Allowed Coulomb branch scaling dimensions of four-dimensional $\mathcal{N} = 2$ SCFTs

A basic datum of a rank-$r$ $\mathcal{N}{=}2$ superconformal field theory (SCFT) is the $r$-tuple of its Coulomb branch scaling dimensions, i.e., the scaling dimensions of a set of special protected scalar operators whose vevs generate the coordinate ring of the Coulomb branch of the theory. It is well known that when the coordinate ring is freely generated these scaling dimensions can only take values in a small set of rational numbers. But there are further constraints on which $r$-tuples of these numbers can appear. The main aim of this work is to clarify what these are. Along the way we also compute explicitly the $r$-tuples of allowed scaling dimensions for theories of ranks $r = 2, 3, 4$.