Triviality proof for mean-field φ⁴ theories in four dimensions

The differential equations of the Wilson renormalization group are a powerful tool to study the Schwinger functions of Euclidean quantum field theory. In particular renormalization theory can be based entirely on inductively bounding their perturbatively expanded solutions. Recently the solutions of these equations for scalar field theory have been analysed rigorously without recourse to perturbation theory, at the cost of restricting to the mean-field approximation [Kopper, Ann. Henri Poincare 23, 3453-3492 (2022)]. In particular it was shown there that one-component φ⁴-theory in four dimensions is trivial if the bare coupling constant of the UV regularized theory is not large. This paper presents progress w.r.t. [Kopper, Ann. Henri Poincare 23, 3453-3492 (2022)]: (1) The upper bound on the bare coupling is sent to infinity and the proof is extended to O(N) vector models. (2) The unphysical infrared cutoff used in [Kopper, Ann. Henri Poincare 23, 3453-3492 (2022)] for technical simplicity is replaced by a physical mass.