One Loop corrections in ee--->ee scattering in QED

[Ace10]

Homework Statement

First of all, Happy New Year! I have to solve the following exercise (xmas gift :P) and some things are a bit vague..Here is it: For the ee--->ee scattering process, draw all amputated and connected graphs that would contribute. The hint is that one should find 10 different contributions.

Homework Equations

QED Feynman rules and graph amputation rules, check S-matrix and LSZ Reduction Formula.

The Attempt at a Solution

My first move was to go for a rigorous computation but as you can imagine it didnt work out. Then I tried ti be more practical and drew as many contributions for one loop correction to te process i could. Finally I ended up with four, as I can't go blind with it, I mean I have to step on a solid basis (ie formula) to start to draw more. The four diagramms I drew can be seen in the following image. How can I find seven more of them?? I suppose using brute force is not the deal and I'm very confused. Thank you in advance for your help. Happy New year!

 

[Chopin]

The two right-hand diagrams have more than one loop in them, so they shouldn't be counted here. As for the other diagrams which do contribute, remember that in identical particle scattering, you have to put in the crossing diagrams as well. That should give you at least a few more of them.

 

[Ace10]

First of all thank you for your reply Chopin. Of course it is obvious that I've made a huge mistake with the loop number, these two graphs cannot contribute indeed. I've made another try which can be seen in the following image. Graph 1 is without corrections, graphs 2 and 4 are with simple vertex corrections, 3 and 5 are the crossed symmetric ones and 8,9,10,11 have propagator corrections for ingoing, outgoing electrons and the crossed symmetric ones. One could say that the 8,9,10 and 11 are forbidden due to amputation, is it correct?? I can't find any others unfortunately..Any ideas? Thank you very much in advance for your help!

 

[Chopin]

You are correct that loop corrections to external lines are forbidden due to amputation, so that gets rid of 8-11. However, loop corrections to internal lines are allowed. You had a diagram like that in your first list, and it is still valid. That should get you two more with crossing. The remaining four are a little trickier to see. Try thinking about what combinations can happen if the electrons exchange more than one photon.

 

[mfb]

Where is "the following image"?

 

[Ace10]

Good afternoon,

First of all thank you Chopin very much for your help, I'll try to make graphs with more photons exchanged tonight, it seems a bit strange as it could lead to two-loop corrections. I'll post here.
Secondly, mfb you are right i must have done something wrong here, I have uploaded it again, hope it works! Sorry!

 

[mfb]

There are diagrams with more photons exchanged at 1-loop level.

 

[Ace10]

Thank you mfb, I'll try it in a few hours...hope to get a decent result!

 

[Ace10]

I think I found all ten of them, i have drew them in the following image..

 

[Chopin]

Very close, but not quite. In your bottom two, the arrows on the electron lines don't match up, so the diagram is not a valid one. The arrows at each vertex must continue out in the same direction they came in.

The remaining two diagrams are similar to the two-photon diagrams you have now, but with a slight twist. Try seeing if you can find other ways to connect up the four vertices with photon lines.

 

[Ace10]

Thank you guys i found it! I think the two missing are the ones of the image, right?

 

[mfb]

You had those two in your previous post already, right?

 

[Ace10]

No no, these 2 are new! They are not very clear though. Or am i totally wrong?

 

[mfb]

Lepton from p1 to q1, lepton from p2 to q2, photon from p1 to q1 and from "in between" to p2/q2 is the same as the lower left diagram here (upper left if you rotate it for a correct letter orientation).
The other new diagram is the diagram to the right of (below) that.

 

[Ace10]

wow right. i'll be back with smth new..