Schwinger parametrization

Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. It is named after Julian Schwinger, who introduced the method in 1951 for quantum electrodynamics.

Description

Using the observation that

${\frac {1}{A^{n}}}={\frac {1}{(n-1)!}}\int _{0}^{\infty }du\,u^{n-1}e^{-uA},$

one may simplify the integral:

$\int {\frac {dp}{A(p)^{n}}}={\frac {1}{\Gamma (n)}}\int dp\int _{0}^{\infty }du\,u^{n-1}e^{-uA(p)}={\frac {1}{\Gamma (n)}}\int _{0}^{\infty }du\,u^{n-1}\int dp\,e^{-uA(p)},$

for ${\textstyle \mathrm {Re} (n)>0}$.

Alternative parametrization

Another version of Schwinger parametrization is:

${\frac {i}{A+i\epsilon }}=\int _{0}^{\infty }du\,e^{iu(A+i\epsilon )},$

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

See also

• Feynman parametrization