On the imaginary part of the next-to-leading-order static gluon self-energy in an anisotropic plasma

TL;DR

This paper investigates the imaginary part of the static gluon self-energy at next-to-leading order in an anisotropic plasma, revealing that correct real-time formalism shows this part to be zero, contrary to naive imaginary-time calculations.

Contribution

It demonstrates that the static gluon self-energy's imaginary part vanishes at next-to-leading order when using the proper real-time formalism, correcting previous conjectures based on naive approaches.

Findings

Naive imaginary-time calculations suggest a nonzero imaginary part.

Proper real-time formalism shows the static gluon self-energy is real at NLO.

Results impact understanding of plasma instabilities and their regulation.

Abstract

Using hard-loop (HL) effective theory for an anisotropic non-Abelian plasma, which even in the static limit involves nonvanishing HL vertices, we calculate the imaginary part of the static next-to-leading-order gluon self energy in the limit of a small anisotropy and with external momentum parallel to the anisotropy direction. At leading order, the static propagator has space-like poles corresponding to plasma instabilities. On the basis of a calculation using bare vertices, it has been conjectured that, at next-to-leading order, the static gluon self energy acquires an imaginary part which regulates these space-like poles. We find that the one-loop resummed expression taken over naively from the imaginary-time formalism does yield a nonvanishing imaginary part even after including all HL vertices. However, this result is not correct. Starting from the real-time formalism, which is required in a non-equilibrium situation, we construct a resummed retarded HL propagator with correct causality properties and show that the static limit of the retarded one-loop-resummed gluon self-energy is real. This result is also required for the time-ordered propagator to exist at next-to-leading order.