Abstract
We present a detailed study of the vacua of the S 3-symmetric three-Higgs-doublet potential, specifying the region of parameters where these minimisation solutions occur. We work with a CP conserving scalar potential and analyse the possible real and complex vacua with emphasis on the cases in which the CP symmetry can be spontaneously broken. Results are presented both in the reducible-representation framework of Derman, and in the irreducible-representation framework. Mappings between these are given. Some of these implementations can in principle accommodate dark matter and for that purpose it is important to identify the residual symmetries of the potential after spontaneous symmetry breakdown. We are also concerned with constraints from vacuum stability.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
30 August 2016
An erratum to this article has been published.
References
J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, The Higgs Hunter’s Guide, Front. Phys. 80 (2000) 1 [INSPIRE].
G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher and J.P. Silva, Theory and phenomenology of two-Higgs-doublet models, Phys. Rept. 516 (2012) 1 [arXiv:1106.0034] [INSPIRE].
T.D. Lee, A Theory of Spontaneous T Violation, Phys. Rev. D 8 (1973) 1226 [INSPIRE].
G.C. Branco, Spontaneous CP Nonconservation and Natural Flavor Conservation: A Minimal Model, Phys. Rev. D 22 (1980) 2901 [INSPIRE].
G.C. Branco, A.J. Buras and J.M. Gerard, CP Violation in Models With Two and Three Scalar Doublets, Nucl. Phys. B 259 (1985) 306 [INSPIRE].
L. Lavoura and J.P. Silva, Fundamental CP-violating quantities in a SU(2) × U(1) model with many Higgs doublets, Phys. Rev. D 50 (1994) 4619 [hep-ph/9404276] [INSPIRE].
F.J. Botella and J.P. Silva, Jarlskog-like invariants for theories with scalars and fermions, Phys. Rev. D 51 (1995) 3870 [hep-ph/9411288] [INSPIRE].
G.C. Branco, L. Lavoura and J.P. Silva, CP Violation, Int. Ser. Monogr. Phys. 103 (1999) 1 [INSPIRE].
G.C. Branco, M.N. Rebelo and J.I. Silva-Marcos, CP-odd invariants in models with several Higgs doublets, Phys. Lett. B 614 (2005) 187 [hep-ph/0502118] [INSPIRE].
J.F. Gunion and H.E. Haber, Conditions for CP-violation in the general two-Higgs-doublet model, Phys. Rev. D 72 (2005) 095002 [hep-ph/0506227] [INSPIRE].
C.C. Nishi, CP violation conditions in N-Higgs-doublet potentials, Phys. Rev. D 74 (2006) 036003 [Erratum ibid. D 76 (2007) 119901] [hep-ph/0605153] [INSPIRE].
E. Accomando et al., Workshop on CP Studies and Non-Standard Higgs Physics, hep-ph/0608079.
M. Maniatis, A. von Manteuffel and O. Nachtmann, CP violation in the general two-Higgs-doublet model: A geometric view, Eur. Phys. J. C 57 (2008) 719 [arXiv:0707.3344] [INSPIRE].
B. Grzadkowski, O.M. Ogreid and P. Osland, Diagnosing CP properties of the 2HDM, JHEP 01 (2014) 105 [arXiv:1309.6229] [INSPIRE].
N.G. Deshpande and E. Ma, Pattern of Symmetry Breaking with Two Higgs Doublets, Phys. Rev. D 18 (1978) 2574 [INSPIRE].
R. Barbieri, L.J. Hall and V.S. Rychkov, Improved naturalness with a heavy Higgs: An alternative road to LHC physics, Phys. Rev. D 74 (2006) 015007 [hep-ph/0603188] [INSPIRE].
K. Olaussen, P. Osland and M.A. Solberg, Symmetry and Mass Degeneration in Multi-Higgs-Doublet Models, JHEP 07 (2011) 020 [arXiv:1007.1424] [INSPIRE].
S. Pakvasa and H. Sugawara, Discrete Symmetry and Cabibbo Angle, Phys. Lett. B 73 (1978) 61 [INSPIRE].
J. Kubo, H. Okada and F. Sakamaki, Higgs potential in minimal S 3 invariant extension of the standard model, Phys. Rev. D 70 (2004) 036007 [hep-ph/0402089] [INSPIRE].
E. Derman, Flavor Unification, τ Decay and b Decay Within the Six Quark Six Lepton Weinberg-Salam Model, Phys. Rev. D 19 (1979) 317 [INSPIRE].
E. Derman and H.-S. Tsao, SU(2) × U(1) × S(n) Flavor Dynamics and a Bound on the Number of Flavors, Phys. Rev. D 20 (1979) 1207 [INSPIRE].
I.P. Ivanov and C.C. Nishi, Symmetry breaking patterns in 3HDM, JHEP 01 (2015) 021 [arXiv:1410.6139] [INSPIRE].
T. Teshima, Higgs potential in S 3 invariant model for quark/lepton mass and mixing, Phys. Rev. D 85 (2012) 105013 [arXiv:1202.4528] [INSPIRE].
