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VV-fusion in CMS: a model-independent way to investigate EWSB
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ORGANISATION EUROPÉENNE POUR LA RECHERCHE NUCLÉAIRE
CERN
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
Workshop on CP Studies and
Non-Standard Higgs Physics
May 2004 – December 2005
Edited bySabine Kraml
1, Georges Azuelos
2,3, Daniele Dominici
4, John Ellis
1,
Gerald Grenier
5, Howard E. Haber
6, Jae Sik Lee
7, David J. Miller
8,
Apostolos Pilaftsis
9and Werner Porod
10GENEVA 2006
1CERN, Geneva, Switzerland. 2
Université de Montréal, Montreal, Canada. 3
TRIUMF, Vancouver, Canada.
4Università di Firenze and INFN, Firenze, Italy. 5IPNL, Université Lyon-1, Villeurbanne, France. 6
University of California, Santa Cruz, USA. 7Seoul National University, Seoul, Korea. 8University of Glasgow, Glasgow, UK. 9
University of Manchester, Manchester, UK. 10
The MSSM with CP phases: M. Boonekamp, M. Carena, S. Y. Choi, J. S. Lee, M. Schumacher SUSY models with an extra singlet: S. Baffioni, J. Gunion, D. Miller, A. Pilaftsis, D. Zerwas The MSSM with R-parity violation: M. Besançon, W. Porod
Extra Gauge groups: P. Langacker, A. Raspereza, S. Riemann Little Higgs models: T. Gregoire, H. Logan, B. McElrath Large extra dimensions: D. Dominici, S. Ferrag
Randall-Sundrum model: A. De Roeck, S. Ferrag, J. L. Hewett, T. G. Rizzo Higgsless models: B. Lillie, J. Terning
Strongly interacting Higgs sector: G. Azuelos, W. Kilian, T. Han Technicolour: G. Azuelos, F. Sannino
Higgs Triplets: J. F. Gunion, C. Hays
Organizing Committee S. Y. Choi (Chonbuk) J. Conway (Davis) R. M. Godbole (Bangalore) J. F. Gunion (Davis) J. Ellis (CERN) J. L. Hewett (SLAC) S. Kraml (CERN) M. Krawczyk (Warsaw) M. Mangano (CERN) D. J. Miller (Glasgow) Y. Okada (KEK) M. Oreglia (Chicago) Meetings 14–15 May 2004 at CERN 2–3 December 2004 at CERN 24–25 March 2005 at SLAC 14–16 December 2005 at CERN
There are many possibilities for new physics beyond the Standard Model that feature non-standard Higgs sectors. These may introduce new sources of CP violation, and there may be mixing between multiple Higgs bosons or other new scalar bosons. Alternatively, the Higgs may be a composite state, or there may even be no Higgs at all. These non-standard Higgs scenarios have important implications for collider physics as well as for cosmology, and understanding their phenomenology is essential for a full com-prehension of electroweak symmetry breaking. This report discusses the most relevant theories which go beyond the Standard Model and its minimal, CP-conserving supersymmetric extension: two-Higgs-doublet models and minimal supersymmetric models with CP violation, supersymmetric models with an extra singlet, models with extra gauge groups or Higgs triplets, Little Higgs models, models in extra dimensions, and models with technicolour or other new strong dynamics. For each of these scenarios, this report presents an introduction to the phenomenology, followed by contributions on more detailed theoretical aspects and studies of possible experimental signatures at the LHC and other colliders.
G. Azuelos6,7, S. Baffioni8, A. Ballestrero1, V. Barger9, A. Bartl10, P. Bechtle11, G. Bélanger12,
A. Belhouari1, R. Bellan1, A. Belyaev13, P. Beneš14, K. Benslama15, W. Bernreuther16, M. Besançon17, G. Bevilacqua1, M. Beyer18, M. Bluj19, S. Bolognesi1, M. Boonekamp20, F. Borzumati21,22, F. Boudjema12, A. Brandenburg16, T. Brauner14, C. P. Buszello23, J. M. Butterworth24, M. Carena25,
D. Cavalli26, G. Cerminara1, S. Y. Choi27,28, B. Clerbaux29, C. Collard8, J. A. Conley11, A. Deandrea30, S. De Curtis31, R. Dermisek32, A. De Roeck33, G. Dewhirst34, M. A. Díaz35, J. L. Díaz-Cruz36, D. D. Dietrich37, M. Dolgopolov3, D. Dominici31, M. Dubinin38, O. Eboli5, J. Ellis33, N. Evans39,
L. Fano40, J. Ferland6, S. Ferrag41,42, S. P. Fitzgerald23, H. Fraas43, F. Franke43, S. Gennai44,45, I. F. Ginzburg46, R. M. Godbole47, T. Grégoire48, G. Grenier30, C. Grojean33,49, S. B. Gudnason37,
J. F. Gunion50, H. E. Haber51, T. Hahn52, T. Han9, V. Hankele53, C. Hays54, S. Heinemeyer55, S. Hesselbach43, J. L. Hewett11, K. Hidaka56, M. Hirsch57, W. Hollik52, D. Hooper25, J. Hošek14, J. Hubisz25, C. Hugonie58, J. Kalinowski59, S. Kanemura60, V. Kashkan1, T. Kernreiter10, W. Khater61,
V. A. Khoze62,63, W. Kilian64,28, S. F. King39, O. Kittel65, G. Klämke53, J. L. Kneur58, C. Kouvaris37, S. Kraml33, M. Krawczyk59, P. Krstonoši´c28, A. Kyriakis66, P. Langacker67, M. P. Le11, H.-S. Lee68,
J. S. Lee69, M. C. Lemaire20, Y. Liao70, B. Lillie71,72, V. Litvin73, H. E. Logan74, B. McElrath50, T. Mahmoud29, E. Maina1, C. Mariotti1, P. Marquard75, A. D. Martin62, K. Mazumdar76, D. J. Miller42,
P. Miné8, K. Mönig77, G. Moortgat-Pick33, S. Moretti39, M. M. Mühlleitner33, S. Munir39, R. Nevzorov39, H. Newman73, P. Nie˙zurawski78, A. Nikitenko34, R. Noriega-Papaqui79, Y. Okada2,80,
P. Osland81, A. Pilaftsis82, W. Porod57, H. Przysiezniak83, A. Pukhov38, D. Rainwater84, A. Raspereza28, J. Reuter28S. Riemann77, S. Rindani85, T. G. Rizzo11, E. Ros57, A. Rosado79, D. Rousseau86, D. P. Roy76, M. G. Ryskin62,63, H. Rzehak87, F. Sannino37, E. Schmidt18, H. Schröder18,
M. Schumacher65, A. Semenov88, E. Senaha89, G. Shaughnessy9, R. K. Singh47, J. Terning50, L. Vacavant90, M. Velasco91, A. Villanova del Moral57, F. von der Pahlen43, G. Weiglein62, J. Williams23,92, K. Williams62, A. F. ˙Zarnecki78, D. Zeppenfeld53, D. Zerwas86, P. M. Zerwas28,
4
California Institute of Technology, Pasadena, CA 91125, USA 5
Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, Brazil 6
Département de Physique, Université de Montréal, Montréal, Qué., H3C 3J7, Canada 7TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., V6T 2A3, Canada
8Laboratoire Leprince-Ringuet (LLR), Ecole Polytechnique, IN2P3-CNRS, F-91128 Palaiseau Cedex, France 9Department of Physics, University of Wisconsin, Madison, WI 53706, USA
10
Institut für Theoretische Physik, Universität Wien, A-1090 Vienna, Austria
11Stanford Linear Accelerator Center (SLAC), P.O. Box 20450, Stanford, CA 94309, USA 12LAPTH, 9 Chemin de Bellevue, B.P. 110, F-74941 Annecy-le-Vieux, France
13
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA 14Department of Theoretical Physics, Nuclear Physics Institute, 25068 ˇRež, Czech Republic
15Physics Department, Nevis Laboratories, Columbia University, 136 S Broadway, Irvington, NY 10533, USA 16
Institut für Theoretische Physik, RWTH Aachen, D-52056 Aachen, Germany 17
Centre d’Etudes de Saclay, Orme des Merisiers, F-91191 Gif-sur-Yvette Cedex, France 18Institute of Physics, University of Rostock, D-18051 Rostock, Germany
19Soltan Institute for Nuclear Studies, Ho˙za 69, 00-681 Warsaw, Poland 20
DAPNIA/SPP, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France 21International Center for Theoretical Physics (ICTP), I-34000 Trieste, Italy 22
Scuola Internazionale Superiore di Studi Avanzati (SISSA), I-34000 Trieste, Italy 23
Cambridge University, Dept. of Physics, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK 24Physics and Astronomy Department, University College London, Gower St, London WC1E 6BT, UK 25
Fermi National Accelerator Laboratory (FNAL), P.O. Box 500, Batavia, IL 60510-0500, USA 26
INFN and Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy 27Department of Physics, Chonbuk National University, Jeonju 561-756, Korea 28
Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany 29
Université Libre de Bruxelles (ULB), CP230 Blvd. du Triomphe, B-1050 Brussels, Belgium
30Institut de Physique Nucléaire de Lyon, CNRS/IN2P3, Université Lyon 1, F-69622 Villeurbanne, France 31
Dipartimento di Fisica, Univiversità di Firenze, and INFN, I-50019 Sesto Fiorentino (Firenze), Italy 32
Institute for Advanced Study, Princeton, NJ 08540, USA 33
CERN, CH-1211 Geneva 23, Switzerland 34
Imperial College, Department of Physics, Prince Consort Road, London SW7 2BW, UK 35
Departamento de Física, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile 36FCFM, BUAP. Apdo. Postal 1364, C.P. 72000 Puebla, Pue., México
37
The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark 38
D.V. Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia 39School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK 40
INFN and Università degli Studi di Perugia, Dipartimento di Fisica, via Pascoli, 06100 Perugia, Italy 41
Department of Physics, University of Oslo, P.O. Box 1048 Blindern, NO-0316 Oslo, Norway 42Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
43Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany 44
INFN Pisa, Largo Ponte Corvo 3, Pisa 56126, Italy 45
Centro Studi Enrico Fermi, Compendio Viminale, Roma 00184, Italy 46
Sobolev Institute of Mathematics, Novosibirsk 630090, Russia 47
Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India 48
Boston University, Dept. of Physics, 590 Commonwealth Ave., Boston, MA 02215, USA 49Service de Physique Théorique, CEA Saclay, F-91191 Gif-sur-Yvette, France
50
Department of Physics, University of California at Davis, Davis, CA 95616, USA 51
Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USA 52MPI für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München, Germany
56
Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan
57Instituto de Física Corpuscular / C.S.I.C., Edificio Institutos de Paterna, Apartado 22085, E-46071 Valencia, Spain 58
Laboratoire Physique Théorique et Astroparticules, Univ. Montpellier II, F-34095 Montpellier, France 59
Institute of Theoretical Physics, Warsaw University, Ho˙za 69, 00-681 Warsaw, Poland 60Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan 61Department of Physics, Birzeit University, Birzeit, West Bank, Palestine 62
Department of Physics and Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, UK 63
Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg, 188300, Russia 64Fachbereich Physik, University of Siegen, D-57068 Siegen, Germany 65
Physikalisches Institut der Universität Bonn, Nussallee 12, D-53115 Bonn, Germany 66
Institute of Nuclear Physics, NCSR "Demokritos", Athens, Greece
67Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 68
Department of Physics, University of Florida, Gainesville, FL 32608, USA 69
Center for Theoretical Physics, School of Physics, Seoul National University, Seoul 151-747, Korea 70Department of Physics, Nankai University, Tianjin 300071, China
71
High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA 72
Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA
73Department of Physics, California Institute of Technology, MS356-48, Pasadena, CA 91125, USA 74
Ottawa Carleton Institute for Physics, Carleton University, Ottawa K1S 5B6, Canada 75
Institut für Theoretische Teilchenphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany 76
Tata Institute of Fundamental Research, Homi Bhabha Rd., Mumbai 400005, India 77
DESY, Platanenallee 6, D-15738 Zeuthen, Germany 78
Institute of Experimental Physics, Warsaw University, Ho˙za 69, 00-681 Warsaw, Poland 79Instituto de Física, BUAP. Apdo. Postal J-48, C.P. 72570 Puebla, Pue., México 80
Graduate University for Advanced Studies, Tsukuba, Ibaraki 305-0801, Japan 81
Department of Physics and Technology, University of Bergen, Allegt. 55, N-5007 Bergen, Norway 82School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK
83
LAPP, 9 Chemin de Bellevue, B.P. 110, F-74941 Annecy-le-Vieux, France 84
Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 85Physical Research Lab, Navrangpura, Ahmedabad 380009, Gujarat, India
86
LAL Orsay, Université de Paris-Sud et IN2P3-CNRS, F-91898 Orsay Cedex, France 87
Paul Scherrer Institut, Würenlingen und Villigen, CH-5232 Villigen PSI, Switzerland 88Joint Institute for Nuclear Research (JINR), 141980, Dubna, Russia
89
Department of Physics, National Central University, Chungli, Taiwan 320, R.O.C. 90
CPPM, CNRS/IN2P3/Univ. Méditerranée, 163 Av. de Luminy, Case 902, F-13288 Marseille Cedex 09, France 91Northwestern University, Evanston, IL 60201, USA
92
Physics and Electronics Department, Rhodes University, Grahamstown, 6140, South Africa 93Instituto de Física, Universidad Austral de Chile, Casilla 567, Valdivia, Chile
1 Introduction 1
2 The CP-Violating Two-Higgs Doublet Model 5
2.1 Theory review . . . 5
2.2 Overview of phenomenology . . . 17
2.3 Basis-independent treatment of Higgs couplings in the CP-violating 2HDM . . . 25
2.4 Symmetries of 2HDM and CP violation . . . 31
2.5 Textures and the Higgs boson-fermion couplings . . . 36
2.6 Electroweak baryogenesis and quantum corrections to the Higgs potential . . . 41
2.7 Neutral Higgs bosons with (in)definite CP: decay distributions for τ+τ− and t¯t final states . . . 45
2.8 CP-violating top Yukawa couplings in the 2HDM . . . 51
2.9 Higgs CP measurement via t¯tφ partial reconstruction at the LHC . . . 54
2.10 Higgs + 2 jets as a probe for CP properties . . . 58
2.11 CP-violating Higgs bosons decaying via H → ZZ → 4 leptons at the LHC . . . 62
2.12 Testing the spin and CP properties of a SM-like Higgs boson at the LHC . . . 67
2.13 Study of the CP properties of the Higgs boson in the Φ → ZZ → 2e2µ process in CMS 73 2.14 Higgs-boson CP properties from decays to W W and ZZ at the Photon Linear Collider 78 References . . . 87
3 The Minimal Supersymmetric Standard Model with CP Phases 97 3.1 Introduction . . . 97
3.2 Search for CP-violating neutral Higgs bosons in the MSSM at LEP . . . 106
3.3 The ATLAS discovery potential for Higgs bosons in the CPX scenario . . . 110
3.4 Higgs phenomenology with CPsuperH . . . 117
3.5 Higgs phenomenology in the Feynman-diagrammatic approach / FeynHiggs . . . 124
3.6 Self-couplings of Higgs bosons in scenarios with mixing of CP-even/CP-odd states . . 133
3.7 Production of neutral Higgs bosons through b-quark fusion in CP-violating SUSY sce-narios . . . 139
3.8 CP-violating Higgs in diffraction at the LHC . . . 144
3.9 CP violation in supersymmetric charged Higgs production at the LHC . . . 150
3.10 Exploring CP phases of the MSSM at future colliders . . . 155
3.11 Probing CP-violating Higgs contributions in γγ → f ¯f . . . 159
3.12 Resonant H and A mixing in the CP-noninvariant MSSM . . . 164
3.13 Higgs boson interferences in chargino and neutralino production at a muon collider . . 169
3.14 Impact of Higgs CP mixing on the neutralino relic density . . . 174
3.15 Decays of third generation sfermions into Higgs bosons . . . 178
4.2 The NMSSM Higgs mass spectrum . . . 200
4.3 Low fine-tuning scenarios in the NMSSM and LHC/ILC implications . . . 204
4.4 Di-photon Higgs signals at the LHC as a probe of an NMSSM Higgs sector . . . 210
4.5 Dark matter in the NMSSM and relations to the NMSSM Higgs sector . . . 215
4.6 Relic density of neutralino dark matter in the NMSSM . . . 222
4.7 Comparison of Higgs bosons in the extended MSSM models . . . 226
4.8 Distinction between NMSSM and MSSM in combined LHC and ILC analyses . . . 232
4.9 Moderately light charged Higgs bosons in the NMSSM and CPV-MSSM . . . 236
References . . . 240
5 The MSSM with R-Parity Violation 245 5.1 Introduction . . . 245
5.2 The Higgs sector in models with explicitly broken R-parity . . . 253
5.3 Phenomenology of the neutral Higgs sector in a model with spontaneously broken R-parity . . . 257
5.4 Charged-Higgs-boson and charged-slepton radiation off a top quark at hadron colliders 263 References . . . 268
6 Extra Gauge Groups 271 6.1 Introduction . . . 271
6.2 The Higgs sector in a secluded sector U (1)0model . . . 277
6.3 Higgs spectrum in the exceptional supersymmetric standard model . . . 284
6.4 Doubly charged Higgs bosons from the left-right symmetric model at the LHC . . . 289
References . . . 294
7 Little Higgs models 297 7.1 Introduction . . . 297
7.2 Impact of electroweak precision data on the little Higgs models . . . 309
7.3 Couplings of the Littlest Higgs boson . . . 312
7.4 Pseudo-axions in Little Higgs models . . . 317
7.5 Little Higgs with T-parity . . . 319
7.6 Little Higgs studies with ATLAS . . . 326
7.7 Search for new heavy quark T in CMS . . . 336
7.8 Determination of Littlest Higgs model parameters at the ILC . . . 341
References . . . 345
8 Large Extra Dimensions 349 8.1 Introduction . . . 349
8.2 Invisibly decaying Higgs at the LHC . . . 358
8.3 Search for invisible Higgs decays in CMS . . . 362
8.4 Search for heavy resonances . . . 368
9.2 Higgs–radion phenomenology . . . 385
9.3 Radion search in ATLAS . . . 389
9.4 Radion search in CMS . . . 392
9.5 Search for Randall-Sundrum excitations of gravitons decaying into two photons in CMS 395 9.6 SUSY Higgs production from 5-D warped supergravity . . . 400
References . . . 405
10 Higgsless Models 407 10.1 Introduction . . . 407
10.2 Quark and lepton masses . . . 415
10.3 Higgsless electroweak symmetry breaking from moose models . . . 420
References . . . 426
11 Strongly Interacting Higgs Sector and Anomalous Couplings 429 11.1 Introduction . . . 429
