Quasiparticle weight and effective mass of the Landau liquid from the complex-frequency formalism

We present a method for calculating the effective mass and quasiparticle weight of a Landau Fermi liquid using time-ordered Green’s functions and a Wick rotation to the imaginary frequency axis. This technique simplifies the evaluation of self-energy derivatives by ensuring analytic properties compatible with contour deformations. We apply the method to the electron gas in both two and three spatial dimensions within the random phase approximation (RPA), and derive explicit expressions for the static and dynamic contributions to the self-energy. Our results confirm known limits at weak coupling and offer an accurate and transparent route to evaluate effective mass and quasiparticle weight, with clear extensions possible for systems including vertex corrections or local-field effects and band-gap renormalization.