# $2\to 3$ cross-section phase space simplification

Suppose the $2\to 3$ cross-section: $$\sigma = (2\pi)^4\int \frac{d^{3}\mathbf p_{3}}{(2\pi)2E_{3}}\frac{d^{3}\mathbf p_{4}}{(2\pi)^{3}2E_{4}}\frac{d^{3}\mathbf p_{5}}{(2\pi)^{3}2E_{5}}|M(\mathbf p_{1}, \mathbf p_2, \mathbf p_3,\mathbf p_4,\mathbf p_5)|^2\frac{\delta^{4}(p_1+p_2-p_3-p_4-p_5)}{v_{\text{relative}}}$$ Is there some way to simplify the phase space integration, i.e., whether some simplified form of the phase space integral exist, for some new variables with definite integration limits?

Edit. It turns out that relative simple phase space structure, without complicated angles, is given in terms of four kinematic invariants. It is not hard to evaluate the domain of definition for these invariants. Of course, the full cross-section not always can be expressed in terms of elementary functions, but typically the single cross-section, i.e., the differential cross-section with respect to one kinematic invariant, often exists in terms of elementary functions.

• You can answer your own question. It sounds like you now know the answer – innisfree Mar 8 '18 at 23:50

## 1 Answer

I can recommend you to read the paper of Tord and Riemann Phase space integrals for 2,3 and 4 particle production. The Idea is to factor the phase space into 2 2-particle phase spaces times $ds'$ where $s'=(p_1+p_2)^2$.

https://www-zeuthen.desy.de/~riemann/Teaching/ws20132014/script-tord-riemann-sm-2013_12pt-appA.pdf

• Welcome to SE. This answer is close to being basically link only. That's not ideal because links (especially to pdfs on personal homepages) can rot, and because we want self-contained answers here, if possible. So if you can, try to summarize the key points of the linked document – innisfree Mar 8 '18 at 23:51