Schwinger parametrization - Alchetron, the free social encyclopedia

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

1 A n = 1 ( n − 1 ) ! ∫ 0 ∞ d u u n − 1 e − u A ,

Julian Schwinger noticed that one may simplify the integral:

∫ d p A ( p ) n = 1 Γ ( n ) ∫ d p ∫ 0 ∞ d u u n − 1 e − u A ( p ) = 1 Γ ( n ) ∫ 0 ∞ d u u n − 1 ∫ d p e − u A ( p ) ,

for Re(n)>0.

Another version of Schwinger parametrization is:

1 A = − i ∫ 0 ∞ d u e i u A ,

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

References

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