Linear theory of transverse and longitudinal ionization cooling in a quadrupole channel

Ionization cooling is the best-known cooling mechanism for the envisioned muon colliders and neutrino factories. In this paper, using the moment-equation approach, we present a linear theory of ionization cooling dynamics in 6D phase space in a quadrupole focusing channel. A simple set of differential equations that governs the evolution of both the transverse and longitudinal emittances is derived, and closed-form solutions are given. Two new significant heating processes have been identified. This theory is analogous to the standard linear theory in electron storage rings. Multiple scattering integrals and energy straggling integrals, quantities like the synchrotron radiation integrals, are introduced to specify the cooling process and the equilibrium emittances in a periodic cooling channel.