Self Consistent Field Method for Planar phi^3 Theory

TL;DR

This paper advances the self consistent field method for planar phi^3 theory, deriving asymptotic solutions and analyzing string formation, Lorentz invariance, and renormalization within a local world sheet action framework.

Contribution

It introduces a new version of the self consistent field approximation, enabling exact solutions and deeper understanding of string formation in phi^3 theory.

Findings

Derived asymptotic forms of solutions

Obtained exact form of a specific solution

Confirmed formation of an unstable string

Abstract

We continue and extend earlier work on the summation of planar graphs in phi^3 field theory, based on a local action on the world sheet. The present work employs a somewhat different version of the self consistent field (meanfield) approximation compared to the previous work on the same subject. Using this new approach, we are able to determine in general the asymptotic forms of the solutions, and in the case of one solution, even its exact form. This solution leads to formation of an unstable string, in agreement with the previous work. We also investigate and clarify questions related to Lorentz invariance and the renormalization of the solution.