11.2 Anomalous quartic gauge couplings at the ILC . . . 438
11.3 W W scattering at high W W centre-of-mass energy . . . 443
11.4 V V -fusion in CMS: a model-independent way to investigate EWSB . . . 445
References . . . 453
12 Technicolor 457 12.1 Introduction . . . 457
12.2 Extended technicolor . . . 464
12.3 Composite Higgs from higher representations . . . 468
12.4 Minimal walking technicolor: effective theories and dark matter . . . 472
12.5 Associate production of a light composite Higgs at the LHC . . . 477
12.6 Towards understanding the nature of electroweak symmetry breaking at hadron collid-ers: distinguishing technicolor and supersymmetry . . . 480
12.7 Dynamical breakdown of an Abelian gauge chiral symmetry by strong Yukawa couplings487 References . . . 491
13 Higgs Triplets 497 13.1 Introduction . . . 497
13.2 Single and pair production of χ±±at Hadron Colliders . . . 516
13.3 Photonic decays of neutral triplet scalars . . . 519
References . . . 524
The next step in high-energy particle physics will be the exploration of the TeV energy scale, an adven-ture that starts with the LHC and is to be continued with other colliders operating in a similar energy range. The energy scale of the LHC is largely determined by the need to complete the successful Stan-dard Model and, in particular, to understand the origin of the elementary particle masses. Even though the Standard Model is very successful, having survived stringent high-energy tests at the SLC, LEP, HERA and the Tevatron, in particular, it is known to be incomplete. The existence of a Higgs sector is essential for the breaking of the electroweak gauge symmetry, enabling the W and Z gauge bosons and the matter fermions to acquire masses. The presence of a Higgs boson is the only way to avoid having the scattering amplitudes for massive particles grow indefinitely, leading to unrenormalizable divergences in loop diagrams.
General theoretical arguments based on unitarity and lattice calculations imply that an elemen-tary Higgs boson should have a mass less than about a TeV. The Standard Model’s successes in all its experimental tests to date implies that the dangerous loop diagrams must indeed be cut off by some unde-tected ingredient resembling a Higgs boson, with a mass that is likely to be no larger than a few hundred GeV. The discovery of the Standard Model Higgs boson or some equivalent substitute is therefore one of the primary objectives of the experimental programmes at the LHC and other TeV-scale colliders. Its discovery is expected to lead to a flowering of new physics, as illustrated in Fig. 1.
This is because there are many reasons to suspect that the simplest Higgs sector postulated in the original formulations of the Standard Model is unlikely to be the complete solution to the origin of particle masses. It may be supplemented by additional new physics beyond the Standard Model, or the Higgs sector may be more complicated, or it may be replaced by some very different dynamics serving a similar purpose. An elementary Higgs field and its associated Higgs boson are subject to quantum-mechanical instabilities induced by loop diagrams that threaten to subject the electroweak mass scale to large corrections. It is, in principle, possible to maintain a low electroweak scale despite these large corrections, but this would appear to require unnatural fine tuning of the model parameters. One of the favoured solutions to this naturalness problem is to postulate the appearance of supersymmetry at or below the TeV scale. Even the minimal supersymmetric extension of the Standard Model (MSSM) requires two doublets of Higgs fields and hence five physical Higgs bosons. However, there is still no experimental evidence for supersymmetry, and the first collider able to provide evidence for its relevance to particle physics is likely to be the LHC.
The Higgs sectors of the Standard Model and its minimal supersymmetric extension have been discussed extensively in connection with the experimental programmes of the LHC and other TeV-scale colliders. These define the standard options in Higgs physics. The prospects for discovering and charac-terizing the MSSM at these colliders have also been discussed in some detail. The purpose of this report is to explore the other petals of the Higgs flower shown in Fig. 1, by assembling studies of non-standard Higgs models within and beyond the framework of supersymmetry. Particular emphasis is placed on Higgs scenarios that violate CP. In addition to the problem of mass, one of the most puzzling aspects of the Standard Model is flavour physics, and particularly the violation of CP symmetry. The Standard Model accommodates CP violation quite economically via the Kobayashi-Maskawa mechanism, but it does not explain its origin. In principle, CP violation could also be present in the strong interactions, as a result of non-perturbative effects, but this has not been seen. An attractive option for suppressing this strong CP violation is to complicate the minimal Higgs sector of the Standard Model by postulating an axion. On the other hand, non-minimal Higgs sectors introduce in general additional sources of CP violation beyond the Kobayashi-Maskawa mechanism. This is not necessarily unwelcome, since some additional source of CP violation would in any case be required in order to explain the cosmological baryon asymmetry on the basis of elementary particle interactions. Cosmological baryogenesis could be
Fig. 1: The flowering of the Higgs physics that is expected to bloom at the TeV scale.
achieved at any temperature scale at or beyond the electroweak scale. One particularly attractive option is to generate the cosmological baryon asymmetry at the electroweak scale by means of a non-minimal Higgs sector, such as that in the MSSM. The MSSM already contains a plethora of possible CP-violating phases, which may have interesting signatures within and beyond the Higgs sector of the theory. We place particular emphasis on the possible implications of these phases for Higgs phenomenology at colliders, including the LHC, International Linear Collider (ILC) and its photon-photon collider option.
Even more possibilities for CP and flavour violation are offered by modifications of the MSSM in which R parity is violated. These introduce many novel Yukawa-like interactions that possess, in gen-eral, additional CP-violating phases. We give particular emphasis in this report to the possible mixing of Higgs bosons with sleptons and the corresponding phenomenological signatures. Yet another possi-bility is to augment the MSSM framework, for example by postulating its extension to include a singlet superfield that expands the Higgs sector of the theory. One of the motivations for such an addition is to avoid introducing a priori a Higgs mixing parameter with a magnitude similar to the electroweak scale, instead replacing it with a vacuum expectation value generated dynamically. In addition to enriching the possibilities for Higgs phenomenology at colliders, such scenarios also have interesting cosmological implications, e.g., for the nature of cold dark matter.
A more radical expansion of the field content of the Standard Model is to postulate an extension of the gauge group. Various such extensions have been considered in previous studies of collider physics. Among the motivations for such models are grand-unified and string models that contain supplementary U(1) gauge groups, and left-right symmetric models. Aspects of these have been studied previously: the new thrust here is to consider in more detail the phenomenology of Higgs bosons in such scenarios. An extra stimulus to such models has recently been provided by little Higgs models. Their central idea is to interpret the Higgs boson of the Standard Model as a pseudo-Goldstone boson of a higher electroweak gauge symmetry after its breakdown into the SU(2) × U(1) of the Standard Model. In such little Higgs
models, the low scale of the electroweak vacuum relative to the breaking of the larger gauge symmetry is protected by the pseudo-Goldstone character of the Higgs boson. This protection is provided in specific models by additional matter, gauge and Higgs fields with relatively low masses that might be accessible to TeV-scale colliders. We discuss here the possible phenomenology of such extra degrees of freedom, as well as the phenomenology of the little Higgs boson itself.
