# Why does Spontaneous Symmetry Breaking Depend on Energy Scale?

In the Standard Model we tend to argue that electroweak symmetry is broken below a certain (large) energy scale, yielding the $W$ and $Z$ bosons and the photon. The usual argument for spontaneously symmetry breaking relies on a Higgs potential of the form

$V(\varphi)= \varphi^4 - \mu^2\varphi^2$

and the argument follows from the degeneracy of the lowest energy state.

Remark: Strictly speaking we really ought to be talking about the effective potential, which takes into account radiative corrections. The lowest energy state would then be the vacuum expectation value of the field. I’ll treat the rigorous foundations of SSB in a future post, a topic which is often overlooked in lectures and textbooks!

Critically the standard argument makes no mention of energy scale. So why should we expect it to play a role. The answer is twofold.

Firstly we must remember that we can never observe a full theory. In fact our observations are at best approximations to our theory. This slightly backwards way of looking at things aids our understanding of spontaneous symmetry breaking. At low energies we see the “Mexican hat” clearly, and observe it’s effects in experiments. But as we go to higher energies we “zoom out” on the $V$ axis. This means that the Mexican hat effect is no longer “visible”. From an experimental perspective it’s effects are dominated by other parameters such that they are unobservable. To all intents and purposes the symmetry remains unbroken.

In some theories there’s something more fundamental which averts the SSB scenario at high energies. Recall that renormalization causes the coupling constants to run. In particular our mass pseudoparameter $\mu^2$ can actually change sign at high energies. This eliminates the vacuum degeneracy in the (effective) potential. To see an example, look at Figure 13 in Ben Allanach’s SUSY notes.

I’m grateful to Zac Kenton for a fruitful discussion over a (much-needed) coffee.