Composite Higgs models

In particle physics, composite Higgs models (CHM) are speculative extensions of the Standard Model (SM) where the Higgs boson is a bound state of new strong interactions. These scenarios are the leading alternative to supersymmetric models for physics beyond the SM presently tested at the Large Hadron Collider (LHC) in Geneva.

According to CHM the recently discovered Higgs boson is not an elementary particle (or point-like) but has finite size, typically around 10−18 meters. This dimension is related to the Fermi scale (100 GeV) that determines the strength of the weak interactions such as in β-decay. Microscopically the composite Higgs will be made of smaller constituents in the same way as nuclei are made protons and neutrons.

The main prediction of CHM are new particles with mass around TeV that are excitations of the composite Higgs. This is analogous to the resonances in nuclear physics. The new particles could be produced and detected in collider experiments if the energy of the collision exceeds their mass or could produce deviations from the SM predictions in low energy observables. Within the most compelling scenarios each Standard Model particle has a partner with equal quantum numbers but heavier mass. For example, the photon, W and Z bosons have heavy replicas with mass determined by the compositeness scale, expected around 1012 eV.

CHM are motivated by the so-called naturalness or hierarchy problem of the SM,[1] the difficulty to explain the different energy scales that appear in the fundamental interactions of particle physics. CHM can solve the naturalness problem because the Higgs boson is not an elementary particle so that a new energy scale exists that can be explained dynamically similarly to the mass of the proton. Naturalness requires that new particles exist with mass around TeV and these could be discovered at LHC or future experiments. As of 2015, no direct or indirect signs that the Higgs or other SM particles are composite has been detected.

History

CHM were introduced in the early '80s as an extension of technicolor theories to allow for the presence of a physical Higgs boson. At the time this was not required by the data but recent discoveries have shown the necessity of a physical Higgs doublet to break the electro-weak symmetry. This differs from ordinary technicolor theories where strong dynamics directly breaks the electro-weak symmetry without the need of a physical Higgs boson. The first CHM proposed by Georgi and Kaplan were based on known gauge theory dynamics[2] that produces the Higgs doublet as a Goldstone boson. These models are very constrained and it is difficult to include fermion masses. The subject remained dormant for several years until it was realized that this type of models naturally arise in 5 dimensional theories known as Randall–Sundrum scenarios. Within these constructions enough freedom exists to build realistic models at the price of abandoning a full microscopic description. It was soon understood that these scenarios can also be realized in hypothetical strongly coupled conformal field theories (CFT) also providing a dynamical explanation for the generation of hierarchies. The connection between Randall-Sundrum models and CFTs is rooted in the AdS-CFT correspondence. This spurred a great deal of activity in the field. At first the Higgs was a generic scalar bound state. In the highly influential work[3] the Higgs as a Goldstone boson was realized in Randall-Sundrum models and in CFTs. Detailed phenomenological studies showed that within this framework agreement with experimental data can be obtained with a mild tuning of parameters.

Models

CHM can be characterized by the mass (M) of the lightest new particles and their coupling (g). The latter is expected to be larger than the SM couplings for consistency. Various realizations of CHM exist that differ for the mechanism that generates the Higgs doublet. Broadly they can be divided in two categories:

  1. Higgs is a generic bound state of strong dynamics.
  2. Higgs is a Goldstone boson of spontaneous symmetry breaking[4][5]

In both cases the electro-weak symmetry is broken by the condensation of a Higgs scalar doublet. In the first type of scenarios there is no a priori reason why the Higgs boson is lighter than the other composite states and moreover larger deviations from the SM are expected.

Higgs as Goldstone boson

In this scenario existence of the Higgs boson follows from the symmetries of the theory. This allows to explain why this particle is lighter than the rest of the composite particles whose mass is expected from direct and indirect tests around TeV or higher. It is assumed that the composite sector has a global symmetry G spontaneously broken to a subgroup H where G and H are compact Lie groups. Contrary to technicolor models the unbroken symmetry must contain the SM electro-weak group SU(2)xU(1). According to general theorems of quantum field theory the spontaneous breaking of a global symmetry produces massless scalar particles known as Goldstone bosons. By appropriately choosing the global symmetries it is possible to have Goldstone bosons that correspond to the Higgs doublet in the SM. This can be done in a variety of ways[6] and is completely determined by the symmetries. In particular group theory determines the quantum numbers of the Goldstone bosons. From the decomposition of the adjoint representation one finds,

where R[Π] is the representation of the Goldstone bosons under H. The phenomenological request that a Higgs doublet exists selects the possible symmetries. Typical example is the pattern

that contains a single Higgs doublet as a Goldstone boson.

The physics of the Higgs as a Goldstone boson is strongly constrained by the symmetries and determined by the symmetry breaking scale f that controls their interactions. An approximate relation exists between mass and coupling of the composite states, In CHM one finds that deviations from the SM are proportional to,

where v=246 GeV is the electro-weak vacuum expectation value. By construction these models approximate the SM to arbitrary precision if ξ is sufficiently small. For example, for the model above with SO(5) global symmetry the coupling of the Higgs to W and Z bosons is modified as,

Phenomenological studies suggest f > 1 TeV. However the tuning of parameters required to achieve v < f is inversely proportional to ξ so that viable scenarios require some degree of tuning.

Goldstone bosons generated from the spontaneous breaking of an exact global symmetry are exactly massless. In CHM the Higgs potential is generated by effects that explicitly break the global symmetry G. Minimally these are the SM Yukawa and gauge couplings that cannot respect the global symmetry but other effects can also exist. The top coupling is expected to give a dominant contribution to the Higgs potential as this is the largest coupling in the SM. In the simplest models one finds a correlation between the Higgs mass and the mass M of the top partners,[7]

In models with f~TeV as suggested by naturalness this indicates fermionic resonances with mass around 1 TeV. Spin-1 resonances are expected to be somewhat heavier. This is within the reach of future collider experiments.

