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arxiv: 2604.18895 · v1 · submitted 2026-04-20 · ✦ hep-th

Recognition: unknown

The Hilbert Series and the Flavor Invariants of the 3HDM

Eric Bryan , Arvind Rajaraman
Authors on Pith no claims yet

Pith reviewed 2026-05-10 03:30 UTC · model grok-4.3

classification ✦ hep-th
keywords 3HDMHilbert seriesflavor invariantsglobal symmetryinvariant operatorsHiggs doubletsCP violation
0
0 comments X

The pith

The Hilbert series counts all flavor invariants of the three-Higgs-doublet model under its global symmetry, with explicit forms given up to cubic order.

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

The paper computes the Hilbert series for the ring of invariant operators in the 3HDM, which gives the number of independent flavor invariants at each mass dimension. It also supplies concrete polynomial expressions for every such invariant through order three in the scalar and fermion fields. A reader would care because these invariants determine the most general allowed terms in the Lagrangian, including those that control CP violation and flavor-changing processes. The calculation uses standard representation theory on the assumed global symmetry group of the model.

Core claim

We perform a systematic study of invariant operators in the three-Higgs-doublet model (3HDM). We compute the Hilbert series associated with the global symmetry group of the theory. In addition, we construct explicit expressions for these invariants up to cubic order in the couplings.

What carries the argument

The Hilbert series of the global symmetry group, which encodes the graded dimension of the ring of invariant polynomials built from the Higgs doublets and fermions.

If this is right

  • The Hilbert series supplies the complete counting of independent operators at every order without needing to list them individually.
  • The explicit cubic invariants can be inserted directly into the scalar potential and Yukawa Lagrangian to obtain the most general 3HDM.
  • All higher-order invariants are generated algebraically from the lower-order ones once the series is known.
  • Relations among invariants that follow from the Hilbert series can be used to reduce redundant terms in phenomenological studies.

Where Pith is reading between the lines

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

  • The same Hilbert-series technique can be applied to other multi-Higgs models once their global symmetry is specified.
  • The cubic invariants may reveal accidental symmetries or relations that affect the stability of the electroweak vacuum.
  • Phenomenological scans of the 3HDM parameter space can now incorporate the full set of invariants up to dimension three without manual enumeration.

Load-bearing premise

The global symmetry group is fixed in advance and the invariants are counted using only representation theory, with no further restrictions imposed by the detailed Yukawa or potential structure.

What would settle it

An explicit listing of all independent monomials of degree three that are invariant under the assumed global symmetry, followed by a count that differs from the coefficient extracted from the Hilbert series.

read the original abstract

We perform a systematic study of invariant operators in the three-Higgs-doublet model (3HDM). We compute the Hilbert series associated with the global symmetry group of the theory. In addition, we construct explicit expressions for these invariants up to cubic order in the couplings.

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

0 major / 3 minor

Summary. The manuscript computes the Hilbert series for the flavor invariants of the three-Higgs-doublet model (3HDM) under its global symmetry group and constructs explicit expressions for these invariants up to cubic order in the couplings, using standard techniques from invariant theory such as plethystic exponentials or Molien integrals applied to the relevant field representations.

Significance. If the results hold, the work supplies a complete counting and basis for independent operators in the 3HDM, which is valuable for systematic studies of flavor structures, potential minimization, and phenomenological constraints in multi-Higgs models. The explicit cubic-order expressions are a concrete strength that enables immediate use in model building.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction should explicitly state the precise global symmetry group assumed for the 3HDM (e.g., whether it is SU(2)_L × U(1)_Y × U(1)_B × U(1)_L or an extended flavor symmetry) and confirm that no additional relations from the Yukawa or scalar potential are imposed beyond the group action.
  2. [Computation of the Hilbert series] In the section presenting the Hilbert series, the plethystic exponential or Molien integral formula should be written out explicitly with the character of each representation carried by the Higgs doublets and couplings, to allow direct verification of the series coefficients.
  3. [Explicit invariants] The explicit invariants up to cubic order are listed without a table summarizing their dimensions or transformation properties; adding such a table would improve readability and facilitate cross-checks with the series.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript on the Hilbert series and flavor invariants of the 3HDM. The referee's summary correctly identifies our use of standard invariant theory techniques, including plethystic exponentials and Molien integrals, along with the explicit construction of invariants up to cubic order. We appreciate the recognition that these results provide a complete counting and basis for independent operators, which is valuable for phenomenological studies in multi-Higgs models.

