Feynman Amplitudes in Periodically Compactified Spaces -- Spin 0

TL;DR

This paper extends the Schwinger parametric representation for Feynman amplitudes to spaces with some dimensions compactified via periodic boundary conditions, providing two useful representations for different regimes and illustrating with multi-loop scalar field theory calculations.

Contribution

It introduces a novel extension of the Schwinger parametric representation for compactified spaces, applicable in different length regimes, and demonstrates its use with explicit multi-loop Feynman amplitude calculations.

Findings

Two valid representations for Feynman amplitudes in compactified spaces.

Explicit calculations of multi-loop Feynman amplitudes in scalar field theory.

Applicability near bulk and dimensional reduction regimes.

Abstract

We propose an extension of the Schwinger parametric representation for Feynman amplitudes in $D$ euclidean dimensions to a scenario where $d$ dimensions are compactified ($d<D$) through the introduction of periodic boundary conditions in space. We obtain two valid representations, one useful near the bulk (large compactification length) and another useful near the dimensional reduction (small compactification length). Also, to illustrate, we exhibit some Feynman amplitudes up to three loops in a compactified scalar field theory.