# Phase space suppression

In talking about production/decay processes, I've heard people speaking of decay modes or cross sections being 'phase space' suppressed. For example, a two body final state is more likely to occur than a three body final state, since in the later case the three particles must share the initial energy momentum whereas in the former case only two particles must share.

But, how do I quantify such an argument based purely on the phase space integration that appears in the cross section/decay rate formulas. The two-body phase space integration has dimensions [mass]$^0$ whereas the three-body phase space integration has dimensions [mass]$^2$. How then am I to compare the phase space 'volumes' available to a process if they have different dimensions?

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Hmmm...the "related" question finder has failed us here. This is nearly a duplicate of What does the concept of phase space mean in particle physics?, and you can find a discussion of how the available phase space affects a class of reaction in my answer to Why is the (free) neutron lifetime so long?. –  dmckee Aug 15 '12 at 13:43

The volumes of phase spaces have different dimensions (units) depending on the number of decay products but this is no problem because there exists a natural mass scale, namely $m$, the mass of the unstable particle at the beginning.
If the phase space over which the integrand is comparable to its maximum is of order $m^k$ where $k$ is an exponent determined by the dimensional analysis (i.e. by the number of final particles), we deal with the generic situation. If it is smaller (or much smaller) than $m^k$, the decay rate is phase-space-suppressed.
The neutron decay is a textbook example of the phase-space suppression because the electron's and antineutrino's energies can't go up to $O(m)$ where $m$ is the neutron mass; instead, these energies have to be thousands of times smaller because $m_{n}\approx m_{p}$ and the proton already takes almost all the energy. The "generic" lifetime of a strongly as well as weakly interacting, weakly decaying particle of this mass would be a tiny fraction of a second but because of the phase space suppression, neutrons live for 15 minutes or so (mean lifetime).