On the renormalization of non-polynomial field theories

TL;DR

This paper investigates a scalar field model with non-polynomial interactions, revealing that its renormalized form is effectively non-interacting with finite mass, and discusses its relation to earlier non-polynomial interaction theories.

Contribution

It provides a perturbative renormalization analysis of a non-polynomial scalar model, showing the model's effective non-interacting nature at weak coupling and connecting it to historical non-polynomial interactions.

Findings

Renormalized model is non-interacting with finite mass.

Vertices are suppressed by inverse cut-off in effective theories.

Green functions are approximate solutions with errors proportional to coupling.

Abstract

A class of scalar models with non-polynomial interaction, which naturally admits an analytical resummation of the series of tadpole diagrams is studied in perturbation theory. In particular, we focus on a model containing only one renormalizable coupling that appear as a multiplicative coefficient of the squared field. A renormalization group analysis of the Green functions of the model shows that these are only approximated solutions of the flow equations, with errors proportional to powers of the coupling, therefore smaller in the region of weak coupling. The final output of the perturbative analysis is that the renormalized model is non-interacting with finite mass and vanishing vertices or, in an effective theory limited by an ultraviolet cut-off, the vertices are suppressed by powers of the inverse cut-off. The relation with some non-polynomial interactions derived long ago, as solutions of the linearized functional renormalization group flow equations, is also discussed.