Feynman-Schwinger representation approach to nonperturbative physics

TL;DR

This paper explores the Feynman-Schwinger representation for nonperturbative propagator calculations, demonstrating its application in a solvable scalar QED model and discussing divergence regularization techniques.

Contribution

It provides an analytical illustration of the Feynman-Schwinger formalism in a toy model and discusses divergence regularization methods within this framework.

Findings

Analytic solution for scalar QED in 0+1 dimensions matches perturbative results.

Pauli-Villars regularization effectively removes ultraviolet divergences.

Using an imaginary Feynman parameter avoids spurious divergences.

Abstract

The Feynman-Schwinger representation provides a convenient framework for the cal culation of nonperturbative propagators. In this paper we first investigate an analytically solvable case, namely the scalar QED in 0+1 dimension. With this toy model we illustrate how the formalism works. The analytic result for the self energy is compared with the perturbative result. Next, using a $\chi^2\phi$ interaction, we discuss the regularization of various divergences encountered in this formalism. The ultraviolet divergence, which is common in standard perturbative field theory applications, is removed by using a Pauli-Villars regularization. We show that the divergence associated with large values of Feynman-Schwinger parameter $s$ is spurious and it can be avoided by using an imaginary Feynman parameter $is$.