Wess–Zumino Models

A Wess–Zumino model is a theory of chiral multiplets specified, at the two-derivative level, by a Kähler potential $K$ and a holomorphic superpotential $W$. With canonical $K$, the entire interaction is encoded in $W$: its first derivative determines the scalar potential, its second derivative determines fermion masses and Yukawa couplings, and its zeros identify supersymmetric classical vacua. Around every isolated supersymmetric vacuum, the complex scalar and Weyl fermion have equal mass. This makes the model the cleanest laboratory for component reduction, spontaneous supersymmetry breaking, and the precise content of perturbative nonrenormalization.

Required background. Supersymmetric Action Principles and Component Reduction derives the component formula and the off-shell transformation laws used below.

Helpful background. The 1PI Effective Action and Mean-Field Equations is needed to distinguish a Wilsonian superpotential statement from a claim about physical 1PI vertices.

The model is fixed by K and W

For chiral fields $\Phi^i$, define

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

The model card must state:

• the number of chiral multiplets and any flavor or discrete symmetries;
• the target patch and positive Kähler metric $g_{i\bar j}=K_{i\bar j}$;
• the holomorphic function $W$ and its parameter dimensions and phases;
• the spacetime signature and reality condition;
• the vacuum about which perturbation theory is performed; and
• the regulator and Wilsonian or 1PI destination of any quantum claim.

For canonical $K=\sum_i\Phi^{\dagger\bar i}\Phi^i$, eliminating the auxiliary fields gives

$$\begin{aligned} \mathcal L={}& \partial_\mu\phi^{*\bar i}\partial^\mu\phi^i +i\bar\psi^{\bar i}\bar\sigma^\mu\partial_\mu\psi^i -\sum_i|W_i|^2\\ &-\frac12W_{ij}\psi^i\psi^j -\frac12\bar W_{\bar i\bar j} \bar\psi^{\bar i}\bar\psi^{\bar j}. \end{aligned}$$

The auxiliary order parameter is $F^i=-\bar W_{\bar i}$. A Poincaré-invariant classical configuration preserves supersymmetry precisely when all $F^i=0$; then its energy density from this sector is zero. If the equations $W_i=0$ have no common solution in the allowed field space, supersymmetry is broken at tree level. This criterion assumes a positive Kähler metric and no gauge $D$ terms; both qualifications matter in later models.

The original four-dimensional interacting construction was given in Wess and Zumino 1974, pp. 52–54, while a modern component and superspace treatment is in Weinberg 2000, §26.4, pp. 75–82.

The cubic model has two supersymmetric vacua

Take one chiral superfield with

$$K=\Phi^\dagger\Phi, \qquad W(\Phi)=\frac12m\Phi^2+\frac13y\Phi^3,$$

where $[m]=1$ and $[y]=0$. Field rephasings can remove one phase, but the phase convention must be fixed before comparing Yukawa or domain-wall formulas. The component interactions are

$$\begin{aligned} V(\phi)&=|m\phi+y\phi^2|^2,\\ \mathcal L_{\mathrm{Yukawa}} &=-\frac12(m+2y\phi)\psi\psi+\text{h.c.} \end{aligned}$$

For $m y\neq0$, the supersymmetric vacua are

$$\phi_0=0, \qquad \phi_1=-\frac{m}{y}.$$

Let $\phi=\phi_a+\varphi$ around either vacuum. Since $W'(\phi_a)=0$,

$$W'(\phi_a+\varphi) =M_a\varphi+y\varphi^2, \qquad M_a=W''(\phi_a)=m+2y\phi_a.$$

Thus $M_0=m$ and $M_1=-m$. The quadratic Lagrangian is

$$\mathcal L^{(2)} =\partial_\mu\varphi^*\partial^\mu\varphi +i\bar\psi\bar\sigma^\mu\partial_\mu\psi -|M_a|^2|\varphi|^2 -\frac12M_a\psi\psi -\frac12M_a^*\bar\psi\bar\psi.$$

The two real scalar degrees of freedom and the two fermionic degrees of freedom all have mass $|m|$. The sign change in $M_a$ is removable by a local fermion phase and does not change the pole mass. This spectrum is a more reliable convention check than comparing an isolated sign in the component Lagrangian.

The model also has a finite-tension wall sector connecting the two vacua, but existence, boundary conditions, central charge, and stability belong to BPS Solitons, Walls, Strings, and Defects. The present page establishes only the vacuum data exported to that analysis.

General mass matrices follow from the Hessian of W

At a supersymmetric vacuum $\phi_0$ of a canonical multi-field model, define

$$M_{ij}=W_{ij}(\phi_0).$$

The fermion mass term is $-M_{ij}\psi^i\psi^j/2+\text{h.c.}$, while the scalar quadratic form is

$$V^{(2)} =\delta\phi^i(M^\dagger M)_{i\bar j} \delta\phi^{*\bar j}.$$

A Takagi factorization $M=U^{\mathsf T}\operatorname{diag}(m_a)U$ with $m_a\geq0$ displays the chiral multiplets directly: each singular value $m_a$ is both a Weyl-fermion mass and a complex-scalar mass. Zero singular values indicate flat directions at quadratic order, not automatically exact moduli; higher powers of $W$ or quantum effects may lift them.

