## CMT150

##
**
On Nambu's Fermion-Boson Relations for Superfluid **^{3}He

**Author(s):**
J. A. Sauls and Takeshi Mizushima
- Address: Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208
- Date: November 22, 2016
**Journal:**
Physical Review B 95, 094515 (2017)
[arXiv]
[DOI]
[PDF]
**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}×
`SO``(``3``)`_{L}→
`SO``(``3``)`_{J}.
The Fermions acquire mass, m_{F} ≡ ∆, where ∆ 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 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,J_{z}, there are two
Bosonic partners with C=±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 labeled 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→ 0 and T→ T_{c} limits.
We also discuss experimental results, and theoretical analysis, for the masses of the J^{c}=2^{±}
Higgs modes and the magnitude of the violation of the NSR.

- Comment: 20 pages, 5 figures

**Eprint:**
[PDF]
[arXiv]