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The usually quoted value of the fine structure constant is 1/137.0359... This value holds at low energies. As is well known, this value increases somewhat with energy: it is about 1/128 at (M_Z)^2. Now, IF we imagine that there is no GUT, no physics beyond the standard model, no supersymmetry, and that QED is correct all the time (yes, that is a big IF), what would the value be at Planck energy (10^19 GeV)?

Yes, increases is correct - i edited the text!

The measurements go only up to 1000 GeV at most :-(

The coupling constant of QED is known to diverge at a finite energy scale: https://en.wikipedia.org/wiki/Landau_pole

@RyanThorngren: I am afraid Mayhem means SM rather than QED here.

In the meantime, since there were no answers here, I have asked a few experts on the topic. By using the extrapolated couplings alpha_1, alpha_2 and alpha_3 from the graphs of the running coupling constants, and assuming no other interactions, no GUT, no susy, etc., you get:

The fine structure constant at Planck energy is around 1/105.

any detailed calculations to back up this estimate, or is it just another magic number?

1/105 or 1/100 is still a small number, which says nothing. Do you, Mayhem, know that $\alpha$ is always multiplied by some dimensionless function of the physical problem variables in question and only this product serves as a "small parameter"?

it is difficult to give an estimation because all the negative if. One can take the Bohr Sommerfield equation and derive a value but there is no more garantee of predictability. For the math, a computation on a lattice model may give interesting results at more common energies

There are no predictions for the fine structure constant that I know of. I remmember seeing an image for Grand Unification once where there was SM on one side and MSSM on the other. I've done some work on this topic my self and the value is around 1/100. There's no way to test if it's true however my equation is in full agreement with experimental evidence where available, both for the EM coupling (fine structure constant) and the strong coupling constant.

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