Energy Correlators of Spinning Sources

TL;DR

This work develops a comprehensive framework to retain the full angular structure of $N$-point energy correlators for spinning sources, revealing a universal Euler-angle dependence governed by $D^J$-matrices and dynamical spinning correlators $H^J_{h'-h,m'-m}(z_{ij})$ that depend on internal detector angles $z_{ij}$. Positivity and unitarity bound these correlators into a finite region, with extremal points realized by pure-spin states and interior points as convex mixtures, extending Hofman–Maldacena bounds to arbitrary spin and points. The authors compute spinning two-point correlators in perturbative QCD and extend the framework to energy-charge correlators, showing that ratios to the inclusive correlator isolate hard dynamics and are largely infrared safe, while they preserve sensitivity to polarization and frame changes. Generalized sum rules connect $N$-point and $(N-1)$-point spinning correlators in a way that decouples each spin channel, providing a consistent scaffolding to relate different observables and to understand how angular momentum information propagates through detector configurations. Overall, the spinning correlator program offers new observables, a bridge between collider QCD and light-ray/CFT formalisms, and a path toward probing hadronization and confinement through angular-momentum-resolved energy flows.

Abstract

The $N$-point energy correlator measures the energy flux through $N$ detectors. We present a general framework that characterizes its full angular dependence in a series of \textit{spinning energy correlators}. These spinning correlators resurrect the angular momentum structure of both the source and the detector configuration, lost otherwise in inclusive measurements. We demonstrate that unitarity and energy positivity confine these correlators to a sharply bounded region, with the boundary realized by extremal correlators generated by pure spin states. We present a first calculation of spinning energy correlators in QCD as well as spinning energy-charge correlators. Their enhanced insensitivity to infrared dynamics opens up a new set of observables that directly probe the hard part of the scattering. Finally, we provide generalized sum rules, extended to spinning correlators and to conserved charges beyond energy.

Figures (8)

• Figure 1: Example of the kinematics of a 5 point correlator. The $2\times 5=10$ coordinates that specify the location of the detectors can be split into $2\times 5-3=7$ internal angles fixing the configuration, and 3 Euler angles locating the rigid body of detectors. The 7 internal angles are obtained by a triangulation of the detectors. Measuring the Euler angles requires an external coordinate system to embed the rigid body. The angle $\Theta$ and $\phi$ require an axis and an orientation within the detectors and can always be measured. The azimuthal $\Phi$ requires an external reference frame, as it is the case when the operator exciting the vacuum is boosted.
• Figure 2: Bounds on the normalized coefficients $H_2/H_0$ and $H_4/H_0$ of the spinning energy correlator for various values of the spin-$J$ of the source operator.
• Figure 3: Generic configuration of the two-point energy correlator. The single rigid body angle is given by $\theta$. The remaining angles $\Theta$, $\Phi$ and $\phi$ are Euler angles.
• Figure 4: Allowed space for the $c(z)$ and $b(z)$ functions of the two point correlator. The hadronic tensor at special points are explicitly shown, together with some example of theories that generates them. In the collinear $(z\to 0)$ and back-to-back $(z\to 1)$ limits, the allowed space collapses to the lines at $b(0)=0$ and $c(1)=0$, respectively.
• Figure 5: Spinning Energy-Energy correlators as a function of $z$ in QCD for an unpolarized vector current. The solid line is the analytical result in Eq. \ref{['eq:EEC:bc:QCD']}, while the data points are the result of a simulation of parton shower and hadronization via Pythia8. In pink and blue, the spinning correlators $a^{(2,0)}_{{\mathcal{E}}{\mathcal{E}}}$ and $a^{(2,2)}_{{\mathcal{E}}{\mathcal{E}}}$, respectively.