Double-Trace Deformations of Conformal Correlations

TL;DR

This work analyzes unitary large-N CFTs deformed by double-trace operators and computes the leading 1/N changes in scalar four-point functions from the UV to the IR fixed point. Interpreting the flow as a boundary-condition change in AdS, the authors derive a compact, crossing-symmetric expression for the four-point function difference in terms of bar{D}-functions and extract explicit shifts in double-trace OPE data, including anomalous dimensions γ_{n,ℓ} and OPE-coefficient corrections. A key finding is the sign-definiteness of certain γ_{0} shifts and the universal structure governing leading-twist double-trace operators, with detailed applications to O(N) vector models and non-singlet representations. The results are connected to conformal-harmonic analysis and provide new analytic data for vector-model correlators, including finite-dimension and ε-expansion regimes, as well as broader implications for bulk locality and higher-spin dynamics in holography.

Abstract

Large $N$ conformal field theories often admit unitary renormalization group flows triggered by double-trace deformations. We compute the change in scalar four-point functions under double-trace flow, to leading order in $1/N$. This has a simple dual in AdS, where the flow is implemented by a change of boundary conditions, and provides a physical interpretation of single-valued conformal partial waves. We extract the change in the conformal dimensions and three-point coefficients of infinite families of double-trace composite operators. Some of these quantities are found to be sign-definite under double-trace flow. As an application, we derive anomalous dimensions of spinning double-trace operators comprised of non-singlet constituents in the $O(N)$ vector model.

Figures (5)

• Figure 1: A renormalization group flow triggered by a double-trace deformation of a large-$N$ CFT: the UV fixed point, at which the operator $O$ has $\Delta<d/2$, flows to an IR fixed point, at which $O$ has conformal dimension $d-\Delta$, at leading order in $1/N$. In this paper, we compute the leading order change in four-point functions of single-trace operators, other than $O$, that couple to $O$.
• Figure 2: The triangle diagram determines the three-point coupling $\langle \Phi\Phi\sigma\rangle$, to which the UV coupling $\langle \Phi\Phi O\rangle$ flows. The purple point is integrated over.
• Figure 3: The two-triangle diagram, given in \ref{['2-triangle']}, determines the change in the connected correlator $\langle \Phi\Phi\Phi\Phi\rangle$ in the $s$-channel, to leading order in $1/N$. The total result is a sum of three such diagrams, one from each channel.
• Figure 4: The AdS dual of the two-triangle diagram in CFT. The difference of two exchange diagrams with external $\Phi$ fields -- one with standard quantization $(\Delta)$ of the field dual to $O$, and one with alternate quantization $(d-\Delta)$ -- can be written, using the split representation of the AdS harmonic function, as a pair of boundary three-point functions tied together by a boundary two-point function of dimension $d-\Delta$. This is manifestly equivalent to the two-triangle diagram in CFT.
• Figure 5: In $d=3$, a plot of $\delta\gamma_{\ell}$ evaluated at $\Delta=3/2$, its minimum, as a function of $\Delta_{\Phi}$. We plot $\ell=2,4,\ldots,16$, where red is $\ell=2$ and the spin increases as we move through the rainbow. For $\Delta_{\Phi}>3/4$, the function is positive for all $\ell$.