We report some recent progress in the computation of the n-point correlationfunctions of conserved currents in a class of four dimensional conformal fieldtheories with higher spin symmetry. Global conformal invariance leads to verystrong constraints on both the general form and the singularity structure ofthe correlation functions of conserved currents. Namely, all these functionshave to be rational functions with at most double pole singularities. We showthat this implies that the 4-, 5- and 6-point correlation functions of the(symmetric, conserved and traceless) stress-energy tensor are linearcombinations of the three free field expressions. Hence, in four dimensions,any globally conformal invariant theory is free.
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