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 Lagrangian of the free massless Wess–Zumino model in four-dimensional spacetime with flat metric \( {\mathrm {diag}}(-1,1,1,1) i \) is

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

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:

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

where ϵ {\displaystyle \epsilon } \epsilon is a Majorana spinor-valued transformation parameter and \( \gamma _{{5}} i \) 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 \( \psi \) .

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

Chern–Simons Conformal field theory Ginzburg–Landau Kondo effect Local QFT Noncommutative QFT Quantum Yang–Mills Quartic interaction sine-Gordon String theory Toda field Topological QFT Yang–Mills Yang–Mills–Higgs


Chiral Non-linear sigma Schwinger Standard Model Thirring–Wess Wess–Zumino Wess–Zumino–Witten Yukawa

Four-fermion interactions


BCS theory Fermi's interaction Luttinger liquid Top quark condensate


Gross–Neveu Hubbard Nambu–Jona-Lasinio Thirring Thirring–Wess


History Axiomatic QFT Loop quantum gravity Loop quantum cosmology QFT in curved spacetime Quantum chaos Quantum chromodynamics Quantum dynamics Quantum electrodynamics
links Quantum gravity
links Quantum hadrodynamics Quantum hydrodynamics Quantum information Quantum information science
links Quantum logic Quantum thermodynamics

Physics Encyclopedia



Hellenica World - Scientific Library

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License