Two-particle collisional coordinate shifts and hydrodynamic anomalous Hall effect in systems without Lorentz invariance

We show that electrons undergoing a two-particle collision in a crystal experience a coordinate shift that depends on their single-particle Bloch wave functions, and derive a gauge-invariant expression for such shift, valid for arbitrary band structures, and arbitrary two-particle interaction potentials. As an application of the theory, we consider two-particle coordinate shifts for Weyl fermions in space of three spatial dimensions. We also demonstrate that such shifts in general contribute to the anomalous Hall conductivity of a clean electron liquid.