Wess–Zumino model

Not to be confused with Wess–Zumino–Witten model.

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 ${\mathrm {diag}}(-1,1,1,1)$ is

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

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

\delta _{{\epsilon }}S={\bar {\epsilon }}\psi

\delta _{{\epsilon }}P={\bar {\epsilon }}\gamma _{{5}}\psi

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

where $\epsilon$ is a Majorana spinor-valued transformation parameter and $\gamma _{{5}}$ 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 $S$ , $P$ and $\psi$ .

References

Figueroa-O'Farrill, J. M. (2001). "Busstepp Lectures on Supersymmetry". arXiv:hep-th/0109172.
Wess, J.; Zumino, B. (1974). "Supergauge transformations in four dimensions". Nuclear Physics B. 70 (1): 39–50. Bibcode:1974NuPhB..70...39W. doi:10.1016/0550-3213(74)90355-1.

Quantum field theories

Standard	Chern–Simons model Chiral model Kondo model Conformal field theory Local quantum field theory Noncommutative quantum field theory Non-linear sigma model Quartic interaction Quantum chromodynamics Quantum electrodynamics Quantum hadrodynamics Quantum Yang–Mills theory Schwinger model sine-Gordon Standard Model String theory Toda field theory Topological quantum field theory Wess–Zumino model Wess–Zumino–Witten model Yang–Mills theory Yang–Mills–Higgs model Yukawa model Thirring–Wess model

Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model

Related	History of quantum field theory Axiomatic quantum field theory Quantum field theory in curved spacetime

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