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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.

In a massive two-dimensional quartic scalar field theory, the two-loop sunset diagram is ultraviolet finite but changes the pole mass. Feynman parameters give, below the three-particle threshold,At $s=m^2$ the denominator is positive in the interior of the simplex, and its corners are integrable. Thus the result is finite and nonzero for $g_4\ne 0$. Absence of new divergent counterterms does not imply absence of finite mass counterterms in an on-shell renormalization scheme.

Four quartic Yukawa vertices on a fermion loop with one internal scalar chord give six external scalar legs, two loops and overall logarithmic degree. The coefficient is nonzero: in a constant scalar background $v$, the induced cubic Yukawa coupling is $y=gv$ and the fermion mass is $\mu =m+g{v}^2/2$. The two-loop sunset diagram in the quantum effective potential has a logarithmic pole proportional to $y^2(4{\mu }^2-{\mathcal{B}}^2)$, where ${\mathcal{B}}^2$ is the background scalar squared mass. Its $v^6$ term is $g^4{v}^6$ when the tree sextic coupling vanishes. Thus zero renormalized sextic coupling at one scale does not eliminate the sextic counterterm or make that zero stable under renormalization.

An ultraviolet divergence comes from arbitrarily large loop momentum, equivalently from singular short-distance behavior. Locality makes its regulator dependence cancellable by local counterterms in a renormalizable theory.

An ultraviolet divergence appears as a pole of a dimensionally continued Feynman integral. In $d=4-2ϵ$, a logarithmically divergent one-loop integral typically has a $1/ϵ$ term. Dimensional regularization can also produce infrared divergences with the same formal regulator, so the origin must be identified before interpreting a pole as an ultraviolet counterterm.

In Feynman gauge, with insertion $-i\Sigma (p)$ and $d=4-2ϵ$, the one-loop QED result is ${\Sigma }_{\mathrm{UV}}=e^2(-/p+4m)/(16{\pi }^2ϵ)$. The coefficient of $/p$ is the electron kinetic counterterm ${\delta }_2=-e^2/(16{\pi }^2ϵ)$ when its insertion is $i/p{\delta }_2$. The scalar part fixes a separate mass counterterm in the chosen normalization convention.

For one unit-charge Dirac field in $d=4-2ϵ$, define the insertion as $i{\Pi }_{\mu \nu }=i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu })\Pi$. The one-loop ultraviolet pole is ${\Pi }_{\mathrm{UV}}=-e^2/(12{\pi }^2ϵ)$. The Ward identity cancels any apparent divergent vector mass term. With photon counterterm insertion $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu }){\delta }_3$, this gives ${\delta }_3=-e^2/(12{\pi }^2ϵ)$ in the minimal subtraction scheme.

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.

In a gauge-preserving regularization and consistent renormalization convention, the proper-vertex Ward-Takahashi identity in QED equates the vertex and electron wave-function renormalization constants. With ${\psi }_0=Z_2^{1/2}\psi$ and $A_0=Z_3^{1/2}A$, the vertex coefficient gives $e_0={\mu }_R^{ϵ}{Z}_{1}e/(Z_2\sqrt{Z_3})$, proving the displayed charge relation in $d=4-2ϵ$. Gauge invariance restricts counterterms but does not forbid charge running.

Four scalar Yukawa interactions on a closed fermion loop generate a scalar four-point one-particle-irreducible vertex. At high momentum four dirac propagators contribute $k^{-4}$, leaving a logarithmic four-dimensional ultraviolet divergence. Its local scalar-fourth-power term requires a quartic scalar coupling; scalar mass and field counterterms cannot cancel it.