Causality Constraints in Conformal Field Theory

Abstract

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interactions of low-lying operators. As an application, we use the bootstrap to rederive the well known sign constraint on the $(\partialφ)^4$ coupling in effective field theory, from a dual CFT. We also find constraints on theories with higher spin conserved currents. Our analysis is restricted to scalar correlators, but we argue that similar methods should also impose nontrivial constraints on the interactions of spinning operators.

Figures (6)

• Figure 1: The usual $\psi(0)O(z,\bar{z})$ OPE does not converge, since the dashed red circle contains another operator. But if we expand around the origin of the solid blue circle, it converges. This is implemented by the $\rho$ variable.
• Figure 2: Left: Circles on the $z$ plane. Right: Corresponding paths on the $\rho$ plane. The thick red path, in both plots, is $|z|=1$. The branch cut along $[1,\infty]$ in the $z$ plane maps to $|\rho| = 1$.
• Figure 3: Insertion points in the Lorentzian 4-point function. The 4th insertion $\psi(t=0, y=\infty)$ is not shown. Caveat: This picture does not apply to the shockwave kinematics discussed later in the paper.
• Figure 4: Kinematics of the shockwave 4-point function. The two $\psi$'s are inserted at $t = \pm i \delta$, $\vec{x} = 0$, so the shockwave is spread over a width $\sim 2\delta$.
• Figure 5: The analytic continuation appropriate to the ordering $\psi OO \psi$ is the dashed green line in \ref{['f:pathbetween']}. This same path is shown here in real spacetime (left) and on the $z$ plane (right)..