Can Muonium Be Produced and Observed in Electron-Positron Collider Experiments?

[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 ($e^+μ^-$).

 

[nrqed]

Is your question about what Feynman diagram contributes to this reaction?
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.

 

[Naeem Anwar]

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

 

[nrqed]

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

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

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

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)

 

[nrqed]

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

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)

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.

 

[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.

 

[nrqed]

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.

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