On the Nambu fermion-boson relations for superfluid ${}^3\mathrm{He}$

Superfluid ${}^3\mathrm{He}$ is a spin-triplet $\left(S=1\right)$, $p$-wave $\left(L=1\right)$ 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 subgroup $\mathrm{SO}{\left(3\right)}_S\times \mathrm{SO}{\left(3\right)}_L\to {\mathrm{SO}\left(3\right)}_J$. The fermions acquire mass $m_F\equiv \mathrm{\Delta }$, where $\mathrm{\Delta }$ is the BCS gap. There are also 18 bosonic excitations: 4 Nambu-Goldstone modes and 14 massive amplitude Higgs modes. The bosonic modes are labeled by the total angular momentum $J\in \{0,1,2\}$, and parity under particle-hole symmetry $\mathrm{c}=\pm 1$. For each pair of angular momentum quantum numbers $J,J_z$, there are two bosonic partners with $\mathrm{c}=\pm 1$. Based on this spectrum, Nambu proposed a sum rule connecting the fermion and boson masses for BCS-type theories, which for ${}^3\mathrm{He}\text{−}B$ is $M_{J,+}^2+M_{J,-}^2=4m_F^2$ for each family of bosonic modes labeled by $J$, where $M_{J,c}$ is the mass of the bosonic mode with quantum numbers $(J,\mathrm{c})$. The Nambu 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 the Nambu 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 $T\to 0$ and $T\to T_c$ limits. We also discuss experimental results, and theoretical analysis, for the masses of the $J^{\text{c}}=2^{\pm }$ Higgs modes and the magnitude of the violation of the NSR.

Figure 1

Strong-coupling corrections $({\beta }_i^{\mathrm{sc}}\equiv {\beta }_i-{\beta }_i^{\mathrm{wc}})$ to the GL $\beta$ parameters interpolated from the results of Ref. [27] (data squares). The ${\beta }_i^{\text{sc}}$ are extrapolated below $P=12$ bar to weak coupling $\left({\beta }_i^{\text{sc}}=0\right)$ at $p=0$ bar.

Figure 2

Strong-coupling corrections to the bosonic masses obtained from the TDGL theory for the GL $\beta$ parameters shown in Fig. 1. The dashed lines correspond to the weak-coupling values for the masses.

Figure 3

Masses of the $J^{\text{c}}=2^{\pm }$ Higgs modes vs $\ell =2$ particle-hole $\left(F_2^{s,a}\right)$ and $f$-wave pairing $\left(x_3^{-1}\right)$ interactions at $T=0$. The perturbative results [Eqs. (92) and (113)] for $x_3^{-1}=0$ are shown as the dashed black lines.

Figure 4

Deviation of the Nambu sum from polarization corrections to the the $J=2$ Higgs modes of ${}^3\mathrm{He}\text{−}B$ for a range of interactions in both the Landau and Cooper channels.

Figure 5

$J^{\text{c}}=2^{+}$ Higgs mass. The data are for a pressure of $p=13\text{bar}$ (yellow diamonds) [60] and for $p=0.8$–$3.5\text{bar}$ (red squares) [61]. Theoretical calculations of the mass are for $F_2^a=-0.88$ and values of the $f$-wave interaction in the Cooper channel given in the legend.