Feynman parametrization

Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well.

Richard Feynman observed that:

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

which simplifies evaluating integrals like:

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

More generally, using the Dirac delta function:

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

Even more generally, provided that Re($\alpha_j$)>0 for all 1 ≤ j ≤ n:

$\frac{1}{A_{1}^{\alpha_{1}}\cdots A_{n}^{\alpha_{n}}}=\frac{\Gamma(\alpha_{1}+\dots+\alpha_{n})}{\Gamma(\alpha_{1})\cdots\Gamma(\alpha_{n})}\int_{0}^{1}du_{1}\cdots\int_{0}^{1}du_{n}\frac{\delta(\sum_{k=1}^{n}u_{k}-1)u_{1}^{\alpha_{1}-1}\cdots u_{n}^{\alpha_{n}-1}}{\left[u_{1}A_{1}+\cdots+u_{n}A_{n}\right]^{\sum_{k=1}^{n}\alpha_{k}}} .$

See also Schwinger parametrization.

Derivation

$\frac{1}{AB} = \frac{1}{A-B}\left(\frac{1}{B}-\frac{1}{A}\right)=\frac{1}{A-B}\int_B^A \frac{dz}{z^2}$

now just linearly transform integral to desired bounds

$u=(z-B)/(A-B)$ and $du = dz/(A-B)$ so $z = uA + (1-u)B$

and we get the desired result.

$\frac{1}{AB} = \int_0^1 \frac{du}{\left[uA + (1-u)B\right]^2}$