 Six-dimensional cubic scalar field theory

A canonically normalized real scalar field in six dimensions has mass dimension two. The cubic coupling is classically a marginal coupling, whereas a local quartic coupling has mass dimension minus two. This theory is perturbatively renormalizable by power counting in quantum field theory, but a real cubic Euclidean potential is unbounded below. Its expansion about a massive Gaussian theory is therefore formal perturbation theory, not a convergent positive functional integral at nonzero real $g$.