• vector-like quarks
• little Higgs
• extended top sector
• extended symmetry
• composite Higgs
• Higgs
• pseudo Goldstone boson
• beyond the Standard Model

Naturalness arguments strongly suggest new physics well below LHC energies, and this motivates many attempts to increase the SM symmetry and/or particle content.

Here is presented a ''composite''-Higgs model in which the Higgs-top sector comes from an approximate SO(5) symmetry as a minimal extension of the custodial SO(4), so that the Higgs sector is represented by a scalar 5-plet.
The Higgs boson is realized as a pseudo-Goldstone boson so that quadratically-divergent scalar-loop corrections to $m_h$ disappear, and the top-bottom doublet is extended to a 5-plet of Weyl spinors adding heavy vector-like quarks in order to cancel the top loop quadratic divergence.
New vector bosons should be added for the same reason, however it is phenomenologically meaningful to concentrate on new quarks.

With no worse than 10% finetuning the model can be valid up to 2.5 TeV.

The one-loop impact of the model on EWPT and B-physics has been studied and used as a constraint on the free parameters of the fermion sector.
In particular it comes out that the minimal top-sector extension with only three new quarks has problems in fulfilling experimental requirements, but an extended model with an SU(2)_L singlet and two new doublets of hypercharge 1/6 and 7/6 can be consistent with data. Vector-like quark masses can be relatively light, also down to 200-600 GeV, so in principle they could be detectable at the LHC.
Randomly ``picking up a point'' in the relevant parameter space with all fermion masses below 2.5 TeV the probability of being consistent with data is roughly 1/170.

Finally, at least five new heavy quarks have to be added and moreover the experimentally allowed region is a relatively thin slice in parameter space.
So we can conclude that in principle this SO(5)/SO(4) composite Higgs model can be consistent with data, although the price is not so low.