Stability limits with Landau damping in the FCC-hh

The resistive wall impedance is one of the expected main drivers of transverse beam instabilities in the proposed Future Circular Collider hadron-hadron option (FCC-hh). We obtain the resistive wall impedance for the FCC-hh beam screen from a two-dimensional finite element solver. The impedances and resulting growth rates are compared to the LHC, using similar models for the resistivity of the copper layer. Similar to the LHC and in addition to active feedback, dedicated octupole magnets together with a finite chromaticity should be employed for Landau damping, as a cure against transverse beam instabilities. The stability boundary provided by an LHC-like octupoles configuration in combination with an electron lens is obtained from a dispersion relation including the two-dimensional tune spreads. The prediction from the simple dispersion relation are compared to the corresponding beam transfer function and to the stability boundaries reconstructed using particle tracking with an effective impedance. The electron cloud induced tune spreads and their scaling with higher energy and smaller beam pipe radius are estimated. Besides the important estimation of growth rates and stability threshold for FCC-hh we also try to improve the understanding of the scaling of coherent instabilities and their thresholds with energy towards a possible highest-collider limit, using the example of two high-energy colliders, the existing LHC and the proposed FCC-hh.