  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.