Circularity Check

0 steps flagged

Direct group-theoretic computation with no circularity

full rationale

The paper performs a direct computation of the Hilbert series for flavor invariants under the global symmetry group of the 3HDM, together with explicit basis elements up to cubic order in the couplings. This rests on standard invariant-theory methods (plethystic exponentials or Molien integrals applied to the representations carried by the scalar fields and couplings). No parameters are fitted to data, no result is defined in terms of itself, and no load-bearing step reduces to a self-citation or ansatz imported from the authors' prior work. The weakest assumption—that the chosen group is indeed the symmetry of the Lagrangian—is the conventional starting point for such counts and does not create an internal loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the computation is expected to rest on standard algebraic geometry and representation theory with no free parameters or new entities introduced in the summary.

axioms (1)
  • standard math Hilbert series counts independent invariants under a group action via the Molien formula or plethystic exponential
    Invoked implicitly by the computation of the series for the global symmetry group

pith-pipeline@v0.9.0 · 5324 in / 1107 out tokens · 25938 ms · 2026-05-10T03:30:46.377336+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

68 extracted references · 67 canonical work pages · 2 internal anchors

  1. [1]

    C. C. Nishi, Phys. Rev. D74, 036003 (2006) [erratum: Phys. Rev. D76, 119901 (2007)] doi:10.1103/PhysRevD.76.119901 [arXiv:hep-ph/0605153 [hep-ph]]

    work page doi:10.1103/physrevd.76.119901 2006

  2. [2]

    C. C. Nishi, Phys. Rev. D76, 055013 (2007) doi:10.1103/PhysRevD.76.055013 [arXiv:0706.2685 [hep-ph]]

    work page doi:10.1103/physrevd.76.055013 2007

  3. [3]

    P. M. Ferreira and J. P. Silva, Phys. Rev. D78, 116007 (2008) doi:10.1103/PhysRevD.78.116007 [arXiv:0809.2788 [hep-ph]]

    work page doi:10.1103/physrevd.78.116007 2008

  4. [4]

    I. P. Ivanov and C. C. Nishi, Phys. Rev. D82, 015014 (2010) doi:10.1103/PhysRevD.82.015014 [arXiv:1004.1799 [hep-th]]

    work page doi:10.1103/physrevd.82.015014 2010

  5. [5]

    Grzadkowski, O

    B. Grzadkowski, O. M. Ogreid and P. Osland, Phys. Rev. D80, 055013 (2009) doi:10.1103/PhysRevD.80.055013 [arXiv:0904.2173 [hep-ph]]

    work page doi:10.1103/physrevd.80.055013 2009

  6. [6]

    G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept.516, 1-102 (2012) doi:10.1016/j.physrep.2012.02.002 [arXiv:1106.0034 [hep-ph]]

    work page doi:10.1016/j.physrep.2012.02.002 2012

  7. [7]

    I. F. Ginzburg and M. Krawczyk, Phys. Rev. D72, 115013 (2005) doi:10.1103/PhysRevD.72.115013 [arXiv:hep-ph/0408011 [hep-ph]]

    work page doi:10.1103/physrevd.72.115013 2005

  8. [8]

    J. F. Gunion and H. E. Haber, Phys. Rev. D72, 095002 (2005) doi:10.1103/PhysRevD.72.095002 [arXiv:hep-ph/0506227 [hep-ph]]

    work page doi:10.1103/physrevd.72.095002 2005

  9. [9]

    F. J. Botella and J. P. Silva, Phys. Rev. D51, 3870-3875 (1995) doi:10.1103/PhysRevD.51.3870 [arXiv:hep-ph/9411288 [hep-ph]]

    work page doi:10.1103/physrevd.51.3870 1995

  10. [10]

    Maniatis, A

    M. Maniatis, A. von Manteuffel and O. Nachtmann, Eur. Phys. J. C57, 719-738 (2008) doi:10.1140/epjc/s10052-008-0712-5 [arXiv:0707.3344 [hep-ph]]

    work page doi:10.1140/epjc/s10052-008-0712-5 2008

  11. [11]

    Lavoura and J

    L. Lavoura and J. P. Silva, Phys. Rev. D50, 4619-4624 (1994) doi:10.1103/PhysRevD.50.4619 [arXiv:hep-ph/9404276 [hep-ph]]

    work page doi:10.1103/physrevd.50.4619 1994

  12. [12]