A different set of possible extensions of the Standard Model are those with extra spatial dimen-sions. Such theories have been around for many decades, but recently gained motivation from string theory. This apparently requires such extra dimensions, though they might well be much smaller than the inverse-TeV scale. However, it has been realized that the phenomenological constraints on at least some extra-dimensional scenarios are quite weak, laying them open to experimental tests at colliders. In particular, they provide options for invisible Higgs decays and for other sources of missing trans-verse energy. One particularly interesting possibility is that an extra dimension is warped. An important new scalar degree of freedom in such a model is the radion, which has several potential interfaces with Higgs physics. Higgs-radion mixing must be taken into account, since several of the radion decay modes mimic those of a conventional Higgs boson, such as those to γγ and ZZ(∗), and radion decays into pairs
of Higgs bosons are also of potential interest.
The imminent exploration of the Higgs sector by the LHC and other colliders has prompted new questions whether Higgsless models are viable. In their original four-dimensional formulations, they lead to strong W W scattering at relatively low energies, and run into related problems with the preci-sion electroweak data. However, these difficulties may be alleviated by postulating an extenpreci-sion to five dimensions, where electroweak symmetry may be broken by appropriate boundary conditions. From a theoretical point of view, the absence of a Higgs boson would be a very interesting outcome from the LHC, even if experimentalists might be disappointed. However, they should be encouraged by the fact that, even in such Higgsless models, there are possible experimental probes of the mechanism of electroweak symmetry breaking.
Strongly-interacting Higgs sectors arise in a number of other scenarios, in addition to Higgsless models. One general approach to these is provided by effective Lagrangian techniques modeled on those used in QCD at low energies. As well as probing strong W W scattering and possible massive resonances via the production of pairs of weak gauge bosons, it may also be possible to study anomalous quartic gauge-boson couplings via triple weak-boson production. There is also an interesting class of models in which the elementary Higgs field of the Standard Model is replaced by a composite field in a theory of new strong ‘technicolour’ interactions. Models in which the technicolour dynamics is closely modeled on that in QCD have problems with precision electroweak data and the generation of fermion masses. However, the first problem may be mitigated in ‘walking’ technicolour models whose dynamics is not related to that of QCD by simple rescaling. The fermion mass problem may be solved in extended technicolour models, which offer interesting possibilities for light composite Higgs bosons as well as predicting complex strong dynamics at higher energies.
A final class of scenarios to consider is that with higher-dimensional Higgs representations. These arise in generic little Higgs scenarios of the type mentioned above, but may also arise in other models. These may give rise to distinctive signatures due to doubly-charged Higgs bosons, as well as interesting effects in the physics of neutral and singly-charged Higgs bosons.
This brief summary gives an impressionistic survey of the different non-standard Higgs scenarios that should be considered in preparations for collider experiments. It demonstrates that one should not allow one’s attention to be dominated by the single weakly-interacting Higgs boson of the Standard Model, nor even by its modest extension to the MSSM. One should keep in mind, in particular, the possibility that there may be a close link between the Higgs sector and CP violation, and one should be open to the possible appearance of non-standard Higgs representations such as singlets and triplets, as well as novel decay patterns, including invisible modes. Revealing the full details of the underlying mechanism of electroweak symmetry breaking may be considerably more complex than in the Standard
Model or the MSSM.
This report is the product of a Workshop which extended from May 2004 to December 2005, with significant support from the CERN Theory Division and elsewhere. It consists of chapters discussing each of the principal non-standard Higgs scenarios mentioned above and shown in the Higgs flower. Each chapter starts with a pedagogical introduction to the corresponding scenario, which is followed by a set of individual contributions describing specific studies made in the context of the Workshop. In addition to many phenomenological studies, this report reviews several studies made of LHC capabilities using detailed simulations of the ATLAS and CMS detectors. It is encouraging that, although many of the non-standard Higgs scenarios were not considered in the designs of ATLAS and CMS, nevertheless they have excellent capabilities for revealing such scenarios. This indicates that the ATLAS and CMS detector designs and trigger concepts are sufficiently robust to respond to new challenges. Thus one may also hope that they will also be sensitive to Nature’s choice for new physics, even if it extends beyond the options considered here. This volume also contains many studies of non-standard Higgs signatures at linear e+e−colliders. It is also encouraging that the ILC, in particular, also offers excellent prospects for
exploring more aspects of non-standard Higgs scenarios, thanks to its very clean experimental conditions. The LHC will soon start revealing what physics lies at the TeV scale, and in particular what the Higgs sector holds in store for us. We do not know in advance whether it will reveal a single elementary Standard Model Higgs boson, something more complicated, or even a Higgsless model. One must ap-proach LHC physics in general, and Higgs physics in particular, with an open mind. The Higgs sector may be not only the completion of the Standard Model, but also the first window on physics beyond it. In addition to answering one of the key open questions in the Standard Model, for example, it pro-vides some of the key motivation for supersymmetry and is deeply implicated in the linked problems of flavour and CP violation. This volume provides a hitchhiker’s guide to these and other possible aspects of non-standard Higgs physics at the LHC and other colliders.
Howard E. Haber and Maria Krawczyk
The Standard Model (SM) of electroweak physics is an SU(2)L×U(1) gauge theory coupled to quarks, leptons and one complex hypercharge-one, SU(2)L doublet of scalar fields. Due to the form of the scalar potential, one component of the complex scalar field acquires a vacuum expectation value, and the SU(2)L×U(1) electroweak symmetry is spontaneously broken down to the U(1)EM gauge
symme-try of electromagnetism. Hermiticity requires that the parameters of the SM scalar potential are real. Consequently, the resulting bosonic sector of the electroweak theory is CP-conserving.
The SM, with its minimal Higgs structure, provides an extremely successful description of ob-served electroweak phenomena. Nevertheless, there are a number of motivations to extend the Higgs sector of this model by adding a second complex doublet of scalar fields [1–10]. Perhaps the best motivated of these extended models is the minimal supersymmetric extension of the Standard Model (MSSM) [11–13], which requires a second Higgs doublet (and its supersymmetric fermionic partners) in order to preserve the cancellation of gauge anomalies. The Higgs sector of the MSSM is a two-Higgs-doublet model (2HDM), which contains two chiral Higgs supermultiplets that are distinguished by the sign of their hypercharge. The theoretical structure of the MSSM Higgs sector is constrained by the supersymmetry, leading to numerous relations among Higgs masses and couplings. In particular, as in the case of the SM, the tree-level MSSM Higgs sector is CP-conserving. However, the supersymmetric relations among Higgs parameters are modified by loop-corrections due to the effects of supersymmetry-breaking that enter via the loops. Thus, the Higgs-sector of the (radiatively-corrected) MSSM can be described by an effective field theory consisting of the most general CP-violating two-Higgs-doublet model.
The 2HDM Lagrangian contains eight real scalar fields. After electroweak symmetry breaking, three Goldstone bosons (G±and G0) are removed from the spectrum and provide the longitudinal modes
of the massive W±and Z. Five physical Higgs particles remain: a charged Higgs pair (H±) and three
neutral Higgs bosons. If experimental data reveals the existence of a Higgs sector beyond that of the SM, it will be crucial to test whether the observed scalar spectrum is consistent with a 2HDM interpre-tation. In order to be completely general within this framework, one should allow for the most general CP-violating 2HDM when confronting the data. Any observed relations among the general 2HDM pa-rameters would surely contribute to the search for a deeper theoretical understanding of the origin of electroweak symmetry breaking.
2.1.1 The general Two-Higgs-Doublet Model (2HDM)
The 2HDM is governed by the choice of the Higgs potential and the Yukawa couplings of the two scalar-doublets to the three generations of quarks and leptons. Let Φ1 and Φ2 denote two complex
hypercharge-one, SU(2)Ldoublet scalar fields. The most general gauge-invariant renormalizable Higgs scalar potential is given by
V = m211Φ†1Φ1+ m222Φ†2Φ2− [m212Φ†1Φ2+ h.c.] +12λ1(Φ†1Φ1)2+12λ2(Φ†2Φ2)2+ λ3(Φ†1Φ1)(Φ†2Φ2) + λ4(Φ†1Φ2)(Φ†2Φ1) +n12λ5(Φ†1Φ2)2+ λ6(Φ†1Φ1) + λ7(Φ†2Φ2) Φ†1Φ2+ h.c. o , (2.1) where m2
2.1.1.1 Covariant notation with respect to scalar field redefinitions
In writing Eq. (2.1), we have implicitly chosen a basis in the two-dimensional “flavor” space of scalar fields. To allow for other basis choices, it will be convenient to rewrite Eq. (2.1) in a covariant form with respect to global U(2) transformations, Φa→ Ua¯bΦb(and Φa¯† → Φ¯b†Ub¯†a), where the 2 × 2 unitary matrix
U satisfies Ub¯†aUa¯c = δb¯c. In our index conventions, replacing an unbarred index with a barred index is
equivalent to complex conjugation (for further details see section 2.3). Thus, Eq. (2.1) can be expressed in U(2)-covariant form as [10, 14]:
V = Ya¯bΦ†¯aΦb+12Za¯bc ¯d(Φ†a¯Φb)(Φ†c¯Φd) , (2.2)
where the indices a, ¯b, c and ¯drun over the two-dimensional Higgs “flavor” space and Za¯bc ¯d = Zc ¯da¯b.
