Holomorphy argument for superpotential non-renormalization

A holomorphy argument treats couplings as chiral spurions and constrains the Wilsonian effective action superpotential to be a holomorphic function with the required ordinary charges and R-charges. Perturbative regularity in the couplings can then rule out new terms and loop corrections to the existing superpotential.

For quadratic and cubic interactions, assigning the couplings chiral spurion transformations makes each monomial formally covariant. The ordinary $U(1)$ leaves $\theta$ neutral, while the R-symmetry gives it charge one and the superpotential charge two. The neutral ratio $g\Phi /m$ and the covariant factor $m{\Phi }^2$ give the symmetry-allowed holomorphic form $m{\Phi }^{2}f(g\Phi /m)$. Perturbative regularity and tree matching are additionally needed for superpotential non-renormalization; symmetry alone leaves an arbitrary function. Fixed numerical couplings need not enjoy these formal symmetries as actual field symmetries.

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 $\Phi$-${\Phi }^{†}$ 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 Seiberg's non-renormalization paper.

The holomorphic coupling is $\tau =\vartheta /(2\pi )+4\pi i/g^2$. Perturbative coefficients are independent of the topological angle $\vartheta$. A holomorphic function invariant under continuous shifts of $\mathrm{Re}\tau$ is constant by the Cauchy-Riemann equations, hence a perturbative local superpotential coefficient cannot depend on $\tau$. Terms in $e^{2\pi i\tau }$ can evade this argument nonperturbatively. This does not forbid renormalization of the separate gauge kinetic F-term, or wave-function renormalization from the Kähler potential.