Scalar excitation with Leggett frequency in ${}^3\mathrm{He}\text{−}\mathrm{B}$ 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 in both condensed matter and relativistic field theory. In ${}^3\mathrm{He}\text{−}\mathrm{B}$ the symmetry breaking gives rise to four Nambu-Goldstone (NG) modes and 14 Higgs modes. At lower energy one of the four NG modes becomes the Higgs boson with a small mass. This is the mode measured in experiments with the longitudinal NMR, and the Higgs mass corresponds to the Leggett frequency $M_{\mathrm{H}}=\hslash {\mathrm{\Omega }}_B$. 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 $\mathrm{\Delta }$ (the gap in the fermionic spectrum) and the Higgs mass scale $M_{\mathrm{H}}$ are highly separated: $M_{\mathrm{H}}\ll \mathrm{\Delta }$. On the particle physics side we consider the model inspired by the models of Refs. Cheng et al. [J. High Energy Phys. 08 (014) 095] and Fukano et al. [Phys. Rev. D 90, 055009 (2014)]. At high energies the SU(3) symmetry is assumed which relates the left-handed top and bottom quarks to the additional fermion ${\chi }_L$. 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 a candidate for the 125 GeV scalar boson. We consider a condensation pattern different from that typically used in top-seesaw models, where the condensate $⟨{\overline{t}}_L{\chi }_R⟩$ is off-diagonal. In our case the condensates are mostly diagonal. Unlike the work of Cheng et al. [J. High Energy Phys. 08 (014) 095] and Fukano et al. [Phys. Rev. D 90, 055009 (2014)], the explicit mass terms are absent and the soft breaking of SU(3) symmetry is given solely by the four-fermion terms. This reveals a complete analogy with ${}^3\mathrm{He}$, where there is no explicit mass term and the spin-orbit interaction has the form of the four-fermion interaction.