Randomness in infinitesimal extent in the McLerran-Venugopalan model

We study the discrepancy between the analytical definition and the numerical implementation of the McLerran-Venugopalan model. The infinitesimal extent of a fast-moving nucleus should retain longitudinal randomness in the color source distribution even when the longitudinal extent approximates zero due to the Lorentz contraction, which is properly taken into account in the analytical treatment. We point out that the longitudinal randomness is lost in numerical simulations because of the lack of the path-ordering of the Wilson line along the longitudinal direction. We quantitatively investigate how much the results with and without longitudinal randomness differ from each other. We finally mention that the discrepancy could be absorbed in a choice of the model parameter in the physical unit, and, nevertheless, it should be an important problem in the McLerran-Venugopalan model as a fully theoretical approach.