SphaJeron-induced baryogenesis
123
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Figure 4.9. Schematic picture of the dependence of the static energy of the gauge-Higgs
system upon the field configuration [A. 4>1. The minima are topologically distinct vacua,
labelled by the integer-valued Chern-Simons numbers. The maxima, on this slice. are
unstable sphaleron configurations.
the barrier height will be the enerfY associated with a field configuration having
Chern-Simons number Ncs = l' (Remember that Ncs is only integral in a
vacuum field configuration.) Manton [36] and Klinkhamer and Manton [43]
constructed just such an object. They assumed a static. spherically symmetric
ansatz in which the Higgs field
,p(r) = h(mwr)U(r)t/Jo
(4.176)
with
,pO =
(4.177)
_I (0)
..(iv
as before. m w = !g2vand
W; = ~ f(mwr)(a;U)U-'
(4.178)
g2
where U has the form (4.140)
U = ~[zh + iyrl + ixr2]
(4.179)
r
and
r = Irl = (x 2 + y2 + ,?)1/2.
(4.180)
hand f are functions to be determined. The energy of the field configuration is
given by
E = f d 3 x [! tr(Wij Wij) + iBij B;j + (D;tP)t (D;,p) + V(,p)] (4.181)