On Nambu's Fermion-Boson Relations for Superfluid $^3$He-B

Superfluid $^3$He is a spin-triplet ($S=1$), p-wave ($L=1$) BCS condensate of Cooper pairs with total angular momentum $J=0$ in the ground state. In addition to the breaking of $U(1)$ gauge symmetry, separate spin or orbital rotation symmetry is broken to the maximal sub-group, $SO(3)_S\times SO(3)_L\rightarrow SO(3)_J$. The Fermions acquire mass, $m_F\equiv\Delta$, where $\Delta$ is the BCS gap. There are also 18 Bosonic excitations - 4 Nambu-Goldstone (NG) modes and 14 massive amplitude Higgs (AH) modes. The Bosonic modes are labelled by the total angular momentum, $J\in\{0,1,2\}$, and parity under particle-hole symmetry, $C=\pm 1$. For each pair of angular momentum quantum numbers, $J,J_z$, there are two Bosonic partners with $C=\pm 1$. Based this spectrum Nambu proposed a sum rule connecting the Fermion and Boson masses for BCS type theories, which for $^3$He-B is $M_{J^+}^2 + M_{J^-}^2 = 4m_F^2$ for each family of Bosonic modes labelled by $J$, where $M_{J^C}$ is the mass of the Bosonic mode with quantum numbers $(J,C)$. Nambu's sum rule (NSR) has recently been discussed in the context of Nambu-Jona-Lasinio models for physics beyond the standard model to speculate on possible partners to the recently discovered Higgs Boson at higher energies. Here we point out that Nambu's Fermion-Boson mass relations are not exact. Corrections to the Bosonic masses from (i) leading order strong-coupling corrections to BCS theory, and (ii) polarization of the parent Fermionic vacuum lead to violations of the sum-rule. Results for these mass corrections are given in both the $T\rightarrow 0$ and $T\rightarrow T_c$ limits. We also discuss experimental results, and theoretical analysis, for the masses of the $J^{C}=2^{\pm}$ Higgs modes and the magnitude of the violation of the NSR.