Moments and saddles of heavy CFT correlators

A bstract 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 ∆ and J 2 ≡ ℓ ( ℓ + 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 ∆ and the covariance between ∆ 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.