Analyticity in Spin in Conformal Theories

TL;DR

The paper develops a conformal Froissart-Gribov inversion that makes OPE data analytic in spin by projecting Lorentzian discontinuities of four-point functions. It derives a convergent, spin-analytic formula expressing each OPE coefficient as a positive integral over double discontinuities, with contributions organized into even/odd spin families and controlled large-spin expansions. The framework clarifies how sparse large-N spectra and AdS bulk locality suppress higher-derivative bulk interactions and yields exact sum rules and generating functions for extracting twist and OPE data, demonstrated in Ising-model tests and AdS-inspired bounds. Overall, it provides a robust, non-perturbative tool linking high-energy Regge behavior, crossing symmetry, and bulk locality to precise CFT data across spins. Its implications span analytic bootstrap, AdS/CFT correspondence, and potential improvements to numerical bootstrap error estimates.

Abstract

Conformal theory correlators are characterized by the spectrum and three- point functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an "imaginary part" which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields 1/J expansions with controlled errors. In large-N theories, the imaginary part is saturated by single-trace operators. For a sparse spectrum, it manifests the suppression of bulk higher-derivative interactions that constitutes the signature of a local gravity dual in Anti-de-Sitter space.

Figures (6)

• Figure 1: Relation between low-energy coefficients and discontinuity. Analyticity in spin holds if the arcs at infinity (the Regge limit) can be dropped.
• Figure 2: (a) Four points in Lorentzian signature, with time running upward. The pairs $x_4-x_1$ and $x_2-x_3$ are timelike separated for $\bar{\rho}>1$. (b) Corresponding configuration of cross-ratios $\rho$ and $\bar{\rho}$.
• Figure 3: (a) Correlator in the Regge limit $\rho\propto 1/E\to 0$, $\bar{\rho}\propto E\to \infty$ (with an overall boost applied to figure \ref{['fig:rindler']} to make the figure more symmetrical). (b) The crossing symmetry path $E\to Ee^{-i\pi}$, which interchanges points $3$ and $4$.
• Figure 4: Scattering amplitudes in the complex $w$-plane, where $w=e^{i\theta}$. Two copies of the $t$-channel appear on the right, related to each other by $w\to 1/w$, and two copies of the $u$-channel cut appear on the left.
• Figure 5: The $w$-plane contours $C$ and $C_\pm$ in the CFT case. The singularities at $w=\pm 1$ are integrable and come only from the measure factor $|z-\bar{z}|^{d-2}$. The branch points at $w=\pm\sigma$ and $w=\pm1/\sigma$ pose no problems, but the arcs at $w=0$ and $w\to\infty$ correspond to the Regge limit where the integrand should vanish at large positive spin.