# Multi-Higgs doublet models: physical parametrization, sum rules and unitarity bounds

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## Abstract

If the scalar sector of the Standard Model is non-minimal, one might expect multiple generations of the hypercharge-1/2 scalar doublet analogous to the generational structure of the fermions. In this work, we examine the structure of a Higgs sector consisting of *N* Higgs doublets (where *N* ≥ 2). It is particularly convenient to work in the so-called *charged Higgs basis*, in which the neutral Higgs vacuum expectation value resides entirely in the first Higgs doublet, and the charged components of remaining *N* − 1 Higgs doublets are mass-eigenstate fields. We elucidate the interactions of the gauge bosons with the physical Higgs scalars and the Goldstone bosons and show that they are determined by an *N* × 2*N* matrix. This matrix depends on (*N* − 1)(2*N* − 1) real parameters that are associated with the mixing of the neutral Higgs fields in the charged Higgs basis. Among these parameters, *N* − 1 are unphysical (and can be removed by rephasing the physical charged Higgs fields), and the remaining 2(*N* − 1)^{2} parameters are physical. We also demonstrate a particularly simple form for the cubic interaction and some of the quartic interactions of the Goldstone bosons with the physical Higgs scalars. These results are applied in the derivation of Higgs coupling sum rules and tree-level unitarity bounds that restrict the size of the quartic scalar couplings. In particular, new applications to three Higgs doublet models with an order-4 CP symmetry and with a \( {\mathbb{Z}}_3 \) symmetry, respectively, are presented.

