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We consider a 3D homogeneous superfluid at low temperature $T$ with 2 typesof excitations, gapless phonons with a linear dispersion relation at lowwavenumber, and gapped quasiparticles with a quadratic dispersion relationaround extrema. We calculate the scattering amplitude of a phonon on aquasiparticle to leading order in $T$ for all subsonic quasiparticlevelocities, with a $S$-matrix formalism between exact asymptotic states dressedby virtual phonons. We then characterize the erratic motion of thequasiparticle in the superfluid due to its unceasing collisions with thermalphonons through mean force $F(k)$, longitudinal and transverse $k$-dependentmomentum diffusion coefficients, and spatial diffusion coefficient. At theminimum location $k_0$ of the dispersion relation, where the velocity vanishes,$F(k)$ varies linearly with velocity with an isotropic friction coefficient; if$k_0=0$, the momentum diffusion is also isotropic and $F(k_0)=0$; if $k_0>0$,it is not, and $F(k_0)$ is nonzero but subleading with respect to friction byone order in $T$. The velocity time correlation function, whose integral is thespatial diffusion coefficient, decays with the mean velocity damping rate if$k_0=0$; if $k_0>0$, it has a second exponential component, with an amplitudeand a damping rate lower by a factor $\propto T$ (it is the velocity directionthermalization rate). We also characterize force and momentum diffusion closeto the stability domain sonic edge. Our general expressions are expected to beexact to leading order in $T$. We illustrate them in the BCS approximation, fora fermionic quasiparticle (an unpaired fermion) in a superfluid of spin 1/2fermions, realisable with cold atoms in flat bottom traps. We also refute thestatement of Lerch, Bartosch and Kopietz (2008), that there would be nofermionic quasiparticle in such a superfluid.

Random walk of a massive quasiparticle in the phonon gas of an ultralow temperature superfluid