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Scalar excitation with Leggett frequency in $^3$ He-B and the $125$ GeV Higgs particle in top quark condensation models as Pseudo - Goldstone bosons

We consider the scenario, in which the light Higgs scalar boson appears as the Pseudo - Goldstone boson. We discuss examples both in condensed matter and in relativistic field theory. In $^3$He-B the symmetry breaking gives rise to 4 Nambu-Goldstone modes and 14 Higgs modes. At lower energy one of the four NG modes becomes the Higgs boson with small mass. This is the mode measured in experiments with the longitudinal NMR, and the Higgs mass corresponds to the Leggett frequency $M_{\rm H}=\hbar Ω_L$. The formation of the Higgs mass is the result of the violation of the hidden spin-orbit symmetry at low energy. In this scenario the symmetry breaking energy scale $Δ$ (the gap in the fermionic spectrum) and the Higgs mass scale $M_{\rm H}$ are highly separated: $M_{\rm H}\ll Δ$. On the particle physics side we consider the model inspired by the models of \cite{dobrescu,Yamawaki}. At high energies the $SU(3)$ symmetry is assumed that relates the left - handed top and bottom quarks to the additional fermion $χ_L$. This symmetry is softly broken at low energies. As a result the only CP - even Goldstone boson acquires a mass and may be considered as the candidate for the role of the $125$ GeV scalar boson. We consider the condensation pattern different from the one typical for the top - seesaw models, where the condensate $< \bar{t}_Lχ_R >$ is off - diagonal. In our case the condensates are mostly diagonal. Unlike \cite{dobrescu,Yamawaki} the explicit mass terms are absent and the soft breaking of $SU(3)$ symmetry is given solely by the four - fermion terms. This reveals the complete analogy with $^3$He, where there is no explicit mass term and the spin - orbit interaction has the form of the four - fermion interaction.