  Landau's description of the excitations in a macroscopic system in terms of
quasiparticles stands out as one of the highlights in quantum physics. It
provides an accurate description of otherwise prohibitively complex many-body
systems, and has led to the development of several key technologies. In this
paper, we investigate theoretically the Landau effective interaction between
quasiparticles, so-called Bose polarons, formed by impurity particles immersed
in a Bose-Einstein condensate (BEC). In the limit of weak interactions between
the impurities and the BEC, we derive rigorous results for the effective
interaction. They show that it can be strong even for weak impurity-boson
interaction, if the transferred momentum/energy between the quasiparticles is
resonant with a sound mode in the BEC. We then develop a diagrammatic scheme to
calculate the effective interaction for arbitrary coupling strengths, which
recovers the correct weak coupling results. Using this, we show that the Landau
effective interaction in general is significantly stronger than that between
quasiparticles in a Fermi gas, mainly because a BEC is more compressible than a
Fermi gas. The interaction is particularly large near the unitarity limit of
the impurity-boson scattering, or when the quasiparticle momentum is close to
the threshold for momentum relaxation in the BEC. Finally, we show how the
Landau effective interaction leads to a sizeable shift of the quasiparticle
energy with increasing impurity concentration, which should be detectable with
present day experimental techniques.
