We study the renormalization problem for the Hartree-Fock approximation of the ON invariant 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.

The standard Hartree-Fock approximation of the ON-invariant 4 model suffers from serious renormalization problems. In addition, when the symmetry is spontaneously broken, another shortcoming appears in relation to the Goldstone bosons: they fail to be massless in the intermediate states. In this work, within the framework of out-of-equilibrium Quantum Field Theory, we propose a class of systematic improvements of the Hartree-Fock resummation which overcomes all the above mentioned difficulties while ensuring also exact Renormalization-Group invariance to one loop.