The Landau damping mechanism plays a crucial role in providing single-bunch stability in the LHC, high-luminosity LHC, and other existing as well as previous and future circular hadron accelerators. In this paper, the thresholds for the loss of Landau damping (LLD) in the longitudinal plane are derived analytically using the Lebedev matrix equation (1968) and the concept of the emerged van Kampen modes (1983). We have found that for the commonly used particle distribution functions from a binomial family, the LLD threshold vanishes in the presence of the constant inductive impedance $\mathrm{Im}Z/k$ above transition energy. Thus, the effect of the cutoff frequency or the resonant frequency of a broadband impedance on beam dynamics is studied in detail. The findings are confirmed by direct numerical solutions of the Lebedev equation as well as using the Oide-Yokoya method (1990). Moreover, the characteristics, which are important for beam operation, as the amplitude of residual oscillations and the damping time after a kick (or injection errors) are considered both above and below the threshold. Dependence of the threshold on particle distribution in the longitudinal phase space is also analyzed, including some special cases with a nonzero threshold for $\mathrm{Im}Z/k=\text{const}$. All main results are confirmed by macroparticle simulations and consistent with available beam measurements in the LHC.

中文翻译：

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在纵向平面上损失Landau阻尼的阈值

朗道阻尼机制在为大型强子对撞机，高发光度大型强子对撞机以及其他现有的以及以前和将来的圆形强子加速器中提供单束稳定性方面起着至关重要的作用。在本文中，利用Lebedev矩阵方程（1968）和出现的van Kampen模式（1983）的概念，分析得出了纵向平面中Landau阻尼（LLD）的损失阈值。我们发现，对于来自二项式族的常用粒子分布函数，在存在恒定电感阻抗的情况下，LLD阈值消失了$林ž/ķ$高于过渡能量。因此，详细研究了宽带阻抗的截止频率或谐振频率对光束动力学的影响。Lebedev方程的直接数值解以及使用Oide-Yokoya方法（1990年）证实了这一发现。此外，对于梁的运行而言很重要的特性是，残余振动的幅度和突跳后的阻尼时间（或注入误差）都被认为在阈值之上和之下。还分析了阈值对纵向相空间中粒子分布的依赖性，包括一些特殊情况，其中阈值非零$林ž/ķ=\text{const}$。所有主要结果均通过宏观粒子模拟得到证实，并且与大型强子对撞机中可用的光束测量结果一致。