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Consider a real 4-dimensional Minkowski space-time with the usual inner product: $p\cdot q = p_0\cdot q_0 - p_1\cdot q_1 - p_2\cdot q_2 - p_3\cdot q_3$

$p^2 = p_0^2 - p_1^2 - p_2^2 - p_3^2$

Let $S_{12}, S_{23}, S_{31}$ be 3 real constants. The sum of these constants is called $S$:

$S_{12} + S_{23} + S_{31} = S$

I should want the result of the following integral (if it is not infinite):

$ I = \int (d^4p_1) (d^4p_2) (d^4p_3) δ(p_1^2) δ(p_2^2) δ(p_3^2) δ(p_1.p_2 -S_{12}) δ(p_2\cdot p_3 -S_{23}) δ(p_3\cdot p_1 -S_{31}) $

I am interested in the result, first when $S=0$, then in the general case $(S \neq 0)$

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