What determines the specific value of the order parameter in spontaneous symmetry breaking? Three examples in the spontaneous symmetry breaking that occurs at a phase transitions:


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*A ferromagnet below the Curie temperature chooses an axis of quantisation along which all the spins align, giving rise to a macroscopic magnetisation in that direction. Symmetry broken is $SO(3)$ (isotropy of space) to $SO(2)$ (only symmetry about magnetisation axis).

*Bose-Einstein condensation: below $T_c$, bosons amass in the same ground state, described by the same wavefunction and with a physical (non-gaugeable) phase. Symmetry broken is $U(1)$.

*Higgs mechanism. The complex Higgs doublet chooses a phase, a non-zero vacuum expectation value (VEV) $\propto \mu^2/\nu$. This then determines the mass of the Higgs, the $W^\pm$ and $Z^0$ bosons, and the coupling to fermions. Symmetry broken is $SU(2)_L \times U(1)_Y$ to $U(1)_{em}$

in the absence of any decoherence, coupling to environments and measurements, how is the phase chosen? Why is not a superposition of all possible ones.
The phase controls the masses of particles, for the higgs. So it's quite important. What caused the field to choose that particular phase during the Higgs phase transition?
 A: You don't need symmetry breaking for a symmetry broken phase.  What is relevant is that all ground states -- be they symmetric or symmetry broken -- have a long-range order in the relevant order parameter.  This long-range order is present regardless of the presence or absence of any external fluctuations which explicitly break that symmetry.  
Note that there are examples where this is arguably the case, such as superconducivity: The U(1) symmetry -- corresponding to particle number conservation -- will not be explicitly broken, since (at least at the relevant energies) electron number is in fact a conserved quantity. Rather, what is important is the long-range order in the superconducting order parameter, and the relative value of the order parameter of two coupled superconductors.  However, assuming that the symmetry is broken is convenient for calculations.  
A similar scenario is encountered when talking of coherent lasers with a fixed phase: What is actually fixed is the relative phase of different beams to each other, while the global phase is constantly fluctuating. However, assuming a fixed global phase simplifies calculations and has no negative consequences. 
