Schwinger parametrization

Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops.

Using the well-known observation that


 * $$\frac{1}{A^n}=\frac{1}{(n-1)!}\int^\infty_0 du \, u^{n-1}e^{-uA},$$

Julian Schwinger noticed that one may simplify the integral:


 * $$\int \frac{dp}{A(p)^n}=\frac{1}{\Gamma(n)}\int dp \int^\infty_0 du \, u^{n-1}e^{-uA(p)}=\frac{1}{\Gamma(n)}\int^\infty_0 du \, u^{n-1} \int dp \, e^{-uA(p)},$$

for Re(n)>0.

Another version of Schwinger parametrization is:


 * $$\frac{i}{A+i\epsilon}=\int^\infty_0 du \, e^{iu(A+i\epsilon)},$$

which is convergent as long as $$\epsilon >0$$ and $$A \in \mathbb R$$. It is easy to generalize this identity to n denominators.