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.

The Stueckelberg Field

Superfield formulation of extended BRS symmetry

The principle of BRS symmetry: An alternative approach to Yang-Mills theories

Slavnov transformations and supersummetry

The QCD running coupling

We discuss renormalization for a class of Lagrangians containing an explicit mass term for the vector field and a four-vertex for the Faddeev-Popov ghost field. The action is shown to be invariant under nonlinear gauge transformations, which are a generalization of those introduced by Becchi, Rouet and Stora. This invariance guarantees the validity of the Slavnov-Taylor identities, which are then used to discuss the renormalization of the theory. The problem of unitarity is not discussed here. Canonical and Lorentz-covariant formalism is employed.

G. Curci

Papers

On a class of Lagrangian models for massive and massless Yang-mills fields