Operator regularization and multiloop Green's functions

We present in this paper the evaluation of multiloop Green's functions in the context of a recently proposed regulating scheme called operator regularization. We show that, in contrast with other schemes, imposing the requirement of unitarity (rather than finiteness) is crucial in obtaining the (perturbative) renormalized effective action and Green's functions of a given theory. We demonstrate how an evaluation of these quantities may be carried out using this regulating technique in a manner which preserves the unitarity of the $S$ matrix. This method is then applied to a two-loop calculation in the ${\left({\phi }^4\right)}_4$ theory, yielding agreement with results obtained in other schemes. Specifically, we calculate the $\beta$ function, ${\gamma }_m$ function, and the anomalous dimension of the field without having to look at the relationship between bare and renormalized quantities. Indeed, they directly come from the finite sector of the Green's functions.