# Are there winding-number vacua in Weinberg-Salam (Or are they a gauge artifact)?

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.

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