Abstract
Soft breaking of a symmetry requires an invariance of the dimension-4 part of the Lagrangian and non-trivial variation of the lower-dimensional part. However, in general, separation between the dim-4 and lower-dimensional Lagrangian is not invariant with respect to basis transformations of fields. Therefore, a natural question of the physical meaning of soft symmetry breaking arises. This problem is addressed here in the framework of two-Higgs-Doublet Models (2HDM). It has been shown, within these models, that in spite of the ambiguity corresponding to the separation between dim-4 and the lower-dimension Lagrangian, implications of the soft symmetry breaking could be formulated in terms of observables, i.e., they are physical and measurable. There are six global symmetries that can be imposed on the scalar sector of the generic 2HDM. Necessary and sufficient tree-level conditions for soft breaking of all of them have been formulated in terms of observables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T.D. Lee, A Theory of Spontaneous T Violation, Phys. Rev. D 8 (1973) 1226 [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].
S.L. Glashow and S. Weinberg, Natural Conservation Laws for Neutral Currents, Phys. Rev. D 15 (1977) 1958 [INSPIRE].
R.D. Peccei and H.R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].
J. Velhinho, R. Santos and A. Barroso, Tree level vacuum stability in two Higgs doublet models, Phys. Lett. B 322 (1994) 213 [INSPIRE].
I.P. Ivanov, Minkowski space structure of the Higgs potential in 2HDM, Phys. Rev. D 75 (2007) 035001 [hep-ph/0609018] [INSPIRE].
M. Maniatis, A. von Manteuffel, O. Nachtmann and F. Nagel, Stability and symmetry breaking in the general two-Higgs-doublet model, Eur. Phys. J. C 48 (2006) 805 [hep-ph/0605184] [INSPIRE].
C.C. Nishi, CP violation conditions in N-Higgs-doublet potentials, Phys. Rev. D 74 (2006) 036003 [hep-ph/0605153] [INSPIRE].
I.P. Ivanov, Minkowski space structure of the Higgs potential in 2HDM. II. Minima, symmetries, and topology, Phys. Rev. D 77 (2008) 015017 [arXiv:0710.3490] [INSPIRE].
C.C. Nishi, The Structure of potentials with N Higgs doublets, Phys. Rev. D 76 (2007) 055013 [arXiv:0706.2685] [INSPIRE].
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].
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. A 26 (2011) 769 [arXiv:1010.0935] [INSPIRE].
S. Davidson and H.E. Haber, Basis-independent methods for the two-Higgs-doublet model, Phys. Rev. D 72 (2005) 035004 [hep-ph/0504050] [INSPIRE].
B. Grzadkowski, O.M. Ogreid and P. Osland, Measuring CP violation in Two-Higgs-Doublet models in light of the LHC Higgs data, JHEP 11 (2014) 084 [arXiv:1409.7265] [INSPIRE].
B. Grzadkowski, O.M. Ogreid and P. Osland, Spontaneous CP violation in the 2HDM: physical conditions and the alignment limit, Phys. Rev. D 94 (2016) 115002 [arXiv:1609.04764] [INSPIRE].
B. Grzadkowski, H.E. Haber, O.M. Ogreid and P. Osland, Heavy Higgs boson decays in the alignment limit of the 2HDM, JHEP 12 (2018) 056 [arXiv:1808.01472] [INSPIRE].
O.M. Ogreid, Invariants and CP violation in the 2HDM, PoS CORFU2017 (2018) 065 [arXiv:1803.09351] [INSPIRE].
P.M. Ferreira, B. Grzadkowski, O.M. Ogreid and P. Osland, Symmetries of the 2HDM: an invariant formulation and consequences, JHEP 02 (2021) 196 [arXiv:2010.13698] [INSPIRE].
P.M. Ferreira, R. Santos and A. Barroso, Stability of the tree-level vacuum in two Higgs doublet models against charge or CP spontaneous violation, Phys. Lett. B 603 (2004) 219 [hep-ph/0406231] [INSPIRE].
A. Barroso, P.M. Ferreira and R. Santos, Charge and CP symmetry breaking in two Higgs doublet models, Phys. Lett. B 632 (2006) 684 [hep-ph/0507224] [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].
P.M. Ferreira and J.P. Silva, A Two-Higgs Doublet Model With Remarkable CP Properties, Eur. Phys. J. C 69 (2010) 45 [arXiv:1001.0574] [INSPIRE].
M. Maniatis and O. Nachtmann, On the phenomenology of a two-Higgs-doublet model with maximal CP symmetry at the LHC, JHEP 05 (2009) 028 [arXiv:0901.4341] [INSPIRE].
M. Maniatis and O. Nachtmann, On the phenomenology of a two-Higgs-doublet model with maximal CP symmetry at the LHC. II. Radiative effects, JHEP 04 (2010) 027 [arXiv:0912.2727] [INSPIRE].
P. Draper, A. Ekstedt and H.E. Haber, A natural mechanism for approximate Higgs alignment in the 2HDM, JHEP 05 (2021) 235 [arXiv:2011.13159] [INSPIRE].
J.F. Donoghue and L.F. Li, Properties of Charged Higgs Bosons, Phys. Rev. D 19 (1979) 945 [INSPIRE].
H. Georgi and D.V. Nanopoulos, Suppression of Flavor Changing Effects From Neutral Spinless Meson Exchange in Gauge Theories, Phys. Lett. B 82 (1979) 95 [INSPIRE].
