Hadron beam evolution in microbunched electron cooling

The technique of microbunched electron cooling (MBEC) is a coherent cooling scheme with possible applications in high-energy hadron and electron-ion machines. In our previous work we analyzed the cooling of the hadron energy spread and transverse emittance using a one-dimensional (1D) technique that tracked the microscopic fluctuations in the hadron and electron beams. However, in order to obtain analytical expressions for our key quantities, we limited ourselves to calculating and optimizing only the initial values of the cooling rates. In this paper, we extend our approach so that it properly addresses the issue of the long-term, dynamic evolution of the hadron beam. In order to do so, it becomes necessary to consider the synchrotron motion of the hadron beam, in conjunction with the effects of diffusion and intrabeam scattering (IBS). With these modifications, our formalism allows us to develop a simple numerical tool that can effectively model the final state of hadron beam after many passages through the MBEC cooler.