mike372
Feb15-10, 10:38 AM
I want to calculate the partial wave amplitudes for various processes but get divergent results in certain cases which I assume is related to the propagator going on shell - the forward scattering amplitude is infinite. I guess I have to introduce some sort of cut but don't know how to do this.
As an example consider electron - electron scattering in QED with both electrons having the same spin. The full amplitude goes as:
\mathcal{M}=-e^2 4 m^2 \left(\frac{1}{t}-\frac{1}{u}\right)
Now calculating the J=1 partial wave...
\mathcal{M}^{(1)}=\frac{1}{32\pi}\int_{-1}^1\: \mathcal{M} \cos{\theta} \: d(\cos{\theta}) =\frac{e^2 m^2}{8\pi p^2}\int_{-1}^1\: \frac{\cos^2{\theta}}{1-\cos^2{\theta}}d(\cos{\theta})
which diverges. The partial wave amplitude cannot be infinite so how do I get around the problem?
As an example consider electron - electron scattering in QED with both electrons having the same spin. The full amplitude goes as:
\mathcal{M}=-e^2 4 m^2 \left(\frac{1}{t}-\frac{1}{u}\right)
Now calculating the J=1 partial wave...
\mathcal{M}^{(1)}=\frac{1}{32\pi}\int_{-1}^1\: \mathcal{M} \cos{\theta} \: d(\cos{\theta}) =\frac{e^2 m^2}{8\pi p^2}\int_{-1}^1\: \frac{\cos^2{\theta}}{1-\cos^2{\theta}}d(\cos{\theta})
which diverges. The partial wave amplitude cannot be infinite so how do I get around the problem?