Beam-envelope theory of ionization cooling

Linear beam-envelope theory of ionization cooling in 6D phase space has been systematically established in the past few years. In this paper, we briefly review the general formalism as well as the specific theories for a quadrupole channel and a bent-solenoidal channel with symmetric focusing. These channels play important roles in the design of cooling channels for the envisioned neutrino factories and muon colliders. The analytical solutions of these channels are relatively simple yet provide good understanding of cooling and heating mechanisms in both transverse and longitudinal phase spaces. Furthermore, the resulting formulae can be used to evaluate cooling channel designs the same way as the radiation integrals are used in storage ring designs