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:

$\frac{1}{A}=-i\int^\infty_0 du \, e^{iuA},$

and it is easy to generalize this identity to n denominators.

See also Feynman parametrization.