## Keywords

Beyond Standard Model Higgs Physics CP violation Discrete Symmetries## Notes

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## References

- [1]ATLAS collaboration,
*Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC*,*Phys. Lett.***B 716**(2012) 1 [arXiv:1207.7214] [INSPIRE]. - [2]CMS collaboration,
*Observation of a new boson at a mass of*125*GeV with the CMS experiment at the LHC*,*Phys. Lett.***B 716**(2012) 30 [arXiv:1207.7235] [INSPIRE]. - [3]ATLAS and 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]. - [4]Particle Data Group collaboration, C. Patrignani et al.,
*Review of particle physics*,*Chin. Phys.***C 40**(2016) 100001 [INSPIRE]. - [5]G.A. White,
*A pedagogical introduction to electroweak baryogenesis*, IOP Publishing, Bristol U.K., (2016).CrossRefGoogle Scholar - [6]L. Susskind,
*The gauge hierarchy problem, technicolor, supersymmetry, and all that*,*Phys. Rept.***104**(1984) 181 [INSPIRE].ADSCrossRefGoogle Scholar - [7]J.F. Gunion, H.E. Haber, G. Kane and S. Dawson,
*The Higgs hunter’s guide*, Westview Press, Boulder CO U.S.A., (2000).Google Scholar - [8]I.P. Ivanov,
*Building and testing models with extended Higgs sectors*,*Prog. Part. Nucl. Phys.***95**(2017) 160 [arXiv:1702.03776] [INSPIRE].ADSCrossRefGoogle Scholar - [9]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]. - [10]C.C. Nishi,
*The structure of potentials with N Higgs doublets*,*Phys. Rev.***D 76**(2007) 055013 [arXiv:0706.2685] [INSPIRE].ADSGoogle Scholar - [11]S.L. Glashow and S. Weinberg,
*Natural conservation laws for neutral currents*,*Phys. Rev.***D 15**(1977) 1958 [INSPIRE].ADSGoogle Scholar - [12]
- [13]M.P. Bento, H.E. Haber, J.C. Romão and J.P. Silva, preprint in preparation.Google Scholar
- [14]C.H. Llewellyn Smith,
*High-energy behavior and gauge symmetry*,*Phys. Lett.***B 46**(1973) 233 [INSPIRE].ADSCrossRefGoogle Scholar - [15]J.M. Cornwall, D.N. Levin and G. Tiktopoulos,
*Derivation of gauge invariance from high-energy unitarity bounds on the S matrix*,*Phys. Rev.***D 10**(1974) 1145 [*Erratum ibid.***D 11**(1975) 972] [INSPIRE]. - [16]H.A. Weldon,
*The effects of multiple Higgs bosons on tree unitarity*,*Phys. Rev.***D 30**(1984) 1547 [INSPIRE].ADSGoogle Scholar - [17]J.F. Gunion, H.E. Haber and J. Wudka,
*Sum rules for Higgs bosons*,*Phys. Rev.***D 43**(1991) 904 [INSPIRE].ADSGoogle Scholar - [18]B. Grinstein, C.W. Murphy, D. Pirtskhalava and P. Uttayarat,
*Theoretical constraints on additional Higgs bosons in light of the*126*GeV Higgs*,*JHEP***05**(2014) 083 [arXiv:1401.0070] [INSPIRE].ADSCrossRefGoogle Scholar - [19]B.W. Lee, C. Quigg and H.B. Thacker,
*The strength of weak interactions at very high-energies and the Higgs boson mass*,*Phys. Rev. Lett.***38**(1977) 883 [INSPIRE].ADSCrossRefGoogle Scholar - [20]B.W. Lee, C. Quigg and H.B. Thacker,
*Weak interactions at very high-energies: the role of the Higgs boson mass*,*Phys. Rev.***D 16**(1977) 1519 [INSPIRE].ADSGoogle Scholar - [21]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].ADSCrossRefGoogle Scholar - [22]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]. - [23]G.C. Branco, L. Lavoura and J.P. Silva,
*CP violation*, Oxford University Press, Oxford U.K., (1999).Google Scholar - [24]S. Davidson and H.E. Haber,
*Basis-independent methods for the two-Higgs-doublet model*,*Phys. Rev.***D 72**(2005) 035004 [*Erratum ibid.***D 72**(2005) 099902] [hep-ph/0504050] [INSPIRE]. - [25]H.E. Haber and D. O’Neil,
*Basis-independent methods for the two-Higgs-doublet model. II. The significance of*tan*β*,*Phys. Rev.***D 74**(2006) 015018 [*Erratum ibid.***D 74**(2006) 059905] [hep-ph/0602242] [INSPIRE]. - [26]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]. - [27]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]. - [28]I.P. Ivanov and C.C. Nishi,
*Properties of the general NHDM. I. The orbit space*,*Phys. Rev.***D 82**(2010) 015014 [arXiv:1004.1799] [INSPIRE]. - [29]M. Maniatis and O. Nachtmann,
*Stability and symmetry breaking in the general N -Higgs-doublet model*,*Phys. Rev.***D 92**(2015) 075017 [arXiv:1504.01736] [INSPIRE].ADSGoogle Scholar - [30]W. Grimus and L. Lavoura,
*Soft lepton flavor violation in a multi Higgs doublet seesaw model*,*Phys. Rev.***D 66**(2002) 014016 [hep-ph/0204070] [INSPIRE]. - [31]W. Grimus, L. Lavoura, O.M. Ogreid and P. Osland,
*A precision constraint on multi-Higgs-doublet models*,*J. Phys.***G 35**(2008) 075001 [arXiv:0711.4022] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [32]W. Grimus, L. Lavoura, O.M. Ogreid and P. Osland,
