The Stueckelberg Field

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.

Explaining dark matter and B decay anomalies with an L μ − L τ model

“…The dark matter phenomenology is dictated by gauge interactions mediated by the massive new Z gauge boson. The mass of the Z can be generated either through a Stueckelberg mechanism, thereby leaving the L µ − L τ symmetry unbroken [44,45], or by a spontaneous symmetry breaking mechanism with the introduction of a scalar field, as described in ref. [14].…”

Localization of abelian gauge fields on thick branes

“…Here A M is the 5D gauge vector field and B is a dynamical scalar field (see [24] for a review of the Stueckelberg field).…”

Emergence of special and doubly special relativity

“…Similar field-enlarging transformations are also found useful in non-Abelian YangMills theories [64] or in Gravity [65].…”