Yukawa theory in non-perturbative regimes: towards confinement, exact $\beta$-function and conformal phase

We study possible hints towards confinement in a Z$_2$-invariant Yukawa system with massless fermions and a real scalar field in the strongly-coupled regime. Using the tools developed for studying non-perturbative physics via Jacobi elliptical functions, for a given but not unique choice of the vacuum state, we find the exact Green's function for the scalar sector so that, after integrating out the scalar degrees of freedom, we are able to recover the low-energy limit of the theory that is a fully non-local Nambu-Jona-Lasinio (NJL) model. We provide an analytical result for the Renormalization Group (RG) running of the self-interaction coupling in the scalar sector and critical indexes in the strongly-coupled regime. In the fermion sector, we provide some clues towards confinement, after deriving the gap equation with the non-local NJL model, a property which is well-known to not emerge in the local limit of this model. We conclude that, for the scalar-Yukawa theory in the non-perturbative domain with our choice of the vacuum state, the fundamental fermions of the theory form bound states and cannot be observed as asymptotic states.