Partial compositeness

One of the key ingredients of modern realizations of CHM is the hypothesis of partial compositeness originally proposed by D. B. Kaplan.[8] This hypothesis is automatically realized in Randall-Sundrum scenarios. Every SM particle has a heavy partner that can mix with it. In practice the SM particles are linear combinations of elementary and composite states:

where α denotes the mixing angle. Partial compositeness is naturally realized in the gauge sector where an analogous phenomenon happens quantum chromodynamics and is known as photon–ρ mixing. For fermions it is an assumptions that in particular requires the existence of heavy fermion with equal quantum numbers as SM quarks and leptons. These interact with the Higgs through the mixing. One schematically finds the formula for the SM fermion masses,

where L and R refer to the left and right mixings, Y is a composite sector coupling.

The composite particles are multiplets of the unbroken symmetry H. For phenomenological reasons this should contain the custodial symmetry SU(2)xSU(2) extending the electro-weak symmetry SU(2)xU(1). Composite fermions often belong to representations larger than the SM particles. For example, a strongly motivated representation for left-handed fermions is the (2,2) that contains particles with exotic electric charge 5/3 or –4/3 with special experimental signatures.

Partial compositeness ameliorates the phenomenology of CHM providing a logic why no deviations from the SM have been measured so far. In the so-called anarchic scenarios the hierarchies of SM fermion masses are generated through the hierarchies of mixings and anarchic composite sector couplings. The light fermions are almost elementary while the third generation is strongly or entirely composite. This leads to a structural suppression of all effects that involve first two generations that are the most precisely measured. In particular flavor transitions and corrections to electro-weak observables are suppressed. Other scenarios are also possible[9] with different phenomenology.

Experiments

The main experimental signatures of CHM are:

  1. New Resonances with SM quantum numbers and masses around TeV
  2. Modified SM couplings
  3. New contributions to flavor observables

All the deviations from the SM are controlled by the tuning parameter ξ. The mixing of the SM particles determine the coupling with the known particles of the SM. The detailed phenomenology depends strongly on the flavor assumptions and is in general-model-dependent. The Higgs and the top quark typically have the largest coupling to the new particles. For this reason third generation partners are the most easy to produce and top physics has the largest deviations from the SM. Top partners have also special importance given their role in the naturalness of the theory.

After the first run of the LHC direct experimental searches exclude third generation fermionic resonances up to 800 GeV.[10][11] Bounds on gluon resonances are in the multi-TeV range[12][13] and somewhat weaker bounds exist for electro-weak resonances.

Deviations from the SM couplings is proportional to the degree of compositeness of the particles. For this reason the largest departures from the SM predictions are expected for the third generation quarks and Higgs couplings. The first have been measured with per mille precision by the LEP experiment. After the first run of the LHC the couplings of the Higgs with fermions and gauge bosons agree with the SM with a precision around 20%. These results pose some tension for CHM but are compatible with a compositeness scale f~TeV.

The hypothesis of partial compositeness allows to suppress flavor violation beyond the SM that is severely constrained experimentally. Nevertheless, within anarchic scenarios sizable deviations from the SM predictions exist in several observables. Particularly constrained is CP violation in the Kaon system and lepton flavor violation for example the rare decay μ->eγ. Overall flavor physics suggests the strongest indirect bounds on anarchic scenarios. This tension can be avoided with different flavor assumptions.

References

  1. ↑ G. F. Giudice, Naturalness after LHC8, PoS EPS HEP2013, 163 (2013) http://arxiv.org/pdf/1307.7879.pdf.
  2. ↑ M. J. Dugan, H. Georgi and D. B.Kaplan, Anatomy of a Composite Higgs Model, Nucl. Phys. B254, 299 (1985).
  3. ↑ K. Agashe, R. Contino and A. Pomarol, The Minimal composite Higgs model, Nucl. Phys. B719, 165 (2005) http://arxiv.org/pdf/hep-ph/0412089.pdf.
  4. ↑ R. Contino, The Higgs as a Composite Nambu-Goldstone Boson, http://arxiv.org/pdf/1005.4269.pdf.
  5. ↑ M. Redi, https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxyZWRpNzZ8Z3g6YjZlZDk5NDU4NTBjMmU5
  6. ↑ J. Mrazek, A. Pomarol, R. Rattazzi, M. Redi, J. Serra and A. Wulzer, The Other Natural Two Higgs Doublet Model, Nucl. Phys. B853, 1 (2011) http://arxiv.org/pdf/1105.5403.pdf.
  7. ↑ M. Redi and A. Tesi, Implications of a Light Higgs in Composite Models, JHEP 1210, 166 (2012) http://arxiv.org/pdf/1205.0232.pdf.
  8. ↑ D. B. Kaplan, Flavor at SSC energies: A New mechanism for dynamically generated fermion masses, Nucl. Phys. B 365, 259 (1991).
  9. ↑ M. Redi and A. Weiler, Flavor and CP Invariant Composite Higgs Models, JHEP 1111, 108 (2011) http://arxiv.org/pdf/1106.6357.pdf.
  10. ↑ ATLAS, https://cds.cern.ch/record/1557777/files/ATLAS-CONF-2013-060.pdf
  11. ↑ CMS, https://cds.cern.ch/record/1524087/files/B2G-12-012-pas.pdf
  12. ↑ ATLAS, https://cds.cern.ch/record/1547568/files/ATLAS-CONF-2013-052.pdf
  13. ↑ CMS, https://cds.cern.ch/record/1545285/files/B2G-12-005-pas.pdf
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