Suppose you have a field theory with a real scalar field $\phi$ and a potential term of the form $\lambda \phi^4 - \mu \phi^2$ that breaks the symmetry $\phi \to - \phi$ in the ground state. Is this symmetry restored in a scattering with high momentum transfer in any physically meaningful way? My problem is that there is no background field for which I could take a mean field approximation, so the usual argument for phase transitions does not work.

One thing that came to my mind is that the potential will have higher order corrections, so maybe one could ask which way these corrections work in the limit $\Lambda \to \infty$, where $\Lambda$ is a UV scale? I know the first order correction for small $\phi$ does actually contribute to the breaking rather than to restoration, but then that doesn't seem to be the right limit for the scattering. If you don't know the answer I'd also be grateful for a reference. (Please do not point me to references that are for in-medium effects, nuclear matter etc, as I have plenty of these and it's not addressing my problem.)


First of all, the vev of $\phi$ scales like a positive power of $\mu$ which has the units of mass. So all the effects of the symmetry breaking scale like a positive power of the mass scale $\mu$. At energies $E$ satisfying $E\gg \mu$, i.e. much higher than $\mu$, the value of $\mu$ itself as well as vev and other things may simply be neglected relatively to $E$: all the corrections from the symmetry breaking go to zero, relatively speaking.

In the UV, i.e. at short distances, the dimensionless couplings (and interactions) such as $\lambda$ are much more important than the positive-mass-dimension dimensionful couplings such as $\mu$.

It's still true that the high-energy theory wasn't exactly symmetric in the treatment above; it was just approximately symmetric. However, in fact, sometimes it is exactly symmetric under the sign flip of $\phi$. That's because the value of $\mu$ runs, as in the renormalization group, and in many interesting theories, $\mu$ actually switches the sign at energies exceeding some critical energy scale $E_0$. So the high energy theory may have a single symmetry-preserving minimum of the potential and the symmetry breaking may be a result of the flow to low energies.

It's legitimate you want to know the higher-loop corrections to the Higgs potential but the subtlety you should appreciate is that all these calculations proceed relatively to some renormalization (mass) scale. If the scale is low, the behavior of the theory at much higher energies may only be deduced by the (inverted) RG flows. However, the deduction of the short-distance theory from its long-distance effective field theory approximation is never exact; it's the high-energy theory that is the legitimate starting point and the low-energy physics is its consequence.

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  • $\begingroup$ Ok, thanks. I see the point that the operator becomes less relevant at high energies. Let me think about that. Can you give me a keyword for the "many interesting theories" in which $\mu$ switches the sign? $\endgroup$ – WIMP May 30 '12 at 10:18
  • $\begingroup$ Dear WIMP, maybe I overstated by using the word "many", but you may want to search for radiative electroweak symmetry breaking, e.g. arxiv.org/abs/hep-ph/0510366 - the whole potential, especially the quadratic symmetry-breaking part, is generated from loops in the UV theory. $\endgroup$ – Luboš Motl May 30 '12 at 10:33
  • $\begingroup$ Okay, thanks. I've thought about it. I think it makes sense. Though, it would mean, as you say, symmetry isn't really restored, just that the deviation of the minima from zero becomes less and less relevant. I'll check the reference. $\endgroup$ – WIMP May 31 '12 at 6:21

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