# Why is a superposition of vacuum states possible in QCD, but not in electroweak theory?

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There are two standard stories floating around in modern particle physics:

1. Spontaneous symmetry breaking can only happen in a QFT, like in the electroweak theory, because no tunneling between the degenerate vacuum states of the scalar field are possible. Otherwise we the ground state would be a superposition of the degenerate ground states. The reason for the non-tunneling is that we assume that the spatial volume is infinite an thus the tunneling amplitude is zero.
2. When we investigate the vacuum of QCD, we observe that there are infinitely many degenerate vacua. However here the correct vacuum state is a superposition of all these possible degenerate vacua.

How does 2.) fit together with 1.)? Why is tunneling suddenly allowed in QCD while otherwise it is stated strongly that there is no tunneling between degenerate ground states in a QFT?

(My guess would be that the tunneling in QCD is localized (= hence the name instantons) and thus the tunneling amplitude is non-zero. However, I can't see why the same argument wouldn't hold in the electroweak theory. Shouldn't it be equally possible that there is localized tunneling? Is the reason that we haven't found any electroweak instanton solutions that could describe such tunneling?)

This post imported from StackExchange Physics at 2017-04-10 21:55 (UTC), posted by SE-user JakobH

edited Apr 10, 2017
Aren't these vacua related by unitary generators of $SU(3)$?

This post imported from StackExchange Physics at 2017-04-10 21:55 (UTC), posted by SE-user Solenodon Paradoxus
You can look at this book, in particular around equation (27.4.1) : books.google.es/…

This post imported from StackExchange Physics at 2017-04-10 21:55 (UTC), posted by SE-user user40085
Which type of "many different vacua" are you talking about in the second point? The ones associated to spontaneous chiral breaking, the ones associated to instantons, something different? You should find that the vacua you're superposing, regardless of the QCD vacuum model, are not the same "type" of vacua as in your first point, i.e. are not associated to SSB.

This post imported from StackExchange Physics at 2017-04-10 21:55 (UTC), posted by SE-user ACuriousMind
@ACuriousMind I'm talking about the vacua that are connected by instanton (which is also wrote in the OP). Yes it is a different kind of vacuum, but the question still remains unanswered

This post imported from StackExchange Physics at 2017-04-10 21:55 (UTC), posted by SE-user JakobH
@user40085 I actually have Volume 2 of Leader Predazzi on my desk :D however as far as I see, they only talk about the standard QCD vacuum picture and do not draw conclusions to the scalar vacuum of the electroweak theory

This post imported from StackExchange Physics at 2017-04-10 21:55 (UTC), posted by SE-user JakobH

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