Third-order corrections to the ground-state energy of the gas of spin-s fermions with arbitrary densities of different spin projections

Recently, we computed third-order corrections to the ground-state energy of an arbitrarily polarized diluted gas of spin-21 fermions interacting through a spin-independent repulsive two-body potential. Here we extend this result to the gas of spin-s fermions [a system whose Hamiltonian has an accidental SU(2s + 1) symmetry] with arbitrary densities of fermions having different spin projections. The corrections are computed semianalytically using the effective-field-theory approach and are parametrized by the s- and p-wave scattering lengths a0 and a1 and the s-wave effective radius r0, measurable in the low-energy fermion-fermion elastic scattering. The result is used to study the impact the higher-order corrections can have on the characteristics of the phase transition (at zero temperature) to the ordered phase (on the emergence of the itinerant ferromagnetism).