Event shapes in N=4 super-Yang-Mills theory

TL;DR

The paper develops a unified, symmetry-driven framework to compute event shapes in N=4 SYM by relating infrared-safe weighted cross sections to Wightman correlation functions of flow operators. Central to the approach is a master formula expressing double-detector observables as convolutions of a universal Mellin amplitude, encoding Euclidean data, with a coupling-independent detector kernel that depends only on the observable pair. The authors demonstrate this formalism at weak and strong coupling, showing exact matches with amplitude-based results at leading order and providing strong consistency checks via AdS/CFT inputs. They also reveal deep SU(4) and superconformal constraints that link different observables and channels, offering a path to higher-loop and nonperturbative analyses. The work lays groundwork for systematic, symmetry-based predictions of angular energy and charge distributions in highly supersymmetric gauge theories, with potential insights applicable to broader conformal field theories.

Abstract

We study event shapes in N=4 SYM describing the angular distribution of energy and R-charge in the final states created by the simplest half-BPS scalar operator. Applying the approach developed in the companion paper arXiv:1309.0769, we compute these observables using the correlation functions of certain components of the N=4 stress-tensor supermultiplet: the half-BPS operator itself, the R-symmetry current and the stress tensor. We present master formulas for the all-order event shapes as convolutions of the Mellin amplitude defining the correlation function of the half-BPS operators, with a coupling-independent kernel determined by the choice of the observable. We find remarkably simple relations between various event shapes following from N=4 superconformal symmetry. We perform thorough checks at leading order in the weak coupling expansion and show perfect agreement with the conventional calculations based on amplitude techniques. We extend our results to strong coupling using the correlation function of half-BPS operators obtained from the AdS/CFT correspondence.

Figures (6)

• Figure 1: Final states in $e^+e^-$ annihilation in QCD. The electron and positron annihilate to produce a virtual photon $\gamma^*(q)$ that decays into an arbitrary number of quarks and gluons which go through a hadronization process (shaded rectangle) to become hadrons (double lines). The dot denotes the electromagnetic QCD current.
• Figure 2: The Feynman diagram contributing to $\sigma_{\rm tot}(q)$. The thin line stands for the unitarity cut.
• Figure 3: Cross-talk between the two detectors. The thin line stands for the unitarity cut. The shaded blobs (vertices 1 and 4) stand for the source and sink. The crosses denote the two detectors (vertices 2 and 3) oriented along the vectors $\vec{n}$ and $\vec{n}'$. The detectors interact with each other by exchanging a particle with zero momentum.
• Figure 4: Graphical representation of the double energy correlation: particles produced out of the vacuum by the source are captured by the two detectors located at spatial infinity in the directions of the unit vectors $\vec{n}$ and $\vec{n}'$.
• Figure 5: The relation between the weighted cross section and Wightman correlation function. The operators at points $1$ and $4$ describe the source and sink, respectively. The operators at points $2$ and $3$ define the flow operators shown by crosses. 'Limit${}_{2,3}$' stands for the detector limit which amounts to sending the operators at point $2$ and $3$ at null infinity with subsequent integration over their light-cone coordinates (see Eqs. (\ref{['E-new']}) and (\ref{["4.4'"]}) below).