On the Nambu fermion-boson relations for superfluid He 3

Superfluid He3 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 subgroup SO(3)S×SO(3)L→SO(3)J. The fermions acquire mass mF≡Δ, where Δ 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{0,1,2}, and parity under particle-hole symmetry c=±1. For each pair of angular momentum quantum numbers J,Jz, there are two bosonic partners with c=±1. Based on this spectrum, Nambu proposed a sum rule connecting the fermion and boson masses for BCS-type theories, which for He3-B is MJ,+2+MJ,-2=4mF2 for each family of bosonic modes labeled by J, where MJ,c is the mass of the bosonic mode with quantum numbers (J,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→0 and T→Tc limits. We also discuss experimental results, and theoretical analysis, for the masses of the Jc=2± Higgs modes and the magnitude of the violation of the NSR.