# Momentum in breit wigner cross section

The Breit Wigner cross section derived in my lecture notes is

$$\sigma = \frac{g\pi}{p_i^2}\frac{\Gamma_{Z\rightarrow i}\Gamma_{Z \rightarrow f}}{(E-E_0)^2+\frac{\Gamma^2}{4}}$$

where $$g$$ is the spin degeneracy factor, $$\Gamma$$ is the resonance width for the intermediate particle $$Z$$, and $$\Gamma_{Z \rightarrow i/f}$$ are the partial decay widths.

The $$p_i$$ factor corresponds to the density of states due to the incident momentum, and is supposedly the momentum in the centre of mass frame. But this can't be true, as in head-on collisions the momentum in the centre of mass frame is zero. While the decay rates correspond to decays to a specified state (e.g. electron and positron), the density of states was derived as if the initial and final states were a single particle- a single plane wave. I am not sure how to consider this for the case of two colliding particles, say an electron and positron with equal and opposite momenta in the lab frame, compared with stationary target collisions in he lab frame. What is the form of $$p_i$$ then?

• The spatial part of the sum of the COM momentna is zero, but the momentum is not just a 3D vector, it has four components, one of which is always positive representing the energy of the particle. – Triatticus Apr 30 at 21:56