Moments and saddles of heavy CFT correlators

TL;DR

This work recasts heavy scalar CFT four-point functions as a double Stieltjes moment problem for the OPE data, and builds a toolkit of principal-series operators to extract global averages over scaling dimension $\Delta$ and spin-related $J_2$ in the heavy limit. It derives two-sided bounds on moments and the covariance between $\Delta$ and $J_2$ from crossing and unitarity, and identifies extremal, saddle-point solutions that correspond to generalized free field (GFF) limits organized by higher-spin conformal blocks. The authors then introduce weight-interpolating functions (WIFs), notably Gaussian interpolants, to approximate OPE coefficients of heavy double-twist families from a few low moments, and show these approximations persist under perturbations from bulk contact interactions, a phenomenon they term Gaussianization. The results illuminate how heavy CFT data cluster into fixed-length operator families, offering a coarse-grained, tractable description of high-dimension spectra with potential applications to holography and the large- $N$ regime, and they point to semidefinite optimization as a path to exact moment constraints.

Abstract

We study the operator product expansion (OPE) of identical scalars in a conformal four-point correlator as a Stieltjes moment problem, and use Riemann-Liouville type fractional differential operators to generate classical moments from the correlation function. We use crossing symmetry to derive leading and subleading relations between moments in $Δ$ and $J_2 \equiv \ell(\ell+d-2)$ in the ``heavy" limit of large external scaling dimension, and combine them with constraints from unitarity to derive two-sided bounds on moment sequences in $Δ$ and the covariance between $Δ$ and $J_2$. The moment sequences which saturate these bounds produce ``saddle point" solutions to the crossing equations which we identify as particular limits of correlators in a generalized free field (GFF) theory. This motivates us to study perturbations of heavy GFF four-point correlators by way of saddle point analysis, and we show that saddles in the OPE arise from contributions of fixed-length operator families encoded by a decomposition into higher-spin conformal blocks. To apply our techniques, we consider holographic correlators of four identical single scalar fields perturbed by a bulk interaction, and use their first few moments to derive Gaussian weight-interpolating functions that predict the OPE coefficients of interacting double-twist operators in the heavy limit.

Figures (4)

• Figure 1: Operators in a four-point correlator placed on a cylinder. When the operators are sufficiently heavy, the conformal block resembles a Gibbs measure with inverse temperature $\beta \sim -\tau$ and 1d spin fugacity $\vec{\Omega} \sim \theta$. Macroscopic observables are produced by generating "twists" and "pulls" of the operator pairs. Under the integral transformation $\mathcal{T}$ introduced in section \ref{['sec:operators']}, this relation becomes exact in 1, 2, and 4 spatial dimensions.
• Figure 2: Weight-interpolating functions at $z=1/2$ for correlators in generalized free field theories with $\Delta_\varphi = \Delta_{\phi^6} = 200$. The $x$-axis is labeled by a rescaled scaling dimension $\hat{\Delta} = \Delta/\left(2\sqrt{2}\Delta_{\phi/\varphi}\right)$, so that saddles are located at $\hat{\Delta} = K$. The black dots mark the exact weights of operators in the OPE distribution, including the $s$-channel identity at $\Delta=0$. The saddle of largest scaling dimension is associated with the t-channel identity.
• Figure 3: Gaussian weight-interpolating functions (WIFs) at $z= 1/2$, plotted against the exact weights of operators in $\mathcal{G}_1$ coupled to the interaction (\ref{['cinteraction']}) with $L=2$ at different values of $\Delta_\phi$ and $g$. Blue: $g=0$, Green: $g = -1$, Red: $g=1$.
• Figure 4: The regulated Error $=\frac{\nu_k}{\nu_0} -\left.\partial^k_t e^{\mu t + \frac{\sigma^2}{2} t ^ 2 }\right|_{t = 0}$ away from Gaussian moments for $k=3$, plotted as a function of $\Delta_\phi$. The left plot is computed with $g=-1$, while the right plot is computed with $g= 1$. The dashed line is a reference bound showing that saddles in either interaction tend to Gaussianize with errors of order $\Delta_\phi^{k-2}$.