Physics:Schwinger parametrization

Short description: Loop integral parametrization

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

[math]\displaystyle{ \frac{1}{A^n}=\frac{1}{(n-1)!}\int^\infty_0 du \, u^{n-1}e^{-uA}, }[/math]

Julian Schwinger noticed that one may simplify the integral:

[math]\displaystyle{ \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)}, }[/math]

for Re(n)>0.

Another version of Schwinger parametrization is:

[math]\displaystyle{ \frac{i}{A+i\epsilon}=\int^\infty_0 du \, e^{iu(A+i\epsilon)}, }[/math]

which is convergent as long as [math]\displaystyle{ \epsilon \gt 0 }[/math] and [math]\displaystyle{ A \in \mathbb R }[/math]. It is easy to generalize this identity to n denominators.

See also

• Feynman parametrization