Higgs as Goldstone boson

These are essentially Little Higgs theories. In this scenario the 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 to be around a 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 Goldstone's theorem 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^[13] 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

${\displaystyle {\rm {{Adj}[G]={\rm {Adj}}[H]+{\rm {R}}[\Pi ]}}}$,

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

${\displaystyle {\frac {SO(5)}{SU(2)_{L}\times SU(2)_{R}}}\rightarrow GB=(2,2)}$

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, ${\displaystyle M=gf}$ In CHM one finds that deviations from the SM are proportional to

${\displaystyle \xi ={\frac {v^{2}}{f^{2}}}}$,

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

${\displaystyle {\frac {h_{VV}}{h_{VV}^{SM}}}\approx 1-{\frac {\xi }{2}}}$.

Phenomenological studies suggest f > 1 TeV and thus at least a factor of a few larger than v. 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. Therefore, if the Higgs boson is a Goldstone boson the global symmetry cannot be exact. 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,^[14]

${\displaystyle m_{h}^{2}\sim {\frac {3}{2\pi ^{2}}}{\frac {M^{2}}{f^{2}}}v^{2}}$

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 ingredient of modern CHM is the hypothesis of partial compositeness proposed by D.B. Kaplan.^[15] This is similar to a (deconstructed) extra dimension, in which every Standard Model particle has a heavy partner(s) that can mix with it. In practice, the SM particles are linear combinations of elementary and composite states:

${\displaystyle |{\mathsf {Std.Mod.}}\rangle ~=~\cos {\alpha }~|{\mathsf {Elem.}}\rangle +\sin {\alpha }~|{\mathsf {Comp.}}\rangle }$

where ${\displaystyle \ \alpha \ }$ denotes the mixing angle.

Partial compositeness is naturally realized in the gauge sector, where an analogous phenomenon happens quantum chromodynamics and is known as γ–ρ mixing (after the photon and rho meson – two particles with identical quantum numbers which engage in similar intermingling). For fermions it is an assumption that in particular requires the existence of heavy fermions with equal quantum numbers to S.M. quarks and leptons. These interact with the Higgs through the mixing. One schematically finds the formula for the S.M. fermion masses,

${\displaystyle {\frac {\ m_{\mathsf {f}}\ }{v}}~\approx ~\sin \alpha _{\mathsf {L}}~Y~\sin \alpha _{\mathsf {R}}\ ,}$

where subscripts L and R mark the left and right mixings, and 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)×SU(2) extending the electro-weak symmetry SU(2)×U(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 S.M. have been measured so far. In the so-called anarchic scenarios the hierarchies of S.M. 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^[16] with different phenomenology.