# Wess–Zumino model

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In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with 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 Lagrangian of the free massless Wess–Zumino model in four-dimensional spacetime with flat metric ${\displaystyle \mathrm {diag} (-1,1,1,1)}$ is

${\displaystyle {\mathcal {L}}=-{\frac {1}{2}}(\partial S)^{2}-{\frac {1}{2}}(\partial P)^{2}-{\frac {1}{2}}{\bar {\psi }}\partial \!\!\!/\psi }$

with ${\displaystyle S}$ a scalar field, ${\displaystyle P}$ a pseudoscalar field and ${\displaystyle \psi }$ a Majorana spinor field. The action is invariant under the transformations generated by the superalgebra. The infinitesimal form of these transformations is:

${\displaystyle \delta _{\epsilon }S={\bar {\epsilon }}\psi }$
${\displaystyle \delta _{\epsilon }P={\bar {\epsilon }}\gamma _{5}\psi }$
${\displaystyle \delta _{\epsilon }\psi =\partial \!\!\!/(S+P\gamma _{5})\epsilon }$

where ${\displaystyle \epsilon }$ is a Majorana spinor-valued transformation parameter and ${\displaystyle \gamma _{5}}$ 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 superpotential for the chiral superfield containing the fields ${\displaystyle S}$, ${\displaystyle P}$ and ${\displaystyle \psi }$.