I'm working on calculating the cross section for a scattering process that results in three bodies in the final state. My goal is to express the cross section in terms of the invariant masses $s_{ij}$. Is this approach feasible? If so, could someone provide guidance or references on how to achieve this? Is the quadri momentum conservation is $p +p'=k+k'+h=q$ how i can write $k \cdot p$, $k'\cdot p'$, $k \cdot p'$ and $k' \cdot p$ in terms of $s_{ij}$?
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I don't see what's wrong with writing a scattering cross section in terms of Mandelstam variables and masses. Defining $s_{ij} = (p_i + p_j)^2$, any dot product $p_i \cdot p_j$ is just $\frac{1}{2} s_{ij} - m_i^2 - m_j^2$ where we have used the mass shell condition.
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$\begingroup$ You mean $\frac{s_{ij} -m^2_i -m^2_j}{2}?$ @Connor Behan $\endgroup$– AndreaCommented Jun 12 at 17:59
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$\begingroup$ Are there 5 independent Mandelstam variables, it is correct? @Connor Behan $\endgroup$– AndreaCommented Jun 12 at 18:01
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$\begingroup$ Yes on both counts. I forgot the $1/2$ and the number of 5-point invariants is 5. $\endgroup$ Commented Jun 12 at 18:06
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$\begingroup$ Thanks! So, when I' ll integrate in the phase space I'll have a cross section that depends from 3 of them?@Connor Behan $\endgroup$– AndreaCommented Jun 12 at 18:13
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$\begingroup$ I'd like to have a idea of the behavior of the cross section depending of my Mandelstam variables, have any suggestions how I can see it? With Mathematica for example? I mean for integrate it and for see it in a imagine@ConnorBehan $\endgroup$– AndreaCommented Jun 12 at 18:22