In the real-time formalism of the finite-temperature field theory, we compute the one-loop gluon self-energy in a semiquark-gluon plasma (QGP) where a background field $\mathcal{Q}$ has been introduced for the vector potential, leading to a nontrivial 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 $\mathcal{Q}=0$ 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 nonvanishing and reproduce the correct limit as $\mathcal{Q}\to 0$. 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 $\mathcal{Q}\ne 0$. Given the above results, we explicitly verify that the Kubo-Martin-Schwinger condition can be satisfied in a semi-QGP with a nonzero background field.

• Received 25 July 2022
• Accepted 14 September 2022

DOI:https://doi.org/10.1103/PhysRevD.106.054033

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Published by the American Physical Society