Schwinger parametrization

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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.