The renormalized and Renormalization-Group invariant Hartree-Fock approximation

We study the renormalization problem for the Hartree–Fock approximation of the $O(N)-$invariant ${\varphi }^4$ 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.