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As I understand the relevant popular sources (example), since the end of the Electroweak epoch, the Higgs field became fixed in the whole known Universe. Thus, we are somewhere on the deep blue part (which is really a 4-sphere) of the Mexican hat potential, and this is a constant value of the Higgs field everywhere. Of course $\rm{|\phi|=246GeV}$.

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This should mean that we have 2 complex (or 4 scalar) universal physical constants, from which we know that their square sum is 246GeV.

But, is the induvidual values of these contants known? What is the value of the 4 real (2 complex) components of the Higgs field now?

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If "we" are the vacuum, then, yes, we are on the blue minimum of a 4D Goldstone sombrero. The blue minimum is not a circle as in the 2D picture, but, instead, a three sphere, a hypersphere of the three goldstons eaten up in the Higgs mechanism.

The expectation values are, by convention, $$ \langle \phi^0 \rangle = v = \langle \Re \phi^0 \rangle \qquad \langle \Im \phi^0 \rangle = 0= \langle \phi^+ \rangle. $$ These are at the minimum of the sombrero. Oscillations around the minimum are the h field and the would-be-goldstons, which have been rearranged into pieces of the gauge fields.

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    $\begingroup$ Thus, being on another (minimal energy) point, we would have exactly the same rest masses? $\endgroup$
    – peterh
    Commented Oct 13, 2020 at 14:39
  • $\begingroup$ Yes, any point (vacuum choice) on said O(4)/O(3) hypersphere will result in an identical theory, provided we redefine our manifold coordinates to describe the same entities. $\endgroup$
    – Cosmas Zachos
    Commented Oct 13, 2020 at 14:50

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