Plasma Dynamics in Higher-Derivative Electrodynamics: A Renormalised Two-Loop Framework

We present a finite-temperature study of Bopp-Podolsky electrodynamics, following electron-proton plasmas through one- and two-loop order with dimensional regularisation and hard-thermal-loop resummation. The higher-derivative operator is found to generate no new ultraviolet divergences; all counter-terms reduce to the single photon wave-function factor of ordinary QED. The static inter-particle force acquires a double-Yukawa profile, the familiar Debye term plus an opposite-signed contribution from the heavy Podolsky pole that removes the Coulomb singularity at sub-femtometre distances, providing an intrinsic ultraviolet completion of electrostatics. Gauge symmetry drives the transverse photon self-energy to zero at vanishing momentum, so no magnetic screening mass appears at any perturbative order. In a covariantly constant background the full two-loop sunset diagram yields a single, dimension-eight operator suppressed by T^2/M^2, implying permille-level shifts in thermodynamic quantities for realistic plasmas. The exact Debye mass and a leading-log calculation show the dc electrical conductivity exceeds its QED value by less than 10^-4. Conditions for observable Podolsky plasmons and cosmological constraints are identified, supplying precise benchmarks for future strong-field, collider and lattice investigations.