Abstract
In composite Higgs models the Higgs is a pseudo-Goldstone boson of a high-energy strong dynamics. We have constructed the effective chiral Lagrangian for a generic symmetric coset, restricting to CP-even bosonic operators up to four momenta which turn out to depend on seven parameters, aside from kinetic terms. Once the same sources of custodial symmetry breaking as in the Standard Model are considered, the total number of operators in the basis increases up to ten, again aside from kinetic terms. Under these assumptions, we have then particularised the discussion to three distinct frameworks: the original SU(5)/SO(5) Georgi-Kaplan model, the minimal custodial-preserving SO(5)/SO(4) model and the minimal SU(3)/(SU(2) × U(1)) model, which intrinsically breaks custodial symmetry. The projection of the high-energy electroweak effective theory into the bosonic sector of the Standard Model is shown to match the low-energy chiral effective Lagrangian for a dynamical Higgs, and it uncovers strong relations between the operator coefficients. Finally, the relation with the bosonic basis of operators describing linear realisations of electroweak symmetry breaking is clarified.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys. Rev. Lett. 13 (1964) 321 [INSPIRE].
P.W. Higgs, Broken symmetries, massless particles and gauge fields, Phys. Lett. 12 (1964) 132 [INSPIRE].
P.W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13 (1964) 508 [INSPIRE].
D.B. Kaplan and H. Georgi, SU(2) × U(1) Breaking by Vacuum Misalignment, Phys. Lett. B 136 (1984) 183 [INSPIRE].
D.B. Kaplan, H. Georgi and S. Dimopoulos, Composite Higgs Scalars, Phys. Lett. B 136 (1984) 187 [INSPIRE].
T. Banks, Constraints on SU(2) × U(1) breaking by vacuum misalignment, Nucl. Phys. B 243 (1984) 125 [INSPIRE].
H. Georgi, D.B. Kaplan and P. Galison, Calculation of the Composite Higgs Mass, Phys. Lett. B 143 (1984) 152 [INSPIRE].
H. Georgi and D.B. Kaplan, Composite Higgs and Custodial SU(2), Phys. Lett. B 145 (1984) 216 [INSPIRE].
M.J. Dugan, H. Georgi and D.B. Kaplan, Anatomy of a Composite Higgs Model, Nucl. Phys. B 254 (1985) 299 [INSPIRE].
N. Arkani-Hamed, A.G. Cohen and H. Georgi, Electroweak symmetry breaking from dimensional deconstruction, Phys. Lett. B 513 (2001) 232 [hep-ph/0105239] [INSPIRE].
N. Arkani-Hamed, A.G. Cohen, E. Katz and A.E. Nelson, The Littlest Higgs, JHEP 07 (2002) 034 [hep-ph/0206021] [INSPIRE].
R. Contino, Y. Nomura and A. Pomarol, Higgs as a holographic pseudoGoldstone boson, Nucl. Phys. B 671 (2003) 148 [hep-ph/0306259] [INSPIRE].
K. Agashe, R. Contino and A. Pomarol, The Minimal composite Higgs model, Nucl. Phys. B 719 (2005) 165 [hep-ph/0412089] [INSPIRE].
R. Contino, L. Da Rold and A. Pomarol, Light custodians in natural composite Higgs models, Phys. Rev. D 75 (2007) 055014 [hep-ph/0612048] [INSPIRE].
B. Gripaios, A. Pomarol, F. Riva and J. Serra, Beyond the Minimal Composite Higgs Model, JHEP 04 (2009) 070 [arXiv:0902.1483] [INSPIRE].
A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234 (1984) 189 [INSPIRE].
R. Contino, The Higgs as a Composite Nambu-Goldstone Boson, arXiv:1005.4269 [INSPIRE].
O. Matsedonskyi, G. Panico and A. Wulzer, Light Top Partners for a Light Composite Higgs, JHEP 01 (2013) 164 [arXiv:1204.6333] [INSPIRE].
A. Pomarol and F. Riva, The Composite Higgs and Light Resonance Connection, JHEP 08 (2012) 135 [arXiv:1205.6434] [INSPIRE].
M. Redi and A. Tesi, Implications of a Light Higgs in Composite Models, JHEP 10 (2012) 166 [arXiv:1205.0232] [INSPIRE].
D. Marzocca, M. Serone and J. Shu, General Composite Higgs Models, JHEP 08 (2012) 013 [arXiv:1205.0770] [INSPIRE].
G. Panico, M. Redi, A. Tesi and A. Wulzer, On the Tuning and the Mass of the Composite Higgs, JHEP 03 (2013) 051 [arXiv:1210.7114] [INSPIRE].
D. Pappadopulo, A. Thamm and R. Torre, A minimally tuned composite Higgs model from an extra dimension, JHEP 07 (2013) 058 [arXiv:1303.3062] [INSPIRE].
R. Contino, C. Grojean, D. Pappadopulo, R. Rattazzi and A. Thamm, Strong Higgs Interactions at a Linear Collider, JHEP 02 (2014) 006 [arXiv:1309.7038] [INSPIRE].
O. Matsedonskyi, F. Riva and T. Vantalon, Composite Charge 8/3 Resonances at the LHC, JHEP 04 (2014) 059 [arXiv:1401.3740] [INSPIRE].
