Moments and saddles of heavy CFT correlators

TL;DR

This paper analyzes heavy scalar correlators in conformal field theories using moment problems and saddle point techniques, revealing bounds and structures of the operator product expansion, with applications to holographic models.

Contribution

It introduces a novel moment problem approach to conformal correlators, deriving bounds and saddle point solutions, and applies these methods to holographic and generalized free field theories.

Findings

Derived bounds on moments and covariances in heavy CFT correlators.

Identified saddle point solutions corresponding to generalized free field limits.

Predicted OPE coefficients for double-twist operators in holographic models.

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 $\Delta$ 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 $\Delta$ and the covariance between $\Delta$ 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.