Hermiticity of V implies that Ya¯b = (Yb¯a)∗ and Za¯bc ¯d = (Zb¯ad¯c)∗. Explicitly, the coefficients of the
quadratic terms are
Y11= m211, Y12=−m212,
Y21=−(m212)∗, Y22= m222, (2.3)
and the coefficients of the quartic terms are
Z1111 = λ1, Z2222 = λ2,
Z1122 = Z2211 = λ3, Z1221 = Z2112 = λ4,
Z1212 = λ5, Z2121 = λ∗5,
Z1112 = Z1211 = λ6, Z1121 = Z2111 = λ∗6,
Z2212 = Z1222 = λ7, Z2221 = Z2122 = λ∗7. (2.4)
Under the global U(2) transformation, the tensors Y and Z transform covariantly: Ya¯b→ Ua¯cYc ¯dUd¯b† and
Za¯bc ¯d → Ua¯eUf¯b† Uc¯gUh ¯†dZe ¯f g¯h. Indices can only be summed over using the U(2)-invariant tensor δa¯b.
The advantage of introducing the U(2)-covariant notation is that one can immediately identify U(2)-invariant quantities as basis-independent; such quantities do not depend on the original choice of the Φ1–Φ2 basis. In particular, any physical observable must be independent of the basis choice and
hence can be identified as some U(2)-invariant quantity. For example, the well-known tan β parameter of the general 2HDM is not a physical quantity [14–16].
2.1.1.2 Counting the degrees of freedom
The 2HDM scalar potential depends on six real parameters and four complex parameters, for a total of fourteen degrees of freedom. However, these parameters depend on the choice of the Φ1–Φ2
ba-sis. In order to determine the number of physical degrees of freedom, one must take into account the possibility that unphysical degrees of freedom can be removed by redefining the scalar fields via the global U(2) “flavor” transformations. However, note that the global U(2) group can be decomposed as U(2) ∼= SU(2)×U(1), where the global hypercharge U(1) transformation has no effect on the scalar potential parameters. In contrast, the scalar potential parameters will be modified by a general SU(2)-“flavor” transformation. Since an SU(2) transformation is specified by three parameters, three degrees of freedom can be removed by a redefinition of the scalar fields. Thus, the scalar potential provides eleven physical degrees of freedom that govern the properties of the 2HDM scalar sector [14, 15, 17].
2.1.1.3 Discrete symmetries and the 2HDM potential
The general 2HDM is not phenomenologically viable over most of its parameter space. In particular, if we allow for the most general fermion Yukawa couplings, the model exhibits tree-level Higgs-mediated flavor-changing neutral currents (FCNCs), which may contradict the experimental bounds on
FCNCs. This can be ameliorated by either avoiding the untenable regions of parameter space or by introducing additional structure into the model. For example, in the Higgs sector of the MSSM, tree-level Higgs-mediated FCNCs are absent due to the supersymmetric structure of the Higgs-fermion Yukawa couplings. Tree-level Higgs-mediated FCNCs can also be eliminated by invoking appropriate discrete symmetries [18]. Here, we focus on discrete symmetries imposed on the scalar fields. Consider a discrete Z2 symmetry realized for some choice of basis: Φ1 → Φ1, Φ2→ −Φ2. This discrete symmetry implies
that m2
12= λ6 = λ7= 0. A basis-independent characterization of this discrete symmetry has been given
in [14,19]. In practice, the discrete symmetry must also be extended to the fermion sector. By specifying the transformation properties of the fermions with respect to the discrete symmetry, one can constrain the form of the Higgs-fermion Yukawa interactions. In fact, removing the possibility of dangerous FCNC effects can also be achieved if the symmetry of the Z2 discrete transformation of the Higgs potential is
softly broken; i.e., there exists a basis in which λ6 = λ7 = 0but m2126= 0 [15, 17]. A basis-independent
characterization of the softly-broken discrete symmetry can also be given [14]. Finally, hard-breaking of the discrete Z2 symmetry corresponds to the case in which no basis exists in which λ6 = λ7 = 0.
Additional implications of the broken Z2symmetry can be found in section 2.4.
2.1.1.4 The scalar field vacuum expectation values
Electroweak symmetry breaking arises if the minimum of the scalar potential occurs for nonzero expec-tation values of the scalar fields. The condition for extrema of the scalar potential
∂V ∂Φ1 Φ1=hΦ1i, Φ2=hΦ2i = 0, ∂V ∂Φ2 Φ1=hΦ1i, Φ2=hΦ2i = 0 (2.5)
yields the vacuum expectation values (vevs) hΦ1,2i. The scalar fields will develop non-zero vevs if the
mass matrix constructed from the quadratic squared-mass parameters of the Higgs potential (m2
ij) has at
least one negative eigenvalue. By employing an appropriate weak isospin and U(1)Y transformation, it
is always possible to write the scalar field vevs in the following form hΦ1i = 1 √ 2 0 v1 , hΦ2i = 1 √ 2 u v2eiξ , (2.6)
where v1and v2are real and positive, and 0 ≤ ξ < 2π. Depending on the parameters of Higgs potential,
the extremum for u 6= 0 describes either saddle point or a minimum of the potential, called the charged vacuum, where the U(1)EM symmetry is spontaneously broken [15, 20–22]. The vacuum solution with
u = 0preserves the U(1)EMsymmetry; it corresponds to a local minimum of potential if its parameters
are such that the physical Higgs squared-masses are non-negative. In this case, one can show that the energy of the charged vacuum is larger than energy of the U(1)EM preserving vacuum [20, 22].
Henceforth, we assume that the global minimum of the scalar potential respects the U(1)EMgauge symmetry. In this case u = 0 and it is convenient to write:
v1 ≡ v cos β , v2 ≡ v sin β , (2.7)
where v2≡ v2
1+ v22= (
√
2GF)−1/2= (246 GeV)2and 0 ≤ β ≤ π/2.
One is always free to rephase Φ2in order to set ξ = 0. In the following, we shall always work in
a basis in which the two neutral Higgs field vevs are real and positive (corresponding to a real vacuum). The scalar minimum conditions (2.5) then yield:
m211 = m212tβ−12v2 λ1c2β+ (λ3+ λ4+ λ5s2β+ (2λ6+λ∗6)sβcβ+ λ7s2βtβ (2.8) m222 = (m212)∗t−1β −1 2v 2hλ 2s2β+ (λ3+ λ4+ λ∗5c2β+ λ∗6c2βt−1β + (λ7eiξ+ 2λ∗7e−iξ)sβcβ i , (2.9)
where sβ = sin β, cβ = cos βand tβ = tan β. Since m211and m222are both real, the imaginary part of
either Eq. (2.8) or Eq. (2.9) yields one independent equation:
Im (m212) = 12v2Im (λ5)sβcβ + Im (λ6)c2β + Im (λ7)s2β . (2.10) The quantities δ ≡ Im (m 2 12) v2s βcβ , η≡ Re (m 2 12) v2s βcβ , (2.11)
will be useful in our discussion of the Higgs mass eigenstates and the mixing of CP-even and CP-odd states. Note that Re (m2
12)is not determined by the scalar potential minimum conditions.
2.1.1.5 Theoretical constraints on the Higgs potential parameters
The parameters of Higgs potential are constrained by various conditions. To have a stable vacuum, the potential must be positive at large quasi–classical values of the magnitudes of the scalar fields for an arbitrary direction in the (Φ1, Φ2) plane. These are the positivity constraints [23–26]. The minimum
constraints are the conditions ensuring that the extremum is a minimum for all directions in (Φ1, Φ2)
space, except for the direction of the Goldstone modes. It is realized when the squared-masses of the five physical Higgs bosons are all positive.
The tree-level amplitudes for the scattering of longitudinal gauge bosons at high energy can be related via the equivalence theorem [27] to the corresponding amplitudes in which the longitudinal gauge bosons are replaced by Goldstone bosons. The latter can be computed in terms of quartic couplings λi
that appear in the Higgs potential. By imposing tree-level unitarity constraints on these amplitudes, one can derive upper bounds on the values of certain combinations of Higgs quartic couplings [28–34].
