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.

中文翻译：

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半夸克胶子等离子体中的实时硬热环胶子自能

在有限温度场理论的实时形式中，我们计算了半夸克胶子等离子体（QGP）中的单环胶子自能，其中背景场$\mathcal{问}$已经为矢量势引入，导致在去限制阶段的 Polyakov 环具有非平凡的期望值。获得了在硬热回路近似中胶子自能达到次领先级的显式结果。我们发现对于迟滞/先进胶子自能，在次领先阶的相应贡献在形式上类似于众所周知的结果$\mathcal{问}=0$其中德拜质量上的背景场修改完全编码在第二个伯努利多项式中。对称胶子自能中的主要顺序贡献具有相同的特征，其中背景场修改变得更加复杂，包括三角函数和伯努利多项式。这些贡献是不消失的，并重现了正确的极限$\mathcal{问}\to 0$. 此外，对迟滞/超前胶子自能的主要贡献和对对称胶子自能的次要贡献是全新的，因为它们只存在于$\mathcal{问}\ne 0$. 鉴于上述结果，我们明确地验证了 Kubo-Martin-Schwinger 条件可以在具有非零背景场的半 QGP 中得到满足。