  The Charge Convexity Conjecture (CCC) states that in a unitary conformal
field theory in $d\geq 3$ dimensions with a global symmetry, the minimal
dimension of operators in certain representations of the symmetry, as a
function of the charge $q$ of the representation (or a generalized notion of
it), should be convex. More precisely, this was conjectured to be true when $q$
is restricted to positive integer multiples of some integer $q_0$. The CCC was
tested on a number of examples, most of which are in $d<4$ dimensions, and its
version in which $q_0$ is taken to be the charge of the lowest-dimension
positively-charged operator was shown to hold in all of them.
  In this paper we test the conjecture in a non-trivial example of a $d=4$
theory, which is the family of Caswell-Banks-Zaks IR fixed points of $SU(N_c)$
gauge theory coupled to $N_f$ massless fermions and $N_s$ massless scalars. In
these theories, the lowest-dimension gauge-invariant operators that transform
non-trivially under the global symmetry are mesons. These may consist of two
scalars, two fermions or one of each.
  We find that the CCC holds in all applicable cases, providing significant new
evidence for its validity, and suggesting a stronger version for non-simple
global symmetry groups.
