In
field theory, the Stueckelberg action (named after
Ernst Stueckelberg
Ernst Carl Gerlach Stueckelberg (baptised as Johann Melchior Ernst Karl Gerlach Stückelberg, full name after 1911: Baron Ernst Carl Gerlach Stueckelberg von Breidenbach zu Breidenstein und Melsbach; 1 February 1905 – 4 September 1984) was a S ...
) describes a massive spin-1 field as an R (the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s are the
Lie algebra of
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
)
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
coupled to a real
scalar field φ. This scalar field takes on values in a real 1D
affine representation In mathematics, an affine representation of a topological Lie group ''G'' on an affine space ''A'' is a continuous (smooth) group homomorphism from ''G'' to the automorphism group of ''A'', the affine group Aff(''A''). Similarly, an affine represen ...
of R with'' m'' as the
coupling strength.
:
This is a special case of the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other be ...
, where, in effect, and thus the mass of the Higgs scalar excitation has been taken to infinity, so the Higgs has decoupled and can be ignored, resulting in a nonlinear, affine representation of the field, instead of a
linear representation
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
— in contemporary terminology, a U(1) nonlinear -model.
Gauge-fixing φ=0, yields the
Proca action.
This explains why, unlike the case for non-abelian vector fields,
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
with a massive photon ''is'', in fact,
renormalizable
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
, even though it is not manifestly
gauge invariant
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
(after the Stückelberg scalar has been eliminated in the Proca action).
Stueckelberg extension of the Standard Model
The Stueckelberg extension of the Standard Model (''StSM)'' consists of a
gauge invariant
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
kinetic term for a massive
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
gauge field. Such a term can be implemented into the Lagrangian of the
Standard Model
without destroying the renormalizability of the theory and further provides a mechanism for
mass generation that is distinct from the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other be ...
in the context of
Abelian gauge theories.
The model involves a non-trivial
mixing of the Stueckelberg and the Standard Model sectors by including an additional term in the effective Lagrangian of the 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 of gauge boson dubbed
which inherits a very distinct narrow
decay width
Decay may refer to:
Science and technology
* Bit decay, in computing
* Software decay, in computing
* Distance decay, in geography
* Decay time (fall time), in electronics
Biology
* Decomposition of organic matter
* Tooth decay (dental caries ...
in this model. The St sector of the StSM decouples from the SM in limit
.
Stueckelberg type couplings arise quite naturally in theories involving
compactifications of higher-dimensional
string theory, in particular, these couplings appear in the dimensional reduction of the ten-dimensional N = 1
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
coupled to
supersymmetric
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
Yang–Mills gauge fields in the presence of internal gauge fluxes. In the context of intersecting
D-brane
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polch ...
model building, products of U(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 of
Kaluza–Klein 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.
See also
*
Higgs mechanism#Affine Higgs mechanism
References
The edited PDF files of the physics course of Professor Stueckelberg, openly accessible, with commentary and complete biographical documents.Review:Stueckelberg Extension of the Standard Model and the MSSM* Boris Kors, Pran Nath
* Daniel Feldman, Zuowei Liu, Pran Nath
{{Quantum field theories
Gauge theories
Physics beyond the Standard Model
Symmetry
Theoretical physics