Cubic scalar box contribution to a quartic coupling ($\delta g_4=-3g^4{I}_4$)

In a six-dimensional cubic scalar field theory, three labelled box Feynman diagrams give the one-loop local quartic vertex at zero external momentum. With an effective-action term $g_4{\varphi }^4/4!$, their contribution is $\delta g_4=-3g^4{I}_4$. To check its sign and multiplicity, expand the one-loop scalar effective action as $\frac{1}{2}\mathrm{Tr}\log (D+g\varphi )$, where $D=-{\partial }^2+m^2$. Its fourth-order term is $-g^4\mathrm{Tr}(D^{-1}\varphi {)}^4/8$, giving the stated coefficient. The associated amputated connected diagram insertion has the opposite sign.