The functional renormalization group method is applied for a scalar theory in Minkowski space-time. It is argued that the appropriate choice of the subtraction point is more important in Minkowski than in Euclidean space-time. The parameters of the cutoff theory, defined by a subtraction point in the quasi-particle domain, are complex due to the mass-shell contributions to the blocking and the renormalization group flow becomes more involved. The Landau poles are avoided when the parameters are complexified. The absence of the UV pole owing to the marginal parameters makes the scalar theory asymptotically free in four dimensions. However the continuation of the trajectory beyond the regularized IR pole at the UV-IR crossover in the phase with spontaneously broken symmetry is possible only if the non-trivial saddle points to the blocking are taken into account.