# What is the value of the fine structure constant at Planck energy?

At low energy, 511 keV, the value of the fine structure constant is 1/137.03599...

At Planck energy $\sqrt{\frac{\hbar c^5}{G}}$, or 1.956 $\times$ 109 Joule, or 1.22 $\times$ 1028 eV, it has a value around 1/100 .

Does anybody know a more precise value, maybe with error bars?

Clarification:

The idea is to take the famous "semi-official" plot for the standard model: go to page http://www-ekp.physik.uni-karlsruhe.de/~deboer/ then click on "our research" then go towards the end of the page: the graph is the link "Unification plot.ps (update from Phys. Lett. B260 (1991) 447)". Now take the left graph (not the right one for supersymmetry). The upper curve shows the fine structure constant, but scaled with some funny numbers suggested by GUTs. The precise question is: what value for the fine structure constant would this normal QED/standard model graph show at Planck energy if these funny GUT numbers would not be in the graph? My estimate is about 1/100, but I would be interested in a more precise value. It is clear that this question assumes, like the graph itself, that both QED and the standard model are correct all the way up to Planck energy.

• There might be an ansatz to deduct it from some renormalization approach. But I think its pretty save to say its not known for the usual reasons - no experimental data, no GUT. Greets – Robert Filter May 11 '11 at 8:13
• Possibly related: physics.stackexchange.com/q/5552/2451 – Qmechanic May 12 '11 at 15:01
• Dear @Claude, I assure you that my answer does answer your question. Read the discussion below my answer to catch at least many of the numerous mistakes you are making in your reasoning and which also prevent you from seeing that my answer does answer your question. – Luboš Motl May 12 '11 at 15:56
• Yes, as I write below, you are correct. Thank you! – Claude May 13 '11 at 7:23
• it was a pleasure although it took some time. Cheers, LM – Luboš Motl May 13 '11 at 9:08

## 1 Answer

Dear Claude, you are extrapolating electromagnetism way too high. You're going from low energies to the Planck scale, assuming that nothing qualitatively changes, but this assumption is wrong.

The fine-structure constant is essentially constant below the mass of the electron - the lightest charged particle - which is 511,000 eV or so. You are extrapolating the running of the electromagnetic fine-structure constant $\alpha = 1/137.03599$ all the way up to the Planck scale, about 10,000,000,000,000,000,000,000,000,000 eV. I chose to avoid the scientific notation to make it more explicit how far you have extrapolated.

However, at the electroweak scale, about 247,000,000,000 eV, which is much lower than the Planck scale, the electromagnetic force is no longer the right description. The weak nuclear force gets as strong and important as electromagnetism and in fact, they start to mix in nontrivial ways. The right theory doesn't use $U(1)_{\rm electromagnetism}$ but $SU(2)_{\rm weak}\times U(1)_{\rm hypercharge}$. Note that the electromagnetic $U(1)$ is not the same thing as the hypercharge $U(1)$.

So instead of the fine-structure for the electromagnetic $U(1)$, one must express physics in terms of the fine-structure constants for the new $SU(2)$ and $U(1)$ electroweak gauge groups. These fine-structure constants are not as tiny as the electromagnetic one.

The hypercharge $U(1)$ fine-structure constant gets stronger as the energy grows - much like the electromagnetic one would - while the $SU(2)$ fine-structure constant gets weaker as the energy grows (however, it would be also getting stronger, like electromagnetism, if we included new $SU(2)$-charged particles such as superpartners).

At the same time, the $SU(3)$ QCD fine-structure constant is getting weaker as we raise the energy. Ultimately, all the three fine-structure constants, when properly normalized, become close near $10^{16}$ GeV which is the GUT scale - they exactly cross if one includes the superpartners of the known particles.

The common value of all these three fine-structure constants at the GUT scale is something like $1/24$ or $1/25$ - in fact, there exist somewhat preliminary arguments based on F-theory (in string theory) and similar frameworks that suggest that the number could be exactly $1/24$ or $1/25$ when certain conventions are carefully followed.

If your question was not one about the real world, but one about a fictitious world that only contains QED up to the Planck scale, then indeed, the fine-structure constant would increase just "somewhat", perhaps to $1/100$ or so; indeed, if the world were pure QED, the Landau pole with $\alpha=\infty$ would occur at much higher energies than the Planck scale.

The number can be calculated as a function of $\alpha_{E=0}$. However, it is meaningless to assign error margins to a theory with adjustable parameters that doesn't describe the real world - and QED above the electroweak scale doesn't. A theory that doesn't describe the real world is always in error, even when you're within any error margins, and you can't just "match" it to the real world because different ways of "matching" two different theories would yield different results.

• Lubos, see the comments I added to the question. The question assumes that QED is valid for all energies and is a specific question about the standard graph that shows how the three coupling constants change with energy. – Claude May 11 '11 at 8:35
• Dear Claude, what do you exactly mean by QED? Just photons, electrons, and positrons? Or do you include muons, tau, and quarks? When you do, why don't you also include W-bosons which will destroy your "QED" at the electroweak scale, anyway? The incorporation or not incorporation of muon - and all other things - has a profound impact on all the numbers and running. It's straightforward to compute the running in any of these (wrong) toy models, as long as they are consistent, but why would you do it? – Luboš Motl May 11 '11 at 8:39
• BTW one can't even say that it's possible to find an electron-positron-photon QED that agrees with the real world up to the electron mass scale. It's because the real world also has neutrinos and gravitons that are lighter, and maybe axions. – Luboš Motl May 11 '11 at 8:42
• Dear @Claude, you're confused. The left graph on www-ekp.physik.uni-karlsruhe.de/~deboer/html/Forschung/… shows the standard $U(1)\times SU(2)\times SU(3)$ couplings of the Standard Model I was talking about. None of these three constants ever becomes $1/137$: none of them is the electromagnetic fine-structure constant. The fact that the $U(1)_Y$ coupling in the graph starts at $1/60$ is no mistake while $1/137$ would be mistake. As I explained, you can't extrapolate "the" $1/137$ coupling because it's electromagnetism which breaks down above 100 GeV. – Luboš Motl May 12 '11 at 15:53
• Dear Lubos, you are correct, and I was confused. The graph indeed shows the hypercharge coupling. – Claude May 13 '11 at 7:22