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Stueckelberg actionFrom Wikipedia, the free encyclopedia

In field theory, the Stueckelberg action (named after Ernst Stueckelberg, (1938), "Die Wechselwirkungskrfte i

der Elektrodynamik und in der Feldtheorie der Krfte",Helv. Phys. Acta.11: 225) describes a massive spin-1

field as an R (the real numbers are the Lie algebra ofU(1)) Yang-Mills theory coupled to a real scalar field. Thi

scalar field takes on values in a real 1D affine representation ofR with m as the coupling strength.

This is a special case ofthe Higgs mechanism, where, in effect, the mass ofthe Higgs scalar excitation has been

taken to infinity, so the Higgs has decoupled and is ignorable, resulting in a nonlinear, affine representation ofthe

field, instead ofa linear representation in contemporary terminology, a U(1) nonlinear-model.

Gauge-fixing=0, yields the Proca action.

This explains why, unlike the case for non-abelian vector fields, quantum electrodynamics with a massive photon iin fact, renormalizable, even though it is not manifestly gauge invariant (after the Stckelberg scalar has been

eliminated in the Proca action).

The Stueckelberg Extension ofthe Standard Model

The Stueckelberg Lagrangian ofthe StSM(Stueckelberg extension ofthe Standard Model) consists ofa gauge

invariant kinetic term for a massive U(1) gauge field. Such a term can be implemented into the Lagrangian ofthe

Standard Model without destroying the renormalizability ofthe theory and further provides a mechanism for mass

generation that is distinct fromthe Higgs mechanism in the context ofAbelian gauge theories.

The model involves a non-trivial mixing ofthe Stueckelberg and the Standard Model sectors by including an

additional term in the effective Lagrangian ofthe Standard Model given by

The first term above is the Stueckelberg field strength, and are topological mass parameters and is the

axion. After symmetry breaking in the electroweak sector the photon remains massless. The model predicts a new

type ofgauge boson dubbed which inherits a very distinct narrow decay width in this model. The St sector o

the StSM decouples fromthe SM in limit .

Stueckelberg type couplings arise quite naturally in theories involving compactifications ofhigher dimensional string

theory, in particular, these couplings appear in the dimensional reduction ofthe ten dimensional N = 1 supergravity

coupled to supersymmetric Yang-Mills gauge fields in the presence ofinternal gauge fluxes. In the context of

intersecting D-brane model building, products ofU(N) gauge groups are broken to their SU(N) subgroups via the

Stueckelberg couplings and thus the Abelian gauge fields become massive. Further, in a much simpler fashion one

may consider a model with only one extra dimension (a type ofKaluzaKlein model) and compactify down to a

four dimensional theory. The resulting Lagrangian will contain massive vector gauge bosons that acquire masses

through the Stueckelberg mechanism.

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See also

Higgs mechanism #Affine Higgs mechanism

References

The edited PDF files ofthe physics course ofProfessor Stueckelberg, openly accessible, with commentaryand complete biographical documents. (http://www.stuckelberg.org/)

Review:Stueckelberg Extension ofthe Standard Model and the MSSM (http://arxiv.org/abs/hep-

ph/0503208)

Boris Kors, PranNath: [1] (http://arxiv.org/abs/hep-ph/0402047), [2] (http://arxiv.org/abs/hep-

ph/0406167), [3] (http://arxiv.org/abs/hep-ph/0503208)

Daniel Feldman, Zuowei Liu, Pran Nath: [4] (http://arxiv.org/abs/hep-ph/0603039), [5]

(http://arxiv.org/abs/hep-ph/0606294)

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