A Conformal Dispersion Relation: Correlations from Absorption

TL;DR

This work derives a conformal dispersion relation for four-point scalar correlators, reconstructing the correlator from its double discontinuity via a universal two-variable kernel. The kernel, computed by resumming Lorentzian inversion data, splits into a bulk term and a contact term, and is shown to be dimension-independent by explicit $d=2$ and $d=4$ analyses; differential equations govern unequal-dimension cases. A direct contour-deformation proof establishes the theorem beyond reliance on the inversion formula, and a subtracted version ensures convergence in any unitary CFT. The authors validate the approach with generalized free fields, holographic correlators, and the 3D Ising model, and uncover an integral relation between conformal blocks, highlighting the method’s bootstrap and analytic-functional potential for constraining CFT data.

Abstract

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its 'absorptive part', defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the 'inverted' conformal block with the ordinary conformal block.

Figures (7)

• Figure 1: Left: The 4-particle scattering amplitude $\mathcal{M}(s,t)$, with external momenta $p_i$. Right: $t$-channel tree level exchange diagram of particle with spin $J$ .
• Figure 2: Left: The amplitude can be written as a contour integral by using Cauchy's theorem. Right: Upon deforming the contour, there will be contributions from the branch cuts and from the arcs at infinity.
• Figure 3: Left: The original integration contour of the principle series representation. One can close the contour either to the left or to the right, depending on the behaviour of the integrand at $|\Delta| \to \infty$. The integrand has poles on the real $\Delta$ axis.
• Figure 4: The integration contour used in eq. (\ref{['0 contour']}) to prove the dispersion relation is a product of a keyhole and a circle. (a) $\sigma_w$-plane: the keyhole $\mathcal{C}_\sigma$ starts and ends at $\sigma_w=0$. (b) $\eta_w$-plane: the integral over the circle $|\eta_w|=1$ vanishes and is equal to the sum of the pole and cuts in its interior. The pole gives minus the correlator $\mathcal{G}(z,{\bar{z}})$ and the cuts give the dispersion integral. Values of $x$ are shown in light gray above the axis.
• Figure 5: Keyhole contour in the ${\bar{\rho}_w}$ variable to avoid cross-channel singularity (at ${\bar{w}}=1$). The angle runs over $\theta\in [0,\pi]$. The radius of the circle shrinks as $\rho_{\rm max}\to 1$.