Energy Correlator Conformal Blocks and Positivity

Abstract

Correlation functions of energy flow operators (energy-energy correlators) are one of the simplest observables in quantum field theory and gravity, with diverse applications ranging from real world collider physics to constraining the space of consistent theories. In this paper we further develop the conformal block decomposition of energy-energy correlators in conformal field theories (CFTs), focusing on the source-detector operator product expansion (OPE). We compute the general conformal blocks in this channel for traceless symmetric operators of arbitrary spin in the background of a scalar source, considering both parity-even and parity-odd contributions. Motivated by the availability of data from the conformal bootstrap, we analyze the convergence of this source-detector OPE, taking a tensor product of two decoupled CFTs as an elementary example. Finally, we use positivity of energy correlators to derive novel bounds on OPE coefficients involving the stress-energy tensor in generic CFTs, and demonstrate the application of these bounds in the specific example of the 3d Ising CFT, obtaining new constraints for both parity-even and parity-odd operators.

Figures (7)

• Figure 1: Schematic representation of the conformal block decomposition of the two-point energy correlator \ref{['eq:SumStates']}. Each conformal block is constructed from three-point functions of the external operator $\phi$, the energy operator ${\mathcal{E}}$ obtained from $T_{\mu\nu}$ via \ref{['eq:Edef']}, and the exchanged primary operator ${\mathcal{O}}$.
• Figure 2: Exact results for the mixed energy correlator $\langle{\mathcal{E}}_1\mspace{1mu}{\mathcal{E}}_2\rangle$ in a $d=3$ tensor product CFT for different values of the source scaling dimension $\Delta_\phi$, as a function of $z = \sin^2\frac{\theta}{2}$ (left) and decomposed into partial waves $\widehat{C}^{(\frac{1}{2})}_\ell(\cos\theta) \sim \cos\ell\theta$ (right).
• Figure 3: Cumulative sum of conformal block partial wave coefficients for double-twist operators $[\phi_1\phi_2]_{0,J}$ with $J\leq J_{\max}$ as a function of $J_{\max}$ in a $d=3$ tensor product CFT, normalized by the exact energy correlator partial wave coefficient from \ref{['eq:ProductCFTExactL']}, for different values of $\ell$ and the source scaling dimension $\Delta_\phi$.
• Figure 4: Top: Sum of energy correlator conformal blocks for double-twist operators $[\phi_1\phi_2]_{0,J}$ with $J\leq J_{\max}$ as a function of $z$ in a $d=3$ tensor product CFT, normalized by the exact correlator from \ref{['eq:ProductCFTExactZ']}, for different values of $J_{\max}$ and the source scaling dimension $\Delta_\phi$. Bottom: Averaged version of the sum of conformal blocks, again normalized by the exact correlator. The result at each value of $z$ was obtained by averaging over the window $z \pm \frac{\epsilon}{2}$ with $\epsilon \propto z(1-z)$.
• Figure 5: Positivity bounds on $|\lambda_{\varepsilon T{\mathcal{O}}}|$ for any parity-even (left) or parity-odd (right) primary operator ${\mathcal{O}}$ with spin $J=2$ in the 3d Ising CFT (other than the stress tensor $T_{\mu\nu}$) as a function of its scaling dimension $\Delta$. The shaded region indicates the allowed OPE coefficient magnitudes, using the numerical bootstrap data \ref{['eq:BootstrapData']} obtained in Chang:2024whx. Similar bounds can be derived for operators with spin $J\geq 3$.