*The oblique parameters in multi-Higgs-doublet models*,*Nucl. Phys.***B 801**(2008) 81 [arXiv:0802.4353] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [33]W. Grimus and H. Neufeld,
*Radiative neutrino masses in an*SU(2) × U(1)*model*,*Nucl. Phys.***B 325**(1989) 18 [INSPIRE].ADSCrossRefGoogle Scholar - [34]J.C. Romao and J.P. Silva,
*A resource for signs and Feynman diagrams of the Standard Model*,*Int. J. Mod. Phys.***A 27**(2012) 1230025 [arXiv:1209.6213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [35]A. Djouadi,
*The anatomy of electro-weak symmetry breaking. I: the Higgs boson in the Standard Model*,*Phys. Rept.***457**(2008) 1 [hep-ph/0503172] [INSPIRE]. - [36]A. Djouadi,
*The anatomy of electro-weak symmetry breaking. II: The Higgs bosons in the minimal supersymmetric model*,*Phys. Rept.***459**(2008) 1 [hep-ph/0503173] [INSPIRE]. - [37]H.E. Haber and O. Stål,
*New LHC benchmarks for the CP-conserving two-Higgs-doublet model*,*Eur. Phys. J.***C 75**(2015) 491 [*Erratum ibid.***C 76**(2016) 312] [arXiv:1507.04281] [INSPIRE]. - [38]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]. - [39]
- [40]M.P. Bento, in preparation.Google Scholar
- [41]A. Arhrib,
*Charged Higgs production through W-Higgs fusion*, talk given at the*Workshop on multi-Higgs models*, Lisbon Portugal, 6-9 September 2016.Google Scholar - [42]A. Arhrib, K. Cheung, J.S. Lee and C.-T. Lu,
*Enhanced charged Higgs production through W-Higgs fusion in W-b scattering*,*JHEP***05**(2016) 093 [arXiv:1509.00978] [INSPIRE].ADSCrossRefGoogle Scholar - [43]I.F. Ginzburg and I.P. Ivanov,
*Tree level unitarity constraints in the*2*HDM with CP-violation*, hep-ph/0312374 [INSPIRE]. - [44]I.F. Ginzburg and I.P. Ivanov,
*Tree-level unitarity constraints in the most general*2*HDM*,*Phys. Rev.***D 72**(2005) 115010 [hep-ph/0508020] [INSPIRE]. - [45]S. Kanemura and K. Yagyu,
*Unitarity bound in the most general two Higgs doublet model*,*Phys. Lett.***B 751**(2015) 289 [arXiv:1509.06060] [INSPIRE].ADSCrossRefGoogle Scholar - [46]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]. - [47]M. Lüscher and P. Weisz,
*Is there a strong interaction sector in the standard lattice Higgs model?*,*Phys. Lett.***B 212**(1988) 472 [INSPIRE].ADSCrossRefGoogle Scholar - [48]I.P. Ivanov and J.P. Silva,
*CP-conserving multi-Higgs model with irremovable complex coefficients*,*Phys. Rev.***D 93**(2016) 095014 [arXiv:1512.09276] [INSPIRE].ADSGoogle Scholar - [49]P.M. Ferreira and J.P. Silva,
*Discrete and continuous symmetries in multi-Higgs-doublet models*,*Phys. Rev.***D 78**(2008) 116007 [arXiv:0809.2788] [INSPIRE].ADSMathSciNetGoogle Scholar - [50]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].ADSCrossRefGoogle Scholar - [51]I.F. Ginzburg, M. Krawczyk and P. Osland,
*Two Higgs doublet models with CP-violation*, hep-ph/0211371 [INSPIRE]. - [52]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].ADSCrossRefGoogle Scholar - [53]A. Barroso, P.M. Ferreira, R. Santos and J.P. Silva,
*Probing the scalar-pseudoscalar mixing in the*125*GeV Higgs particle with current data*,*Phys. Rev.***D 86**(2012) 015022 [arXiv:1205.4247] [INSPIRE].ADSGoogle Scholar - [54]D. Fontes, J.C. Romão and J.P. Silva,
*h*→*Z*γ*in the complex two Higgs doublet model*,*JHEP***12**(2014) 043 [arXiv:1408.2534] [INSPIRE]. - [55]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]. - [56]
- [57]S. Caracciolo, A.D. Sokal and A. Sportiello,
*Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians*,*Adv. Appl. Math.***50**(2013) 474 [arXiv:1105.6270].MathSciNetCrossRefzbMATHGoogle Scholar - [58]J. Gallier and D. Xu,
*Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices*,*Int. J. Robotics Automat.***17**(2002) 1.zbMATHGoogle Scholar - [59]H.K. Dreiner, H.E. Haber and S.P. Martin,
*Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry*,*Phys. Rept.***494**(2010) 1 [arXiv:0812.1594] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [60]A. Baker,
*Matrix groups: an introduction to Lie group theory*, Springer-Verlag, London U.K., (2002).CrossRefzbMATHGoogle Scholar - [61]F.D. Murnaghan,
*The unitary and rotation groups*, Spartan Books, Washington DC U.S.A., (1962).zbMATHGoogle Scholar - [62]L.-F. Li,
*Group theory of the spontaneously broken gauge symmetries*,*Phys. Rev.***D 9**(1974) 1723 [INSPIRE].ADSMathSciNetGoogle Scholar - [63]V. Elias, S. Eliezer and A.R. Swift,
*Comment on ‘group theory of the spontaneously broken gauge symmetries’*,*Phys. Rev.***D 12**(1975) 3356 [INSPIRE].ADSGoogle Scholar - [64]D.-D. Wu,
*The symmetry breaking pattern of scalars in low dimension representations*,*Nucl. Phys.***B 199**(1982) 523 [*Erratum ibid.***B 213**(1983) 545] [INSPIRE].