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It is known that dyagonal correlation functions (say propagators) can at most have single poles in their spectrum. I am wondering if the existence of double poles in mixed correlators in a QFT (say for example the correlators between the stress energy tensor and a scalar operator in a conformal field theory $\langle T^{\mu \nu}(q) \phi(q') \rangle$) has any bad implication for the theory.

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    $\begingroup$ In a CFT such correlator will vanish identically unless $\phi$ is a descendant of $T$ in which case this correlator is a derivative of the diagonal one (so it has no more poles than $\langle TT\rangle$ in the momentum space). $\endgroup$ – Peter Kravchuk Oct 26 '16 at 7:03

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