Consider a two body scattering process with four momenta. There are N particles in the final state with four momenta.
The general expression for the cross section is given by
In the laboratory frame, where, say, particle “b” is at rest (i.e. vb = 0, Eb = mb)
va = c = 1, if the incident particle is relativistic, that is, Ea ≫ ma.
In the center of mass frame, where particles “a” and “b” approach each other from exactly opposite direction, that is, θab = 180o, with the same magnitude of three momenta, that is, such that
where is the center of mass energy.
More conveniently, this is also written as
Two body scattering
For a reaction, where “a” and “b” are particles in the initial state and “1” and “2” are particles in the final state, that is,
the general expression for the differential scattering cross section is given by
In any experiment, one observes either particle “1” or particle “2”; therefore, the kinematical quantities of the particle which is not to be observed are fixed by doing phase space integration. For example, if particle “2” is not to be observed, then
which gives the constraint on where is the three momentum transfer and, which results in
Integrating over the energy of particle “1”, using the delta function integration property, we Get
Which result in,
If the scattering takes place in the lab frame, and, it result in
If the scattering takes place in the center of mass frame, where, is the center of mass energy, we write
Energy distribution of the outgoing particle “1”
Here, we evaluate energy distribution in the lab frame.