Destabilising Effect of the LHC Transverse Damper

Three questions motivated this study for the CERN Large Hadron Collider in terms of beam stability: (i) why a chromaticity close to zero seemed more critical than predicted during Run 1 (in 2011 and 2012) and during Run 2 (in 2015)?; (ii) why some past simulations with a chromaticity close to zero revealed a more critical situation with the transverse damper than without?; (iii) what should be the minimum operational chromaticity in the future in the LHC and High-Luminosity LHC? A new Vlasov solver (called GALACTIC) was developed to shed light on the destabilising mechanism of the transverse damper, which is a potential contributor to explain the LHC observation. Due to the features, which are discussed in this paper, the name 'ISR (for Imaginary tune Split and Repulsion) instability' is suggested for this new kind of single-bunch instability with zero chromaticity.