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The leading mass correction is the one-vertex tadpole diagram: two legs of the quartic $g_0({\mathbf{\varphi }}^2{)}^2$ vertex are external and the remaining two form one fast loop. One contraction has a freely summed component index and contributes $N$, while two exchange contractions have their index fixed by the external component. The result is proportional to $N+2$.

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.

With canonical field $\phi =\varphi /\beta$, the quartic interaction has coupling $-m^2{\beta }^2$. Cancelling the vacuum tadpole diagram requires the divergent mass counterterm $\delta m^2=(m^2{\beta }^2/4){\int }_{-\Lambda }^{\Lambda }dk/[2\pi \sqrt{k^2+1}]$. Its kink energy is $\Delta M_{\mathrm{ct}}=4\delta m^2/(m{\beta }^2)$ because $\int (1-\cos {\varphi }_K)dx=4$. This cancels the logarithmic ultraviolet divergence in the one-loop soliton mass correction. Finite parts depend on the renormalization condition.

A linear counterterm can be chosen so that the renormalized one-point function, or equivalently the chosen vacuum expectation value, is zero. This cancels attached tadpole diagrams in the perturbative expansion about that vacuum. In a massive six-dimensional cubic scalar field theory, the one-loop proper tadpole is $t=g{I}_1/2$; without subtraction the stationary background shifts by $v=-t/m^2+O(g^3)$ at one-loop order. One must specify this convention when comparing reducible exchange diagrams or mass corrections.

An ultraviolet subdivergence arises when loop momenta internal to a proper subgraph tend to infinity while the momenta entering that subgraph are held fixed. It can occur even if simultaneous scaling of all momenta of the full Feynman diagram is convergent. Recursive local counterterms subtract these subgraph divergences before the remaining overall divergence is analyzed. A tadpole diagram inserted into an otherwise convergent six-point phi-fourth theory Feynman diagram is a simple massive example.

With interaction $-{\lambda }_0{\varphi }^4/4!$, the tadpole diagram has symmetry factor one half. After Wick rotation and a rotationally invariant Euclidean cutoff, its self-energy is $({\lambda }_0/2){\int }_{|k|<\Lambda }{d}^{4}k(2\pi {)}^{-4}(k^2+m_0^2{)}^{-1}$. The sphere area $2{\pi }^2$ and radial integral ${\int }_0^{\Lambda }{k}^{3}dk/(k^2+m_0^2)=\frac{1}{2}[{\Lambda }^2-m_0^2\log (1+{\Lambda }^2/m_0^2)]$ 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.