Photon self-energy at all temperatures and densities in all of phase space

TL;DR

This paper develops comprehensive analytic approximations for the photon self-energy across all temperatures, densities, and phase space, revealing significant differences between on-shell and off-shell behaviors in plasma environments.

Contribution

It introduces accurate, universal analytic formulas for the photon self-energy off-shell, extending previous on-shell approximations and connecting with the Lindhard response function.

Findings

Off-shell self-energy differs qualitatively from on-shell cases.

Derived formulas are accurate across all phase space regions.

Reproduces the Lindhard response function for longitudinal modes.

Abstract

In an isotropic background comprised of free charges, the transverse and longitudinal modes of the photon acquire large corrections to their dispersion relations, described by the in-medium photon self-energy. Previous work has developed simple approximations that describe the propagation of on-shell photons in plasmas of varying temperatures and densities. However, off-shell excitations can also receive large medium-induced corrections, and the on-shell approximations have often been used in an effort to capture these effects. In this work we show that the off-shell self-energy can be qualitatively very different than the on-shell case. We develop analytic approximations that are accurate everywhere in phase space, especially in classical and degenerate plasmas. From these, we recover the on-shell expressions in the appropriate limit. Our expressions also reproduce the well-known Lindhard response function from solid-state physics for the longitudinal mode.