Why aren't all possible VEVs of the Higgs field in (quantum) superposition? It is is usually said the the Higgs field chooses a Vacuum Expectation Value (VEV), around which we now expand our field operators. From this a mass term emerges, and one or more gauge fields acquire mass.
Actually the Higgs VEV is one of the free parameters of the Standard Model, i.e. one that needs to measured experimentally.
Shouldn't all possible VEVs be in quantum superposition though, until we make a measurment of it?
Does the fact the other parts of the Standard Model Lagrangian couple to the Higgs field constitute a measurement, in the quantum sense?
 A: If you will allow me to make your question slightly more precise, I think you are asking the following:
In spontaneous symmetry breaking (SSB), generally speaking, we say that the system has a range (either continuous or discrete) of possible degenerate values, and as a result it picks at random one of these configurations, resulting in a state without the symmetry of the underlying Hamiltonian. But why should it do this? After all, generally speaking in quantum mechanics if several states are degenerate, one expects a quantum system at that energy to be in some undetermined superposition over these states.
I agree with you that people are usually rather careless about discussing this, but the resolution is something like what you propose. Some nice thoughts on this can be found in the blog of Gen Zhang; I will more or less reproduce them here.
One does not necessarily need a measurement to make SSB make sense, but one does need decoherence. A superposition over the degenerate states of a system in SSB is necessarily a macroscopic superposition, and as a result we can expect that it will decohere along the relevant basis very quickly.
A useful model, as usual, is of a magnetic material. Let's imagine a Heisenberg magnet, which is just a bunch of spins that can each rotate freely in 3D but have some nearest-neighbor dipole interactions. Below some critical temperature, the preferential state is for most of the dipoles to be aligned with each other and a net magnetization to develop. But without an external field it doesn't matter which direction they all end up pointing along.
So let's imagine starting with a bunch of these spins, and cooling them below the magnetization temperature. Also, imagine doing this in such a way that we extract heat, but no information about the overall magnetization leaves the system. Then we would indeed have to expect that the system is in some superposition over magnetized states in every direction. However, as soon as anything is coupled to the magnetization, which will almost always be very quickly for a macroscopic system, it will become entangled and decohere along the basis of different magnetizations, so any given observer will see the system as having picked, at random, one of the possible orientations.
The Higgs case specifically is a little harder to think about, and also somewhat outside my expertise, so I certainly invite any remarks from high-energy people. That said, I think it should end up similarly. The Higgs field couples to fermions with a form like:
$\lambda \bar{\psi}_L \phi \psi_R+h.c.$,
where $\psi$ are the fermionic operators of a given chirality, and $\phi=\phi_0 e^{i \theta}$ is the Higgs field at the Mexican hat minimum $\phi_0$. One normally considers a single definite value for $\phi$, in which case its phase is not observable, but if $\phi$ was in a quantum superposition like: $\frac{1}{\sqrt{2}}\phi_0(e^{i \theta_1}+e^{i \theta_2})$ one would expect this to lead to a superposition of mass states with different phases for any given fermion. Quantum fluctuations about either of $\theta_1$ or $\theta_2$ would then affect the cancellation between these two phases and change the masses of everything, certainly resulting in observable consequences and entanglement with other degrees of freedom. Ultimately, only a configuration with a single $\theta$ would be stable against this decohering mechanism.
Finally, as Gen Zhang also mentions, some nice general discussion of this aspect of SSB, in the context of Bose condensation, can be found in Leggett's textbook Quantum Liquids.
