Sphaleron - Equations

Equations

For an SU(2) gauge theory, neglecting, we have the following equations for the gauge field and the Higgs field in the gauge A₀ = A_r = 0

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

where, 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 . 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, the gauge sector approaches one of the pure gauge transformation, which is the same as the pure gauge transformation to which the BPST instanton approaches as at t=0, hence establishing the connection between the sphaleron and the instanton.