On a possible approach to general field theories with nonpolynomial interactions

TL;DR

This paper explores a novel method to analyze scalar field theories with nonpolynomial interactions by mapping them into polynomial theories and applying both perturbative and nonperturbative techniques, including auxiliary fields and regularization.

Contribution

It introduces a mapping for nonpolynomial scalar theories into polynomial ones and develops a nonperturbative approach using auxiliary fields and dimensional regularization.

Findings

Mapping of nonpolynomial to polynomial theories under Fourier representation

Development of a nonperturbative treatment with auxiliary fields

Regularization and counterterm computation for singularities

Abstract

In this work a class of massive scalar field theories with self-interactions described by a general potential is studied. Under the sole condition that the potential admits the Fourier representation, it is shown that such theories may be mapped into a standard field theory, in which the interaction of the new fields is a polynomial of fourth degree. With some restrictions, this mapping allows the perturbative treatment of models that are otherwise intractable with standard field theoretical methods. A nonperturbative approach to these theories is attempted. The original scalar field is integrated out exactly at the price of introducing auxiliary vector fields. The latter are treated in a mean field theory approximation. The singularities that arise after the elimination of the auxiliary fields are cured using the dimensional regularization. The expression of the counterterms to be subtracted is computed.