# Production of Pure QED Atom

1. May 17, 2015

### Naeem Anwar

Studying the production of the bound state of $μ^+μ^-$ in the scattering experiments of $e^+e^-$ at electron positron colliders $e^+e^-\toμ^+μ^-$ via $e^+e^-\to 2γ\toμ^+μ^-$. This bound state further can decay into $e^+e^-$ or $2γ$.

I am confused about the vertex $2γ\toμ^+μ^-$ in the Feynman diagram, how can I include it in decay amplitude? Of course it need some detail of bound state, wave-function etc., but how to deal with this stuff? Looking for some literature/QFT book which involves this type of vertex for production of positronium ($e^+e^-$) or muonium ($e^+μ^-$).

2. May 19, 2015

### nrqed

First, note that there is no vertex for two photons going directly to a mu+ and mu-. The diagram is a one-loop "box" diagram.

3. May 23, 2015

### Naeem Anwar

Dear nrqed, I know the diagram, I am curious about how to add the details of the "box" in the amplitude?

#### Attached Files:

• ###### Untitled.png
File size:
3.2 KB
Views:
62
4. May 26, 2015

### nrqed

I am a bit confused why you initially mentioned e e+ going to two gammas, I don't see this in the diagram.

For the box, it depends if the
Hi,

Ah ok. I am confused why you mentioned e- e+ going to two photons in your original post, this does not occur in your diagram.

Well, it depends on whether the muon and antimuon are produced with a small relative speed or not. If it is large, you can just do the usual Feynman diagram expansion. But at small speeds, the muon and antimuon can interact for a long time, being essentially bound through the Coulomb interaction. In that case, one must sum up an infinite number of Coulomb photon exchanges between the two. This can be done. I don't know if my PhD thesis is available online easily (Patrick Labelle, Cornell University, 1994) but I do it there. Or look for old papers by Peter Lepage (on positronium) or my Toichiro Kinoshita (on muonium)

5. May 26, 2015

### nrqed

Now that I think of it, the easiest reference to find is a paper on pionium by Labelle and Buckley. It is on the archives. I am pretty sure we give the formula needed in that paper.

Last edited: May 26, 2015
6. May 26, 2015

### Naeem Anwar

Thanks dear P. Labelle, I acknowledge your help. Sorry for my poor understanding & language used in question. I found all of your papers on HEP-INSPIRE, I also found many of my answers there. The paper you mentioned (with Buckley), I just downloaded & reading now. I am not able to get your thesis, it is not available in PDF form online.

7. May 27, 2015

### nrqed

You are very welcome. And there is no reason to apologize for anything!

If anything is unclear in my papers, please don't hesitate to ask!!

Regards,

Patrick