The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry,[2] and to some extent of Tong.[3]
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.
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.
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)
Wess–Zumino action (preliminary treatment, alternative expression)
Upon substituting in , one finds that this is a theory with a massive complex scalar and a massive Majorana spinor of the same mass. The interactions are a cubic and quartic interaction, and a Yukawa interaction between and , which are all familiar interactions from courses in non-supersymmetric quantum field theory.
Superspace consists of the direct sum of Minkowski space with 'spin space', a four dimensional space with coordinates , where are indices taking values in More formally, superspace is constructed as the space of right cosets of the Lorentz group in the super-Poincaré group.
The fact there is only 4 'spin coordinates' means that this is a theory with what is known as supersymmetry, corresponding to an algebra with a single supercharge. The dimensional superspace is sometimes written , and called super Minkowski space. The 'spin coordinates' are so called not due to any relation to angular momentum, but because they are treated as anti-commuting numbers, a property typical of spinors in quantum field theory due to the spin statistics theorem.
A superfield is then a function on superspace, .
Defining the supercovariant derivative
a chiral superfield satisfies The field content is then simply a single chiral superfield.
However, the chiral superfield contains fields, in the sense that it admits the expansion
with Then can be identified as a complex scalar, is a Weyl spinor and is an auxiliary complex scalar.
These fields admit a further relabelling, with and This allows recovery of the preliminary forms, after eliminating the non-dynamical using its equation of motion.
The massless Wess–Zumino model admits a larger set of symmetries, described at the algebra level by the superconformal algebra. As well as the Poincaré symmetry generators and the supersymmetry translation generators, this contains the conformal algebra as well as a conformal supersymmetry generator .
The conformal symmetry is broken at the quantum level by trace and conformal anomalies, which break invariance under the conformal generators for dilatations and for special conformal transformations respectively.
The R-symmetry of supersymmetry holds when the superpotential is a monomial. This means either , so that the superfield is massive but free (non-interacting), or so the theory is massless but (possibly) interacting.
The action generalizes straightforwardly to multiple chiral superfields with . The most general renormalizable theory is
where the superpotential is
,
where implicit summation is used.
By a change of coordinates, under which transforms under , one can set without loss of generality. With this choice, the expression is known as the canonical Kähler potential. There is residual freedom to make a unitary transformation in order to diagonalise the mass matrix .
When , if the multiplet is massive then the Weyl fermion has a Majorana mass. But for the two Weyl fermions can have a Dirac mass, when the superpotential is taken to be
This theory has a symmetry, where rotate with opposite charges
If renormalizability is not insisted upon, then there are two possible generalizations. The first of these is to consider more general superpotentials. The second is to consider in the kinetic term
to be a real function of and .
The action is invariant under transformations : these are known as Kähler transformations.
Considering this theory gives an intersection of Kähler geometry with supersymmetric field theory.
By expanding the Kähler potential in terms of derivatives of and the constituent superfields of , and then eliminating the auxiliary fields using the equations of motion, the following expression is obtained:
where
is the Kähler metric. It is invariant under Kähler transformations. If the kinetic term is positive definite, then is invertible, allowing the inverse metric to be defined.