Wess–Zumino Model

The Wess–Zumino model is the simplest interacting four-dimensional theory of chiral supermultiplets. Supersymmetry ties the scalar potential, fermion masses, and Yukawa interactions to derivatives of one holomorphic function. It makes those constraints concrete without adding gauge fields.

One chiral multiplet

A chiral multiplet $\Phi$ contains a complex scalar $\phi$, a Weyl fermion $\psi$, and a nondynamical auxiliary field $F$. Consider canonical Kähler potential $K=\Phi^\dagger\Phi$ and superpotential

$$W(\Phi)=\frac m2\Phi^2+\frac\lambda3\Phi^3.$$

Here $m$ has mass dimension one and $\lambda$ is dimensionless. The superspace action is

$$S=\int\mathrm d^4x\left[ \int\mathrm d^4\theta\,\Phi^\dagger\Phi +\left(\int\mathrm d^2\theta\,W(\Phi)+\mathrm{h.c.}\right) \right].$$

The Grassmann integrals select the supersymmetric component terms. Eliminating $F$ gives $F=-\overline{W'(\phi)}$, a positive scalar potential $V=\lvert W'(\phi)\rvert^2$, and fermion interaction $-\tfrac12W''(\phi)\psi\psi+\mathrm{h.c.}$ The coefficient and conjugate are essential: scalar and fermion couplings cannot be chosen independently. See Weinberg 2000, §26.4.

Vacua and a mass-degeneracy check

Assume $m\lambda\ne0$. Supersymmetric classical vacua solve

$$W'(\phi)=m\phi+\lambda\phi^2=0,$$

so $\phi_0=0$ and $\phi_0=-m/\lambda$ both have zero potential energy. Near either vacuum, $\phi=\phi_0+\delta\phi$ gives

$$V=\lvert W''(\phi_0)\rvert^2\lvert\delta\phi\rvert^2 +O(\delta\phi^3).$$

Since $W''=m+2\lambda\phi$, its values at the two vacua are $m$ and $-m$. The scalar mass squared is $\lvert m\rvert^2$ and the fermion mass magnitude is $\lvert m\rvert$ in either case. The complex scalar supplies two real bosonic degrees of freedom, matching the two fermionic on-shell degrees of freedom.

What supersymmetry does and does not determine

Vacuum equations, the spectrum near a chosen vacuum, and scattering among multiplet components are useful observables. Perturbation theory at small $\lambda$ gives an explicit laboratory for supersymmetric cancellations.

The perturbative nonrenormalization statement concerns the Wilsonian holomorphic superpotential under its stated assumptions. Wavefunction normalization and the Kähler potential can receive corrections, so physical couplings and amplitudes are not all unrenormalized. Weinberg 2000, §27.6 distinguishes these claims.

This single-field example has supersymmetric vacua; it is not a demonstration of spontaneous supersymmetry breaking. More fields or different superpotentials can change the vacuum equations, and a runaway direction must be distinguished from a stable broken-symmetry vacuum.

Develop the calculation

Wess–Zumino Models develops the general multi-field case. Supersymmetric Action Principles and Component Reduction explains the superspace integrals. Continue to Supersymmetric QCD when vector multiplets and strong gauge dynamics are added.

References

• Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge University Press, 2000, §§26.4 and 27.6. DOI.