Feynman parametrization
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Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops.
Richard Feynman observed that
![\frac{1}{AB}=\int^1_0 \frac{du}{\left[uA +(1-u)B\right]^2}](../../../math/b/0/1/b01075c7b68a9176b1e228b7fcd7bce3.png)
which simplifies evaluating integrals like
![\int \frac{dp}{A(p)B(p)}=\int dp \int^1_0 \frac{du}{\left[uA(p)+(1-u)B(p)\right]^2}=\int^1_0 du \int \frac{dp}{\left[uA(p)+(1-u)B(p)\right]^2}.](../../../math/2/2/1/221fd349d1ff7abd187b5f2088983fdc.png)
More generally,
![\frac{1}{A_1\cdots A_n}=(n-1)!\int^1_0 du_1 \cdots \int^1_0 du_n \frac{\delta(u_1+\dots+u_n-1)}{\left[u_1 A_1+\dots +u_n A_n\right]^n}.](../../../math/1/2/1/121b5e4d4bc67416a3713984a6847fd5.png)
Even more generally,
![\frac{1}{A_1^{\alpha_1}\cdots A_n^{\alpha_n}}=\frac{(\alpha_1+\dots +\alpha_n-1)!}{(\alpha_1-1)!\cdots (\alpha_n-1)!}\int^1_0 du_1 \cdots \int^1_0 du_n \frac{\delta(u_1+\dots+u_n-1)u_1^{\alpha_1-1}\cdots u_n^{\alpha_n-1}}{\left[u_1 A_1+\dots +u_n A_n\right]^{\alpha_1+\dots+\alpha_n}}.](../../../math/b/e/2/be2ce4060d61e0e39253bcc00fafcd1e.png)
See also Schwinger parametrization.
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