Abstract
The high-energy evolution of Wilson line operators, which at leading order is described by the Balitsky-JIMWLK equations, receives large radiative corrections enhanced by single and double collinear logarithms at next-to-leading order and beyond. We propose a method for resumming such logarithmic corrections to all orders, at the level of the Langevin formulation of the JIMWLK equation. The ensuing, collinearly-improved Langevin equation features generalized Wilson line operators, which depend not only upon rapidity (the logarithm of the longitudinal momentum), but also upon the transverse size of the color neutral projectile to which the Wilson lines belong. This additional scale dependence is built up during the evolution, via the condition that the successive emissions of soft gluons be ordered in time. The presence of this transverse scale in the Langevin equation furthermore allows for the resummation of the one-loop running coupling corrections.
Highlights
JHEP08(2016)083 described by the Balitsky-JIMWLK equations1 [4,5,6,7,8,9,10]
The only exact solutions to the Balitsky hierarchy were obtained via numerical simulations of this Langevin process [13,14,15,16,17,18]. (All the other studies of this evolution have involved additional approximations, like the multi-color limit Nc 1, in which the Balitsky-JIMWLK hierarchy reduces to the Balitsky-Kovchegov (BK) equation [4, 19], or various mean field approximations to the CGC weight function [20,21,22,23,24,25,26,27].) Importantly, this Fokker-Planck structure supports the probabilistic interpretation of the CGC effective theory: it ensures that the CGC weight function remains positive definite and properly normalized under the evolution
We have constructed a collinearly-improved version of the leading order JIMWLK evolution which allows for a complete resummation of the large radiative corrections enhanced by double collinear logarithms together with partial resummations of the single collinear logarithms and of the running coupling corrections
Summary
Our starting point is the Langevin formulation [11, 12] of the leading order (LO) JIMWLK equation [5,6,7,8,9,10]. The unitary matrix Un† (x) should not be viewed anymore as a genuine Wilson line — if by a ‘genuine Wilson line’ we understand the path-ordered phase built with the field A− according to eq (2.6) —, but rather as a more complicated scattering operator, which refers to a multi-partonic system built with the original quark at x together with the n soft gluons generated by the evolution. These gluons are strongly ordered in k+ — the longitudinal momentum for a right-mover — which decreases from the projectile towards the target. Where αs ≡ αsNc/π and SY(2) is the average S-matrix for a system of two dipoles, defined as SY(2)(x1, y1; x2, y2) ≡
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