# Question about the vacua of the Standard Model

This question is probably based on a misunderstanding. Please correct me if I'm wrong, and if unclear, I'll try to put it in a clearer language.

In Yang-Mills theory such as the theory of strong interactions, the vacuum is infinitely degenerate with non-zero tunnelling amplitude between different vacua $\{|0\rangle_n\}$ where $n\in \mathbb{Z}$. As a consequence, it turns out that the true vacuum, represented by $|\theta\rangle$, of strong interaction is a superposition given by$$|\theta\rangle=\sum\limits_{n=-\infty}^{n=+\infty}e^{in\theta}|0\rangle_n.$$

On the other hand, the vacuum of the Higgs potential is locked in a definite vacuum. This is because, in any case of spontaneous symmetry breaking in 3-dimensions, the tunnelling amplitude between any two degenerate vacua vanishes (See Quantum field theory by Maggiore, page 255, eq. 11.7 and also page 189 of Srednicki).

Both the strong interactions and Higgs potential are the part of the Standard Model: $$SU(3)_c\times SU(2)_L\times U(1)_Y.$$

1. Am I right in my understanding that the vacuum of the theory of strong interaction i.e., $SU(3)$ sector is different from that of the Higgs sector?

2. The weak interaction is also described by a Yang-Mills action. Does it mean that like strong interaction, there is also a $\theta-$vacuum for the description of weak interaction in the Standard model?

• Can you provide.a reference for the sentence "in any case of spontaneous symmetry breaking in 3-dimensions, the tunnelling amplitude between any two degenerate vacua vanishes"?? – apt45 Dec 25 '17 at 15:53
• Quantum field theory by Maggiore (page 255, eq. 11.7). Also page 189 of Srednicki. @apt45 – SRS Dec 25 '17 at 16:03
• This appears to be essentially a duplicate of physics.stackexchange.com/q/321867/50583 – ACuriousMind Dec 25 '17 at 16:11
• @ACuriousMind Thanks. But I'm confused how many "sectors" does the Standard model have? Are the quarks fields excitations above the $\theta$-vacuum? What about the lepton fields? This all seems very disconnected to me. I can't make a coherent picture. – SRS Dec 25 '17 at 16:19
• I'm afraid you'll have to ask a more specific question for me to be of any help there. I don't know what it means for the quark field to be "excitations above the $\theta$-vacuum" or what you think is "disconnected" here. – ACuriousMind Dec 25 '17 at 16:27