  Beliaev damping provides one of the most important mechanisms for dissipation
of quasiparticles through beyond-mean-field effects at zero temperature. Here
we present the first analytical result of Beliaev damping in low-energy
excitations of spin-$\frac{1}{2}$ interacting bosons with equal Rashba and
Dresslhaus spin-orbit couplings. We identify novel features of Beliaev decay
rate due to spin-orbit coupling, in particular, it shows explicit dependence on
the spin-density interaction and diverges at the interaction-modified phase
boundary between the zero-momentum and plane-wave phases. This represents a
manifestation of the effect of spin-orbit coupling in the beyond-mean-field
regime, which by breaking Galilean invariance couples excitations in the
density- and spin-channels. By describing the Beliaev damping in terms of the
observable dynamic structure factors, our results allow direct experimental
access within current facilities.
