Effective $\sigma$ Model Formulation for Two Interacting Electrons in a Disordered Metal

We derive an analytical theory for two interacting electrons in a $d$--dimensional random potential. Our treatment is based on an effective random matrix Hamiltonian. After mapping the problem on a nonlinear $\sigma$ model, we exploit similarities with the theory of disordered metals to identify a scaling parameter, investigate the level correlation function, and study the transport properties of the system. In agreement with recent numerical work we find that pair propagation is subdiffusive and that the pair size grows logarithmically with time.