Symmetric Contour Integration for Pole Analysis of 2D Correlation Functions: Application to Gaussian-Charge Plasma

Two-dimensional (2D) correlation functions are central to understanding structural crossovers in soft-core fluids; however, their asymptotic analysis is hindered by the Hankel-transform kernel, whose asymptotic representation introduces a term that breaks the natural conjugate symmetry of the poles. To address this, we present a symmetric contour integration scheme that restores symmetry at the level of the integration path. By employing quarter-circle contours in the first and fourth quadrants, the method captures conjugate pole pairs simultaneously and evaluates the sine term from the Bessel-function asymptotic without variable transformation or real-part extraction, yielding closed-form analytic expressions for the long-range decay of the density–density correlation function. The approach is demonstrated for a 2D Gaussian-charge one-component plasma under the random phase approximation at intermediate coupling, where the pole analysis provides direct access to the oscillation wavelength and decay length. In the high-density regime, the pole equations simplify to a form amenable to a Lambert W-function approximation, revealing a logarithmic scaling of correlation lengths even at moderate coupling. These findings establish symmetric contour integration as a transparent and versatile framework for pole-resolved asymptotics in 2D liquids.