Impact of beam coupling impedance on crab cavity noise induced emittance growth

Crab cavities will be deployed as a part of the High Luminosity Large Hadron Collider (HL-LHC) upgrade to mitigate the luminosity reduction induced by the crossing angle at the main experiments (ATLAS and CMS). Two prototype crab cavities have been installed in the CERN Super Proton Synchrotron (SPS) in 2018 for studies with proton beams. An issue of concern is the transverse emittance growth induced by noise in the crab cavity radio frequency (rf) system, which is anticipated to limit the performance of the HL-LHC. In measurements conducted in the SPS in 2018, the crab cavity noise-induced emittance growth was measured to be a factor of 4 lower than predicted from the existing analytical models. In this paper, it is shown that the observed discrepancy is explained by damping effects from the beam coupling impedance, which were not included in the models up to now. Using the van Kampen mode approach, a new theory is developed, suggesting that the impedance can separate the coherent tune from the incoherent spectrum leading to an effective reduction of the crab cavity rf noise-induced emittance growth. This mechanism is validated in tracking simulations using the SPS impedance model as well as in dedicated experimental measurements conducted in the SPS in 2022. The implications for the HL-LHC project are discussed.