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

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, $SO(3{)}_S\times SO(3{)}_L\to SO(3{)}_J$. 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 $T\to 0$ and $T\to T_c$ 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.

arXiv comment20 pages, 5 figures

References in corpus (10)

• Emergent Space-time Supersymmetry at the Boundary of a Topological Phase
• Higgs bosons in particle physics and in condensed matter
• Discovery of the Acoustic Faraday Effect in Superfluid 3He-B
• Comment on the Coupling of Zero Sound to the $J=1^{-}$ Modes of ${}^3$He-B
• Light Higgs channel of the resonant decay of magnon condensate in superfluid ${}^3$He-B
• Strong Coupling Corrections to the Ginzburg-Landau Theory of Superfluid ^{3}He
• The Nambu sum rule and the relation between the masses of composite Higgs bosons
• Collective Modes and f-wave Pairing Interactions in Superfluid 3He
• Thermodynamic Potential for Superfluid 3He in Aerogel
• Nonlinear field-dependence and f-wave interactions in superfluid 3He

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