Scalar triangle Feynman integral ($C=\int \frac{d^4\ell }{(2\pi {)}^4}\frac{1}{D_0{D}_1{D}_2}$)

In mostly-minus signature, take $D_j=(\ell +r_j{)}^2-m^2+i0$. Feynman parameters and a momentum shift giveFor massive internal lines this is ultraviolet finite in four dimensions. Physical external kinematics can still create threshold cuts, encoded by the Feynman i-epsilon prescription.