Abstract
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,\bar l) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0,|l-\bar l|) and one (|l-\bar 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,\bar 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-\bar 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.
Highlights
For each primary operator, in particular, one has to resum the infinite series of associated descendant operators in what is called a Conformal Partial Wave (CPW)
Before the advent of ref. [3], the only known Conformal Blocks (CBs) were those associated to symmetric traceless tensors exchanged in scalar 4-point functions in even number of dimensions [8, 9], denoted for short scalar symmetric CBs in the following
Summarizing, the problem of computing CPWs and CBs associated to the exchange of mixed symmetry operators O(, +p) and O( +p, ) in any 4-point function is reduced to the computation of the p+1
Summary
In 4D CFTs, for a given 4-point function, CBs and CPWs are labelled by the quantum numbers of the exchanged primary operator and they depend on its scaling dimen-. There are several CPWs for each exchanged primary operator Or, depending on the number of allowed 3-point function structures. The correlators (2.9) and (2.10) are the ones with the minimum number of tensor structures and the simplest This is understood by noticing that for any value of δ (and for δ = p) the operators O( , +δ) and their conjugates O( +δ, ) appear in both the (12) and (34) OPE’s with one tensor structure only, since there is only one tensor structure in the corresponding three-point functions:. Summarizing, the problem of computing CPWs and CBs associated to the exchange of mixed symmetry operators O( , +p) and O( +p, ) in any 4-point function is reduced to the computation of the p+1. Where G(ep)(u, v) and G(ep)(u, v) are the 4D CBs entering the r.h.s. of eq (2.7) when expanding the
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