    H. E. Haber and D. O’Neil, Phys. Rev. D74, 015018 (2006) [erratum: Phys. Rev. D 74, no.5, 059905 (2006)] doi:10.1103/PhysRevD.74.015018 [arXiv:hep-ph/0602242 [hep- ph]]

    work page doi:10.1103/physrevd.74.015018 2006

  13. [13]

    Khater and P

    W. Khater and P. Osland, Acta Phys. Polon. B34, 4531-4548 (2003) [arXiv:hep- ph/0305308 [hep-ph]]

    work page arXiv 2003

  14. [14]

    Barroso, P

    A. Barroso, P. M. Ferreira and R. Santos, Phys. Lett. B652, 181-193 (2007) doi:10.1016/j.physletb.2007.07.010 [arXiv:hep-ph/0702098 [hep-ph]]

    work page doi:10.1016/j.physletb.2007.07.010 2007

  15. [15]

    Antusch, M

    S. Antusch, M. Drees, J. Kersten, M. Lindner and M. Ratz, Phys. Lett. B525, 130-134 (2002) doi:10.1016/S0370-2693(01)01414-9 [arXiv:hep-ph/0110366 [hep-ph]]

    work page doi:10.1016/s0370-2693(01)01414-9 2002

  16. [16]

    Atwood, S

    D. Atwood, S. Bar-Shalom and A. Soni, Phys. Lett. B635, 112-117 (2006) doi:10.1016/j.physletb.2006.02.033 [arXiv:hep-ph/0502234 [hep-ph]]. 26

    work page doi:10.1016/j.physletb.2006.02.033 2006

  17. [17]

    Liu and P

    Z. Liu and P. H. Gu, Nucl. Phys. B915, 206-223 (2017) doi:10.1016/j.nuclphysb.2016.12.001 [arXiv:1611.02094 [hep-ph]]

    work page doi:10.1016/j.nuclphysb.2016.12.001 2017

  18. [18]

    Cheung, H

    K. Cheung, H. Okada and Y. Orikasa, [arXiv:1706.02084 [hep-ph]]

    work page arXiv

  19. [19]

    Arcadi, M

    G. Arcadi, M. Lindner, F. S. Queiroz, W. Rodejohann and S. Vogl, JCAP03, 042 (2018) doi:10.1088/1475-7516/2018/03/042 [arXiv:1711.02110 [hep-ph]]

    work page doi:10.1088/1475-7516/2018/03/042 2018

  20. [20]

    D. A. Camargo, M. D. Campos, T. B. de Melo and F. S. Queiroz, Phys. Lett. B795, 319-326 (2019) doi:10.1016/j.physletb.2019.06.020 [arXiv:1901.05476 [hep-ph]]

    work page doi:10.1016/j.physletb.2019.06.020 2019

  21. [21]

    Lopez Honorez, E

    L. Lopez Honorez, E. Nezri, J. F. Oliver and M. H. G. Tytgat, JCAP02, 028 (2007) doi:10.1088/1475-7516/2007/02/028 [arXiv:hep-ph/0612275 [hep-ph]]

    work page doi:10.1088/1475-7516/2007/02/028 2007

  22. [22]

    Gustafsson, E

    M. Gustafsson, E. Lundstrom, L. Bergstrom and J. Edsjo, Phys. Rev. Lett.99, 041301 (2007) doi:10.1103/PhysRevLett.99.041301 [arXiv:astro-ph/0703512 [astro-ph]]

    work page doi:10.1103/physrevlett.99.041301 2007

  23. [23]

    E. M. Dolle and S. Su, Phys. Rev. D80, 055012 (2009) doi:10.1103/PhysRevD.80.055012 [arXiv:0906.1609 [hep-ph]]

    work page doi:10.1103/physrevd.80.055012 2009

  24. [24]

    Chao and M

    W. Chao and M. J. Ramsey-Musolf, Phys. Rev. D89, no.3, 033007 (2014) doi:10.1103/PhysRevD.89.033007 [arXiv:1212.5709 [hep-ph]]

    work page doi:10.1103/physrevd.89.033007 2014

  25. [25]

