Tadpole integral ($T_d(m)=\int \frac{d^{d}k}{(2\pi {)}^d}\frac{1}{k^2+m^2}$)

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.