Schwinger parametrization

{{Short description|Loop integral parametrization}}

{{more citations needed|date=July 2022}}

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

:$\backslash frac\{1\}\{A^n\}=\backslash frac\{1\}\{(n-1)!\}\backslash int^\backslash infty\_0\; du\; \backslash ,\; u^\{n-1\}e^\{-uA\},$

Julian Schwinger noticed that one may simplify the integral:

:$\backslash int\; \backslash frac\{dp\}\{A(p)^n\}=\backslash frac\{1\}\{\backslash Gamma(n)\}\backslash int\; dp\; \backslash int^\backslash infty\_0\; du\; \backslash ,\; u^\{n-1\}e^\{-uA(p)\}=\backslash frac\{1\}\{\backslash Gamma(n)\}\backslash int^\backslash infty\_0\; du\; \backslash ,\; u^\{n-1\}\; \backslash int\; dp\; \backslash ,\; e^\{-uA(p)\},$

for Re(n)>0.

Another version of Schwinger parametrization is:

:$\backslash frac\{i\}\{A+i\backslash epsilon\}=\backslash int^\backslash infty\_0\; du\; \backslash ,\; e^\{iu(A+i\backslash epsilon)\},$

which is convergent as long as $\backslash epsilon\; >0$ and $A\; \backslash in\; \backslash mathbb\; R$.{{cite book|first=M. D.|last=Schwartz|title=Quantum Field Theory and the Standard Model|publisher=Cambridge University Press|date=2014|chapter=33|edition=9|page=705|isbn=9781107034730}} It is easy to generalize this identity to n denominators.