Moments in the CFT Landscape

Abstract

We develop a novel numerical bootstrap for unitary, crossing-symmetric conformal field theories, focusing on moment observables defined as weighted averages over conformal data. Providing a global and coarse-grained probe of the operator spectrum, this framework yields numerically rigorous bounds on the operator distribution using standard semidefinite programming techniques. In the heavy correlator regime, these bounds remain robust and converge rapidly towards analytically-derived power laws. At finite external dimensions, low-lying moments capture corrections to analytic heavy limit results, while reproducing familiar bootstrap solutions such as Ising-model kinks on the boundary of moment space. Most importantly, the moment bootstrap reveals new features in previously unexplored regions of the bootstrap landscape. The lower bounds on moment variables exhibit two continuous families of kinks persisting across $2 < d < 6$, reflecting nontrivial spectral reorganizations connected to underlying operator decoupling phenomena. These results demonstrate that moment variables uncover bootstrap solutions and collective structures that are difficult to access within traditional numerical approaches.

Figures (25)

• Figure 1: Bounds on the first normalized moment (excluding the identity) as a function of the external scaling dimension, computed at $\Lambda = 23$ (see appendix \ref{['appendix:rational_approx']} for other truncation parameters). The shaded regions show the allowed moment space under different gap assumptions, with darker shades indicating stronger gaps. The solid points correspond to moments extracted from $\langle\sigma\sigma\sigma\sigma\rangle$ and $\langle\epsilon\epsilon\epsilon\epsilon\rangle$ in the Ising model, while the black dashed lines indicate GFF solutions. The upper bounds are insensitive to the gap choice. See appendix \ref{['appendix:minimal_model']} for details on the minimal model and vertex operator moment computations.
• Figure 2: Two-sided bounds on the first three normalized moments including the identity operator as a function of the external scaling dimension $\Delta_{\phi}$ in $d = 3$, computed at $\Lambda = 23$. The vertical axes are normalized by $\Delta_{\phi}^k$, so that the asymptotic analytic bounds in \ref{['heavy_moments_power_law_bounds']} appear as straight horizontal lines.
• Figure 3: Gap maximization compared to moment maximization (without the identity contribution) at the same derivative order. The moment bounds converge rapidly toward the linear trajectory $2 \sqrt{2}\,\Delta_{\phi}$ and reproduce the correct asymptotic scaling at very low derivative order. In contrast, gap maximization converges much more slowly and fails to produce any bound at large external scaling dimension. Although the two procedures optimize different observables, this comparison illustrates that, in the heavy correlator regime, the moments provide more natural observables for the bootstrap.
• Figure 4: The maximum-entropy reconstruction of the OPE measure (smooth solid curves) compared to the exact discrete measure of the generalized free field correlator $\langle \phi\phi\phi\phi \rangle$. Different colors represent different external scaling dimensions $\Delta_\phi$. The OPE measure is normalized to integrate to unity against $\Delta_{\phi}^{-1} \Delta$.
• Figure 5: Reproduction of the correlator bounds in $d=3$ as a function of the external scaling dimension $\Delta_\phi$, following Paulos:2021jxx. The sharp leftmost edge, where all bounds converge, corresponds to the free theory, while the red dot on the lower bound marks the 3d Ising correlator, computed from the conformal data of Simmons-Duffin:2016wlq. The bounds were obtained at $\Lambda = 11$.