From correlation functions to event shapes

TL;DR

The authors develop a universal, infrared-safe framework to compute event shapes—specifically charge flow correlations—in conformal field theories by relating Lorentzian Wightman correlators to Euclidean four-point functions through a Mellin-space analytic continuation. They show that the two-detector event shape function $F(z)$ is given by a universal detector kernel convolved with the Euclidean Mellin amplitude, or equivalently by a Lorentzian double discontinuity of the four-point function in cross-ratio space. The formalism is tested in ${ m N}=4$ SYM: at weak coupling the results reproduce known amplitude-based predictions, while at strong coupling they reproduce Hofman–Maldacena-type limits, including a homogeneous energy distribution. They also derive constraints on the four-point functions from physical requirements like IR finiteness, positivity, and regularity, which hold at finite coupling and away from the planar limit. The approach offers a powerful bridge between correlation-function techniques and amplitude methods, with potential implications for QCD jet physics and CFT bootstrap studies.

Abstract

We present a new approach to computing event shape distributions or, more precisely, charge flow correlations in a generic conformal field theory (CFT). These infrared finite observables are familiar from collider physics studies and describe the angular distribution of global charges in outgoing radiation created from the vacuum by some source. The charge flow correlations can be expressed in terms of Wightman correlation functions in a certain limit. We explain how to compute these quantities starting from their Euclidean analogues by means of a non-trivial analytic continuation which, in the framework of CFT, can elegantly be performed in Mellin space. The relation between the charge flow correlations and Euclidean correlation functions can be reformulated directly in configuration space, bypassing the Mellin representation, as a certain Lorentzian double discontinuity of the correlation function integrated along the cuts. We illustrate the general formalism in N=4 SYM, making use of the well-known results on the four-point correlation function of half-BPS scalar operators. We compute the double scalar flow correlation in N=4 SYM, at weak and strong coupling and show that it agrees with known results obtained by different techniques. One of the remarkable features of the N=4 theory is that the scalar and energy flow correlations are proportional to each other. Imposing natural physical conditions on the energy flow correlations (finiteness, positivity and regularity), we formulate additional constraints on the four-point correlation functions in N=4 SYM that should be valid at any coupling and away from the planar limit.

Figures (1)

• Figure 1: Penrose diagram of Minkowski space, $\tan({\tau \pm \theta \over 2})=t \pm |\vec{x}|$. a) We can consider a localized state (red) and first integrate over time and then take the large $r$ limit. This procedure should work in theories of massive and massless particles. b) In CFTs the energy flux (as well as any other flux) from the state created by an insertion of a local operator is carried away to the future null infinity, denoted as $\mathcal{I}^+$. In this figure, the symbols $i^+$ and $i^0$ stand for future time-like and spatial infinities, respectively. It is convenient to work with momentum eigenstates which are not localized. For these reasons we adopt a procedure where we first send the detectors to the future null infinity ${\cal I}^+$ and then integrate over the retarded (working) time.