T. Appelquist and C.W. Bernard, Strongly Interacting Higgs Bosons, Phys. Rev. D 22 (1980) 200 [INSPIRE].
A.C. Longhitano, Heavy Higgs Bosons in the Weinberg-Salam Model, Phys. Rev. D 22 (1980) 1166 [INSPIRE].
A.C. Longhitano, Low-Energy Impact of a Heavy Higgs Boson Sector, Nucl. Phys. B 188 (1981) 118 [INSPIRE].
F. Feruglio, The Chiral approach to the electroweak interactions, Int. J. Mod. Phys. A 8 (1993) 4937 [hep-ph/9301281] [INSPIRE].
T. Appelquist and G.-H. Wu, The Electroweak chiral Lagrangian and new precision measurements, Phys. Rev. D 48 (1993) 3235 [hep-ph/9304240] [INSPIRE].
B. Grinstein and M. Trott, A Higgs-Higgs bound state due to new physics at a TeV, Phys. Rev. D 76 (2007) 073002 [arXiv:0704.1505] [INSPIRE].
R. Contino, C. Grojean, M. Moretti, F. Piccinini and R. Rattazzi, Strong Double Higgs Production at the LHC, JHEP 05 (2010) 089 [arXiv:1002.1011] [INSPIRE].
A. Azatov, R. Contino and J. Galloway, Model-Independent Bounds on a Light Higgs, JHEP 04 (2012) 127 [Erratum ibid. 1304 (2013) 140] [arXiv:1202.3415] [INSPIRE].
R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin and J. Yepes, The Effective Chiral Lagrangian for a Light Dynamical “Higgs Particle”, Phys. Lett. B 722 (2013) 330 [Erratum ibid. B 726 (2013) 926] [arXiv:1212.3305] [INSPIRE].
M.B. Gavela et al., CP violation with a dynamical Higgs, JHEP 1410 (2014) 44 [arXiv:1406.6367] [INSPIRE].
G. Buchalla and O. Catà, Effective Theory of a Dynamically Broken Electroweak Standard Model at NLO, JHEP 07 (2012) 101 [arXiv:1203.6510] [INSPIRE].
R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin and J. Yepes, Minimal Flavour Violation with Strong Higgs Dynamics, JHEP 06 (2012) 076 [arXiv:1201.1511] [INSPIRE].
R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin and J. Yepes, Flavor with a light dynamical “Higgs particle”, Phys. Rev. D 87 (2013) 055019 [arXiv:1212.3307] [INSPIRE].
G. Buchalla, O. Catà and C. Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO, Nucl. Phys. B 880 (2014) 552 [arXiv:1307.5017] [INSPIRE].
I. Brivio et al., Disentangling a dynamical Higgs, JHEP 03 (2014) 024 [arXiv:1311.1823] [INSPIRE].
I. Brivio et al., Higgs ultraviolet softening, arXiv:1405.5412 [INSPIRE].
G. Isidori and M. Trott, Higgs form factors in Associated Production, JHEP 02 (2014) 082 [arXiv:1307.4051] [INSPIRE].
G. Cvetič and R. Kogerler, Fermionic Couplings in an Electroweak Theory With Nonlinear Spontaneous Symmetry Breaking, Nucl. Phys. B 328 (1989) 342 [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Naive Dimensional Analysis Counting of Gauge Theory Amplitudes and Anomalous Dimensions, Phys. Lett. B 726 (2013) 697 [arXiv:1309.0819] [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
M. Frigerio, A. Pomarol, F. Riva and A. Urbano, Composite Scalar Dark Matter, JHEP 07 (2012) 015 [arXiv:1204.2808] [INSPIRE].
D. Marzocca and A. Urbano, Composite Dark Matter and LHC Interplay, JHEP 07 (2014) 107 [arXiv:1404.7419] [INSPIRE].
R. Contino, D. Marzocca, D. Pappadopulo and R. Rattazzi, On the effect of resonances in composite Higgs phenomenology, JHEP 10 (2011) 081 [arXiv:1109.1570] [INSPIRE].
Particle Data Group collaboration, J. Beringer et al., Review of Particle Physics (RPP), Phys. Rev. D 86 (2012) 010001 [INSPIRE].
W. Buchmüller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268 (1986) 621 [INSPIRE].
B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085 [arXiv:1008.4884] [INSPIRE].
K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Low-energy effects of new interactions in the electroweak boson sector, Phys. Rev. D 48 (1993) 2182 [INSPIRE].
K. Hagiwara, T. Hatsukano, S. Ishihara and R. Szalapski, Probing nonstandard bosonic interactions via W boson pair production at lepton colliders, Nucl. Phys. B 496 (1997) 66 [hep-ph/9612268] [INSPIRE].
G.F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, The Strongly-Interacting Light Higgs, JHEP 06 (2007) 045 [hep-ph/0703164] [INSPIRE].
M.B. Gavela, K. Kanshin, P.A.N. Machado and S. Saa, On the renornalization of the electroweak chiral lagrangian with a light higgs, arXiv:1409.1571 [INSPIRE].
A. Azatov, R. Contino, A. Di Iura and J. Galloway, New Prospects for Higgs Compositeness in h → Zγ, Phys. Rev. D 88 (2013) 075019 [arXiv:1308.2676] [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: 1409.1589
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
Alonso, R., Brivio, I., Gavela, M.B. et al. Sigma decomposition. J. High Energ. Phys. 2014, 34 (2014). https://doi.org/10.1007/JHEP12(2014)034
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2014)034