Appendix E - General Expression for the Total Scattering Cross Section and DecayRates

Summary

Cross Section

Consider a two body scattering process with four momenta. There areN particles in the final state with four momenta.

The general expression for the cross section is given by

where

In the laboratory frame, where, say, particle“b” is at rest (i.e.v_b = 0, E_b =m_b)

v_a = c = 1, if the incidentparticle is relativistic, that is, E_a ≫ma.

In the center of mass frame, where particles“a” and “b”approach each other from exactly opposite direction, that is,θ_ab = 180o, withthe 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 stateand “1” and “2” are particles in the finalstate, that is,

the general expression for the differential scattering cross section is givenby

In any experiment, one observes either particle “1” or particle“2”; therefore, the kinematical quantities of the particlewhich is not to be observed are fixed by doing phase space integration. Forexample, if particle “2” is not to be observed, then

which gives the constraint on where is the three momentum transfer and, whichresults in

Integrating over the energy of particle “1”, using the deltafunction integration property, we Get

Thus,

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 thecenter of mass energy, we write

Energy distribution of the outgoing particle“1”

Here, we evaluate energy distribution in the lab frame.