Wess-Zumino model
Encyclopedia
In theoretical physics
, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory
with supersymmetry
, at least in the Western world. In 1974, Julius Wess
and Bruno Zumino
studied, using modern terminology, dynamics of a single chiral superfield
(composed of a complex scalar
and a spinor
fermion
) whose cubic superpotential
leads to a renormalizable theory.
The Lagrangian of the free massless Wess–Zumino model in four-dimensional spacetime with flat metric
is
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
with supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
, at least in the Western world. In 1974, Julius Wess
Julius Wess
Julius Wess was an Austrian theoretical physicist noted as the co-inventor of the Wess–Zumino model and Wess–Zumino–Witten model in the field of supersymmetry...
and Bruno Zumino
Bruno Zumino
Bruno Zumino is an Italian theoretical physicist and emeritus faculty at the University of California, Berkeley. He got his bachelor degree from the University of Rome in 1945...
studied, using modern terminology, dynamics of a single chiral superfield
Chiral superfield
In theoretical physics, one often analyzes theories with supersymmetry in which chiral superfields play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using the notion of superspace...
(composed of a complex scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
and a spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
) whose cubic superpotential
Superpotential
Superpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
leads to a renormalizable theory.
The Lagrangian of the free massless Wess–Zumino model in four-dimensional spacetime with flat metric
is
-
with
a scalar field, 
a pseudoscalar field and 
a Dirac spinor field. The action is invariant under the transformations generated by the superalgebra. The infinitesimal form of these transformations is:
where
is a Majorana spinor-valued transformation parameter and 
is the chirality operator.
Invariance under a (modified) set of supersymmetry transformations remains if one adds mass terms for the fields, provided the masses are equal. It is also possible to add interaction terms under some algebraic conditions on the coupling constants, resulting from the fact that the interactions come from superpotentialSuperpotentialSuperpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
for the chiral superfieldChiral superfieldIn theoretical physics, one often analyzes theories with supersymmetry in which chiral superfields play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using the notion of superspace...
containing the fields
, 
and 
.
-
