A Potential between Photons via Delbruck Scattering

[DuckAmuck]

TL;DR

Delbruck scattering generates potential between photons.

From the Born Approximation, you can relate the potential to the scattering amplitude. So it follows that a potential can be derived from the scattering amplitude from Delbruck scattering. I tried to solve this myself, and get a scattering amplitude with only angular dependence, no momentum dependence. This seems to imply that the potential and force between two photons would be long-range? This is clearly incorrect. I would appreciate any help.

 

[vanhees71]

It's hard to guess, what you did. Of course there's no "potential between photons". Delbrück scattering is described in leading order QED by box diagrams with four external photon lines. It's a pretty cumbersome calculation. See Landau&Lifshitz vol. 4.

 

[topsquark]

On an unrelated note: Thank you! I didn't know the name for this. All I had heard of was "scattering of light by light."

-Dan

 

[dextercioby]

TL;DR Summary: Delbruck scattering generates potential between photons.

From the Born Approximation, you can relate the potential to the scattering amplitude. So it follows that a potential can be derived from the scattering amplitude from Delbruck scattering. I tried to solve this myself, and get a scattering amplitude with only angular dependence, no momentum dependence. This seems to imply that the potential and force between two photons would be long-range? This is clearly incorrect. I would appreciate any help.

Perhaps worthwhile would be to post some if not all of your calculations. As a PDF would be ok.

 

[Vanadium 50]

First, I agree - we should call this "light by light" and not "Delvruck". It's closer to what you mean.

Next, at leading order, scattering is the same for attraction and repulsion. So the first order where it makes a difference is α⁶. I suspect that this is calculated somewhere, probably in someone's thesis. Maybe it's published somewhere.

Massless particles don't form bound states, so what is meant by attraction and repulsion needs to be carefully defined. Without doing the calculation (well beyond my abilities) I suspect there is a dependence on the relative phases of the photons.

 

[vanhees71]

The leading order are box diagrams with four vertices, i.e., the cross section is of order $\alpha^4$. It's a genuinely relativistic effect of course, because it involves the massless photons, and thus you cannot expect that this has anything to do with potentials. It's also a pure quantum effect, i.e., due to quantum fluctuations of the quantum fields involved. As I said, you find the calculation in Landau and Lifshitz vol. IV. The calculation is indeed very cumbersome.

What's important to note is that this four-photon diagram is superficially logarithmically divergent. If it were really divergent, it would be a desaster for the renormalizability of QED, because there is no renormalizable counter term for such a divergence. Fortunately, gauge invariance comes to the rescue, and the Ward-Takashi identities tell you before you have done any calculation that indeed the four-photon vertex is finite. This is not true for any single box diagram but for the sum of all the 6 box diagrams.