  We calculate the finite-temperature r-mode spectrum of a superfluid neutron
star accounting for both muons in the core and the entrainment between neutrons
and protons. We show that the standard perturbation scheme, considering the
rotation rate as an expansion parameter, breaks down in this case. We develop
an original perturbation scheme which circumvents this problem by treating both
the perturbations due to rotation and (weak) entrainment simultaneously.
Applying this scheme, we propose a simple method for calculating the superfluid
r-mode eigenfrequency in the limit of vanishing rotation rate. We also
calculate the r-mode spectrum at finite rotation rate for realistic
microphysics input (adopting, however, the Newtonian framework and Cowling
approximation when considering perturbed oscillation equations) and show that
the normal r-mode exhibits resonances with superfluid r-modes at certain values
of temperatures and rotation frequencies in the parameter range relevant to
neutron stars in low-mass X-ray binaries (LMXBs). This turns the recently
suggested phenomenological model of resonance r-mode stabilization into a
quantitative theory, capable of explaining observations. A strong dependence of
resonance rotation rates and temperatures on the neutron superfluidity model
allows us to constrain the latter by confronting our calculations with the
observations of neutron stars in LMXBs.