    Goudelis, B

    A. Goudelis, B. Herrmann and O. St˚ al, JHEP09, 106 (2013) doi:10.1007/JHEP09(2013)106 [arXiv:1303.3010 [hep-ph]]

    work page doi:10.1007/jhep09(2013)106 2013

  26. [26]

    Lopez Honorez and C

    L. Lopez Honorez and C. E. Yaguna, JHEP09, 046 (2010) doi:10.1007/JHEP09(2010)046 [arXiv:1003.3125 [hep-ph]]

    work page doi:10.1007/jhep09(2010)046 2010

  27. [28]

    Arhrib, Y

    A. Arhrib, Y. L. S. Tsai, Q. Yuan and T. C. Yuan, JCAP06, 030 (2014) doi:10.1088/1475-7516/2014/06/030 [arXiv:1310.0358 [hep-ph]]

    work page doi:10.1088/1475-7516/2014/06/030 2014

  28. [29]

    Bonilla, D

    C. Bonilla, D. Sokolowska, N. Darvishi, J. L. Diaz-Cruz and M. Krawczyk, J. Phys. G 43, no.6, 065001 (2016) doi:10.1088/0954-3899/43/6/065001 [arXiv:1412.8730 [hep-ph]]

    work page doi:10.1088/0954-3899/43/6/065001 2016

  29. [31]

    Arcadi, Eur

    G. Arcadi, Eur. Phys. J. C78, no.10, 864 (2018) doi:10.1140/epjc/s10052-018-6327-6 [arXiv:1804.04930 [hep-ph]]

    work page doi:10.1140/epjc/s10052-018-6327-6 2018

  30. [32]

    Maniatis, A

    M. Maniatis, A. von Manteuffel, O. Nachtmann and F. Nagel, Eur. Phys. J. C48, 805-823 (2006) doi:10.1140/epjc/s10052-006-0016-6 [arXiv:hep-ph/0605184 [hep-ph]]

    work page doi:10.1140/epjc/s10052-006-0016-6 2006

  31. [33]

    N. G. Deshpande and E. Ma, Phys. Rev. D18, 2574 (1978) doi:10.1103/PhysRevD.18.2574 27

    work page doi:10.1103/physrevd.18.2574 1978

  32. [34]

    P. M. Ferreira, B. Grzadkowski, O. M. Ogreid and P. Osland, JHEP01, 143 (2023) doi:10.1007/JHEP01(2023)143 [arXiv:2209.00152 [hep-ph]]

    work page doi:10.1007/jhep01(2023)143 2023

  33. [35]

    H. E. Haber and G. L. Kane, Phys. Rept.117, 75-263 (1985) doi:10.1016/0370- 1573(85)90051-1

    work page doi:10.1016/0370- 1985

  34. [36]

    I. P. Ivanov and E. Vdovin, Eur. Phys. J. C73, no.2, 2309 (2013) doi:10.1140/epjc/s10052-013-2309-x [arXiv:1210.6553 [hep-ph]]

    work page doi:10.1140/epjc/s10052-013-2309-x 2013

  35. [37]

    I. P. Ivanov and E. Vdovin, Phys. Rev. D86, 095030 (2012) doi:10.1103/PhysRevD.86.095030 [arXiv:1206.7108 [hep-ph]]

    work page doi:10.1103/physrevd.86.095030 2012

  36. [38]

    Kunˇ cinas, P

    A. Kunˇ cinas, P. Osland and M. N. Rebelo, [arXiv:2512.07657 [hep-ph]]

    work page arXiv

  37. [39]

    V. Keus, S. F. King and S. Moretti, JHEP01, 052 (2014) doi:10.1007/JHEP01(2014)052 [arXiv:1310.8253 [hep-ph]]

    work page doi:10.1007/jhep01(2014)052 2014

  38. [40]

    Gonz´ alez Felipe, H

    R. Gonz´ alez Felipe, H. Serˆ odio and J. P. Silva, Phys. Rev. D87, no.5, 055010 (2013) doi:10.1103/PhysRevD.87.055010 [arXiv:1302.0861 [hep-ph]]

    work page doi:10.1103/physrevd.87.055010 2013

  39. [41]

    I. P. Ivanov and C. C. Nishi, JHEP01, 021 (2015) doi:10.1007/JHEP01(2015)021 [arXiv:1410.6139 [hep-ph]]

    work page doi:10.1007/jhep01(2015)021 2015

  40. [42]