D. Das and U.K. Dey, Analysis of an extended scalar sector with S 3 symmetry, Phys. Rev. D 89 (2014) 095025 [Erratum ibid. D 91 (2015) 039905] [arXiv:1404.2491] [INSPIRE].
S.-L. Chen, M. Frigerio and E. Ma, Large neutrino mixing and normal mass hierarchy: A discrete understanding, Phys. Rev. D 70 (2004) 073008 [Erratum ibid. D 70 (2004) 079905] [hep-ph/0404084] [INSPIRE].
G. Bhattacharyya, P. Leser and H. Pas, Exotic Higgs boson decay modes as a harbinger of S 3 flavor symmetry, Phys. Rev. D 83 (2011) 011701 [arXiv:1006.5597] [INSPIRE].
N. Chakrabarty, High-scale validity of a model with Three-Higgs-doublets, arXiv:1511.08137 [INSPIRE].
A. Barroso, P.M. Ferreira, R. Santos and J.P. Silva, Stability of the normal vacuum in multi-Higgs-doublet models, Phys. Rev. D 74 (2006) 085016 [hep-ph/0608282] [INSPIRE].
D. Emmanuel-Costa, O. Félix-Beltrán, M. Mondragón and E. Rodríguez-Jáuregui, Stability of the tree-level vacuum in a minimal S(3) extension of the standard model, AIP Conf. Proc. 917 (2007) 390.
G.C. Branco and M.N. Rebelo, The Higgs Mass in a Model With Two Scalar Doublets and Spontaneous CP Violation, Phys. Lett. B 160 (1985) 117 [INSPIRE].
Y. Koide, Permutation symmetry S 3 and VEV structure of flavor-triplet Higgs scalars, Phys. Rev. D 73 (2006) 057901 [hep-ph/0509214] [INSPIRE].
G.C. Branco, J.M. Gerard and W. Grimus, Geometrical T violation, Phys. Lett. B 136 (1984) 383 [INSPIRE].
I. de Medeiros Varzielas and D. Emmanuel-Costa, Geometrical CP-violation, Phys. Rev. D 84 (2011) 117901 [arXiv:1106.5477] [INSPIRE].
I. de Medeiros Varzielas, D. Emmanuel-Costa and P. Leser, Geometrical CP-violation from Non-Renormalisable Scalar Potentials, Phys. Lett. B 716 (2012) 193 [arXiv:1204.3633] [INSPIRE].
I. de Medeiros Varzielas, Geometrical CP-violation in multi-Higgs models, JHEP 08 (2012) 055 [arXiv:1205.3780] [INSPIRE].
G. Bhattacharyya, I. de Medeiros Varzielas and P. Leser, A common origin of fermion mixing and geometrical CP-violation and its test through Higgs physics at the LHC, Phys. Rev. Lett. 109 (2012) 241603 [arXiv:1210.0545] [INSPIRE].
I.P. Ivanov and L. Lavoura, Geometrical CP-violation in the N-Higgs-doublet model, Eur. Phys. J. C 73 (2013) 2416 [arXiv:1302.3656] [INSPIRE].
I. de Medeiros Varzielas and D. Pidt, Geometrical CP-violation with a complete fermion sector, JHEP 11 (2013) 206 [arXiv:1307.6545] [INSPIRE].
M. Fallbacher and A. Trautner, Symmetries of symmetries and geometrical CP-violation, Nucl. Phys. B 894 (2015) 136 [arXiv:1502.01829] [INSPIRE].
E. Ma, Verifiable radiative seesaw mechanism of neutrino mass and dark matter, Phys. Rev. D 73 (2006) 077301 [hep-ph/0601225] [INSPIRE].
E. Ma, Common origin of neutrino mass, dark matter and baryogenesis, Mod. Phys. Lett. A 21 (2006) 1777 [hep-ph/0605180] [INSPIRE].
D. Majumdar and A. Ghosal, Dark Matter candidate in a Heavy Higgs Model — Direct Detection Rates, Mod. Phys. Lett. A 23 (2008) 2011 [hep-ph/0607067] [INSPIRE].
L. Lopez Honorez, E. Nezri, J.F. Oliver and M.H.G. Tytgat, The Inert Doublet Model: An Archetype for Dark Matter, JCAP 02 (2007) 028 [hep-ph/0612275] [INSPIRE].
N. Sahu and U. Sarkar, Predictive model for dark matter, dark energy, neutrino masses and leptogenesis at the TeV scale, Phys. Rev. D 76 (2007) 045014 [hep-ph/0701062] [INSPIRE].
M. Gustafsson, E. Lundstrom, L. Bergstrom and J. Edsjo, Significant Gamma Lines from Inert Higgs Dark Matter, Phys. Rev. Lett. 99 (2007) 041301 [astro-ph/0703512] [INSPIRE].
M. Lisanti and J.G. Wacker, Unification and dark matter in a minimal scalar extension of the standard model, arXiv:0704.2816 [INSPIRE].
