= Single-vertex scalar two-point graph
{title2=$S=2^{k-1}(k-1)!$}
For interaction $\lambda_{2k}\phi^{2k}/(2k)!$, one vertex with two labelled external legs has $k-1$ independently paired <tadpole diagrams>. Counting contractions gives symmetry factor $S=2^{k-1}(k-1)!$. In a normalized <Euclidean path integral>, its amputated contribution is $-\lambda_{2k}T_D(m)^{k-1}/S$, where $T_D$ is the <tadpole integral>. It is linear in the coupling but has $k-1$ loops and a highest four-dimensional <ultraviolet pole> of order $k-1$ for nonzero <mass>.
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