# What is the mean field value of a scalar field with spontaneously broken symmetry in a scattering event?

Consider you have a quantum field theory that undergoes spontaneous symmetry breaking at some critical temperature. It doesn't necessarily have to be a continuous symmetry that's broken, I don't think that matters for my question. For simplicity, think of the field being a scalar field.

I (think I) do understand how symmetry restoration works in a background of some temperature. One looks at the mean field approximation, and the mean field then has statistical properties (temperature, pressure) and contributes to the Lagrangian. That adds to the symmetry breaking potential so that at the critical temperature the nature of the minima changes. That works fine eg in the early universe.

But now consider you look at a highly energetic scattering experiment, high energy meaning a momentum transfer much much larger than the critical temperature (arbitrarily large if you want), with ingoing and outgoing asymptotic states and so on. The mean field approximation doesn't make much physical sense in this case because the scattering event itself is a large fluctuation. But the mean field value itself must still exist, or does it? I am thus wondering what's the mean field value of the field whose symmetry is broken in that scattering event. I do not mean the vacuum expectation value - if there is something that's scattering, it's not vacuum. And again, I know that if you have enough energy for a large perturbation, the mean field approximation isn't useful.

I think I am missing something here. I am not even sure how the mean field would be defined for the scattering event.

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The field value is defined away from the scattering event. Your question would be better phrased as "does a high energy two-particle scattering restore the unbroken phase?" This is a complicated question, and there are disagreements in the literature. – Ron Maimon May 27 '12 at 22:55
I'd rather not ask for a broken or unbroken "phase" as that seems to imply many-particle physics already (it's a phase of what?). I'd be grateful for a reference! I've been through pretty much all QFT and Stat Mech phys books I could get my hands on and couldn't find anything about that point. – WIMP May 28 '12 at 7:12