Scaling dimensions of Coulomb branch operators of 4d N=2 superconformal field theories

Under reasonable assumptions about the complex structure of the set of singularities on the Coulomb branch of $\mathcal N=2$ superconformal field theories, we present a relatively simple and elementary argument showing that the scaling dimension, $\Delta$, of a Coulomb branch operator of a rank $r$ theory is allowed to take values in a finite set of rational numbers$\Delta\in \big[\frac{n}{m}\big|n,m\in\mathbb N, 0<m\le n, gcd(n,m)=1,\ \varphi(n)\le2r\big]$ where $\varphi(n)$ is the Euler totient function. The maximal dimension grows superlinearly with rank as $\Delta_\text{max} \sim r \ln\ln r$. This agrees with the recent result of Caorsi and Cecotti.