The Asymptotic Form of Non-Global Logarithms, Black Disc Saturation, and Gluonic Deserts

TL;DR

This paper develops an asymptotic perturbation theory for non-global logarithms in QCD, revealing a universal gaussian tail in their distribution and connecting jet evolution to saturation physics and the black-disc limit.

Contribution

It introduces a novel asymptotic expansion for the BMS equation, linking non-global logarithms to the black-disc limit of the BK equation in saturation physics.

Findings

Distribution has a gaussian tail at large non-global logs

Decay width is analytically computed and geometry independent

Asymptotics map to the black-disc unitarity limit in saturation physics

Abstract

We develop an asymptotic perturbation theory for the large logarithmic behavior of the non-linear integro-differential equation describing the soft correlations of QCD jet measurements, the Banfi-Marchesini-Smye (BMS) equation. This equation captures the late-time evolution of radiating color dipoles after a hard collision. This allows us to prove that at large values of the control variable (the non-global logarithm, a function of the infra-red energy scales associated with distinct hard jets in an event), the distribution has a gaussian tail. We compute the decay width analytically, giving a closed form expression, and find it to be jet geometry independent, up to the number of legs of the dipole in the active jet. Enabling the asymptotic expansion is the correct perturbative seed, where we perturb around an anzats encoding formally no real emissions, an intuition motivated by the buffer region found in jet dynamics. This must be supplemented with the correct application of the BFKL approximation to the BMS equation in collinear limits. Comparing to the asymptotics of the conformally related evolution equation encountered in small-x physics, the Balitisky-Kovchegov (BK) equation, we find that the asymptotic form of the non-global logarithms directly maps to the black-disc unitarity limit of the BK equation, despite the contrasting physical pictures. Indeed, we recover the equations of saturation physics in the final state dynamics of QCD.