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There ate two types of vacuum of the Standard model-the vacuum of the Higgs potential and that of the vacuum of the Yang-Mills fields labelled by the Chern-Simons number. See the figure 5 here.

The Lagrangian of the Standard electroweak theory contains both the gauge fields and Higgs doublet. Through the gauge covariant derivative the Higgs doublet couples to the gauge fields. So are they really different theories? As I understand, after the electroweak symmetry breaking the Universe is locked at one point/direction of the vacuum manifold of the Higgs potential. But I also hear about the Universe being in one of the vacua labelled by the Chern-Simons number.

My question is which vacuum is the Universe really in?

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A "pure" Higgs theory (i.e. containing only the Higgs field) has a vacuum labeled by the VEV of the Higgs field, a pure YM theory has a vacuum labeled by the $\theta$-angle, and the combined theory, i.e. a YM theory with a Higgs field as we find it in the standard model, has a vacuum labeled by both the Higgs VEV and the $\theta$-angle.

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  • $\begingroup$ Which vacuum does the Standard model belong to? But I also see in literature a plot where there are different vacua labelled by the Chern-Simons number. People talk about the tunnelling between these vacua by instantons and sphalerons. For example, see Fig. 6.7 of the Cosmology book by Kolb and Turner books.google.co.in/… $\endgroup$
    – SRS
    Commented Jan 7, 2018 at 15:55
  • $\begingroup$ @SRS I don't understand the question. The SM vacuum is labeled by the Higgs VEV, the $\theta$-angle, and perhaps even a few other things. In contexts where we don't consider the Higgs VEV to be variable (or indeed of interest at all), we can get away by just caring about the $\theta$-angle, and vice versa. What's the problem with that? If your problem is that I said "$\theta$-angle" instead of "Chern-Simons number", these are just two different bases for the space of vacua, see also this answer of mine. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 7, 2018 at 15:59
  • $\begingroup$ $\theta-$vacuum is the superposition of all the other vacua of the YM theory labelled by definite values of the Chern-Simon's number. Am I right? $\endgroup$
    – SRS
    Commented Jan 7, 2018 at 16:02
  • $\begingroup$ @SRS Yes, you are. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 7, 2018 at 16:03
  • $\begingroup$ But at zero temperature, the tunnelling amplitude between two vacua (labelled by two different CS number) is heavily suppressed. Does it not mean that the SM vacuum is labelled by a definite value of CS number? Moreover, is there a way I can visualize (think pictorially about) the complete vacuum manifold of the SM? $\endgroup$
    – SRS
    Commented Jan 7, 2018 at 16:07

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