Wilsonian Superpotential Nonrenormalization in a Finite Supersymmetric EFT Basis

Let the rigid four-dimensional $\mathcal{N}=1$ Wilsonian effective-action data and finite operator basis be as in the action definition, with formal cutoff $0<\lambda <{\mathrm{\Lambda }}_0$. Let $F_{\lambda }(\mathrm{\Phi },W)$ denote the derivative-free, holomorphic chiral integrand of the Wilsonian action, and define its superpotential part by $W_{\lambda }(\mathrm{\Phi })=F_{\lambda }(\mathrm{\Phi },0)$. Fix a finite ordered basis of retained light-field superpotential operators and a scale-independent linear coordinate projection $P_{\mathrm{h}\mathrm{o}\mathrm{l}}$ onto that basis. Use the same unrescaled chiral fields, quotient relations, and operator coordinates at every scale. At constant coefficient backgrounds put $c_0=F_{{\mathrm{\Lambda }}_0}(0,0)$ and fix $F_{\lambda }(0,0)=c_0$ at every scale. Numerical constants have zero rigid F-action, so this fixes a density representative rather than imposing a Ward-identity consequence. Bare constant FI terms, if present, are kept fixed and are excluded from promotion to unconstrained real coefficient backgrounds. For a gauge sector, select the general F-term theorem's linear Lorenz BRST component graph completion, with its nonminimal doublet, homogeneous gauge/ghost kernels, fixed infrared inverses and zero-mode prescription, and fixed component/source coordinates; in an ungauged model the gauge/ghost sector is absent. Its regulator must satisfy that completion's spurion and Ward compatibility conditions, with the explicitly induced Wess–Zumino component variation, the fixed physical coefficient extraction, and the full gauge-fixed action on the change-of-variables Ward identity's right-hand side; gauge-fixing and ghost variations are retained. Assume the selected formal perturbative cutoff prescription has the local nondynamical-spurion extension and local derivative expansion specified in the action data; its local counterterms obey chirality as an identity in the coefficientwise formal background-jet algebra before constant-background evaluation, with independent first anti-chiral jets of each unconstrained real coefficient spurion; its perturbative vertices are one-particle-irreducible; its rigid supersymmetry and BRST/gauge Ward identities hold coefficientwise for every local insertion and their counterterms; and it has no local gauge anomaly.

Adopt full Axiom of Choice as a sufficient ambient assumption for the selected action data; the regulated equivalence supplier requires no additional full-Choice premise. Then, coefficientwise at every perturbative order,

$P_{\mathrm{h}\mathrm{o}\mathrm{l}}{W}_{\lambda }=P_{\mathrm{h}\mathrm{o}\mathrm{l}}{W}_{{\mathrm{\Lambda }}_0}\mathrm{.}$

Thus every retained holomorphic superpotential coefficient in the fixed unrescaled field basis is unchanged by the formal Wilsonian cutoff evolution. The separately displayed holomorphic gauge coupling may receive a one-loop correction but no higher-loop correction. The real D-term function and Kähler data may change. This does not constrain arbitrary one-particle-irreducible vertices, coefficients after a scale-dependent field rescaling, nonperturbative corrections, or a continuum interacting quantum field theory.

Finite $N=1$ Wilsonian action and fixed EFT coefficient basis

General formal perturbative $N=1$ Wilsonian F-term theorem

Weinberg, equations (1)–(12)

Martin, §§4.7, 4.10, and 6.5