Clarify which assumptions of non-renormalization theorems fail with a nilpotent operator

Identify precisely which assumptions or steps of the standard four-dimensional N=1 non-renormalization theorems (as applied to argue Higgs-branch non-renormalization when the gauge coupling resides in a background vector multiplet) fail in the five-dimensional N=1 setting considered here, where a nilpotent gaugino bilinear S induces perturbative corrections to the Higgs-branch chiral ring along the RG flow.

Background

The results of the paper indicate that the Higgs-branch chiral ring is corrected along the RG flow due to the gaugino bilinear S (at finite coupling) and instantons (at infinite coupling), contradicting the common expectation of Higgs-branch non-renormalization.

The authors explicitly state that it is not clear which assumptions of the non-renormalization theorems break down in the presence of a nilpotent operator like S and point to the need to resolve this conceptual tension.