= Wick-rotated cutoff tadpole mass shift
{c}
{title2=$\delta m^2=\lambda_0[\Lambda^2-m_0^2\log(1+\Lambda^2/m_0^2)]/(32\pi^2)$}
With interaction $-\lambda_0\phi^4/4!$, the <tadpole diagram> has <symmetry factor> one half. After <Wick rotation> and a rotationally invariant Euclidean cutoff, its <self-energy> is $(\lambda_0/2)\int_{|k|<\Lambda}d^4k\,(2\pi)^{-4}(k^2+m_0^2)^{-1}$. The sphere area $2\pi^2$ and radial integral $\int_0^\Lambda k^3dk/(k^2+m_0^2)=\tfrac12[\Lambda^2-m_0^2\log(1+\Lambda^2/m_0^2)]$ give the displayed shift. It is independent of external momentum, so no one-loop momentum-dependent <wave-function renormalization> arises from this graph. Different regulator schemes can treat power divergences differently while defining the same chosen <pole mass>.
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