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In pure SU(2) Yang-Mills the vacua van be grouped in homotopy classes labeled by their winding number. Instantons connect these giving rise to the theta-vacuum.

I’m studying the SU(2) sphaleron in the Weinberg-Salam theory without fermions (Original paper by Manton). Which is the same SU(2) Yang-Mills theory with the addition of an SSB Higgs doublet $\Phi$. The theory looses its instantons (Derricks theorem) and obtains a sphaleron. (The theory has no soliton monopoles.)

In the paper Manton constructs a non-contractible loop starting and ending at the same vacuum. It is important to note that using his gauge fixing $A_r = 0$ (as well as fixing a base point on $\Phi^\infty$) there is only a single vacuum left. Once the construction is over he claims that by letting go of $A_r = 0$ and imposing some additional new gauge constraints the loop can be turned into a path connecting topologically distinct vacuums (very reminiscent of how the instanton connected vacuums with different winding numbers in the pure Yang-Mills theory).

Has the inclusion of the Higgs somehow gotten rid of the pure Yang-Mills theory’s multiple vacuum structure? Or is it that these multiple vacua are somehow gauge artifacts only observable in some gauges and not others ($A_r$ = 0)? This appears strange to me since these many vacuums give rise to real physical effects.

edit Added the remark that I’m studying the model in absence of fermions.

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There are indeed interesting winding number instanton configurations in the $SU(2)$ weak interaction sector of the standard model. There is no analogue of the QCD vacuum angle however, as the instantons have exact fermion zero modes even when the fermions gain mass via the Higgs field. Here are some of the key references:

N.V.Krasnikov, V.A. Rubakov, V.F.Tokarev, "Zero-fermion modes in models with spontaneous symmetry- breaking". J. Phys A 12 (1979) L343-346.

A.A.Anselm, A.A.Johansen, "Baryon nonconservation in standard model and Yukawa interaction", Nuclear Physics B 407 (1993) 313-327

A.A.Anselm, A.A.Johansen, "Can the electroweak $\theta$-term be observable?", Nuclear Physics B 412 (1994)553-573.

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  • $\begingroup$ Thank you for the response, I have not heard about these fermion zero modes. However, I’m studying Weinberg-Salam without fermions for the moment (so only with Higgs) and I’m trying to understand the vacuum structure of that particular theory (and whether the theory has true winding-number vacuums rather than those being a gauge artifact). Are you sure these modes are relevant? I’ll look into them if that is the case. $\endgroup$ – Giovanni Oct 14 at 8:09
  • $\begingroup$ Ah I see the point of Derrick's theorem. There may not be exact classical solutions, but the topology still alows winding number changing fluctuations and so, I suppose, theta vacua. I'm not an expert on this though. I expect, from the condensed-matter analogues that the Higgs expectation would be reduced in the core of the instanton. The actual configuration would be similar to that in the Rubakov paper. $\endgroup$ – mike stone Oct 14 at 12:08

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