How Close Are We to Reaching the Landau Pole in QED?

[RedX]

Has the energy scale (or cutoff) at which the electric charge goes to infinity (the landau pole) been calculated? I sometimes hear that QED has been probed to tiny distance scales, and I was curious how far away we are until we get to distances corresponding to the Landau pole, after which we shouldn't be able to do anymore calculations.

Also I just realized that SU(2) isospin interaction is asymptotically free. I had thought only SU(3) color interaction was asymptotically free, but from the looks of the SU(2) isospin beta function (which is negative), so is SU(2) isospin. It seems strange however that whenever I look up asymptotic freedom, there is only mention of SU(3) color asymptotic freedom and no mention of SU(2) isospin.

 

[Haelfix]

The Landau QED ghost exists many orders of magnitude higher than the Planck scale. I forget the exact value (its easily calculable), but its so ridiculously high that its not really important except as a theoretical proof of concept on the limits of field theory.

In practise you would never be able to probe this scale without doing quantum gravity and/or making black holes with your detector.

 

[RedX]

If, theoretically, quantum field theory breaks down at the Planck scale, then shouldn't all integrals of loop momenta be technically up to the Planck scale, and not to infinity? Having an integral up to infinite allows you to use some fancy mathematics to get analytic answers, but is it legitimate?

The effective field theory approach says that if you want the loop momenta to only go to a cutoff such as the Planck scale, the cost is introducing new terms to the Lagrangian of the order (E/Planck). If E<<<Planck, these terms can be ignored. However, there is cost to doing this in that your mass and charge are changed, and all your renormalization constants are finite - so it seems to me that you can't just say integrating to infinite is numerically almost the same as integrating to the Planck scale: in order to get them numerically almost equal, you have to change a lot of constants such as the mass and charge and renormalization factors in a very specific way. In other words, say I calculate the g-factor of the electron. There are some loop integrals up to infinite I have to calculate. However, if I take those loop integrals, but replace infinite with the Planck energy, then the answer isn't accurate. In order to replace infinite with the Planck energy, I have to make all sorts of other changes. Yet integrating to infinite without using effective field theory gives the right experimental answer, even with the Landau pole.

 

[Haelfix]

Technically, in the modern view, the cutoff should be best placed wherever there is a new physics scale that changes the nature of the effective field theory (read up on relevant and marginal operators and so forth). In practise, QED fails somewhere around the Z Pole, long before the Planck scale and much much before the Landau ghost.

The Landau pole is more of a conceptual problem and a bit of a historical artifact. Back in the early 60s, people were still trying to make relativistic quantum mechanics and field theory truly fundamental. After all, we had the Dirac and Klein Gordon equation, and they naively seem to be valid to arbitrarily high energies (well at least if you ignore the self interaction problem), so it bothered some people that the best theory of the time mathematically broke down at some point. This was long before the effective field theory point of view became fashionable.

Still the Landau ghost remains a problem for many (but not all) field theories, as its rather generic. Some people still use it as an argument, say in favor of asymptotically free theories (where the ghost is located not at high energies, but in the IR) being fundamental.