Linearized renormalization

Abstract

Using an infinitesimal approach, this work addresses the renormalization problem to deal with the ultraviolet divergences arising in quantum field theory. Under the assumption that the action has already been renormalized to yield an ultraviolet-finite effective action that satisfies a certain set of renormalization conditions, we analyze how the action must be adjusted to reproduce a first-order change in these renormalization conditions. The analysis then provides the change that is induced on the correlation functions of the theory. This program is successfully carried out in the case of super-renormalizable theories, namely, a scalar field with cubic interaction in four space-time dimensions and with quartic interaction in three space-time dimensions. Relying on existing results in the theory of perturbative renormalization, we derive explicit renormalized expressions for these theories, each of which involves only a finite number of graphs constructed with full propagators and full n-point vertices. The renormalizable case is analyzed as well; the derived expressions are ultraviolet finite as the regulator is removed but cannot be written without a regulator. In this sense, the renormalization is not fully explicit in the renormalizable case. Nevertheless, a perturbative solution of the equations starting from the free theory provides the renormalized Feynman graphs, similar to the BPHZ program. For compatibility with the preservation of the renormalization conditions, a projective renormalization scheme, as opposed to a minimal one, is also introduced. The ideas developed are extended to the study of the renormalization of composite operators and the Schwinger–Dyson equations.