Integrability of the diffusion pole in the diagrammatic description of noninteracting electrons in a random potential

We discuss restrictions on the existence of the diffusion pole in the translationally invariant diagrammatic treatment of disordered electron systems. We use the Bethe-Salpeter equations for the two-particle vertex in the electron-hole and the electron-electron scattering channels and derive for systems with time reversal symmetry a nonlinear integral equation the two-particle irreducible vertices from both channels must obey. We use this equation to test the existence of the diffusion pole in the two-particle vertex. We find that a singularity of the diffusion pole can exist only if it is integrable, that is only in the metallic phase in dimensions $d>2$.