Notes on Relevant, Irrelevant, Marginal and Extremal Double Trace Perturbations

TL;DR

This paper investigates double trace deformations in conformal field theories, revealing potential pathologies and showing that certain flows to UV fixed points are non-existent due to unphysical modes, impacting holographic interpretations.

Contribution

It introduces a Kallen-Lehmann representation for two-point functions in deformed theories and demonstrates that some expected UV fixed points are non-conformal and unphysical.

Findings

Identifies pathologies at intermediate flow points preventing UV fixed point attainment.

Shows that an extremal deformation leads to a non-conformal UV fixed point with unphysical modes.

Resolves a holographic puzzle by demonstrating the non-existence of certain flows.

Abstract

Double trace deformations, that is products of two local operators, define perturbations of conformal field theories that can be studied exactly in the large-N limit. Even when the double trace deformation is irrelevant in the infrared, it is believed to flow to an ultraviolet fixed point. In this note we define the Kallen-Lehmann representation of the two-point function of a local operator O in a theory perturbed by the square of such operator. We use such representation to discover potential pathologies at intermediate points in the flow that may prevent to reach the UV fixed point. We apply the method to an "extremal" deformation that naively would flow to a UV fixed point where the operator O would saturate the unitarity bound. We find that the UV fixed point is not conformal and that the deformed two-point function propagates unphysical modes. We interpret the result as showing that the flow to the UV fixed point does not exist. This resolves a potential puzzle in the holographic interpretation of the deformation.