Euclidean massive loop integral ($I_{d,n}(m)$)

For $m^2>0$ and $n>d/2$, Schwinger parameterization and a Gaussian integral give $I_{d,n}(m)=\int d^{d}k(2\pi {)}^{-d}(k^2+m^2{)}^{-n}=\Gamma (n-d/2)(m^2{)}^{d/2-n}/[(4\pi {)}^{d/2}\Gamma (n)]$. Its meromorphic continuation supplies dimensional regularization poles.

The one-propagator Euclidean loop momentum integral associated with a tadpole diagram. Dimensional regularization gives $T_d(m)=\Gamma (1-d/2)(m^2{)}^{d/2-1}/(4\pi {)}^{d/2}$. Near $d=3$ this has no pole and $T_3(m)=-m/(4\pi )$ for $m>0$. A momentum-cutoff definition instead contains a linear ultraviolet divergence; the absence of a dimensional-regularization pole here concerns logarithms, not the positivity or convergence of the unregularized integral.

For positive $m^2$, a Feynman parameter and a momentum shift turn the integral into ${\int }_0^1{I}_{d,2}(\sqrt{m^2+x(1-x)p^2})dx$. The Euclidean massive loop integral has factor $\Gamma (ϵ/2)=2/ϵ+O(1)$, while its other factors tend to $(4\pi {)}^{-2}$. The pole is independent of the external momentum and the mass, so it is removable by a local counterterm.