When you met the process of spontaneous symmetry breaking (SSB) for first time (in my case during the standard model course) they tell you the example of a (3D) ferromagnetic substance. In the case where $T>T_{c}$ (critical temperature) the magnetisation: $$\langle\vec M\rangle = 0.$$ Instead, if $T<T_{c}$ we have $$\langle\vec M\rangle \neq 0.$$ In this case $SU(3) \to SU(2)$ and you can observe either the magnetisation “state” after and before the symmetry breaking experimentally (varying the temperature) so we can say it’s physical, in the sense it is a “dynamical” process.

My doubt is: is it the same if we’re taking about SSB in electroweak theory ? Can we reach temperature (or energies) where the $SU(2)_L$ is still a symmetry of vacuum state and observe it? And what’s the physical meaning related to have a $m^2$ term that is negative, is it related in some way to the running of the coupling and renormalization? (I mean do I have to interpret all the masses as functions of energy scale of the process?)


2 Answers 2


One way to think about this question is to consider the higgs vaccum expectation value (vev) $v = \langle \phi \rangle$ (which in conventional basis is $246 \mathrm{GeV}$ in the standard model, at $T = 0$). The vev $v$ acts as an order parameter for spontaneous symmetry breaking, exactly like the magnetisation example you mention. (I don't know why you have $SU(3) \to SU(2)$ listed as the breaking pattern for a 3D ferromagnet - for an ising model its just $\mathbb{Z}_2$ symmetry that is broken)

For the physical standard model parameters, it is expected that the transition is a crossover, meaning that the transition from broken $v \neq 0$ to symmetry restored $v \to 0$ is expected to be smooth. Here is a recent paper with many references to previous studies. As mentioned in the paper, it is believed that if you change the parameters of the standard model or introduce new fields, the phase diagram includes a line of first order phase transitions ending at a second-order phase transition, in the 3D Ising universality class.

The studies I linked rely on EFT-methods and lattice simulations - in other words, it is a theoretical prediction. I am unclear on the experimental status of the electroweak symmetry restoration transition.

Having a first-order transition (through the introduction of new fields) would be interesting because I believe you would have bubble nucleation as the universe cools from the big bang, where you have bubbles of the symmetry broken phase living inside the unbroken vacuum. I believe this is something cosmologists study, because an open problem is the observed matter-antimatter symmetry, which within the standard model could only have come from $B$-violating processes (spharelon). Hopefully a cosmologist can add a few words


It depends on the specific theory you are using.

One hypothesis is that there is a "Mexican hat" potential involved.

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The idea is that the origin, with zero-field values, is the top. That is, the zero-field state that one would naively expect to be the vacuum, is not the lowest energy condition. So the system has a tendency to "slide down" to one of the locations in the ring around the origin. This is the spontaneous symmetry breaking in some theories. When the energy density is locally large enough to bring the system back up to the top of the center hill, then full symmetry is restored. When energy density is too low to reach the peak, then only the symmetry represented by the bottom of the ring remains.

So the idea is, at high enough energy, the full symmetry is moving around in any direction on the surface, the tangent surface at the origin. But at lower energy, only the direction around the ring at the bottom is available.

It's a little difficult to visualize, especially when the surface isn't just 2-D but is something like a Lie-algebra-valued manifold. So the full symmetry might be SU(5) or GL(4) or some such, depending on the specific theory being used. And the remnant in the broken symmetry could be U(1).

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    $\begingroup$ I think OP is well aware of all that, but maybe this was just an impression... $\endgroup$ Jan 15 at 18:47

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