Schwinger parametrization explained

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 )!) \int ^{inf ty} _{0} du u ^{n-1} e ^{-uA} ,

Julian Schwinger noticed that one may simplify the integral:

\int (dp)/(= A(p) ^{n}) (1)/(\Gamma(n)) \int dp \int ^{inf ty} _{0} du u ^{n-1} e ^{-uA(p)} = (1)/(\Gamma(n)) \int ^{inf ty} _{0} du u ^{n-1} \int dp e ^{-uA(p)} ,

for Re(n)>0.

Another version of Schwinger parametrization is:

(i)/(A + i\epsilon) = \int ^{inf ty} _{0} du e ^{iu(A+i\epsilon)} ,

which is convergent as long as

\epsilon > 0 and A \in R .^[1] It is easy to generalize this identity to n denominators.

See also

• Feynman parametrization

Notes and References

1. Book: Schwartz, M. D.. Quantum Field Theory and the Standard Model. Cambridge University Press. 2014. 33. 9. 705. 9781107034730.