Non-linear and weak-coupling expansion in Quantum Field Theory

TL;DR

This paper explores a non-linear expansion method for quantum field theories, revealing how resummation of certain diagrams restores the interacting nature of the theory and connects to weak-coupling results.

Contribution

It demonstrates that resumming specific classes of diagrams in a non-linear expansion recovers the interacting behavior and links to traditional weak-coupling expansions in scalar quantum field theories.

Findings

Finite order expansion appears non-interacting

Resummation of diagrams restores interaction

Connects non-linear expansion to weak-coupling results

Abstract

A formal expansion for the Green's functions of an interacting quantum field theory in a parameter that somehow encodes its "distance" from the corresponding non-interacting one was introduced more than thirty years ago, and has been recently reconsidered in connection with its possible application to the renormalization of non-hermitian theories. Besides this new and interesting application, this expansion has special properties already when applied to ordinary (i.e. hermitian) theories, and in order to disentangle the peculiarities of the expansion itself from those of non-hermitian theories, it is worth to push further the investigation limiting first the analysis to ordinary theories. In the present work we study some aspects related to the renormalization of a scalar theory within the framework of such an expansion. Due to its peculiar properties, it turns out that at any finite order in the expansion parameter the theory looks as non-interacting. We show that when diagrams of appropriate classes are resummed, this apparent drawback disappears and the theory recovers its interacting character. In particular we have seen that with a certain class of diagrams, the weak-coupling expansion results are recovered, thus establishing a bridge between the two expansions.