The perturbativity condition for a validity of a tree approximation in the description of interactions of the lightest Higgs boson may be somewhat less restrictive than the unitarity constraints. For example, by requiring that one-loop corrections to Higgs self-couplings are small compared to tree-level couplings, one expects that |λi|/16π2 1.
Unitarity constraints for the 2HDM were first derived for the potential without a hard violation of the discrete Z2symmetry and for the CP conserving case (e.g., see [32]). Extension to the CP-violating
case can be found in [33], and for the case of hard discrete Z2symmetry violation in [34].
2.1.2 Conditions for Higgs sector CP-violation
Higgs sector CP-violation may be either explicit or spontaneous. Explicit CP conservation1 or violation
refers respectively to the consistent or inconsistent CP transformation properties of the various terms that appear in the Lagrangian. If the scalar Lagrangian is explicitly CP-conserving, but the vacuum state of the theory violates CP, then one says that CP is spontaneously broken [1, 10, 35]. The observable conse-quences of Higgs sector CP-violation (either explicit or spontaneous) include the mixing of neutral Higgs states of opposite CP quantum numbers and/or the existence of (direct) CP-violating Higgs interactions. The CP state mixing and the direct CP-violation in the gauge/Higgs interactions are determined by the properties of the scalar Lagrangian (and the corresponding vacuum state). These CP-violating effects are absent if and only if there exists a basis in which the two neutral Higgs vacuum expectation values and the scalar potential parameters are simultaneously real [36, 37]. Given an arbitrary potential, the existence or non-existence of such a basis may be difficult to determine directly. For this problem, the basis-independent methods are invaluable. In particular, a set of basis-independent conditions can be
1Since CP is violated in the SM via the CKM mixing of the quarks, it is generally unnatural to demand that the Higgs sector of the 2HDM explicitly conserve CP. Nevertheless, one can naturally impose a CP-conserving Higgs sector by employing an appropriate discrete symmetry. In the MSSM, the Higgs sector is CP-conserving at tree-level (due to the supersymmetry), although one finds CP-violation arising at one-loop due to supersymmetry-breaking effects.
found to test for the CP-invariance of the scalar sector. Following [38], we introduce three U(2)-invariant quantities [37]: −1 2v 2J 1 ≡ bv∗¯aYa¯bZ (1) b ¯d bvd, (2.12) 1 4v4J2 ≡ bv¯b∗bv¯c∗Yb¯eYc ¯fZe¯af ¯dbvabvd, (2.13) J3 ≡ bv¯b∗bv¯c∗Z (1) b¯e Z (1) c ¯f Ze¯af ¯dbvabvd, (2.14) where hΦ0 ai ≡ vbva/ √
2, and bva is a unit vector in the complex two-dimensional Higgs flavor space.
Then, the scalar sector is CP-conserving (i.e., no explicit nor spontaneous CP-violation is present) if J1,
J2and J3defined in Eqs. (2.12)–(2.14) are real.2 If the scalar potential is CP-violating, then the CP state
mixing depends only on Im J2 [16, 39], whereas CP-violation in the gauge/Higgs boson interactions is
governed by all three quantities Im Jk, k = 1, 2, 3.
2.1.2.1 Explicit CP-conservation
The general 2HDM scalar potential explicitly violates the CP symmetry. An explicitly CP-conserving scalar potential requires the existence of a Φ1–Φ2 basis in which all the Higgs potential parameters are
real. Such a basis will henceforth be called a real basis. However, given an arbitrary potential, the existence or non-existence of a real basis may be difficult to discern, as already noted. In Ref. [37], the necessary and sufficient basis-independent conditions for an explicitly CP-conserving scalar potential have been established, in terms of the following four potentially complex invariants:
IY 3Z ≡ Im (Za¯(1)cZ (1) e¯b Zb¯ec ¯dYd¯a) , (2.15) I2Y 2Z ≡ Im (Ya¯bYc ¯dZb¯ad ¯fZ (1) f ¯c) , (2.16) I6Z ≡ Im (Za¯bc ¯dZ (1) b ¯f Z (1) d¯hZf ¯aj¯kZk¯jm¯nZn ¯mh¯c) , (2.17)
I3Y 3Z ≡ Im (Za¯cb ¯dZc¯ed¯gZe¯hf ¯qYg¯aYh¯bYq ¯f) , (2.18)
where Z(1)
a ¯d ≡ δb¯cZa¯bc ¯d.
The conditions for a CP-conserving scalar potential depend on the invariant quantity [14, 19]: Z ≡ 2 Tr [Z(1)]2− (Tr Z(1))2= (λ1− λ2)2+ 4|λ6+ λ7|2, (2.19)
Note that if Z vanishes, then Eq. (2.19) implies that λ1 = λ2 and λ7 =−λ6 for all basis choices. Two
distinct cases are possible. If Z 6= 0, then the necessary and sufficient conditions for an explicitly CP-conserving 2HDM scalar potential are given by IY 3Z = I2Y 2Z = I6Z = 0. (A similar result has also
been obtained in [40].) In this case I3Y 3Z = 0is automatically satisfied. If Z = 0, then the
aforemen-tioned first three invariants automatically vanish, in which case the necessary and sufficient condition for an explicitly CP-conserving 2HDM scalar potential is given by I3Y 3Z = 0. Explicit expressions for the
imaginary parts of the four CP-odd invariants above can be found in [37]. The significance of the four conditions above from a group-theoretical perspective has been recently discussed in [19, 41].
Finally, we note that the imposition of the discrete Z2 symmetry Φ1 → Φ1, Φ2 → −Φ2implies
that the scalar potential is CP-conserving. Since λ5 is the only nonzero complex parameter in the basis
where the discrete symmetry is manifest, it is a simple matter to rephase one of the scalar doublets to render λ5 real. Explicit CP-violation can arise if the Φ1 → Φ1, Φ2 → −Φ2 discrete Z2 symmetry
breaking is either hard or soft. In the latter case, e.g., CP violation is a consequence of a nontrivial relative phase in the complex parameters m2
12and λ5.
2One can show that the reality of the J
2.1.2.2 Spontaneous CP-violation
If the scalar Lagrangian is explicitly CP-conserving but the Higgs vacuum is CP-violating, then CP is spontaneously broken. However, both spontaneous and explicit CP-violation yield similar CP-violating phenomenology. To distinguish between the two, one would need to discover CP-violation in the Higgs sector and prove that the fundamental scalar Lagrangian is CP-conserving. In principle, such a distinction is possible. For example, suppose one could verify that IY 3Z = I2Y 2Z = I6Z = I3Y 3Z = 0, whereas
at least one of three invariants J1, J2 and J3 possesses a non-zero imaginary part. In this case, the
CP-symmetry in the Higgs sector is spontaneously broken.3 In practice, distinguishing between explicit and
spontaneous CP-violation by experimental observations and analysis seems extremely difficult.
Spontaneous CP-violation cannot arise in the presence of the Φ1 → Φ1, Φ2 → −Φ2 discrete Z2
symmetry. In particular, in this case the scalar potential minimum condition implies that it is possible to transform to a real basis in which the two neutral vacuum expectation values are real.
2.1.3 The Higgs mass spectrum
2.1.3.1 CP violation and mixing of states We introduce the following field decomposition
Φ1= ϕ+1 v1+ ϕ1+ iχ1 √ 2 , Φ2= ϕ+2 v2+ ϕ2+ iχ2 √ 2 . (2.20)
Then the corresponding scalar squared-mass matrix can be transformed to the block diagonal form by a separation of the massless charged and neutral Goldstone boson fields, G± and G0, and the charged
Higgs boson fields H±:
G± = cos β ϕ±1 + sin β ϕ±2 , (2.21)
G0 = cos β χ1+ sin β χ2. (2.22)
The physical charged Higgs boson is orthogonal to G±:
H±=− sin β ϕ±1 + cos β ϕ±2 . (2.23) The mass of the charged Higgs boson is easily obtained:
MH2± = η−1 2(λ4+ Re λ5+ Re λ67) v2, (2.24)
where λ67 ≡ λ6cot β + λ7tan β and η is defined in Eq. (2.11). The physical neutral Higgs bosons are
mixtures of the two CP-even fields ϕ1, ϕ2and a CP-odd field
A =− sin β χ1+ cos β χ2, (2.25)
that is orthogonal to G0. Consequently, in the general 2HDM, the physical neutral Higgs bosons are
states of indefinite CP.
In the {ϕ1, ϕ2, A} basis, the real symmetric squared-mass matrix M2 for neutral sector is
ob-tained: M2 = M 2 11 M122 M132 M122 M222 M232 M2 13 M232 M332 . (2.26)
3One would also have to prove the absence of explicit CP-violation in the Higgs-fermion couplings. The relevant basis-independent conditions have been given in [10, 38].
