= Cubic scalar box contribution to a quartic coupling
{title2=$\delta g_4=-3g^4I_4$}
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 $g_4\phi^4/4!$, their contribution is $\delta g_4=-3g^4I_4$. To check its sign and multiplicity, expand the <one-loop scalar effective action> as $\tfrac12\operatorname{Tr}\log(D+g\phi)$, where $D=-\partial^2+m^2$. Its fourth-order term is $-g^4\operatorname{Tr}(D^{-1}\phi)^4/8$, giving the stated coefficient. The associated amputated connected diagram insertion has the opposite sign.
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