The attractive (U < 0) and repulsive (U > 0) Hubbard models have been studied using the Fermi liquid perturbation approach. The attractive Hubbard model (U < 0) is an adequate model for 3 He , an incompressible and strongly paramagnetic liquid [Formula: see text], [Formula: see text] for |U|N F = 0.9) with a pseudogap in the charge response. A pairing instability and superfluidity for U < 0 exists in the spin channel only: spin-triplet with l = 0, or spin-singlet with l = 1 (p-wave orbital state j = s + l = 1), where l is the orbital momentum of a pair. The repulsive Hubbard model (U > 0) represents a highly compressible and nearly antiferromagnetic liquid [Formula: see text], [Formula: see text] for UN F = 0.9) with a pseudogap in the spin response. However, for U > 0 a pairing instability and superconductivity exist in the charge channel only: spin-singlet with l = 0 (s-wave), or with l = 2 (d-wave) in the case of an anisotropic Fermi surface.
Analytical calculation for the gluon fragmentation into spin-triplet
S
-wave quarkonium