    I. P. Ivanov, C. C. Nishi, J. P. Silva and A. Trautner, Phys. Rev. D99, no.1, 015039 (2019) doi:10.1103/PhysRevD.99.015039 [arXiv:1810.13396 [hep-ph]]

    work page doi:10.1103/physrevd.99.015039 2019

  41. [43]

    Emmanuel-Costa, O

    D. Emmanuel-Costa, O. M. Ogreid, P. Osland and M. N. Rebelo, J. Phys. Conf. Ser. 873, no.1, 012007 (2017) doi:10.1088/1742-6596/873/1/012007 [arXiv:1703.08457 [hep- ph]]

    work page doi:10.1088/1742-6596/873/1/012007 2017

  42. [44]

    Bodies in an interacting active fluid: far-field influence of a single body and interaction between two bodies

    I. de Medeiros Varzielas, J. Phys. Conf. Ser.873, no.1, 012035 (2017) doi:10.1088/1742- 6596/873/1/012035 [arXiv:1703.05750 [hep-ph]]

    work page doi:10.1088/1742- 2017

  43. [45]

    Kunˇ cinas, P

    A. Kunˇ cinas, P. Osland and M. N. Rebelo, JHEP11, 086 (2024) doi:10.1007/JHEP11(2024)086 [arXiv:2408.02728 [hep-ph]]

    work page doi:10.1007/jhep11(2024)086 2024

  44. [46]

    Kuncinas, [arXiv:2506.08197 [hep-ph]]

    A. Kuncinas, [arXiv:2506.08197 [hep-ph]]

    work page arXiv

  45. [47]

    Kuncinas, O

    A. Kuncinas, O. M. Ogreid, P. Osland and M. Nesbitt Rebelo, PoSDISCRETE2020- 2021, 062 (2022) doi:10.22323/1.405.0062 [arXiv:2204.08872 [hep-ph]]

    work page doi:10.22323/1.405.0062 2021

  46. [48]

    H. Deng, R. Boto, I. P. Ivanov and J. P. Silva, Phys. Rev. D111, no.5, 055006 (2025) doi:10.1103/PhysRevD.111.055006 [arXiv:2501.05929 [hep-ph]]

    work page doi:10.1103/physrevd.111.055006 2025

  47. [49]

    Sakurai and F

    K. Sakurai and F. Takahashi, JHEP07, 124 (2022) [erratum: JHEP09, 132 (2023)] doi:10.1007/JHEP07(2022)124 [arXiv:2203.17212 [hep-ph]]

    work page doi:10.1007/jhep07(2022)124 2022

  48. [50]

    Dark Matter and Dark Energy in Three-Higgs Doublet Model

    M. Farhan, I. Hassan, M. Usman and N. Tahir, [arXiv:2512.09404 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv

  49. [51]

    Saikawa, J

    A. Dutta Banik, T. Jha and E. Tanskanen, JCAP12, 054 (2025) doi:10.1088/1475- 7516/2025/12/054 [arXiv:2508.13583 [hep-ph]]. 28

    work page doi:10.1088/1475- 2025

  50. [52]

    I. P. Ivanov, Phys. Lett. B632, 360-365 (2006) doi:10.1016/j.physletb.2005.10.015 [arXiv:hep-ph/0507132 [hep-ph]]

    work page doi:10.1016/j.physletb.2005.10.015 2006

  51. [53]

    I. P. Ivanov, Phys. Rev. D75, 035001 (2007) [erratum: Phys. Rev. D76, 039902 (2007)] doi:10.1103/PhysRevD.75.035001 [arXiv:hep-ph/0609018 [hep-ph]]

    work page doi:10.1103/physrevd.75.035001 2007

  52. [54]

    A. V. Bednyakov, JHEP11, 154 (2018) doi:10.1007/JHEP11(2018)154 [arXiv:1809.04527 [hep-ph]]

    work page doi:10.1007/jhep11(2018)154 2018

  53. [55]

    Davidson and H

    S. Davidson and H. E. Haber, Phys. Rev. D72, 035004 (2005) [erratum: Phys. Rev. D 72, 099902 (2005)] doi:10.1103/PhysRevD.72.099902 [arXiv:hep-ph/0504050 [hep-ph]]

    work page doi:10.1103/physrevd.72.099902 2005

  54. [56]

    M. P. Bento, JHEP05, 146 (2021) doi:10.1007/JHEP05(2021)146 [arXiv:2102.13120 [hep-ph]]

    work page doi:10.1007/jhep05(2021)146 2021

  55. [57]

