Notes on relevant, irrelevant, marginal and extremal double trace perturbations

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 Källen-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 ∆ = d 2 − 1. 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.

Boundary-to-bulk maps for AdS causal wedges and RG flow

“…This result, when expressed as an integral over a positive mass parameter as is done in the Appendix, is consistent with the Kallen-Lehmann representation found in the Euclidean setting in[29] …”

Double-trace deformation in Keldysh field theory

“…Now we emphasize the key observation: at the O(N 0 ) level, the Keldysh function G K remains positive (free of ghosts) and finite (free of tachyons) and hence physical [36] if and only if γ > 0, a condition which nevertheless has been guaranteed by our general construction of the Keldysh theory. Therefore, Eq.…”

“…Therefore, Eq. ( 10) defines a complete RG trajectoryalong which the theory works at all energy scales and may flow in both way between UV and IR fixed points [36]. In particular, when ∆ > d/2, Eq.…”

“…We see that if is directly added to the two-point vertex function G −1 0 (k 2 ) of the QFT. This is because in the large N limit, a double-trace deformation is plainly additive to the effective action [36].…”

“…The main result we find for our Keldysh action is that, under appropriately renormalized perturbation, the non-unitary terms responsible for driven-dissipative dynamics have the natural form of a double-trace deformation [30][31][32][33] which has been very thoroughly studiedespecially in the literature of conformal field theory (CFT) [34,35]. In particular, in the large N limit, it is argued that the existence of a double-trace deformation guarantees and produces a healthy RG flow from infrared (IR) backwards to ultraviolet (UV), suggesting that the theory is UV complete [36], not merely an EFT. In fact, this argument also holds true here, and we further show that this UV theory is free of ghosts and tachyons, hence a physical relativistic theory.…”