Divergent partial wave amplitudes

[mike372]

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?

 

[AndyW]

I understand your frustration with encountering divergent results in your calculations. This is a common issue in physics, and it often indicates that our current understanding of the system is incomplete. In this case, the infinite result for the J=1 partial wave amplitude is indeed related to the propagator going on shell.

To address this issue, you can introduce a cut in the integral to avoid the singularities. This is known as a regularization technique. There are several different methods for regularization, such as dimensional regularization or introducing a momentum cutoff. These methods essentially modify the integral in a mathematically consistent way to avoid the divergences.

In the specific case of electron-electron scattering in QED, one commonly used method is called the "Feynman gauge." In this gauge, the propagator is modified to include a term that cancels out the singularities. This allows for a finite result for the J=1 partial wave amplitude.

I recommend consulting with your colleagues or a textbook on QED to determine the best approach for your specific calculation. Regularization techniques can be complex and require a good understanding of the underlying theory. However, with the right approach, you should be able to obtain finite and physically meaningful results for your partial wave amplitudes.

Best of luck with your calculations!