= Spurion selection rule for perturbative non-renormalization
{title2=$W_{\mathrm{eff}}^{\mathrm{pert}}=W_{\mathrm{tree}}$}
In the local <Wilsonian effective action>, retain the elementary <chiral superfields> and a nonzero infrared cutoff. Promote each coefficient in $W=\sum_A\lambda_A\mathcal O_A(\Phi)$ to a chiral <spurion>. Holomorphy excludes $\bar\lambda_A$. Assigning $R(\Phi_i)=0$ and $R(\lambda_A)=2$ forces a regular perturbative term of <R-charge> two to be linear in these <spurions>. Ordinary flavor charges constrain its field monomial. The <perturbative gauge-coupling independence of the Wilsonian superpotential> removes gauge-dependent coefficients. With the gauge coupling zero, a single holomorphic vertex cannot close a loop through free $\Phi$-$\Phi^\dagger$ propagators; extra vertices would violate the spurion degree or holomorphy. Its coefficient is therefore the tree coefficient. Gauge anomalies of the formal <R-symmetry> are tracked by a transforming holomorphic gauge coupling. Keeping the infrared cutoff matters: singular one-particle-irreducible terms and elimination of entire massive fields are different operations.
The background-field holomorphy method is discussed in https://arxiv.org/abs/hep-ph/9309335[Seiberg's non-renormalization paper].
Back to article page
