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Choose four of the six fields at one sextic vertex to be slow and contract the remaining two fast fields into a tadpole diagram. There are $\left(\genfrac{}{}{0ex}{}{6}{4}\right)=15$ choices. Ifthen the first term of the cumulant expansion contributesAfter the canonical coordinate and field rescaling, its contribution to the quartic coupling is

At order $g_0^2$, use the two-point sunset diagram: two quartic vertices are joined by three internal propagators, with one external line attached to each vertex. Unlike the one-vertex tadpole diagram, its self-energy $\Sigma (p)$ depends nontrivially on the external momentum $p$. The coefficient of $p^2$ in the expansion of $\Sigma (p)$ changes the kinetic term, so normalizing that term requires wave-function renormalization and gives a nonzero field anomalous dimension.

The leading mass corrections are one-vertex tadpole diagrams. In index notation, one contraction closes a freely summed component loop and is proportional to $N$; the two exchange contractions force the internal component to equal the external one and are not proportional to $N$. Including their multiplicities givesThus the $N$ in $N+2$ comes from the closed index loop, while the $2$ comes from the two same-component contractions.