Difference between instantons and sphalerons

Question

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?

Answer 1

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.