Dense-dilute duality at work: Dipoles of the target

We explore the properties of the QCD high energy evolution in the limit of a dilute target. Using the recently established property of self-duality of the evolution operator, we show how to properly define the target gluon and dipole creation operators in terms of dual Wilson lines (dual eikonal factors). We explain how to expand these operators in terms of the functional derivatives of the color charge density, in the situation when they act on the eikonal factors of the projectile partons. We explicitly calculate the expansion of the high energy evolution operator to fourth order in the functional derivatives. Our result is infrared and ultraviolet finite, but does not coincide with the formula given by Mueller, Shoshi, and Wong. We resolve this discrepancy by showing that the identification of the dipole creation and annihilation operators used in Mueller et al. is incomplete, and provide the required corrections to these definitions. The use of the corrected operators in the calculational framework of Mueller et al. reproduces our result. We also prove that there is no discrepancy between the expansion of the JIMWLK equation and the dualization of the expansion of the weak field limit.