One-loop QED effective action at finite temperature and density beyond HTL limit

We develop a framework for computing the nonlocal one-loop effective action in scalar and spinor QED at finite temperature and chemical potential in arbitrary even spacetime dimensions. Based on the heat kernel expansion to quadratic order in background fields, the framework reduces the effective action form factors to a common fundamental kernel. By resumming the external-momentum expansion, we obtain explicit closed-form expressions for the massless form factors in terms of polylogarithms and their derivatives with respect to the order. The same method determines the coefficients of the small-mass expansion to arbitrary order. The resulting expressions retain the full dependence on external Euclidean momentum, temperature, and chemical potential, without invoking the soft-momentum approximation. We also derive the corresponding photon polarization operators and evaluate their static limits, obtaining the Debye screening masses and electromagnetic susceptibilities.