Quartic closure of scalar effective-potential counterterms ($V_{\mathrm{div}}=-[V^{\prime \prime }(\phi ){]}^2/(32{\pi }^2ϵ)$)

For $d=4-ϵ$, the scalar one-loop determinant has the displayed local divergent potential. If $V$ is a quartic polynomial, then $V^{\prime \prime }$ has degree at most two and $(V^{\prime \prime }{)}^2$ 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.