Finite Callan-Symanzik renormalisation for multiple scalar fields

We study a finite, divergence free approach to renormalisation originally proposed in the early 70's by Blaer and Young, and Callan. It is based on equations similar to the Callan-Symanzik equations, and introduced in the context of $\lambda \phi^4$ theory. We generalise this method to the case of two interacting scalar fields, with obvious generalisation to an arbitrary number of fields.