  Using the effective action formalism, we investigate collective excitations
of a dilute mixture of a Bose gas and a two component Fermi gas when both
bosons and fermions have undergone superfluid transitions. We show that there
is a repulsion between Bogoliubov and Anderson modes, which has important
implications including disappearance of boson superfluidity. We derive an
analytic expression for the long-wavelength dispersion relation of the mixture
at zero temperature and give a condition for the instability. We also
numerically calculate the damping rate due to boson-fermion interaction at
finite temperatures and show that the two modes are stable at sufficiently low
temperatures.
