Wess-Zumino model and superpotentials

The canonical action consists of the full superspace integral of $\Phi^\dagger\Phi$ and the chiral integral of W plus its conjugate. A renormalizable single-field superpotential is at most cubic in four dimensions. Choose $W=m\Phi^2/2+\lambda\Phi^3/3$, with mass parameter m and dimensionless coupling lambda. The factors one half and one third are conventions chosen so that its first derivative is m phi plus lambda phi squared. A source using a cubic factor one sixth has a differently normalized coupling.

The chiral highest component yields the auxiliary coupling and a fermion bilinear with coefficient minus one half times the second derivative of W. Eliminating F gives the positive modulus-squared potential. The same lambda consequently controls the quartic scalar interaction and the Yukawa interaction, with related cubic scalar terms when m is nonzero. Adjusting only one of these component couplings would generally break supersymmetry. This is stronger than requiring matching free-particle masses; interactions must also fit the common superfield construction.

For a canonical ungauged model, a supersymmetric classical vacuum has vanishing first derivative of W. When both m and lambda are nonzero, the cubic example has two roots: zero and minus m over lambda. Both have zero classical potential. Expand around either root as phi equals its vacuum value plus a complex fluctuation. The quadratic potential is the modulus squared of the second derivative at that root times the fluctuation modulus squared. Its two real scalar components therefore have the same mass magnitude as the Weyl fermion.

Writing the complex fluctuation as $(a+ib)/\sqrt2$ gives canonically normalized real kinetic terms and a potential with one half times mass squared multiplying each real square. Forgetting this square-root-two convention can introduce a spurious factor two in a scalar mass. The sign or complex phase of the fermionic mass coefficient can change between vacua without making a physical mass negative. At a supersymmetric root the scalar Hessian and fermion mass magnitude agree; at a general background additional terms involving the nonzero first derivative can split scalar curvatures.

Holomorphy means W depends on chiral fields rather than their conjugates. The scalar potential is nevertheless a real function of both phi and its conjugate because it is a modulus square. A real polynomial potential cannot in general be declared supersymmetric merely by naming it a superpotential. The distinction between W and V is essential for finding vacua, couplings, and dimensions. A constant term in W drops out of this rigid nongravitational component dynamics, although that statement changes when gravity is coupled.

The two-vacuum result assumes the entire complex scalar field is available and no additional interactions or constraints modify the model. Restricting the field by hand to a real line can discard physical solutions in other examples. Quantum corrections may renormalize the Kähler metric even when perturbative superpotential corrections are absent, a distinction developed later. Our calculation is thus a classical controlled illustration of related interactions and vacuum degeneracy. It is neither a realistic particle spectrum nor evidence that supersymmetry describes a measured set of elementary particles.