Do conserved currents have to be primary? In many texts about CFT it is proven that spin-1 conserved currents have the dimension $d-1$. In the proof it is used that, sometimes only implicitly, the current $J^\mu$ is a primary operator. Specifically the proof uses the identity $K_\nu J^\mu|0> = 0$. Is this required ? Why conserved currents have to be primary ?
 A: This is not an complete answer, but hopefully illustrates some issues that arise in this context. I would be curious to know a reference that sorts this out carefully.
You can have conserved vector operators that are not primary. For example, take any two-form operator $V^{\mu\nu}$ (not necessarily a primary) and set $J^\mu=\partial_\nu V^{\mu\nu}$ (I will call such $J$ exact). Moreover, if you have a $J$ that is a primary, you can add an exact term to it without spoiling the conservation or any charges (see below), but the result will not be a primary. So, you need to add extra assumptions about your $J^\mu$ in order to prove what you want.
One condition is that the charge operator $Q$ that you can build out of $J^\mu$ should be non-zero. For example, working in Euclidean, you must have $$\partial_\mu J^\mu=\text{contact terms},$$
for if $\partial_\mu J^\mu$ were identically zero, the integral of $J^\mu$ over any surface would vanish by Stokes' theorem. Requiring non-trivial charges saves you from having an exact current as in the example above, but doesn't save you from having a current which differs from a primary current by an exact term.
So, some problems come from the exact term ambiguity. It is actually worse than this. Suppose that $J_\mu$ is a primary conserved current. Then $J'_\mu=J_\mu+\partial^2 J_\mu$ is also a conserved current, and it has the same charges as $J$, but it is not primary and it does not differ from $J$ by an exact term. (It will have non-standard contact terms, but this will not affect the charges.) This issue can likely be avoided if you require $J$ to have a definite scaling dimension (which the example above doesn't satisfy).
I think the statement that has a chance of bein true is that if you have a conserved vector operator with non-trivial charges, then there exists a primary conserved current with the same charges (but possibly different contact terms). I think I can prove it in a unitary CFT with a discrete spectrum, but the proof is not conceptually satisfying -- it just goes by splitting up the original current according to the conformal multiplets and then analyzing each piece using representation theory of conformla algebra.
