Compactification of gauge theories and the gauge invariance of massive modes

TL;DR

This paper investigates how gauge invariance is maintained for massive modes in compactified gauge theories from five to four dimensions, highlighting the natural emergence of St"uckelberg fields in the process.

Contribution

It demonstrates that St"uckelberg fields naturally arise during compactification, differing from traditional methods where they are artificially introduced, especially in non-Abelian theories.

Findings

St"uckelberg fields appear naturally in compactification

Gauge invariance of massive modes is preserved

Differences in St"uckelberg implementation for non-Abelian theories

Abstract

We study the gauge invariance of the massive modes in the compactification of gauge theories from D=5 to D=4. We deal with Abelian gauge theories of rank one and two, and with non-Abelian ones of rank one. We show that St\"uckelberg fields naturally appear in the compactification mechanism, contrarily to what usually occurs in literature where they are introduced by hand, as a trick, to render gauge invariance for massive theories. We also show that in the non-Abelian case they appear in a very different way when compared with their usual implementation in the non-Abelian Proca model.