One-loop scalar effective potential ($V_1(\phi )=\frac{[V^{\prime \prime }(\phi ){]}^2}{64{\pi }^2}\left[\log \frac{V^{\prime \prime }(\phi )}{{\mu }^2}-\frac{3}{2}\right]$)

For one canonically normalized real scalar field and $V^{\prime \prime }(\phi )>0$, the modified minimal subtraction scheme gives the displayed one-loop term. The Gaussian fluctuation approximation produces $\frac{1}{2}\int d^d{k}_E(2\pi {)}^{-d}\log (k_E^2+V^{\prime \prime }(\phi ))$. Its derivative in $V^{\prime \prime }$ is half the Euclidean massive loop integral with $n=1$. Integrating and subtracting its pole yields the formula, up to a field-independent constant. Other finite counterterms alter the scheme-dependent nonlogarithmic part. Negative curvature requires an analytic continuation and can yield an imaginary part indicating instability of that background.

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.