Why does QED work so well? - a question on EFTs

[metroplex021]

Hi folks: I have a question about effective field theories.

The magnetic moment of the electron calculated in renormalized QED gets it right to 12 decimal places. That suggests that QED gets things right all the way down to the order of 10^-12eV - that is, all the way to the TeV scale. But surely QED must break down at around the 100GeV mark - the scale of electroweak unification.

I'm therefore confused about how it can get the magnetic moment right to all 12 places given that the theory must change so profoundly at a lower energy. Can anyone tell me where I've gone wrong? Thanks!

 

[humanino]

Here is a hint : you say 10^12 eV. Why not 10^12 J ?

 

[metroplex021]

Hmmm... well, if we did that, wouldn't we just have to translate the part of the question concerning the weak interactions into 10^11J... and then we'd still be left with the same problem?! (I suspect I'm not smart enough for your hint!)

 

[tom.stoer]

You talk about accuracy of QED and about the energy range were QED is valid. These are two different questions. Accuracy of QED calculations, e.g. for atomic spectra in the eV range, has nothing to do with the application of QED in the TeV range.

 

[ansgar]

that is WHY one wants to do these measurments, to get information about the physics above current accessible experimental energy limits, i.e. how much the "new heavy" particles contribute to the observables at lower energy.

the g-2 experiment of the muon is a famous example.

So what one does it to calculate observable in the SM, make a fine precision data and compare. Then one includes his/her own favourite model beyond the standard model, or just simply the SM higgs boson, and start to calculate constraints for the mass and couplings of such new particles, in order for the experimental value to be statically accessible from your model calculation.

 

[metroplex021]

You talk about accuracy of QED and about the energy range were QED is valid. These are two different questions. Accuracy of QED calculations, e.g. for atomic spectra in the eV range, has nothing to do with the application of QED in the TeV range.

Sure, but it just surprises me that in this case the accuracy of QED is apparently greater than the energy range in which it is theoretically valid. It's not surprising that a high-energy theory like QED gets right calculations in the eV atomic domain. But it seems much stranger that its numerical accuracy goes into a range that outstrips the energies in which it is applicable (to me anyway!)

 

[humanino]

First, the magnetic moment of the muon is not expressed in eV but in eV per T. My un-explicit "hint" was to check its value : it happens to be around a millionth of eV per T. So there are two reasons why starting with ~ 1 eV is not justified.

Second, we cannot just say that a measurement with 10 digits is sensitive to physics at energies 10 times higher, even if the units where right. When folding in loop corrections, where the mass scale will appear, there is a coupling constant which for QED is of the order of a hundredth (1/137). This will suppress loop corrections. Numerically for instance QED corrections give
$$\frac{g-2}{2}=0.5\frac{\alpha}{\pi}-0.32848\left(\frac{\alpha}{\pi}\right)^2+1.18\left(\frac{\alpha}{\pi}\right)^3+\cdots$$
The numerical coefficients of the series expansion are where the various mass-scales of various physics contributions occur. They themselves are not necessarily small.

Precise mass-dependent QED contributions to leptonic g-2 at order alpha^2 and alpha^3
muon and tau magnetic moments: a theoretical update
[/url]

 

[metroplex021]

Thank you very much - that is helpful of you and kind.

First, the magnetic moment of the muon is not expressed in eV but in eV per T. My un-explicit "hint" was to check its value : it happens to be around a millionth of eV per T. So there are two reasons why starting with ~ 1 eV is not justified.

Second, we cannot just say that a measurement with 10 digits is sensitive to physics at energies 10 times higher, even if the units where right. When folding in loop corrections, where the mass scale will appear, there is a coupling constant which for QED is of the order of a hundredth (1/137). This will suppress loop corrections. Numerically for instance QED corrections give
$$\frac{g-2}{2}=0.5\frac{\alpha}{\pi}-0.32848\left(\frac{\alpha}{\pi}\right)^2+1.18\left(\frac{\alpha}{\pi}\right)^3+\cdots$$
The numerical coefficients of the series expansion are where the various mass-scales of various physics contributions occur. They themselves are not necessarily small.

Precise mass-dependent QED contributions to leptonic g-2 at order alpha^2 and alpha^3
muon and tau magnetic moments: a theoretical update
[/url]