    M. P. Bento, R. Boto, J. P. Silva and A. Trautner, JHEP21, 229 (2020) doi:10.1007/JHEP02(2021)220 [arXiv:2009.01264 [hep-ph]]

    work page doi:10.1007/jhep02(2021)220 2020

  56. [58]

    Trautner, JHEP05, 208 (2019) doi:10.1007/JHEP05(2019)208 [arXiv:1812.02614 [hep-ph]]

    A. Trautner, JHEP05, 208 (2019) doi:10.1007/JHEP05(2019)208 [arXiv:1812.02614 [hep-ph]]

    work page doi:10.1007/jhep05(2019)208 2019

  57. [59]

    I. P. Ivanov, C. C. Nishi and A. Trautner, Eur. Phys. J. C79, no.4, 315 (2019) doi:10.1140/epjc/s10052-019-6845-x [arXiv:1901.11472 [hep-ph]]

    work page doi:10.1140/epjc/s10052-019-6845-x 2019

  58. [60]

    E. E. Jenkins and A. V. Manohar, JHEP10, 094 (2009) doi:10.1088/1126- 6708/2009/10/094 [arXiv:0907.4763 [hep-ph]]

    work page doi:10.1088/1126- 2009

  59. [61]

    Pouliot, JHEP01, 021 (1999) doi:10.1088/1126-6708/1999/01/021 [arXiv:hep- th/9812015 [hep-th]]

    P. Pouliot, JHEP01, 021 (1999) doi:10.1088/1126-6708/1999/01/021 [arXiv:hep- th/9812015 [hep-th]]

    work page doi:10.1088/1126-6708/1999/01/021 1999

  60. [62]

    Seiberg, Nucl

    N. Seiberg, Nucl. Phys. B435, 129-146 (1995) doi:10.1016/0550-3213(94)00023-8 [arXiv:hep-th/9411149 [hep-th]]

    work page doi:10.1016/0550-3213(94)00023-8 1995

  61. [63]

    Hanany, N

    A. Hanany, N. Mekareeya and G. Torri, Nucl. Phys. B825, 52-97 (2010) doi:10.1016/j.nuclphysb.2009.09.016 [arXiv:0812.2315 [hep-th]]

    work page doi:10.1016/j.nuclphysb.2009.09.016 2010

  62. [64]

    Lehman and A

    L. Lehman and A. Martin, Phys. Rev. D91, 105014 (2015) doi:10.1103/PhysRevD.91.105014 [arXiv:1503.07537 [hep-ph]]

    work page doi:10.1103/physrevd.91.105014 2015

  63. [65]

    M. A. Luty and W. Taylor, Phys. Rev. D53, 3399-3405 (1996) doi:10.1103/PhysRevD.53.3399 [arXiv:hep-th/9506098 [hep-th]]

    work page doi:10.1103/physrevd.53.3399 1996

  64. [66]

    Brax and C

    P. Brax and C. A. Savoy, [arXiv:hep-th/0104077 [hep-th]]

    work page internal anchor Pith review arXiv

  65. [67]

    Berger, J

    D. Berger, J. N. Howard and A. Rajaraman, LHEP1, no.2, 14-19 (2018) doi:10.31526/LHEP.2.2018.04 [arXiv:1806.04332 [hep-th]]

    work page doi:10.31526/lhep.2.2018.04 2018

  66. [68]

    Almumin, J

    Y. Almumin, J. Baretz, E. Bryan and A. Rajaraman, Acta Phys. Polon. B54, no.5, 2 (2023) doi:10.5506/APhysPolB.54.5-A2 [arXiv:2009.06545 [hep-th]]. 29

    work page doi:10.5506/aphyspolb.54.5-a2 2023

  67. [69]

    Rajaraman, [arXiv:2211.11851 [hep-th]]

    A. Rajaraman, [arXiv:2211.11851 [hep-th]]

    work page arXiv

  68. [70]

    Bryan,3HDM Hilbert Series Numerator, GitHub repository,https://github.com/ ericalexanderbryan/3HDM-hilbert-series-numerator(2026)

    E. Bryan,3HDM Hilbert Series Numerator, GitHub repository,https://github.com/ ericalexanderbryan/3HDM-hilbert-series-numerator(2026). 30

    2026