Diagonalizing the matrix M2 by using an orthogonal transformation R we obtain the physical neutral
states h1,2,3, with corresponding squared-masses Mi2 that are the eigenvalues of the matrix M2:
h1 h2 h3 = R ϕ1 ϕ2 A , with RM2RT = diag(M2 1, M22, M32) . (2.27)
The diagonalizing matrix R can be written as a product of three rotation matrices Ri, corresponding to
rotations by three angles αi ∈ (0, π) about the z, y and x axes, respectively:
R = R3R2R1 = c1c2 c2s1 s2 −c1s2s3−c3s1 c1c3−s1s2s3 c2s3 −c1c3s2+s1s3 −c1s3−c3s1s2 c2c3 . (2.28)
Here, we define ci = cos αi, si = sin αi and adopt the convention for masses that M1 ≤ M2≤ M3.
One can first diagonalize the upper left 2 × 2 block of the matrix M2. This partial diagonalization
[15] results in the neutral, CP-even Higgs fields which we denote as h and (−H),
H = cos α ϕ1+ sin α ϕ2, h =− sin α ϕ1+ cos α ϕ2, (2.29)
where α ≡ α1 − π/2 is the mixing angle that renders the 2 × 2 CP-even submatrix diagonal.4 At this
stage the CP–odd field A remains unmixed: −Hh A = R1 ϕϕ12 A , with R1M2RT1 =M21 ≡ M 2 h 0 M130 2 0 MH2 M230 2 M130 2 M230 2 M2 A , (2.30) where MA2 = η− Re (λ5−12λ67)v2, (2.31) Mh,H2 = 12 M11+ M22∓ q (M11− M22)2+ 4M122 . (2.32)
The off-diagonal squared-masses M0 2
13 and M230 2are given by
M130 2= c1M132 + s1M232 =−12 h 2δ cos(β + α)− Im ˜λ67cos(β− α) i v2, (2.33) M230 2=−s1M132 + c1M232 = 12 h 2δ sin(β + α) + Im ˜λ67sin(β− α) i v2, (2.34) where ˜λ67≡ λ6cot β− λ7tan β and δ is defined in Eq. (2.11).
In the general CP-violating 2HDM, the states h, H and A are useful intermediaries, which do not directly correspond to physical objects. In the case of CP conservation (realized for M0 2
13 = M230 2 = 0),
the fields h, H and A represent physical Higgs bosons: h1= h, h2 =−H, h3 = A. If at least one of the
off diagonal terms differs from zero, an additional diagonalization is necessary, and the mass eigenstates, which are now admixtures of CP–even and CP–odd states, violate the CP symmetry. In this case we express the physical Higgs boson states h1,2,3as linear combinations of h, H, A:
hh12 h3 = R3R2 −Hh A with RM2RT = R 3R2M21RT2RT3 = M 2 1 0 0 0 M22 0 0 0 M2 3 . (2.35)
4The appearance of the minus sign in
−H and the shift by π/2 in the definition of α is needed in order to match the standard convention used for CP-conserving case [8].
The following mass sum rule holds: M2
1 + M22+ M32= Mh2+ MH2 + MA2 = M112 + M222 + M332 . (2.36)
In general, the Higgs mass-eigenstates hi[Eq. (2.27)] are not states of definite CP parity since they
are mixtures of fields ϕ1,2and A, which possess opposite CP parities. Such CP-state mixing is absent if
and only if M2
13 = M232 = 0. In particular, for sin 2β 6= 0, the absence of CP-state mixing implies that
Im ˜λ67 = 0and δ ∝ Im (m212) = 0. In this latter case, h, H and A are the physical Higgs bosons, with
masses given by eqs. (2.31) and (2.32), and α2 = α3= 0.
2.1.3.2 Various cases of CP mixing
We consider a number of possible interesting patterns of CP-even/CP-odd scalar state mixing [15]: • If ε13 ≡ |M130 2/(MA2−Mh2)| 1, then α2 ≈ 0 and the Higgs boson h1practically coincides
with the lighter CP-even state, h. In addition, the CP-violating couplings of h are very small, typically of O(ε13). The diagonalization of the residual h23i corner of the squared-mass matrix (2.30) using the
rotation matrix R3 yields the mass eigenstates h2 and h3. These are superpositions of H and A with a
potentially large mixing angle α3:
tan 2α3 ≈ −2M 0 2 23 M2 A− MH2 . (2.37)
If MA≈ MH, then the CP-violating state mixing can be strong even at small but nonzero |M230 2|/v2. For
large values of MH ≈ MAthe proper widths of H and A become large and the H and A mass peaks
strongly overlap. Here, one should include a (complex) matrix of Higgs polarization operators [42, 43]. • If ε23 ≡ |M230 2/(MA2 − MH2)| 1, then α3 ≈ 0 and the Higgs boson h2 practically
coincides with the heavier CP-even state, −H. Similarly to the previous case, the diagonalization of the h13i part of squared-mass matrix (2.30), using the rotation matrix R2yields the mass eigenstates h1and
h3. These are superpositions of h and A states, which can strongly mix with large mixing angle α2:
tan 2α2≈ −2M 0 2 13 M2 A− Mh2 . (2.38)
As in the previous case, if MA≈ Mh, the CP-violating state mixing can be strong even at small M130 2/v2.
• The case of weak CP-violating state mixing combines both cases above. That is ε13, ε23 1,
which imply that α2, α3 ≈ 0, in which case the CP–even states h, H are weakly mixed with the CP–odd
state A. The corresponding physical Higgs masses are given by M12' M2
h− s22(MA2 − Mh2), M22' MH2 − s23(MA2 − MH2), (2.39)
with M2
3 given by the sum rule (2.36) . In the particular case of soft-violation of the discrete Z2symmetry
we also have s2'δ cos(β + α) M2 A−Mh2 v2, s3'−δ sin(β + α) M2 A−MH2 v2. (2.40)
• The case of the intense coupling regime with MA ≈ Mh ≈ MH [44] may also yield strong
CP-violating state mixing even when both δ and Im ˜λ67are small.
2.1.4 Higgs boson couplings
In the investigation of phenomenological aspects of 2HDM it is useful to introduce relative couplings, defined as the couplings of each neutral Higgs boson hi(i = 1, 2, 3) to gauge bosons W+W−or ZZ,
Higgs bosons H+H− and h
jhk, quarks ¯qq (q = u, d) and charged leptons `+`−, normalized to the
corresponding couplings of the SM Higgs boson:
χ(i) = g(i)j /gjSM, j = W±, Z, H±, u, d, ` . . . , (2.41) where g(i)
j denotes the jjhi coupling. Note that for bosonic j, the relative couplings are real. In the case
of neutral Higgs boson (hi) couplings to fermions pairs f ¯f, the Yukawa couplings take the form
−LY = ¯f (gRi+ igIiγ5)f hi = ¯fL(gRi+ igIi)fRhi+ h.c. , (2.42)
where the right and left-handed fermion fields are defined as usual: fR ≡ PRf and fL ≡ PLf, with
PR,L ≡ 12(1± γ5). Hence, we shall compute the Higgs–fermion relative coupling in Eq. (2.41) by
employing the complex couplings gi = gRi+ igIi.
One can also make use of basis-independent techniques to obtain expressions for Higgs couplings to gauge bosons, Higgs bosons and fermions that are invariant under U(2) field redefinitions of the two complex scalar doublet fields [16]. Further details of this procedure and a complete collection of 2HDM couplings can be found in section 2.3.
2.1.4.1 Bosonic sector
The gauge bosons V (W and Z) couple only to the CP–even fields ϕ1, ϕ2. In terms of the relative
couplings defined in Eq. (2.41), the couplings of gauge bosons to the physical Higgs bosons hiare:
χ(i)V = cos β Ri1+sin β Ri2, V = W or Z. (2.43)
In particular, in the case of weak CP-violating state mixing considered above, we obtain
χ(1)V ' sin(β − α), χ(2)V ' − cos(β − α), χ(3)V ' −s2sin(β− α) + s3cos(β− α). (2.44)
The cubic and quartic Higgs self-couplings as functions of the Higgs potential parameters and the elements of mixing matrix were obtained in [15,16,45–47]. In the case of soft Z2symmetry violation in
the CP-conserving case, these latter results simplify. The Higgs self-couplings can be expressed in terms of the Higgs masses and the mixing angles α and β. Moreover, if the Higgs-fermion Yukawa interactions are of type-II [as defined below Eq. (2.46)], the trilinear couplings can be given in terms of the Higgs masses, the relative couplings to gauge bosons and quarks, and the parameter η [15]. As an important example, in the case of weak CP-violating state mixing and soft Z2symmetry-violation, the coupling of
the neutral scalar hi to a charged Higgs boson pair (normalized to 2MH2±/v) can be expressed in terms
of the relative neutral Higgs couplings to the gauge bosons and fermions as follows: χ(i)H± = 1− M 2 i 2M2 H± χ(i)V +M 2 i − ηv2 2M2 H± Re (χ(i)u + χ(i)d ). (2.45)
Deviations of the cubic Higgs boson self-couplings from the corresponding Standard Model value would also provide insight into the dynamics of the 2HDM. In particular, as emphasized in section 2.6, there is a strong correlation between the loop-corrected hhh coupling and successful electroweak baryogenesis (that makes critical use of the CP-violation from the Higgs sector).
