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Essentially I have a D dimensional loop integral $ \int d^dx_1 d^dx_2 \hspace{5pt} f(x_1,x_2)$ where f is a scalar function. I wish to convert it into an integral with the following parameterization

$\int_0^\infty ds_{11} \int_0^\infty ds_{22} \int_{-\sqrt{s_{11}s_{22}}}^{\sqrt{s_{11}s_{22}}}ds_{12} A(s_{ii},s_{12}) f(x_1,x_2)$ where

$s_{ij} := x_i.x_j$ and A is a scalar function that appears due to this variable change. I know that this is possible but I am getting really confused while deriving this. Any reference will be helpful too.

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    Post on math? I never met $d^dx$ format, what is the second superscript d? Post your attempt? – 2017-01-16
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    the integral is in d dimensional space. – 2017-01-16

1 Answers 1

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The trick is to treat one of the vectors as the z axis and then perform the parameterization in polar co-ordinates after fixing that vector. At the end we integrate over the radial co-ordinate i.e. the length of the fixed vector. For a D dimensional integral the function A turns out to be $(s_{11}s_{22}-s_{12}^2)^{\frac{D-3}{2}}$