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 too.

Richard Feynman observed that:

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

which simplifies evaluating integrals like:

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

More generally, using the Dirac delta function:

:$frac\{1\}\{A\_1cdots\; A\_n\}=(n-1)!int^1\_0\; du\_1\; cdots\; int^1\_0\; du\_n\; frac\{delta(u\_1+dots+u\_n-1)\}\{left\; [u\_1\; A\_1+dots\; +u\_n\; A\_n\; ight]\; ^n\}.$

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

:$frac\{1\}\{A\_1^\{alpha\_1\}cdots\; A\_n^\{alpha\_n=frac\{Gamma(alpha\_1+dots\; +alpha\_n)\}\{Gamma(alpha\_1)cdots\; Gamma(alpha\_n)\}int^1\_0\; du\_1\; cdots\; int^1\_0\; du\_n\; frac\{delta(u\_1+dots+u\_n-1)u\_1^\{alpha\_1-1\}cdots\; u\_n^\{alpha\_n-1\{left\; [u\_1\; A\_1+dots\; +u\_n\; A\_n\; ight]\; ^\{alpha\_1+dots+alpha\_n.$

See also Schwinger parametrization.