Goldstone Bosons and Convexity

We study the spectrum of scalar charged operators in Conformal Field Theories (CFTs) with a $U(1)$ global symmetry. The charged operators are dual, by the state-operator correspondence, to homogenous charged states on the sphere. Such states can break the $U(1)$ symmetry, and we define what we call the large $f$ regime in the CFT as one where the symmetry breaking scale is much higher than the scale of the CFT sphere. In such a regime, there is (an approximate) Goldstone boson associated to the breaking. We show that consistency of the Goldstone boson physics implies that the spectrum of states, and therefore of operators, must be convex in charge. More precisely, we show that any family of operators of different charges, which are lowest dimension of their charge, and which additionally share the same realisation of the Goldstone boson in terms of the degrees of freedom of the CFT, must be convex.