= Two-loop sextic counterterm in quartic Yukawa theory
{title2=$\delta\lambda_6\propto g^4$}
Four quartic Yukawa vertices on a <fermion loop> with one internal scalar chord give six external scalar legs, two loops and overall logarithmic degree. The coefficient is nonzero: in a constant scalar background $v$, the induced cubic <Yukawa coupling> is $y=gv$ and the fermion mass is $\mu=m+gv^2/2$. The two-loop <sunset diagram> in the <quantum effective potential> has a logarithmic pole proportional to $y^2(4\mu^2-\mathcal B^2)$, where $\mathcal B^2$ is the background scalar squared mass. Its $v^6$ term is $g^4v^6$ when the tree sextic coupling vanishes. Thus zero renormalized sextic coupling at one scale does not eliminate the sextic <counterterm> or make that zero stable under <renormalization>.
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