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

TL;DR

This paper examines Nambu's fermion-boson mass relations in superfluid helium-3 B-phase, showing they are approximate and analyzing the corrections and violations due to strong-coupling effects and vacuum polarization.

Contribution

It demonstrates that Nambu's sum rule for fermion-boson masses in superfluid helium-3 B-phase is not exact and quantifies the corrections from strong-coupling and polarization effects.

Findings

Nambu's sum rule is approximately valid but has measurable violations.

Corrections to bosonic masses are significant near the critical temperature.

Experimental and theoretical analysis of Higgs mode masses supports the violation of the sum rule.

Abstract

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.