2.1.4.2 Fermion–Higgs boson Yukawa couplings
The Higgs couplings to fermions are model dependent. The most general structure for the Higgs-fermion Yukawa couplings, often referred to as the type-III model [48, 49], is given in the generic basis by:
where eΦi ≡ iσ2Φ∗i, Q0Lis the weak isospin quark doublet, and UR0, DR0 are weak isospin quark singlets.
Here, Q0
L, UR0, D0Rdenote the interaction basis states, which are vectors in the quark flavor space, and
Γ1, Γ2, ∆1, ∆2 are Yukawa coupling matrices in quark flavor space.5 We have omitted the leptonic
couplings in Eq. (2.46); these follow the same pattern as the down-type quark couplings.
In some models, not all the terms in Eq. (2.46) are present at tree-level [50]. For example, in a type-I model (2HDM-I) [51], there exists a basis where Γ2 = ∆2 = 0.6 Similarly, in a type-II model
(2HDM-II) [52], there exists a basis where Γ1 = ∆2 = 0. The vanishing of certain Higgs-fermion
couplings at tree-level can be enforced by imposing a discrete Z2 symmetry under which Φ1 → Φ1,
Φ2 → −Φ2, and the fermion fields are either invariant or change sign according to whether one wishes
to preserve either the type-I or type-II Higgs-fermion couplings while eliminating the other possible terms in Eq. (2.46). Another well-known example is the MSSM Higgs sector, which exhibits a type-II Higgs-fermion coupling pattern that is enforced by supersymmetry.
The fermion–Higgs boson Yukawa couplings can be derived from Eq. (2.46) (see, e.g., chapter 22 of [10]). Without loss of generality, we choose a basis corresponding to a real vacuum (i.e., ξ = 0). The fermion mass eigenstates are related to the interaction eigenstates by bi-unitary transformations:
PLU = VLUPLU0, PRU = VRUPRU0, PLD = VLDPLD0, PRD = VRDPRD0, (2.47)
and the Cabibbo-Kobayashi-Maskawa matrix is defined as K ≡ VU LV
D†
L . It is also convenient to define
“rotated” linear combinations of the Yukawa coupling matrices:
κU ≡ VLU(Γ1cβ+ Γ2sβ)VRU†, ρU ≡ VLU(−Γ1sβ+ Γ2cβ)VRU†, (2.48)
κD ≡ VLD(∆1cβ+ ∆2sβ)VRD†, ρD ≡ VLD(−∆1sβ+ ∆2cβ)VRD†. (2.49)
The quark mass terms are identified by replacing the scalar fields with their vacuum expectation values. The unitary matrices VU
L, VLD, VRU and VRD are chosen so that κD and κU are diagonal with real
non-negative entries. These quantities are proportional to the diagonal quark mass matrices: MD = v √ 2κ D, M U = v √ 2κ U. (2.50)
In a general model, the matrices ρDand ρUare independent complex non-diagonal matrices.
It is convenient to rewrite Eq. (2.46) in terms of the CP-even Higgs fields H and h and the CP-odd fields A (and the Goldstone boson G0). The end result is:
−LY = 1 vD MDsβ−α+ v √ 2(ρ DP R+ ρD†PL)cβ−α Dh + i vDMDγ5DG 0 +1 vD MDcβ−α− v √ 2(ρ DP R+ ρD†PL)sβ−α DH +√i 2D(ρ DP R− ρD†PL)DA +1 vU MUsβ−α+√v 2(ρ UP R+ ρU †PL)cβ−α U h− i vU MDγ5U G 0 +1 vU MUcβ−α− v √ 2(ρ UP R+ ρU †PL)sβ−α U H−√i 2U (ρ UP R− ρU †PL)U A + ( UhKρDPR− ρU †KPL i DH++ √ 2 v U [KMDPR− MUKPL] DG ++ h.c. ) ,(2.51)
5We have reversed the lettering conventions for these coupling matrices as compared to [10] since ∆ is more naturally associated with the coupling to down-type quarks.
6A type-I model can also be defined as a model in which Γ
1 = ∆1 = 0in some basis. Clearly, the two definitions are equivalent, since the difference in the two conditions is simply an interchange of Φ1and Φ2which can be viewed as a change of basis.
where sβ−α= sin(β−α) and cβ−α= cos(β−α). In the most general CP-violating 2HDM, the physical
Higgs fields are linear combinations of h, H and A. As advertised, since ρD and ρU are non-diagonal,
Eq. (2.51) exhibits tree-level Higgs-mediated FCNCs.7See section 2.5 for a study of the implications of
flavor-changing fermion–Higgs boson couplings for a variety of neutral current processes.
The fermion–Higgs boson Yukawa couplings simplify considerably in type-I and type-II models. In particular, ρD and ρUare no longer independent parameters. For example, in a one-generation type-II
model, Γ1 = ∆2 = 0, which implies that [14]
tan β = −ρ
D
κD =
κU
ρU . (2.52)
These two equations are consistent, since the type-II condition is equivalent to κUκD + ρUρD = 0.
Moreover, using Eqs. (2.50) and (2.52), it follows that: ρD =− √ 2md v tan β , ρ U = √ 2mu v cot β . (2.53)
Inserting this result into Eq. (2.51) yields the well-known Feynman rules for the type-II Higgs-quark interactions. For example, in the case of weak CP-violating state mixing, one finds the expected form for the relative couplings of the neutral Higgs bosons to the up and down-type quarks:
χ(1)d =−sin α cos β = sβ−α− tan β cβ−α, χ (1) u = cos α sin β = sβ−α+ cot β cβ−α, (2.54) −χ(2)d = cos α cos β = cβ−α+ tan β sβ−α, −χ (2) u = sin α sin β = cβ−α− cot β sβ−α,(2.55)
χ(3)d =−i tan β , χ(3)u =−i cot β . (2.56)
Note the extra minus sign in χ(2)i which arises due to the identification of h2 ' −H in this limiting case.
A similar analysis can be given for models of type-I. In the same CP-conserving limiting case considered above, χ(i)
u is identical to the corresponding type-II values given above, but χ(i)d = χ(i)u ∗.
2.1.4.3 The decoupling limit and implications for a SM-like Higgs boson
Suppose that all the coefficients of the quartic terms are held fixed [with values that are not allowed to exceed O(1)]. Then, in the limit that MH± v = 246 GeV, we find that one neutral Higgs boson
has mass of O(v), while the other two neutral Higgs bosons have mass of O(MH±). In this decoupling limit, one can formally integrate out the heavy Higgs states from the theory [53–58]. The resulting Higgs effective theory yields precisely the SM Higgs sector up to corrections of of O(v2/M2
H±). Thus, the
properties of the light neutral Higgs boson of the model, h1, are nearly identical to those of the
CP-even SM Higgs boson. Note that the CP-violating couplings of the lightest neutral Higgs boson to the fermions, gauge bosons and to itself are suppressed by a factor of O(v2/M2
H±). In contrast, the two
heavy neutral Higgs bosons will generally be significant admixtures of the CP-even and CP-odd states Hand A.
In the approach to the decoupling limit, cβ−α ' O(v2/MH2±)[58]. Then, Eqs. (2.44) and (2.54)
yields χ(1)
V ' χ
(1)
d ' χ
(1)
u ' 1, as expected. The flavor structure of the Higgs-quark interactions in the
decoupling limit is also noteworthy. Eq. (2.51) yields approximately flavor-diagonal QQh1 couplings,
since the contribution of the non-diagonal ρQ is suppressed by c
β−α. The heavier neutral Higgs bosons
possess unsuppressed flavor non-diagonal Yukawa interactions, and thus can mediate FCNCs at tree-level. Of course, such FCNC effects would be suppressed by a factor of O(v2/M2
H±)due to the heavy 7Note that even in the case of a CP-conserving Higgs potential, where h, H and A are physical mass eigenstates, Eq. (2.51) exhibits CP-violating Yukawa couplings proportional to the complex matrices ρDand ρU.
