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 a mean-field type but includes memory effects which are generically nonlocal also in space.