Critical line of the ${\mathrm{\Phi }}^4$ scalar field theory on a 4D cubic lattice in the local potential approximation

We establish the critical line of the one-component ${\mathrm{\Phi }}^4$ (or Landau-Ginzburg) model on a simple four dimensional cubic lattice. Our study is performed in the framework of the non-perturbative renormalization group in the local potential approximation with a soft infra-red regulator. The transition is found to be of second order even in the Gaussian limit where first order would be expected according to some recent theoretical predictions.