Vanishing short multiplets in rank one 4d/5d SCFTs

We study the short multiplet spectrum in 4d $\mathcal{N}=2$ superconformal theories of low rank using the full superconformal indices and the selection rules from the superconformal representation theory. We find a universal expression for the leading terms for the superconformal index of rank one $H_0, H_1, H_2, D_4, E_6, E_7$ theories. From this result, we argue that certain short multiplets appear in the operator product expansions involving stress-tensor, conserved current, and Coulomb branch operator vanish. We also apply the same procedure to 5d superconformal theories and find that $E_1$ theory has vanishing short multiplets analogous to that of the $H_1$ theory.