Nambu sum rule and the relation between the masses of composite Higgs bosons

We review the known results on the bosonic spectrum in various Nambu-Jona-Lasinio models both in condensed matter physics and in relativistic quantum field theory including ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{B}$, ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{A}$, the thin films of superfluid ${}^3\mathrm{He}$, and QCD (Hadronic phase and the color-flavor locking phase). Next, we calculate the bosonic spectrum in the relativistic model of top quark condensation suggested in [ Phys. Lett. B 221, 177 (1989)]. In all considered cases, the sum rule appears, which relates the masses (energy gaps) $M_{\mathrm{boson}}$ of the bosonic excitations in each channel with the mass (energy gap) of the condensed fermion $M_f$ as $\sum _{}^{}{M}_{\mathrm{boson}}^2=4M_f^2$. Previously, this relation was established by Nambu for ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{B}$ and for the $s$-wave superconductor. We generalize this relation to the wider class of models and call it the Nambu sum rule. We discuss the possibility to apply this sum rule to various models of top quark condensation. In some cases, this rule allows us to calculate the masses of extra Higgs bosons that are the Nambu partners of the 125 GeV Higgs.