N.G. Deshpande and E. Ma, Pattern of Symmetry Breaking with Two Higgs Doublets, Phys. Rev. D 18 (1978) 2574 [INSPIRE].
A. Barroso, P.M. Ferreira, I.P. Ivanov, R. Santos and J.P. Silva, Evading death by vacuum, Eur. Phys. J. C 73 (2013) 2537 [arXiv:1211.6119] [INSPIRE].
A. Barroso, P.M. Ferreira, I.P. Ivanov and R. Santos, Metastability bounds on the two Higgs doublet model, JHEP 06 (2013) 045 [arXiv:1303.5098] [INSPIRE].
ATLAS, CMS collaborations, Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at \( \sqrt{s} \) = 7 and 8 TeV, JHEP 08 (2016) 045 [arXiv:1606.02266] [INSPIRE].
I.F. Ginzburg and K.A. Kanishev, Two-Higgs-doublet model in terms of observable quantities, Phys. Rev. D 92 (2015) 015024 [arXiv:1502.06346] [INSPIRE].
R.A. Battye, G.D. Brawn and A. Pilaftsis, Vacuum Topology of the Two Higgs Doublet Model, JHEP 08 (2011) 020 [arXiv:1106.3482] [INSPIRE].
A. Pilaftsis, On the Classification of Accidental Symmetries of the Two Higgs Doublet Model Potential, Phys. Lett. B 706 (2012) 465 [arXiv:1109.3787] [INSPIRE].
K. Symanzik, Renormalizable models with simple symmetry breaking. 1. Symmetry breaking by a source term, Commun. Math. Phys. 16 (1970) 48 [INSPIRE].
K. Symanzik, Renormalization of theories with broken symmetry, Cargese Lect. Phys. 5 (1972) 179 [INSPIRE].
H.E. Haber and Y. Nir, Multiscalar Models With a High-energy Scale, Nucl. Phys. B 335 (1990) 363 [INSPIRE].
J.F. Gunion and H.E. Haber, The CP conserving two Higgs doublet model: The Approach to the decoupling limit, Phys. Rev. D 67 (2003) 075019 [hep-ph/0207010] [INSPIRE].
P.M. Ferreira, L. Lavoura and J.P. Silva, A Soft origin for CKM-type CP violation, Phys. Lett. B 704 (2011) 179 [arXiv:1102.0784] [INSPIRE].
I.F. Ginzburg, M. Krawczyk and P. Osland, Two Higgs doublet models with CP violation, in International Workshop on Linear Colliders (LCWS 2002), (2002), pp. 703–706 [hep-ph/0211371] [INSPIRE].
W. Khater and P. Osland, CP violation in top quark production at the LHC and two Higgs doublet models, Nucl. Phys. B 661 (2003) 209 [hep-ph/0302004] [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].
A.W. El Kaffas, P. Osland and O.M. Ogreid, CP violation, stability and unitarity of the two Higgs doublet model, Nonlin. Phenom. Complex Syst. 10 (2007) 347 [hep-ph/0702097] [INSPIRE].
A. Wahab El Kaffas, P. Osland and O.M. Ogreid, Constraining the Two-Higgs-Doublet-Model parameter space, Phys. Rev. D 76 (2007) 095001 [arXiv:0706.2997] [INSPIRE].
P. Osland, P.N. Pandita and L. Selbuz, Trilinear Higgs couplings in the two Higgs doublet model with CP violation, Phys. Rev. D 78 (2008) 015003 [arXiv:0802.0060] [INSPIRE].
B. Grzadkowski and P. Osland, Tempered Two-Higgs-Doublet Model, Phys. Rev. D 82 (2010) 125026 [arXiv:0910.4068] [INSPIRE].
A. Arhrib, E. Christova, H. Eberl and E. Ginina, CP violation in charged Higgs production and decays in the Complex Two Higgs Doublet Model, JHEP 04 (2011) 089 [arXiv:1011.6560] [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].
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].
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].
H.E. Haber and J.P. Silva, Exceptional regions of the 2HDM parameter space, Phys. Rev. D 103 (2021) 115012 [arXiv:2102.07136] [INSPIRE].
P.S. Bhupal Dev and A. Pilaftsis, Maximally Symmetric Two Higgs Doublet Model with Natural Standard Model Alignment, JHEP 12 (2014) 024 [arXiv:1408.3405] [INSPIRE].
S.R. Coleman and E.J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7 (1973) 1888 [INSPIRE].
A. Pilaftsis, Symmetries for standard model alignment in multi-Higgs doublet models, Phys. Rev. D 93 (2016) 075012 [arXiv:1602.02017] [INSPIRE].
N. Darvishi and A. Pilaftsis, Natural Alignment in Multi-Higgs Doublet Models, PoS CORFU2019 (2020) 064 [arXiv:2004.04505] [INSPIRE].
N. Darvishi and A. Pilaftsis, Quartic Coupling Unification in the Maximally Symmetric 2HDM, Phys. Rev. D 99 (2019) 115014 [arXiv:1904.06723] [INSPIRE].
F. Arco, S. Heinemeyer and M.J. Herrero, Sensitivity and constraints to the 2HDM soft-breaking Z2 parameter m12, Phys. Lett. B 835 (2022) 137548 [arXiv:2207.13501] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2209.00152
Rights and permissions
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.
About this article
Cite this article
Ferreira, P.M., Grzadkowski, B., Ogreid, O.M. et al. Softly broken symmetries in the 2HDM — an invariant formulation. J. High Energ. Phys. 2023, 143 (2023). https://doi.org/10.1007/JHEP01(2023)143
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2023)143