= One-loop massive phi-fourth counterterms
{title2=$\delta m^2=\lambda m^2/(16\pi^2\epsilon),\quad\delta\lambda=3\lambda^2/(16\pi^2\epsilon)$}
With interaction $-\mu^\epsilon\lambda\phi^4/4!$ and $d=4-\epsilon$, the two-point <tadpole diagram> has <symmetry factor> $1/2$. The <Euclidean massive loop integral> gives its pole through $I_{d,1}=-m^2/(8\pi^2\epsilon)+O(1)$. The three four-point <bubble diagrams> each have symmetry factor $1/2$, and the <massive scalar bubble pole in four dimensions> gives the quartic counterterm. The tadpole is independent of external <momentum>, so no one-loop <wave-function renormalization> is required. A vacuum-energy counterterm is additionally needed if vacuum diagrams are retained.
Back to article page
