A spacetime derivation of the Lorentzian OPE inversion formula

TL;DR

The paper addresses the problem of extracting OPE data from a CFT four-point function via a Lorentzian inversion formula, reframing Caron-Huot's cross-ratio integral as a spacetime-based construction.Its main approach uses shadow representations and Wick rotations to convert Euclidean five-point integrals into Lorentzian double-commutator integrals, then rewrites the result in cross-ratio variables and demonstrates agreement with Caron-Huot's formula, first in 2D and then in general dimensions, including a 1D specialization related to SYK chaos.Key contributions include a detailed, gauge-fixed path from a five-point shadow integral to a cross-ratio Lorentzian inversion with explicit normalization and kernel structures, an explicit treatment of null-direction isolation and averaging, and a clean demonstration of spin analyticity and the double-discontinuity structure underpinning the inversion.The work provides a robust framework for extending Lorentzian inversion to external spins and higher-point functions, with potential applications to Regge physics, large-spin perturbation theory, and chaos bounds in holographic and SYK-like CFTs.

Abstract

Caron-Huot has recently given an interesting formula that determines OPE data in a conformal field theory in terms of a weighted integral of the four-point function over a Lorentzian region of cross-ratio space. We give a new derivation of this formula based on Wick rotation in spacetime rather than cross-ratio space. The derivation is simple in two dimensions but more involved in higher dimensions. We also derive a Lorentzian inversion formula in one dimension that sheds light on previous observations about the chaos regime in the SYK model.

Figures (5)

• Figure 1: The continuation of $v_3,v_4$ in the case where $0<u_3<u_4<1$. We begin by integrating both variables over the real axis. We deform the $v_3$ contour in the lower half-plane to pick up the discontinuity across the branch cut associated with the $3\sim 2$ singularity, giving the $[O_3,O_2]$ commutator. We deform the $v_4$ contour in the upper half-plane and pick up the $[O_4,O_1]$ singularity.
• Figure 2: We show typical configurations for points 3 and 4 within regions $R_1$ and $R_2$. The dotted line is not fixed in place, it is only to emphasize that points 3 and 4 must be spacelike separated. Time goes up.
• Figure 3: The configuration of points that we choose, with $(u,v)$ coordinates indicated. The grey region is spacelike separated from the four points. The 2$d$ slice shown is the plane where the four points are located. As we move the slice outwards in the transverse directions away from this plane, the inner and outer grey regions grow and eventually merge, see figure \ref{['threedintegrationregion']}.
• Figure 4: The region of integration for $x_5$ is the exterior of the lightcones of the four operators. In the limit of small cross-ratios, the integral is dominated by the region inside the black-outlined box. The width of the box in the transverse directions is large enough to detect the curvature of the lightcones, but small enough not to detect their full geometry.
• Figure 5: (a) after the replacement (\ref{['rep1']}), the contour for $\eta$ circles the branch cut of the $B_J$ function. The arrows indicate the direction in which the contour passes the branch points at $\eta = \pm 1$ as we increase $\tau - \tau'$. After Wick rotation we end up deforming the contour as in (b). The dashed parts of the contour are on the second sheet. Note that the arcs at infinity are shown at finite radius for clarity. The arrows indicate the direction in which the contour passes the branch points as we increase $v-v'$, with $u-u'$ fixed. Finally, when we deform the contour over $v,v'$ we pull the dashed portions of the contour on the second sheet back through the cut to the first sheet, giving the contour in (c). We further drop the arcs at infinity, so that we have just the integrals along the real axes, picking up discontinuities across branch cuts from the four point function.