Real-time hard-thermal-loop gluon self-energy in a semiquark-gluon plasma

TL;DR

This paper calculates the one-loop gluon self-energy in a semi-quark-gluon plasma with a background field, revealing how the background modifies the self-energy and confirming the Kubo-Martin-Schwinger condition in this context.

Contribution

It provides explicit next-to-leading order results for gluon self-energy in a semi-QGP with background field, extending known results to non-zero background scenarios.

Findings

Next-to-leading order contributions are analogous to the zero-background case for retarded/advanced self-energy.

Background field modifies the Debye mass through Bernoulli polynomials.

New contributions at non-zero background field are identified and verified to satisfy KMS condition.

Abstract

In the real time formalism of the finite-temperature field theory, we compute the one-loop gluon self-energy in a semi-quark-gluon plasma (QGP) where a background filed ${\cal Q}$ has been introduced for the vector potential, leading to a non-trivial expectation value for the Polyakov loop in the deconfined phase. Explicit results of the gluon self-energies up to the next-to-leading order in the hard-thermal-loop approximation are obtained. We find that for the retarded/advanced gluon self-energy, the corresponding contributions at next-to-leading order are formally analogous to the well-known result at ${\cal Q}=0$ where the background field modification on the Debye mass is entirely encoded in the second Bernoulli polynomials. The same feature is shared by the leading order contributions in the symmetric gluon self-energy where the background field modification becomes more complicated, including both trigonometric functions and the Bernoulli polynomials. These contributions are non-vanishing and reproduce the correct limit as ${\cal Q} \rightarrow 0$. In addition, the leading order contributions to the retarded/advanced gluon self-energy and the next-to-leading order contributions to the symmetric gluon self-energy are completely new as they only survive at ${\cal Q}\neq0$. Given the above results, we explicitly verify that the Kubo-Martin-Schwinger condition can be satisfied in a semi-QGP with non-zero background field.