The HTL Lagrangian at NLO: the photon case

TL;DR

This paper computes the two-loop correction to the photon self-energy in an electron-positron plasma, completing the NLO HTL effective Lagrangian and providing insights into soft photon propagators and plasmon dispersion relations.

Contribution

It provides the missing two-loop correction to the photon self-energy in the HTL framework at NLO, enabling more precise calculations of photon propagators in hot plasmas.

Findings

Finite NLO correction to the Debye mass.

Discovery of a new non-local structure in the photon propagator.

Calculation of plasmon dispersion relations at NLO.

Abstract

We calculate the two loop hard correction to the photon self-energy in an electron-positron plasma (EPP) for arbitrary soft momenta. This provides the only missing ingredient to obtain the Hard Thermal Loop (HTL) effective Lagrangian at next-to-leading order (NLO), and the full photon propagator at the same order. This result can be easily extended to obtain the soft photon propagator in a quark gluon plasma. We use the Keldysh representation of the real time formalism in the massless fermion limit, and dimensional regularization (DR) to regulate any ultraviolet (UV), infrared (IR) or collinear divergences that appear in the intermediate steps of the calculation. In the limit of soft photon momenta, our result is finite. It not only provides an ${\cal O}(\alpha)$ correction to the Debye mass, but also a new non-local structure. A consistent regularization of radial and angular integrals is crucial to get this new structure. As an application we calculate the plasmon dispersion relations at NLO.