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Landau damping mechanism plays a crucial role in providing single-bunchstability in LHC, High-Luminosity LHC, other existing as well as previous andfuture (like FCC) circular hadron accelerators. In this paper, the thresholdsfor the loss of Landau damping (LLD) in the longitudinal plane are derivedanalytically using the Lebedev matrix equation (1968) and the concept of theemerged van Kampen modes (1983). We have found that for the commonly-usedparticle distribution functions from a binomial family, the LLD thresholdvanishes in the presence of the constant inductive impedance Im$Z/k$ abovetransition energy. Thus, the effect of the cutoff frequency or the resonantfrequency of a broad-band impedance on beam dynamics is studied in detail. Thefindings are confirmed by direct numerical solutions of the Lebedev equation aswell as using the Oide-Yokoya method (1990). Moreover, the characteristics,which are important for beam operation, as the amplitude of residualoscillations and the damping time after a kick (or injection errors) areconsidered both above and below the threshold. Dependence of the threshold onparticle distribution in the longitudinal phase space is also analyzed,including some special cases with a non-zero threshold for Im$Z/k = const$. Allmain results are confirmed by macro-particle simulations and consistent withavailable beam measurements in the LHC.

Thresholds for loss of Landau damping in longitudinal plane