Abstract
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 = frac{d}{2} + i r, with non-vanishing r ∈ 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 AdSd+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.
Highlights
From an AdS/conformal field theories (CFTs) point of view, this corresponds to a well-known tachyonic instability, associated to a violation of the Breitenlohner-Freedman bound in AdSd+1; we derive it here directly for generic ddimensional 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
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 =
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 AdSd+1; we derive it here directly for generic ddimensional 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
Summary
As we will see in the following, such roots typically arise when, by tuning a parameter of the theory, a physical root and its shadow move along the real axis and merge at h = d/2, to split again along the principal series, as depicted in figure 2 It would look like the transition is precisely of the type described above, giving a double pole at the merging point. We should point out that while in the SYK model such zero mode in the discrete series representation leads to a problem when inverting the Hessian to write the four-point function, and one is forced to move away from the conformal IR limit, this is not the case for the zero mode in the principal series, as the integral in (2.41) will in general be defined by a slight contour deformation, as we explained above
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