Primaries, descendents and transformation properties in CFT

[Kurret]

I want to clarify the relations between a few different sets of operators in a conformal field theory, namely primaries, descendants and operators that transform with an overall Jacobian factor under a conformal transformation. So let us consider the the following four sets of operators:Primaries: Defined by the fact that they are annihilated (commute with) the lowering operators of the theory.

Descendants: Obtained by acting on primaries with raising operators.

Set 3: Operators that scale as $\mathcal{O'}(x')=\lambda^{-\Delta/d}\mathcal{O}(x)$ under a scaling (dilatation).

Set 4: Operators that transform as $\mathcal{O'}(x')=|\partial x'/\partial x|^{-\Delta/d}\mathcal{O}(x)$ under any conformal transformation.

What are the relation between the above sets? I think all descendants are also part of set 3 (but not the converse, since there could probably also exist linear combinations of descendants that are part of set 3). All primaries should also be part of set 4, but I am not sure about the converse.

 

[LightHero]

The primary operators are all contained within Set 4, since they are annihilated by the lowering operators and thus must transform with a Jacobian factor under any conformal transformation. All descendants are also contained within Set 4, since they can be obtained from the primary operators by acting with the raising operators. However, not all operators in Set 4 are necessarily primaries or descendants, as there could be other operators that transform with a Jacobian factor under a conformal transformation but do not satisfy the criteria for being a primary or descendant. Set 3 consists of all operators that scale as \mathcal{O'}(x')=\lambda^{-\Delta/d}\mathcal{O}(x) under a scaling (dilatation). Any primary or descendant is part of this set, as they must obey the scaling relation, but it is possible for other operators to be part of this set without being primaries or descendants.