Resonant Scattering

The Kramers-Heisenberg photon scattering cross section, below, has unphysical infinities if an intermediate state is on the mass shell.

$$\bgroup\color{black} {d\sigma\over d\Omega} = \left({e^2\over... ...angle \over \omega_{ji}+\omega'} \right] \right\vert^2 \egroup$$

In reality, the cross section becomes large but not infinite. These infinities come about because we have not properly accounted for the finite lifetime of the intermediate state when we derived the second order perturbation theory formula. If the energy width of the intermediate states is included in the calculation, as we will attempt below, the cross section is large but not infinite. The resonance in the cross section will exhibit the same shape and width as does the intermediate state.

These resonances in the cross section can dominate scattering. Again both resonant terms in the cross section, occur if an intermediate state has the right energy so that energy is conserved.