= One-loop scalar effective potential
{title2=$V_1(\varphi)=\frac{[V''(\varphi)]^2}{64\pi^2}\left[\log\frac{V''(\varphi)}{\mu^2}-\frac32\right]$}
For one canonically normalized <real scalar field> and $V''(\varphi)>0$, the <modified minimal subtraction scheme> gives the displayed one-loop term. The <Gaussian fluctuation approximation> produces $\tfrac12\int d^dk_E\,(2\pi)^{-d}\log(k_E^2+V''(\varphi))$. Its derivative in $V''$ 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.
Back to article page
