= Quartic closure of scalar effective-potential counterterms
{title2=$V_{\rm div}=-[V''(\varphi)]^2/(32\pi^2\epsilon)$}
For $d=4-\epsilon$, the scalar one-loop determinant has the displayed local divergent potential. If $V$ is a quartic <polynomial>, then $V''$ has degree at most two and $(V'')^2$ has degree at most four. Its subtraction therefore uses the original polynomial couplings and a vacuum-energy term. A polynomial potential of degree $r>4$ instead produces degree $2r-4>r$, requiring new couplings. This is a direct potential-level illustration of <power counting in quantum field theory> and <renormalizability>.
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