# Difference between instantons and sphalerons

What is the difference between instantons and sphalerons? If they are different, how do they violate baryon and lepton number in the standard electroweak theory?

## 1 Answer

The sphaleron is kind of the opposite of the instanton, and kind of the same. Let's make that statement precise:

An instanton is a local minimum of the action that mediates vacuum tunneling (link to an answer of mine how and why instantons do that). The sphaleron sits in-between the vacua, in a certain sense, it is the instanton "in the middle of tunneling":

Given an instanton configuration $A_\text{inst}$ on a 4-cylinder $(-\infty,\infty) \times S^3$, representing tunneling between two vacua at the spatial slices at $t = \pm \infty$ with Chern-Simons winding numbers differing by 1, the sphaleron is vaguely the field configuration $A(t = 0)$, and precisely the field configuration for which, at some time $t_0$, the winding number as the integral of the Chern-Simons form

$$\int_{{t_0} \times S^3}\omega \equiv \int_{{t_0} \times S^3} \mathrm{tr}(F\wedge A - \frac{1}{3}A \wedge A \wedge A)$$

is exactly the mean of the winding numbers of the vacua at the ends.

The idea is that an instanton of winding number 1 mediates between vacua of winding numbers $k$ and $k+1$, and that the sphaleron is the field configuration "in the middle", with winding number $\frac{2k+1}{2}$. The reason this is interesting is because, for example, in the electroweak theory, the bosonic potential part of the action is (of course) minimal for the vacuum/instanton configurations, and maximal for the sphaleron configuration. That's why the sphaleron is called that (it's Greek for slippery thing) - it may slip down into either vacuum configuration because it is sitting in a metastable extremum of the potential.

• Interesting, I've never heard of a sphaleron. So, following your conceptual explanation, would a Coleman de Lucia instanton (e.g. tunneling from de Sitter to Minkowski) be a sphaleron at some stage? – JamalS Jan 12 '15 at 17:07
• @JamalS: I have no idea, because the CdL instanton is in a slightly different setting (string theory/theories of gravity). – ACuriousMind Jan 12 '15 at 17:21
• @ ACuriousMind - The physical mechanism by which instantons violate fermion number is tunneling. But what is the mechanism by which sphalerons violate the fermion number? – SRS Jan 16 '15 at 7:03
• @SRS: I've never understood this kind of phrasing. Quantumly, you find that the symmetry that usually would conserve baryon/lepton number is anomalous, and that its non-conservation is exactly the Chern class/winding number. I honestly do not understand how one can meaningfully talk about the process that creates that anomaly, since it is simply the result of the transformation of the path integral measure under the symmetry transformation. – ACuriousMind Jan 16 '15 at 14:01
• @SRS in sphaleron processes the energy barrier is crossed from above the barrier (as opposed to tunneling) through thermal fluctuations. – Optimus Prime Aug 2 '17 at 6:17