Dynamics of the Electron Pinch and Incoherent Tune Shift Induced by Electron Cloud

When a proton bunch passes through an electron cloud, the cloud electrons are attracted by the beam electric field and their density strongly increases near the beam centre. This gives rise to an incoherent proton tune shift, which depends on the longitudinal and radial position within the bunch. We present an analytical description of the ’electron pinch’ and the resulting proton tune shift for a circular symmetry and a Gaussian cloud, considering a linear transverse force and various longitudinal beam profiles. Benchmarking and extending the results by computer simulations, we can also explore the effects of a non-linear transverse force.