On the link between finite QFT and standard RG approaches

A finite formulation of quantum field theory based on a system of differential equations reminiscent of the Callan-Symanzik equations is discussed. This system of equations was previously formulated in the bare language. We rederive it in a fully renormalized language. For the latter, within a simple $\phi^4$ toy model, it is shown that with a specific choice of renormalization conditions - namely, the on-shell scheme for the renormalized mass - this class of finite renormalization prescriptions is equivalent to the standard renormalization-group equation written in the Callan-Symanzik-Ovsyannikov form.