Seed conformal blocks in 4D CFT

We compute in closed analytical form the minimal set of “seed” conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation (ℓ, ℓ) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0, |ℓ − ℓ|) and one (|ℓ − ℓ|, 0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any (ℓ, ℓ), by using an educated ansatz and reducing the problem to an algebraic linear system. Various details on the form of the ansatz have been deduced by using the so called shadow formalism. The complexity of the conformal blocks depends on the value of p = |ℓ − ℓ| and grows with p, in analogy to what happens to scalar conformal blocks in d even space-time dimensions as d increases. These results open the way to bootstrap 4-point functions involving arbitrary spinor/tensor operators in four dimensional conformal field theories.