 Two-loop sextic counterterm in quartic Yukawa theory

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 $\mu =m+g{v}^2/2$. 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 $g^4{v}^6$ 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.