The Stueckelberg Field

TL;DR

Stueckelberg’s mechanism introduces a compensating scalar to restore gauge invariance for a massive vector field, enabling a BRST-invariant, renormalizable Abelian theory and providing a framework to massify the U(1)_Y sector of the electroweak theory while preserving BRST structure. The paper analyzes the Abelian Proca vs Stueckelberg formulations, develops BRST invariance, and extends to the electroweak theory with a massive photon, detailing current structure, anomalies, and phenomenology. It also surveys the historical influence of Stueckelberg’s ideas on renormalization, hidden symmetries, and non-Abelian generalizations, noting that while Stueckelberg-inspired models yield insights and infrared regulators, the Higgs mechanism remains the established renormalizable route for non-Abelian vector masses.

Abstract

In 1938, Stueckelberg introduced a scalar field which makes an Abelian gauge theory massive but preserves gauge invariance. The Stueckelberg mechanism is the introduction of new fields to reveal a symmetry of a gauge--fixed theory. We first review the Stueckelberg mechanism in the massive Abelian gauge theory. We then extend this idea to the standard model, stueckelberging the hypercharge U(1) and thus giving a mass to the physical photon. This introduces an infrared regulator for the photon in the standard electroweak theory, along with a modification of the weak mixing angle accompanied by a plethora of new effects. Notably, neutrinos couple to the photon and charged leptons have also a pseudo-vector coupling. Finally, we review the historical influence of Stueckelberg's 1938 idea, which led to applications in many areas not anticipated by the author, such as strings. We describe the numerous proposals to generalize the Stueckelberg trick to the non-Abelian case with the aim to find alternatives to the standard model. Nevertheless, the Higgs mechanism in spontaneous symmetry breaking remains the only presently known way to give masses to non-Abelian vector fields in a renormalizable and unitary theory.