= Six-dimensional cubic scalar field theory
{title2=$\mathcal L=\tfrac12(\partial\phi)^2+\tfrac12m^2\phi^2+g\phi^3/3!$}
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 $g$.
Back to article page
