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Concept
Wess–Zumino model
Summary
In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry. 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 treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry, and to some extent of Tong.
The model is an important model in supersymmetric quantum field theory. It is arguably the simplest supersymmetric field theory in four dimensions, and is ungauged.
In a preliminary treatment, the theory is defined on flat spacetime (Minkowski space). For this article, the metric has mostly plus signature. The matter content is a real scalar field , a real pseudoscalar field , and a real (Majorana) spinor field .
This is a preliminary treatment in the sense that the theory is written in terms of familiar scalar and spinor fields which are functions of spacetime, without developing a theory of superspace or superfields, which appear later in the article.
The Lagrangian of the free, massless Wess–Zumino model is
where
The corresponding action is
Supersymmetry is preserved when adding a mass term of the form
Supersymmetry is preserved when adding an interaction term with coupling constant :
The full Wess–Zumino action is then given by putting these Lagrangians together:
There is an alternative way of organizing the fields. The real fields and are combined into a single complex scalar field while the Majorana spinor is written in terms of two Weyl spinors: . Defining the superpotential
the Wess–Zumino action can also be written (possibly after relabelling some constant factors)
Upon substituting in , one finds that this is a theory with a massive complex scalar and a massive Majorana spinor of the same mass.
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