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.


2 Answers 2


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.

  • $\begingroup$ 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
    Commented Oct 14, 2019 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
    Commented Oct 14, 2019 at 12:08

I've also been puzzling over this question and I believe I've found the answer --- it is the latter, that the vacua the sphaleron path connects are topologically distinct only in certain gauges.

It's clear that for the incontractible loop detailed in Manton's paper that at $\mu = \pi$ the vacuum is topologically trivial. However, the path itself is topologically non-trivial, as it cannot be contracted to a point --- more specifically there is a non-zero value of $\int d^4 x \, \mathrm{Tr}\, F_{\mu \nu} \tilde{F}^{\mu \nu}$ over the path if $\mu$ is promoted to a function of time.

In Klinkhamer and Manton's follow-up paper where they study the sphaleron configuration itself, they calculate the sphaleron Chern-Simons number and note just above Equation (39) that the sphaleron can be considered as connecting two topologically distinct vacua in a gauge where the the gauge field vanishes faster than $1/r$ at spatial infinity. Note that this is more stringent than the condition $A_r = 0$.

There's a more detailed calculation in sections 2 and 3 of this paper where they show that to recover the Chern-Simons number of the field configuration along Manton's path you either need to introduce time-dependence or transform to a different gauge. With either of these choices the Chern-Simons number is

$$N_\mathrm{CS} = \frac{2\mu - \sin 2\mu}{2 \pi}.$$

For vacuum configurations $\mu = \pi n$ for integer $n$, so $N_\mathrm{CS} \in \mathbb{Z}$.

  • 1
    $\begingroup$ I’ve recently discovered an essay called “Demystifying the QCD Vacuum“ by J. Schwichtenberg. He discusses this precise problem and its unclear history. As far as I have been able to tell, your answers definitely goed in the right direction. The vacuum structure appears to be gauge dependent, but the conclusion (That Yang-Mills has a vacuum angle) does not change. $\endgroup$
    – Giovanni
    Commented Dec 20, 2019 at 11:38

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