Supersymmetric gauge actions and D terms

In Wess-Zumino gauge the surviving components are $A_\mu$, $\lambda_\alpha$, and D. The vector has four off-shell real components, one ordinary gauge redundancy, and the real D supplies one additional bosonic component, matching the four real Grassmann gaugino components. On shell a massless vector has two transverse polarizations, matched by its gaugino states. Wess-Zumino gauge removes extra superfield components but does not itself fix the remaining ordinary gauge freedom or impose the vector field equation.

A supersymmetry transformation generally leaves this superfield gauge slice. A compensating gauge transformation restores it, so the component algebra closes up to gauge transformations. This feature is a bookkeeping consequence of gauge fixing rather than a failure of supersymmetry. For an Abelian vector choose the chiral field strength $\mathcal W_\alpha=-\bar D^2D_\alpha V/4$. Its gauge-invariant action supplies the Maxwell term, gaugino kinetic term, and positive auxiliary square. We distinguish this spinor field strength from the scalar superpotential W by its calligraphic symbol and index.

For scalar backgrounds the D-dependent matter term is g D times the charge-weighted sum of scalar moduli squared. Add xi D and define $J=g\sum_iq_i|\phi_i|^2+\xi$. The auxiliary Lagrangian is one half D squared plus D J. Its equation gives D equal to minus J; substitution gives minus one half J squared. Thus the static potential is positive one half J squared. The sign of D is convention dependent, but its potential and the explicitly defined flatness equation are unambiguous together.

Unlike an electric field propagating through space, D has no derivative kinetic term and no independent particle polarization. It enforces a local relation between scalar backgrounds and a charge moment map. Oppositely charged fields contribute with opposite signs even though each modulus square is nonnegative. Their contributions can cancel. In a non-Abelian canonical theory the same construction uses Hermitian generators in each matter representation, one real auxiliary per generator, and a sum of auxiliary squares. An arbitrary FI term is not available for every simple non-Abelian generator.

A charged scalar expectation can Higgs the gauge field while preserving supersymmetry if both F and D vanish. Conversely, a nonzero D expectation in a genuine stable vacuum breaks rigid supersymmetry through the gaugino transformation. A nonzero value at an arbitrary background proves neither a stable vacuum nor a broken ground state. Finding the vacuum requires minimizing the full potential and respecting gauge equivalence. This distinction prevents a convenient charge assignment or one incomplete square from being mistaken for a full mechanism of symmetry breaking.

For a vectorlike pair with charges plus one and minus one, the Abelian cubic and mixed gravitational gauge anomalies cancel between the fermions. Such a pair is a useful consistent toy matter sector, though a realistic model may need additional fields and interactions. A positive FI parameter can be cancelled by the negatively charged scalar in our sign convention; reversing the FI sign reverses that role. The next lesson uses the zero-F and zero-D equations to identify inequivalent vacuum configurations rather than treating gauge-related field values as distinct vacua.