T. Hambye and M.H.G. Tytgat, Electroweak symmetry breaking induced by dark matter, Phys. Lett. B 659 (2008) 651 [arXiv:0707.0633] [INSPIRE].
Q.-H. Cao, E. Ma and G. Rajasekaran, Observing the Dark Scalar Doublet and its Impact on the Standard-Model Higgs Boson at Colliders, Phys. Rev. D 76 (2007) 095011 [arXiv:0708.2939] [INSPIRE].
S. Andreas, T. Hambye and M.H.G. Tytgat, WIMP dark matter, Higgs exchange and DAMA, JCAP 10 (2008) 034 [arXiv:0808.0255] [INSPIRE].
E. Lundstrom, M. Gustafsson and J. Edsjo, The Inert Doublet Model and LEP II Limits, Phys. Rev. D 79 (2009) 035013 [arXiv:0810.3924] [INSPIRE].
E.M. Dolle and S. Su, The Inert Dark Matter, Phys. Rev. D 80 (2009) 055012 [arXiv:0906.1609] [INSPIRE].
T. Hambye, F.S. Ling, L. Lopez Honorez and J. Rocher, Scalar Multiplet Dark Matter, JHEP 07 (2009) 090 [Erratum ibid. 05 (2010) 066] [arXiv:0903.4010] [INSPIRE].
C. Bonilla, D. Sokolowska, J.L. Diaz-Cruz, M. Krawczyk and N. Darvishi, IDMS: Inert Dark Matter Model with a complex singlet, arXiv:1412.8730 [INSPIRE].
A. Ilnicka, M. Krawczyk and T. Robens, The Inert Doublet Model in the light of LHC and astrophysical data — An Update, arXiv:1508.01671 [INSPIRE].
B. Grzadkowski, O.M. Ogreid and P. Osland, Natural Multi-Higgs Model with Dark Matter and CP-violation, Phys. Rev. D 80 (2009) 055013 [arXiv:0904.2173] [INSPIRE].
B. Grzadkowski, O.M. Ogreid, P. Osland, A. Pukhov and M. Purmohammadi, Exploring the CP-Violating Inert-Doublet Model, JHEP 06 (2011) 003 [arXiv:1012.4680] [INSPIRE].
V. Keus, S.F. King, S. Moretti and D. Sokolowska, Dark Matter with Two Inert Doublets plus One Higgs Doublet, JHEP 11 (2014) 016 [arXiv:1407.7859] [INSPIRE].
A.C.B. Machado and V. Pleitez, A model with two inert scalar doublets, Annals Phys. 364 (2016) 53 [arXiv:1205.0995] [INSPIRE].
E.C.F.S. Fortes, A.C.B. Machado, J. Montaño and V. Pleitez, Scalar dark matter candidates in a two inert Higgs doublet model, J. Phys. G 42 (2015) 105003 [arXiv:1407.4749] [INSPIRE].
E.C.F.S. Fortes, A.C.B. Machado, J. Montaño and V. Pleitez, Prediction of h → γZ from h → γγ at LHC for the IMDS 3 Model, J. Phys. G 42 (2015) 115001 [arXiv:1408.0780] [INSPIRE].
F. González Canales, A. Mondragón, M. Mondragón, U.J. Saldaña Salazar and L. Velasco-Sevilla, Quark sector of S3 models: classification and comparison with experimental data, Phys. Rev. D 88 (2013) 096004 [arXiv:1304.6644] [INSPIRE].
D. Das, U.K. Dey and P.B. Pal, S 3 symmetry and the quark mixing matrix, Phys. Lett. B 753 (2016) 315 [arXiv:1507.06509] [INSPIRE].
D. Wyler, Discrete Symmetries in the Six Quark SU(2) × U(1) Model, Phys. Rev. D 19 (1979) 3369 [INSPIRE].
G.C. Branco and I.P. Ivanov, Group-theoretic restrictions on generation of CP-violation in multi-Higgs-doublet models, JHEP 01 (2016) 116 [arXiv:1511.02764] [INSPIRE].
A.W. El Kaffas, W. Khater, O.M. Ogreid and P. Osland, Consistency of the two Higgs doublet model and CP-violation in top production at the LHC, Nucl. Phys. B 775 (2007) 45 [hep-ph/0605142] [INSPIRE].
M. Maniatis and O. Nachtmann, Stability and symmetry breaking in the general three-Higgs-doublet model, JHEP 02 (2015) 058 [Erratum ibid. 10 (2015) 149] [arXiv:1408.6833] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1601.04654
An erratum to this article can be found online at http://dx.doi.org/10.1007/JHEP08(2016)169.
An erratum to this article is available at https://doi.org/10.1007/JHEP08(2016)169.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Emmanuel-Costa, D., Ogreid, O.M., Osland, P. et al. Spontaneous symmetry breaking in the S 3-symmetric scalar sector. J. High Energ. Phys. 2016, 154 (2016). https://doi.org/10.1007/JHEP02(2016)154
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2016)154