Drude model and Lifshitz formula

Since nearly 10 years, it is known that inserting the permittivity of the Drude model into the Lifshitz formula for free energy causes a violation of the third law of thermodynamics. In this paper we show that the standard Matsubara formulation for free energy contains a contribution that is non-perturbative in the relaxation parameter. We argue that the correct formula must have a perturbative expansion and conclude that the standard Matsubara formulation with the permittivity of the Drude model inserted is not correct. We trace the non-perturbative contribution in the complex frequency plane, where it shows up as a self-intersection or a bifurcation of the integration path.