Fast High-Intensity Single-Bunch Transverse Coherent Instability in Synchrotrons due to a Broad-Band Resonator Impedance

The instability rise-time is computed when it is faster than the synchrotron period, using the mode-coupling formalism. The case is treated of a bunch interacting with a broad-band resonator impedance, and whose length is greater than the inverse of twice the resonance frequency. The formula is compared to the one obtained by Brandt and Gareyte in a beam break-up approach, and to the one first obtained by Ruth and Wang in a fast blow-up theory, and later re-derived by Kernel et al. in a post-head-tail formalism. Stabilisation by synchrotron oscillation is also discussed.