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My question is why the electroweak vacuum of the Standard Model have to electroweak charge and QCD color neutral? What goes wrong if electroweak vacuum has either non-zero charge or color quantum number?

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Explanation for the electromagnetism aspect:

If the vacuum carried charge under some generator, that would mean that the generator would not annihilate the vacuum. That would mean that even if such a generator corresponds to a symmetry of the theory, the vacuum however is not symmetric under that operation. Then the gauge boson corresponding to this generator would become massive via the Higgs mechanism, and the force would be short ranged -- which is clearly in disagreement with what we observe for EM.

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Well, the Logic is reversed compared to the one you are implicitly using. It is the direction of the Higgs'vev that's defining what we call electric charge. In practice SU(2)xU(1) is broken to a certain U(1) that we can always choose to point in a certain direction, and accordingly assign the electric charges afterwards. Since the Higgs can't carry color it will not break it either.

The real hard question at this point would be why the spontaneous breaking of the chiral SU(2) in QCD is aligned with the Higgs vev and does not break the electric charge (or color) as previously defined by the Higgs vev.

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  • $\begingroup$ +1 for the "real hard question". Is there a formal phase rotation that could undo any apparent misalignment? $\endgroup$ – Mitchell Porter Apr 13 '14 at 9:35
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Well, as regards the second part of the first question, "and QCD color neutral", the if the vacuum state of any Standard Model field had a colour charge surely it would be in opposition to the Confinement property of QCD that we observe.

That is, we never see colour charge in an experiment, we only ever see colour singlet (white) field states.

I'm not sure how much detail you're looking for here potentially an awful lot could be written about confinement. I suggest you look it up, maybe starting here, depending on your level of maths you could read any of (in increasing order of dificulty) Martin & Shaw, Griffiths or Peskin & Schroeder who all have well know books on Particle Physics and/ or QFT.

If you give me a better idea of how detailed an answer you want I can add (or subtract) more details.

As for the first part of your first question, "electroweak charge", I was led to believe that indeed it does have weak-hypercharge. Namely, the Higgs scalar field $\phi$ is a complex-valued SU(2) doublet, with a non-vanishing hypercharge.

I think that you are inccorrect here.

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The consequence if the vacuum (i.e. the VEV of the scalar field) was not $SU(2)_L\times U(1)_Y$ charged is that it will not break this symmetry and also the consequence if the vacuum was $SU(3)$ charged is that it will break this symmetry which is believed to be an exact symmetry in the Standard Model (SM).

More precisely, for your first question, check my answer to the question : Higgs mechanism and neutral fields. The question is almost the same so does the answer. It will show you why if the scalar field is not charged you can't obtain a mass for your gauge fields.

For your second question, why, if the vacuum was $SU(3)$ charged, it will break this symmetry ? It's because the vacuum will not be invariant under a general $SU(3)$ transformation when this vacuum will be VEVed. Some generators (from the Lie Algebra $\mathfrak{su}(3)$ to be more precise) will annihilate the vacuum, these are the unbroken generators, but you will also have broken generators and those are responsible for the breaking of $SU(3)$. Finally, if the vacuum is not $SU(3)$ charged then it's a singlet under an $SU(3)$ transformation. If it's a singlet, it means that the vacuum is invariant under $SU(3)$ and thus no broken generators and no broken symmetry !

Hope this answers your question !

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