Operator bases,S-matrices, and their partition functions,JHEP10(2017) 199, [1706.08520]

    2017 arXiv

  65. [76]

    Trautner,Systematic construction of basis invariants in the 2HDM,JHEP05 (2019) 208, [1812.02614]

    A. Trautner,Systematic construction of basis invariants in the 2HDM,JHEP05 (2019) 208, [1812.02614]

    2019 arXiv

  66. [77]

    M. P. Bento,The invariant space of multi-Higgs doublet models,JHEP05(2021) 146, [2102.13120]

    2021 arXiv

  67. [78]

    Bryan and A

    E. Bryan and A. Rajaraman,Hilbert series and the flavor invariants of the 3HDM, Phys. Rev. D114(2026) 015047, [2604.18895]. 41

    2026 arXiv

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.