Feynman parametrization

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The Feynman parametrization is a nifty technology for evaluating loop integrals which crop up when evaluating Feynman diagrams with one or more loops.

Richard Feynman came up with the amazing observation that

$\frac{1}{AB}=\int^1_0 \frac{du}{\left[uA +(1-u)B\right]^2}$

and suddenly, evaluating integrals like

$\int \frac{dp}{A(p)B(p)}=\int dp \int^1_0 \frac{du}{\left[uA(p)+(1-u)B(p)\right]^2}=\int^1_0 du \int \frac{dp}{\left[uA(p)+(1-u)B(p)\right]^2}$

is a breeze!

More generally,

$\frac{1}{A_1\cdots A_n}=(n-1)!\int^1_0 du_1 \cdots \int^1_0 du_n \frac{\delta(u_1+\dots+u_n-1)}{\left[u_1 A_1+\dots +u_n A_n\right]^n}$

Even more generally,

$\frac{1}{A_1^{\alpha_1}\cdots A_n^{\alpha_n}}=\frac{(\alpha_1+\dots +\alpha_n-1)!}{(\alpha_1-1)!\cdots (\alpha_n-1)!}\int^1_0 du_1 \cdots \int^1_0 du_n \frac{\delta(u_1+\dots+u_n-1)u_1^{\alpha_1-1}\cdots u_n^{\alpha_n-1}}{\left[u_1 A_1+\dots +u_n A_n\right]^{\alpha_1+\dots+\alpha_n}}$