The non-linear evolution of jet quenching

We construct a generalization of the JIMWLK Hamiltonian, going beyond the eikonal approximation, which governs the high-energy evolution of the scattering between a dilute projectile and a dense target with an arbitrary longitudinal extent (a nucleus, or a slice of quark-gluon plasma). Different physical regimes refer to the ratio L/τ between the longitudinal size L of the target and the lifetime τ of the gluon fluctuations. When L/τ ≪1, meaning that the target can be effectively treated as a shockwave, we recover the JIMWLK Hamiltonian, as expected. When L/τ ≫1, meaning that the fluctuations live inside the target, the new Hamiltonian governs phenomena like transverse momentum broadening and radiative energy loss, which accompany the propagation of an energetic parton through a dense QCD medium. Using this Hamiltonian, we derive a non-linear equation for the dipole amplitude (a generalization of the BK equation), which describes the high-energy evolution of jet quenching. As compared to the original BK-JIMWLK evolution, the new evolution is remarkably different: the plasma saturation momentum evolves much faster with increasing energy (or decreasing Bjorken’s x) than the corresponding scale for a shockwave. This widely opens the transverse phase-space for the evolution in the vicinity of the saturation line and implies the existence of large radiative corrections, enhanced by the double logarithm ln2(LT ), with T the temperature of the medium. This confirms from a wider perspective a recent result by Liou, Mueller, and Wu (arXiv:1304.7677). The dominant, double-logarithmic, corrections to the dipole amplitude are smooth enough to be absorbed into a renormalization of the jet quenching parameter q^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{q} $$\end{document}. This renormalization is universal: it applies to all the phenomena, like the transverse momentum broadening or the radiative energy loss, which can be computed from the dipole amplitude.