At a nonsupersymmetric stationary point, the scalar Hessian also contains terms proportional to $W_{ijk}\bar W_{\bar k}$. Boson–fermion mass degeneracy need not hold, and stationarity $\partial_iV=0$ is weaker than $W_i=0$. Confusing those conditions is a common error in supersymmetry-breaking models.

Symmetries constrain but do not finish the quantum argument

For the massless cubic theory $W=y\Phi^3/3$, one may assign $R(\Phi)=2/3$, so the superpotential has $R$ charge two. A quadratic mass term is incompatible with that continuous assignment, while discrete subgroups may remain. Such charge counting sharply restricts possible local terms, but it is not by itself a nonrenormalization proof: anomalous symmetries, spurionic transformation of couplings, locality, holomorphy, and the Wilsonian infrared cutoff must all be included.

Perturbatively, a supersymmetric regulator organizes the Wilsonian action so that loop corrections renormalize D-terms such as the Kähler potential while the superpotential is not perturbatively renormalized. Physical couplings can still run through wavefunction renormalization. The 1PI functional can also contain infrared-sensitive nonlocal terms that admit misleading chiral representations. The theorem, its assumptions, and those counterexamples are owned by Nonrenormalization Theorems: Wilsonian and 1PI Scope. Weinberg 2000, §27.6, pp. 148–154 gives a perturbative treatment in a convention that must be translated to the present sigma matrices.

Noncanonical K changes the metric, not holomorphy of W

For a positive Kähler metric $g_{i\bar j}$, the auxiliary equations become

$$F^i=-g^{i\bar j}\bar W_{\bar j}+\text{fermion terms},$$

and the bosonic potential is

$$V_F=g^{i\bar j}W_i\bar W_{\bar j}.$$

The vacuum condition is still $W_i=0$ when the metric is nonsingular, but physical masses require canonical normalization with $g_{i\bar j}(\phi_0)$. A singular metric can invalidate a naive inference from $W$ alone. The full connection, curvature, and four-fermion terms are derived on Kähler Sigma Models.

Controlled uses and limits

The Wess–Zumino model is especially useful for:

• checking superspace and component signs;
• separating off-shell and on-shell closure;
• studying vacuum equations and elementary F-term breaking;
• testing supergraph power counting and wavefunction renormalization; and
• providing a local description near weakly coupled chiral fixed points.

It does not establish that every supersymmetric QFT has elementary chiral fields or a global Kähler potential. Nor does equality of perturbative masses prove nonperturbative stability, absence of solitons, or completeness of the particle description.

Common pitfalls

Solving $V'=0$ and calling every solution supersymmetric. Supersymmetry requires $W_i=0$, not merely stationarity of $V$.

Reading a fermion mass from $W$ instead of $W''$. The quadratic fermion operator is the Hessian evaluated at the chosen vacuum. Changing the vacuum can change its phase and rank.

Calling the superpotential a physical observable. $W$ is part of a field-coordinate and normalization description. Pole masses and S-matrix elements require the Kähler metric, wavefunction renormalization, and the appropriate quantum functional.

Exercises

1. A linear model. Let $W=f\Phi$ with canonical $K$. Determine the classical vacuum energy and whether supersymmetry is preserved.

Solution

$W'=f$ has no zero when $f\neq0$. The auxiliary solution is $F=-f^*$ and $V=|f|^2$, independent of $\phi$. Supersymmetry is broken and the classical scalar direction is flat. The model alone does not stabilize that direction; boundary conditions or additional interactions are needed to define a vacuum state globally.

2. Check the cubic vacua. Show that the scalar masses agree at the two vacua even though $W''$ changes sign.

Solution

At $0$, $W''=m$; at $-m/y$, $W''=-m$. In both cases the scalar quadratic term is $|W''|^2|\varphi|^2=|m|^2|\varphi|^2$. The fermion pole mass is likewise $|W''|=|m|$. The sign is a phase convention, while the singular value is invariant.

Where the model leads

Supergraphs, D-Algebra, and Quantum Effective Actions computes its loop organization. Supersymmetry Breaking and Controlled Deformations uses multi-field descendants to study stabilized breaking, and Supersymmetric Vacua, Moduli Geometry, and BPS Sectors studies the geometric and solitonic consequences of its vacuum equations.

References

• Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000, §§26.4 and 27.6, pp. 75–82 and 148–154. DOI.
• Wess, Julius, and Bruno Zumino. “A Lagrangian Model Invariant under Supergauge Transformations.” Physics Letters B 49, no. 1 (1974): 52–54. DOI.

Further reading

• Wess, Julius, and Jonathan Bagger. Supersymmetry and Supergravity. 2nd ed. Princeton, NJ: Princeton University Press, 1992, chs. 5 and 9.