  Cooling of muon beams for the next-generation lepton collider is necessary to
achieve its higher luminosity with fewer muons. In this paper we present an
idea to combine ionization cooling with parametric resonances that is expected
to lead to muon beams with much smaller transverse sizes. We describe a linear
magnetic transport channel where a half integer resonance is induced such that
the normal elliptical motion of particles in x-x' phase space becomes
hyperbolic, with particles moving to smaller x and larger x' at the channel
focal points. Thin absorbers placed at the focal points of the channel then
cool the angular divergence of the beam by the usual ionization cooling
mechanism where each absorber is followed by RF cavities. We present a theory
of Parametric-resonance Ionization Cooling (PIC), starting with the basic
principles in the context of a simple quadrupole-focused beam line. Then we
discuss detuning caused by chromatic, spherical, and non-linear field
aberrations and the techniques needed to reduce the detuning. We discuss the
requirement that PIC be accompanied by emittance exchange in order to keep the
momentum spread sufficiently small. Examples of PIC channel are presented,
along with computer simulations aimed at practical implementation of the
described theoretical concept.
