Sphaleron

A sphaleron (Greek: σφαλερός "slippery") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and it is involved in processes that violate baryon and lepton numbers. Such processes cannot be represented by Feynman diagrams, and are therefore called non-perturbative. Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy (in the infinite-dimensional field space), much like the saddle point of the surface z(x,y)=x2−y2 in three-dimensional analytic geometry.

In the standard model, processes violating baryon number convert three baryons to three antileptons, and related processes. This violates conservation of baryon number and lepton number, but the difference B−L is conserved. In fact, a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, and hence a quark may be converted to 2 anti-quarks and an anti-lepton, and an anti-quark may be converted to 2 quarks and a lepton. A sphaleron is similar to the midpoint (τ=0) of the instanton, so it is non-perturbative. This means that under normal conditions sphalerons are unobservably rare. However, they would have been more common at the higher temperatures of the early universe.

Sphalerons and baryogenesis

Since a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, if the density of sphalerons was at some stage high enough, they would wipe off any baryon asymmetry (i.e. any net excess of baryons or anti-baryons). This has two important implications in any theory of baryogenesis within the Standard Model [1][2]

• Any baryon net excess existing before the electroweak symmetry breaking was wiped off because sphalerons were abundant due to the high temperatures existing in the universe.
• A baryon net excess can be created during the electroweak symmetry breaking can be later preserved only if this phase transition was first-order. This is because in a second-order phase transition sphalerons would wipe off any baryon asymmetry as it is created, while in a first-order phase transition sphalerons would wipe off baryon asymmetry only in the unbroken phase.

In some theories of baryogenesis an imbalance of the number of leptons and antileptons is formed first by leptogenesis and sphaleron transitions then convert this to an imbalance in the numbers of baryons and antibaryons.

Equations

For an SU(2) gauge theory, neglecting $\theta_W$, we have the following equations for the gauge field and the Higgs field in the gauge A0 = Ar = 0 [3]

$\bold{A} = \nu \frac{f(\xi)}{\xi}\hat{\bold{r}}\times\bold{\sigma} \, , \, \phi = \frac{\nu}{\sqrt{2}}h(\xi)\hat{\bold{r}}\cdot\bold{\sigma}\phi_0$

where $\xi = rg\nu$, $\phi_0 = \begin{bmatrix}1 \\ 0\end{bmatrix}$, the σ-s are the SU(2) generators, g is the electroweak coupling constant ν is the Higgs VEV absolute value.

h(ξ) and f(ξ) are functions going from 0 to 1 as ξ goes from 0 to $\infty$. These functions are found numerically.

For a sphaleron in the background of a non-broken phase, the Higgs field must obviously fall off eventually to zero as ξ goes to infinity.

Note that in the limit $\xi\rightarrow\infty$, the gauge sector approaches one of the pure-gauge transformation $\frac{\bold{r}\cdot\bold{\sigma}}{r}$, which is the same as the pure gauge transformation to which the BPST instanton approaches as $r\rightarrow\infty$ at t=0, hence establishing the connection between the sphaleron and the instanton.