Hydrodynamic Cooperons in Electron Fluids: Schwinger--Keldysh Derivation and Quantum Corrections to Magnetoresistance

We develop a Schwinger--Keldysh effective theory for quantum-interference corrections in a two-dimensional electron system in the hydrodynamic regime. Starting from the clean hydrodynamic fixed point, we introduce a minimal random-friction disorder model that generates a finite momentum-relaxation time within the self-consistent Born approximation. The disorder-averaged theory then allows us to construct a hydrodynamic Cooperon and to compute the associated self-energy corrections to the collective modes. Conservation laws protect the density and momentum sectors, so that the leading quantum-coherence correction is forced into the spin-two stress sector. The associated stress self-energy renormalizes the shear viscosity and modifies both the Gurzhi response and its low-field magnetohydrodynamic signatures.