Stueckelberg FieldEdit
The Stueckelberg field is a real scalar field introduced to preserve gauge invariance when a gauge boson is given a mass within an abelian gauge theory. Named after Ernst Stückelberg, it provides a minimal, gauge-invariant way to endow a vector field with mass without invoking a Higgs-like mechanism in the simplest U(1) context. In its standard form, the mechanism uses a gauge-invariant combination of the gauge field Aμ and the derivative of the scalar ϕ, so that the would-be mass term respects the underlying symmetry. In a particular gauge, the scalar can be gauged away, leaving a massive vector boson with three physical polarizations; in other gauges, the scalar persists as a compensating degree of freedom that ensures unitarity and gauge invariance.
Historically, the Stueckelberg construction emerged in the 1930s as a way to reconcile a massive vector field with the principles of gauge symmetry, before the widespread adoption of the Higgs mechanism in the Standard Model. The approach was recognized as a useful tool in situations where one needed to introduce mass for a gauge boson while keeping the theory renormalizable in a minimal sense, particularly for abelian gauge theories. The idea has since found its place in modern model-building, especially in theories with extra U(1) gauge factors or hidden sectors, where a Stueckelberg mass term can be introduced without necessarily populating the theory with a new scalar that acquires a vacuum expectation value.
Historical background
The foundational idea is that a gauge-invariant mass term for a gauge field can be generated by coupling the gauge field to a scalar field that shifts under gauge transformations. This scalar, the Stueckelberg field (sometimes discussed as a compensator), transforms in such a way that the combination Aμ − (1/m)∂μσ stays invariant when Aμ → Aμ + ∂μα and σ → σ + mα. The result is a theory in which the gauge field acquires a mass m without breaking the local gauge symmetry in the manifest Lagrangian. This contrasts with the Higgs mechanism, where symmetry breaking and a physical scalar (the Higgs boson) play central roles in providing mass to gauge bosons of the electroweak sector.
In contemporary practice, the Stueckelberg idea is particularly useful when physicists want to introduce masses for extra abelian gauge fields, such as a hidden U(1) sector that communicates weakly with the Standard Model, without complicating the scalar sector. The mechanism also appears in string-inspired constructions and in discussions of kinetic mixing between U(1) gauge fields, where a mass term for a hidden photon can be generated in a gauge-invariant way via the Stueckelberg field.
Formalism and interpretation
At its core, the Stueckelberg mechanism augments an abelian gauge theory with a scalar field σ that transforms to compensate gauge transformations. The basic idea is to assemble the gauge field and the scalar into a gauge-invariant combination, ensuring that the theory contains a mass term for the gauge field without explicitly breaking gauge invariance. The resulting Lagrangian typically includes:
- A kinetic term for the gauge field FμνF^μν.
- A mass term that, through the Stueckelberg field, preserves gauge invariance.
- A kinetic term for the Stueckelberg field, which in some gauges can be removed or reinterpreted as providing the longitudinal polarization of the massive gauge boson.
In the unitary gauge, the Stueckelberg field can be gauged away, and the vector field appears as a massive particle with its three physical polarizations. In other gauges, the scalar degree of freedom remains explicit, illustrating how the longitudinal mode of the massive vector boson is distributed between the gauge field and the compensating scalar.
The Stueckelberg construction works cleanly for a single abelian gauge field. Extending it to non-abelian gauge theories introduces technical challenges: the straightforward non-abelian generalization can lead to issues with renormalizability and unitarity unless embedded in a broader framework. For this reason, the Stueckelberg approach is most transparent and widely used in abelian contexts or as a component in larger model-building efforts that also incorporate other symmetry-breaking mechanisms.
See also the Stueckelberg mechanism and Stueckelberg field entries for related discussions, including how these ideas interface with gauge theory and renormalization.
Non-abelian generalizations and limitations
While the abelian Stueckelberg method is conceptually straightforward, extending it to non-abelian gauge theories is subtler. A naive, straightforward extension can jeopardize renormalizability and complicate the ultraviolet behavior of the theory. In many physical applications, especially within the Standard Model, masses for non-abelian gauge bosons (like the W and Z) arise from spontaneous symmetry breaking through the Higgs mechanism rather than a direct Stueckelberg mass term. However, the Stueckelberg idea remains valuable when constructing theories with extra abelian factors or hidden sectors where one wants to provide a mass scale for a gauge boson without introducing a new scalar vev that couples to the visible sector.
In modern model-building, the Stueckelberg mechanism often appears in combination with kinetic mixing between U(1) factors. This arrangement can yield observable phenomenology such as a light hidden photon with small couplings to Standard Model fields, while keeping the visible sector relatively unmodified. The resulting frameworks are frequently discussed in connection with searches for new physics in accelerator experiments and precision tests, and they are linked to ideas about portals between the visible sector and hidden sectors.
Implications and applications
The Stueckelberg field has found practical use in several contexts:
- Extra U(1) gauge bosons: Providing mass to a new abelian gauge boson without a Higgs-like scalar in the hidden sector, preserving gauge invariance.
- Hidden sectors and dark portals: Generating mass terms for hidden photons and enabling kinetic mixing with the Standard Model gauge fields to produce potentially observable signatures.
- String theory and higher-dimensional models: Appearing in constructions where gauge fields acquire mass through compensating scalar degrees of freedom that arise naturally in the effective field theory.
- Phenomenological constraints: Precision measurements and collider searches place bounds on the mass and couplings of Stueckelberg-origin gauge bosons, especially when kinetic mixing with the hypercharge sector is present.
Throughout these applications, the central appeal for a segment of model-builders is the ability to introduce mass scales with minimal changes to the scalar sector, avoiding a proliferation of scalar particles in the visible spectrum while maintaining a consistent gauge-invariant framework. The approach remains complementary to the conventional Higgs mechanism and is frequently discussed alongside it in the broader landscape of beyond-Standard-Model theories.