The simplest event shape in $\mathcal{N}=4$ super Yang--Mills

Problem Statement
Event shapes are fundamental Lorentzian observables in conformal field theories (CFTs), describing correlations between detectors surrounding a source and sink. While they are physically significant, calculating them analytically at finite coupling is notoriously difficult; for instance, the energy–energy correlation in $\mathcal{N}=4$ super Yang–Mills (sYM) is known only to the first few perturbative orders. While the conformal bootstrap offers numerical progress, analytic control remains limited. Previous work established that making operators "heavy" (large charge) facilitates analytic control for local correlators and certain energy correlators. This paper investigates a specific, simpler event shape in planar $\mathcal{N}=4$ sYM where the source, sink, and detectors are all equally heavy (large charge), aiming to achieve exact analytic results at finite coupling.

Methodology
The authors study the Wightman correlation function of two scalar detectors, $O_k(n)$ and $O_k(n')$, in the presence of a source and sink of the same dimension $k$. The detectors are defined as light transforms of half-BPS scalars. The analysis proceeds through the following steps:

1. Perturbative Expansion: The event shape is expanded in the 't Hooft coupling $\lambda$. The authors compute the tree-level, one-loop, and two-loop contributions for arbitrary charge $k$ using Mellin space techniques and discontinuity methods.
2. Large Charge Limit: They take the limit $k \to \infty$ order by order in perturbation theory. In this limit, the event shape factorizes into the square of a fundamental object, the "octagon event shape," analogous to how the large-charge local correlator factorizes into the square of the octagon correlator.
3. Commutation of Limits: The authors demonstrate that the large-charge limit commutes with the detector integrals. This allows them to evaluate the light transform directly on the factorized octagon correlator.
4. Finite Coupling Formulation: Leveraging the known exact results for the octagon correlator (expressed as a Fredholm determinant of a modified Bessel kernel), they propose a finite-coupling formulation for the octagon event shape.
5. Asymptotic Analysis: The proposed formulation is analyzed in the weak, finite, and strong coupling regimes, as well as in the collinear ($\theta \to 0$) and back-to-back ($\theta \to \pi$) limits.

Key Contributions and Results

• Factorization and Definition: The primary result is the demonstration that in the large-charge limit, the two-point event shape $G_k(\gamma)$ factorizes as $G(\gamma)^2$, where $G(\gamma)$ is the "octagon event shape."
• Exact Relation to Octagon Correlator: The authors derive that the octagon event shape is proportional to the octagon correlator evaluated on the unit circle, with the phase encoding the angle $\theta$ between the detectors:
  $$G(\theta) = \frac{\sin(\theta/2)}{4\pi^2} \mathcal{O}(e^{-i\theta}, e^{i\theta})$$
  This relation holds order by order in perturbation theory and is proposed to hold at finite coupling.
• Finite Coupling Proposal: The paper proposes that the finite-coupling octagon event shape is given by the Fredholm determinant of a Bessel kernel:
  $$G(\theta) = \frac{\sin(\theta/2)}{4\pi^2} \det(1 - K)$$
  where the matrix elements depend on the coupling $\lambda$ and the angle $\theta$ via a specific symbol $\chi(t; \theta)$.
• Perturbative Checks: The authors explicitly compute the tree-level, one-loop, and two-loop corrections for the large-charge event shape. They show that expanding the proposed finite-coupling formula reproduces these perturbative results, confirming the commutation of the large-charge and detector limits.
• Finiteness and Convergence: The paper addresses the finiteness of scalar event shapes. It argues that for any fixed loop order $\ell$, there exists a threshold charge $k^*(\ell)$ such that the event shape is finite for all $k \geq k^*(\ell)$. This is due to the suppression of diagrams crossing "bridges" of width $k$ in the planar limit, leading to factorization.
• Physical Interpretation: In the integrability framework, the octagon event shape (normalized by its tree-level value) is interpreted as the probability of finding no fermions in a disk of radius $2\sqrt{\lambda}$, where the disk size is related to the Zhukovsky cut. This provides a probabilistic interpretation where the symbol $\chi(t; \theta)$ acts as an occupation number.
• Limits:
  • Weak Coupling: The expansion matches known perturbative results involving Clausen functions.
  • Strong Coupling: The event shape is exponentially suppressed, approaching zero from above, consistent with the behavior of the octagon correlator.
  • Collinear Limit ($\theta \to 0$): The shape vanishes linearly with $\theta$, with a subleading logarithmic term that is one-loop exact.
  • Back-to-Back Limit ($\theta \to \pi$): Unlike momentum-space energy correlators which probe the $u,v \to 0$ region (governed by Sudakov factors), the position-space back-to-back limit probes the regular point $u=v=1/4$, reducing to a constant value determined by the octagon at specific cross-ratios.

Significance and Claims
The paper claims to identify the "simplest event shape" in $\mathcal{N}=4$ sYM, characterized by the simplification of the light transform in the large-charge limit. This simplification collapses the non-local detector integrals into a local evaluation of the octagon correlator on the unit circle. The significance lies in providing an exact, finite-coupling analytic formula for an event shape, a class of observables that is typically intractable beyond low-order perturbation theory. The authors assert that this result bridges the gap between local correlators and event shapes, offering a new window into the non-perturbative structure of the theory. They note that while the back-to-back limit in momentum space remains an open problem requiring a double-scaling limit, the current work establishes a robust foundation for understanding event shapes via integrability.