A Note on Bounds of Scalar Operators in Perturbative SCFTs

Bounds on anomalous dimensions of scalar operators in 4d superconformal field theory are explored through perturbative viewpoint. Following the recent work of Green and Shih, in which a conjecture involved this issue is verified at the NLO, we consider the NNLO corrections to the bounds, which are important in some situations and can be divided into two cases where $\mathcal{O}(\lambda^{4})$ or $\mathcal{O}(y^2)$ effects dominate respectively. In the former case, we find that the conjecture is maintained at NNLO, while in the later case, the statement still holds due to the null corrections arising from $\mathcal{O}(y^{2})$.