  We prove the instability of $d$-dimensional conformal field theories (CFTs)
having in the operator-product expansion of two fundamental fields a primary
operator of scaling dimension $h=d/2+i r$, with non-vanishing $
r\in\mathbb{R}$. From an AdS/CFT point of view, this corresponds to a
well-known tachyonic instability, associated to a violation of the
Breitenlohner-Freedman bound in AdS${}_{d+1}$; we derive it here directly for
generic $d$-dimensional CFTs that can be obtained as limits of multiscalar
quantum field theories, by applying the harmonic analysis for the Euclidean
conformal group to perturbations of the conformal solution in the two-particle
irreducible (2PI) effective action. Some explicit examples are discussed, such
as melonic tensor models and the biscalar fishnet model.
