OPE statistics from higher-point crossing

TL;DR

This work derives new high-point crossing-based asymptotic formulas for conformal field theory data, revealing non-Gaussian structures in OPE coefficient distributions. Starting from four-point crossing in arbitrary dimensions and extending to six-point crossing, the authors obtain a universal density for $\overline{C_{LLH}^3C_{HHH}}$ and, in 2D, refined results for quasi-primaries and Virasoro primaries via the Virasoro crossing kernel. The results show controlled, exponential suppression with the heavy energy scale $\Delta_H$ and illuminate the interplay between light data and heavy exchanges, including a potential Wick-like structure for quasi-primaries but not for Virasoro primaries. These findings have implications for the statistical landscape of CFT data, Tauberian window considerations, and holographic gravity interpretations, while also pointing to rich directions in Lorentzian regimes and higher-point crossing.

Abstract

We present new asymptotic formulas for the distribution of OPE coefficients in conformal field theories. These formulas involve products of four or more coefficients and include light-light-heavy as well as heavy-heavy-heavy contributions. They are derived from crossing symmetry of the six and higher point functions on the plane and should be interpreted as non-Gaussianities in the statistical distribution of the OPE coefficients. We begin with a formula for arbitrary operator exchanges (not necessarily primary) valid in any dimension. This is the first asymptotic formula constraining heavy-heavy-heavy OPE coefficients in $d>2$. For two-dimensional CFTs, we present refined asymptotic formulas stemming from exchanges of quasi-primaries as well as Virasoro primaries.

Figures (7)

• Figure 1: The kinematics considered for the four-point function in the $z$ plane. The black circle is the unit circle and the pairs of operators are inserted at an angle separation $\theta$, with one pair centered at 0 and one pair centered at $\pi$.
• Figure 2: The kinematics considered for the six-point function in the $z$ plane. The black circle is the unit circle and the pairs of operators are inserted at an angle separation $\theta$, with one pair centered at 0, one pair centered at $2\pi/3$ and the third pair centered at $4\pi/3$.
• Figure 3: The sequence of elementary crossing moves relating two different star channel decompositions. We show the full (top) and simplified (bottom) versions of this diagram.
• Figure 4: The six-point OPE density associated with the channel on the left can be approximated by the four-point density on the right multiplied by a crossing kernel.
• Figure 5: The OPE diagram on the left can be reduced to the one on the right by an application of two crossing moves.