  In a one-dimensional weakly interacting Bose-Fermi mixture one branch of
elementary excitations is well described by the Bogoliubov spectrum. Here we
use the microscopic theory to study the decay of such quasiparticle
excitations. The main scattering process which leads to their decay is the
backscattering of a Bogoliubov quasiparticle off the Fermi sea, where a
particle-hole pair is excited. For a low-momentum quasiparticle (phonon) of
momentum $q$, we find that the decay rate scales as $q^3$ provided $q$ is
smaller than the Fermi momentum $k_F$, while in the opposite case the decay
behaves as $q^2$. If the ratio of the masses of fermions and bosons equals to
the ratio of the boson-fermion and the boson-boson interaction strengths, the
decay rate changes dramatically. It scales as $q^7$ for $q<k_F$ while we find
$q^6$ scaling at $q>k_F$. For a high momentum Bogoliubov quasiparticle, we find
a constant decay rate for $q<k_F$, while it scales as $1/q$ for $q>k_F$. We
also find an analytic expression for the decay rate in the crossover region
between low and high momenta. The decay rate is a continuous, but nonanalytic
function of the momentum at $q=k_F$. In the special case when the parameters of
our system correspond to the integrable model, we observe that the decay rate
vanishes.
