Renormalization-group study of a superconducting phase transition: Asymptotic behavior of higher expansion orders and results of three-loop calculations

We use quantum-field renormalization group methods to study the phase transition in an equilibrium system of nonrelativistic Fermi particles with the ‘`density–density" interaction in the formalism of temperature Green’s functions. We especially attend to the case of particles with spins greater than $1/2$ or fermionic fields with additional indices for some reason. In the vicinity of the phase transition point, we reduce this model to a $\phi^4$-type theory with a matrix complex skew-symmetric field. We define a family of instantons of this model and investigate the asymptotic behavior of quantum field expansions in this model. We calculate the $\beta$-functions of the renormalization group equation through the third order in the $(4{-}\epsilon)$-scheme. In the physical space dimensions $D=2,3$, we resum solutions of the renormalization group equation on trajectories of invariant charges. Our results confirm the previously proposed suggestion that in the system under consideration, there is a first-order phase transition into a superconducting state that occurs at a higher temperature than the classical theory predicts.