The renormalized and Renormalization-Group invariant Hartree-Fock approximation

We study the renormalization problem for the Hartree--Fock approximation of the $O(N)-$invariant $\phi^4$ model in the symmetric phase and show how to systematically improve the corresponding diagrammatic resummation to achieve the correct renormalization properties of the effective field equations, including Renormalization--Group invariance with the one--loop beta function. These new Hartree--Fock dynamics is still of mean field type but includes memory effects which are generically nonlocal also in space.